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# Homology and Bisimulation of Asynchronous Transition Systems and Petri Nets
111This work was performed as a part of the Strategic Development Program at
the National Educational Institutions of the Higher Education, N 2011-PR-054
Ahmet A. Husainov
###### Abstract
Homology groups of labelled asynchronous transition systems and Petri nets are
introduced. Examples of computing the homology groups are given. It is proved
that if labelled asynchronous transition systems are bisimulation equivalent,
then they have isomorphic homology groups. A method of constructing a Petri
net with given homology groups is found.
2000 Mathematics Subject Classification 18G35, 18B20, 55U10, 55U15, 68Q85
Keywords: bisimulation, homology groups, simplicial complex, trace monoid,
partial action, asynchronous system, Petri net.
## Introduction
The paper is devoted to the application of algebraic topology methods for
classification and studying the mathematical models of concurrency. We
consider asynchronous transition systems with label functions on events. Our
purpose is to construct a homology theory of labelled asynchronous transition
systems for which any bisimulation equivalent asynchronous transition systems
have isomorphic homology groups.
We consider a categorical notion of the bisimulation defined by open maps [1].
It was proved in [1], that in the case of labelled transition systems this
definition coincides with a strong bisimulation of R. Milner [2]. A
characterization of the bisimilation equivalence for asynchronous transition
systems was given in [3].
Homology groups have no less than important for the classification and
studying the properties of concurrent systems. In particular, they have been
applied in the work [4] to characterize the condition of solvability for some
classes of problems in parallel distribution systems.
In [5], E. Goubault and T. P. Jensen applied homology groups for studying
higher dimensional automata. There were obtained some signs of bisimulation
equivalence for the higher dimensional automata in terms of the homology
groups [5, Prop. 10]. The results were developed in the [6]. In a survey [7],
open questions were marked on the relationship of the Goubault homology [6]
with directed homotopy. The Goubault homology have been applied also to prove
of homotopy properties for higher dimensional automata in the [8].
Communications between homotopy and bisimilarity of higher dimensional
automata was researched in [9].
These groups were used to find signs of parallelizable asynchronous systems in
[11] and were regarded as the homology groups of a topological space of
intermediate states for an asynchronous system in [12]. An algorithm for
computing the homology groups was developed in [13].
In this paper, we study the homology of the labelled asynchronous transition
systems and Petri nets.
We work in the category of asynchronous transition systems considered in [14].
But we call them simply asynchronous systems. Note that M.A. Bednarczyk [15]
studied the broader category of asynchronous systems. Using results of M.
Nielsen and G. Winskel [3], we study open morphisms. We introduce homology
groups for labelled asynchronous transition systems and Petri nets. We prove
that $Pom_{L}$-bisimilar asynchronous transition systems have isomorphic
homology groups (Theorem 3.1 and Corollary 3.2). We give some examples of
computing the homology groups of asynchronous transition systems and Petri
nets. We prove that for an arbitrary finite sequence of finitely generated
Abelian groups $A_{0}$, $A_{1}$, $A_{2}$, …where $A_{0}$ is free and not equal
$0$ there exists a labelled Petri net the $i$th homology groups of which are
isomorphic to $A_{i}$ for all $i\geqslant 0$.
###### Contents
1. 1 Asynchronous systems and trace monoid actions
1. 1.1 State spaces and asynchronous systems
2. 1.2 Asynchronous systems and partial actions of trace monoids
3. 1.3 Open morphisms
2. 2 Bisimulation equivalence of labelled asynchronous systems
1. 2.1 Labelled asynchronous systems
2. 2.2 Open maps and surjectivity
3. 3 Homology groups of asynchronous systems
1. 3.1 Computing homology groups of simplicial schemes
2. 3.2 Homology groups of labelled asynchronous systems
4. 4 Homology groups of labelled Petri nets
1. 4.1 Petri nets
2. 4.2 Labelled asynchronous system for a Petri net and its homology groups
## 1 Asynchronous systems and trace monoid actions
Let us recall some facts on the mathematical models of concurrency [3], [14],
[15]. We study asynchronous systems as trace monoids with partial action on
sets.
### 1.1 State spaces and asynchronous systems
###### Definition 1.1
A state space $(S,E,I,{\rm Tran})$ consists of a set $S$ of states, a set $E$
of events with a symmetric irreflexive relation $I\subseteq E\times E$ of
independence, and a transition relation ${\rm Tran}\subseteq S\times E\times
S$. The following axioms must be satisfied:
1. (i)
If $(s,a,s^{\prime})\in{\rm Tran}$ $\&$ $(s,a,s^{\prime\prime})\in{\rm Tran}$,
then $s^{\prime}=s^{\prime\prime}$.
2. (ii)
If $(a,b)\in I~{}\&~{}(s,a,s^{\prime})\in{\rm
Tran}~{}\&~{}(s^{\prime},b,s^{\prime\prime})\in{\rm Tran}$, then there exists
$s_{1}\in S$ such that $(s,b,s_{1})\in{\rm Tran}$ $\&$
$(s_{1},a,s^{\prime\prime})\in{\rm Tran}$. (See Fig. 1)
---
$\textstyle{s^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{s\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{b}$$\textstyle{s^{\prime\prime}}$$\textstyle{s_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$
Figure 1: To Axiom (ii)
.
Triples $(s,e,s^{\prime})\in{\rm Tran}$ are denoted by
$s\stackrel{{\scriptstyle e}}{{\to}}s^{\prime}$ and called transitions .
###### Definition 1.2
Asynchronous system ${\mathcal{A}}=(S,s_{0},E,I,{\rm Tran})$ is a state space
$(S,E,I,{\rm Tran})$ with a distinguished initial state $s_{0}\in S$.
Moreover, for every $a\in E$, there must be $s_{1},s_{2}\in S$ satisfying
$(s_{1},a,s_{2})\in Tran$.
###### Definition 1.3
A morphism between state spaces
$(\sigma,\eta):(S,E,I,{\rm Tran})\to(S^{\prime},E^{\prime},I^{\prime},{\rm
Tran}^{\prime})$
is a pair consisting of a partial map $\eta:E\rightharpoonup E^{\prime}$ and a
map $\sigma:S\to S^{\prime}$ satisfying the following conditions
1. (i)
for any triple $(s_{1},e,s_{2})\in{\rm Tran}$, there is the following
alternative
$\left\\{\begin{array}[]{cl}(\sigma(s_{1}),\eta(e),\sigma(s_{2}))\in{\rm
Tran}^{\prime},&\mbox{ if the value }\eta(e)\mbox{ is defined},\\\
\sigma(s_{1})=\sigma(s_{2}),&\mbox{ if }\eta(e)\mbox{ is not
defined};\end{array}\right.$
2. (ii)
for all $(e_{1},e_{2})\in I$, if $\eta(e_{1})$ and $\eta(e_{2})$ both are
defined, then $(\eta(e_{1}),\eta(e_{2}))\in I^{\prime}$.
Let ${\mathcal{A}}=(S,s_{0},E,I,{\rm Tran})$ and
${\mathcal{A}}^{\prime}=(S^{\prime},s^{\prime}_{0},E^{\prime},I^{\prime},{\rm
Tran}^{\prime})$ be asynchronous systems. A morphism of asynchronous systems
$(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ is a mophism
$(\sigma,\eta):(S,E,I,{\rm Tran})\to(S^{\prime},E^{\prime},I^{\prime},{\rm
Tran}^{\prime})$ between the state spaces such that
$\sigma(s_{0})=s^{\prime}_{0}$.
### 1.2 Asynchronous systems and partial actions of trace monoids
Below, throughout the paper, we will denote ${\mathcal{A}}=(S,s_{0},E,I,{\rm
Tran})$ and
${\mathcal{A}}^{\prime}=(S^{\prime},s^{\prime}_{0},E^{\prime},I^{\prime},{\rm
Tran}^{\prime})$.
For an arbitrary category $\cal C$, let ${\cal C}^{op}$ be the opposite
category.
Denote by $PSet$ the category of sets and partial maps. Let $M$ be a monoid
considered as the category with a single object. A partial right action of a
monoid $M$ on a set $S$ is a functor $M^{op}\to PSet$, the value of which on
the single object is equal to $S$. The functor assigns to each morphism
$\mu\in M$ a partial map $S\rightharpoonup S$ the values of which defined on
$s\in S$ are denoted by $s\cdot\mu$. The category $PSet$ is equivalent to the
category of pointed sets and pointed maps [14]. If we leave pointed sets,
whose distinguished points are equal to a fixed common point $*$, then we
obtain a category isomorphic to the category $PSet$. We denote this category
by ${\rm Set}_{*}$. The isomorphism allows us to consider a partial right
action of $M$ on $S$ as a functor $M^{op}\to{\rm Set}_{*}$. We denote this
functor by $(M,S_{*})$. For each $\mu\in M$, its value $(M,S_{*})(\mu)$ is the
map denoted by $s\mapsto s\cdot\mu$ for all $s\in S_{*}$.
In particular, the state space can be considered as a set with a partial
action of a trace monoid. Let us recall the definition of a trace monoid [16].
Let $E$ be a set with a symmetric irreflexive relation $I\subseteq E\times E$.
Denote by $E^{*}$ a free monoid of words with the letters of $E$. Elements
$a,b\in E$ are independent if $(a,b)\in I$. We define an equivalence relation
on $E^{*}$ assuming $w_{1}\equiv w_{2}$ if the word $w_{2}$ can be obtained
from $w_{1}$ by a finite sequence permutations of adjacent independent
elements. Let $[w]$ be the equivalence class of $w\in E^{*}$. It is easy to
see that the operation $[w_{1}][w_{2}]=[w_{1}w_{2}]$ transforms the set of
equivalence classes $E^{*}/\equiv$ in a monoid. This monoid is called a trace
monoid $M(E,I)$.
Let $(S,E,I,{\rm Tran})$ be a state space. For any $s\in S$ and $e\in E$,
there exists at most one $s^{\prime}\in S$ for which $(s,e,s^{\prime})\in{\rm
Tran}$. In this case, we set $s\cdot e=e^{\prime}$. If ${\rm Tran}$ does not
contain such a triple, then let $s\cdot e=*$. Now we can assign to each state
space $(S,E,I,{\rm Tran})$ the partial action $(M(E,I),S_{*})$ defined as
$(s,[e_{1}\cdots e_{n}])\mapsto(\ldots((s\cdot e_{1})\cdot e_{2})\ldots\cdot
e_{n})$. Any asynchronous system can be considered as a partial action
$(M(E,I),S_{*})$ of the trace monoid on $S$ with initial element $s_{0}\in S$.
It follows from the definition of action that the formula $s\cdot e\in S$ is
equivalent to $(\exists t\in S)(s,e,t)\in{\rm Tran}$. This formula means that
the value $s\cdot e$ is defined, but $s\cdot e=*$ means that this value is not
defined. The morphism between asynchronous systems
${\mathcal{A}}\to{\mathcal{A}}^{\prime}$ can be defined as a pair of maps
$\sigma:S\to S^{\prime}$, $\eta:E\to E^{\prime}\cup\\{1\\}$ for which
* •
the map $\eta$ can be extended to a homomorphism of monoids $M(E,I)\to
M(E^{\prime},I^{\prime})$;
* •
for every $s\in S$ and $e\in E$ satisfying $s\cdot e\in S$, it is true that
$\sigma(s)\cdot\eta(e)\in S~{}\&~{}\sigma(s)\cdot\eta(e)=\sigma(s\cdot e)$;
* •
$\sigma(s_{0})=s^{\prime}_{0}$.
### 1.3 Open morphisms
A state $s\in S$ of asynchronous system ${\mathcal{A}}$ is reachable if there
exists a finite sequence of transitions $s_{0}\stackrel{{\scriptstyle
e_{1}}}{{\to}}s_{1}\stackrel{{\scriptstyle e_{2}}}{{\to}}s_{2}\to\cdots\to
s_{n-1}\stackrel{{\scriptstyle e_{n}}}{{\to}}s$.
If we want to emphasize that the map $f:X\to Y$ is defined on all elements of
$X$, then we call it total.
###### Definition 1.4
A morphism of asynchronous systems
$(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ is open, if it has the
following properties:
1. (i)
$\eta:E\to E^{\prime}$ is total;
2. (ii)
for all a state $s\in S$ and transition
$(\sigma(s),e^{\prime},u^{\prime})\in{\rm Tran}^{\prime}$, there exists
$(s,e,u)\in{\rm Tran}$ for which $\eta(e)=e^{\prime}$ and
$\sigma(u)=u^{\prime}$;
3. (iii)
for any reachable $s\in S$, if $(s,e_{1},u)\in{\rm Tran}$ and
$(u,e_{2},v)\in{\rm Tran}$ and $(\eta(e_{1}),\eta(e_{2}))\in I^{\prime}$, then
$(e_{1},e_{2})\in I$.
The property (ii) can be shown visually by drawing
---
$\textstyle{s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists\,e}$$\textstyle{\sigma(s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\forall\,e^{\prime}}$$\scriptstyle{\eta}$$\textstyle{u}$$\textstyle{u^{\prime}}$
For any asynchronous system ${\mathcal{A}}=(S,s_{0},E,I,{\rm Tran})$ and a
reachable $s\in S$, we let ${\mathcal{A}}(s)=(S,s,E,I,{\rm Tran})$. In
particular, ${\mathcal{A}}(s_{0})={\mathcal{A}}$.
###### Proposition 1.1
For any open morphism $(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$
of asynchronous systems and a reachable state $s\in S$, the morphism
$(\sigma,\eta):{\mathcal{A}}(s)\to{\mathcal{A}}^{\prime}(\sigma(s))$ is open.
## 2 Bisimulation equivalence of labelled asynchronous systems
In this section, we consider $Pom_{L}$-bisimilar labelled asynchronous
systems.
### 2.1 Labelled asynchronous systems
A labelled asynchronous system $({\mathcal{A}},\lambda,L)$ consists of an
asynchronous system ${\mathcal{A}}$ with an arbitrary set $L$ of labels and a
map $\lambda:E\to L$ called label function. Each asynchronous system can be
considered as labelled where the set $L=pt$ consists of a single label. In
this sense, according to [3, Prop. 16], open morphisms are precisely
$Pom_{pt}$-open morphisms.
Let $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ be labelled asynchronous
systems. A morphism $(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$
preserves labels , if for all $e\in E$, it satisfies to equality
$\lambda(e)=\lambda^{\prime}(\eta(e))$. In this case, the pair $(\sigma,\eta)$
is called a morphism of labelled asynchronous systems
$({\mathcal{A}},\lambda,L)\to({\mathcal{A}}^{\prime},\lambda^{\prime},L)$.
The following statement is a reformulation of the characterization of
$Pom_{L}$-morphisms given in [3, Prop.16].
###### Proposition 2.1
A morphism
$(\sigma,\eta):({\mathcal{A}},\lambda,L)\to({\mathcal{A}}^{\prime},\lambda^{\prime},L)$
between labelled asynchronous systems is $Pom_{L}$-open if and only if the
morphism $(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ is open and
preserves labels.
This proposition allows us to mean by $Pom_{L}$-open morphisms the open
morphisms, preserving labels.
###### Definition 2.1
[3] Let $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ be labelled asynchronous
systems. If there exists a labelled asynchronous system
$({\mathcal{A}}^{\prime\prime},\lambda^{\prime\prime},L)$ with $Pom_{L}$-open
morphisms
$({\mathcal{A}}^{\prime\prime},\lambda^{\prime\prime},L)\stackrel{{\scriptstyle(\sigma,\eta)}}{{\to}}({\mathcal{A}},\lambda,L)$
and
$({\mathcal{A}}^{\prime\prime},\lambda^{\prime\prime},L)\stackrel{{\scriptstyle(\sigma^{\prime},\eta^{\prime})}}{{\to}}({\mathcal{A}}^{\prime},\lambda^{\prime},L)$,
then $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ are called $Pom_{L}$-bisimilar.
###### Proposition 2.2
Let $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ be $Pom_{L}$-bisimilar labelled
asynchronous systems. For every $a_{1}\in E$ satisfying $s_{0}\cdot a_{1}\in
S$, there exists $a^{\prime}_{1}\in E^{\prime}$ such that the following two
properties hold:
* •
$s^{\prime}_{0}\cdot a^{\prime}_{1}\in S^{\prime}$;
* •
labelled asynchronous systems $({\mathcal{A}}(s_{1}),\lambda,L)$ and
$({\mathcal{A}}(s^{\prime}_{0}\cdot a^{\prime}_{1}),\lambda^{\prime},L)$ are
$Pom_{L}$-bisimilar.
Proof. Given labelled asynchronous systems are $Pom_{L}$-bisimilar. Hence,
there are $({\mathcal{A}}^{\prime\prime},\lambda^{\prime\prime},L)$ and
$Pom_{L}$-open morphisms
$({\mathcal{A}},\lambda,L)\stackrel{{\scriptstyle(\sigma,\eta)}}{{\longleftarrow}}({\mathcal{A}}^{\prime\prime},\lambda^{\prime\prime},L)\stackrel{{\scriptstyle(\sigma^{\prime},\eta^{\prime})}}{{\longrightarrow}}({\mathcal{A}}^{\prime},\lambda^{\prime},L).$
Morphism $(\sigma,\eta)$ is open. It follows by property (ii) of Definition
1.4 that there exists a transition
$(s^{\prime\prime}_{0},a^{\prime\prime}_{1},s^{\prime\prime}_{1})$ satisfying
conditions $\eta(a^{\prime\prime}_{1})=a_{1}$ and
$\sigma(s^{\prime\prime}_{1})=s_{1}$ (Fig. 2). In other words, there exists
$a^{\prime\prime}_{1}\in E^{\prime\prime}$ such that
$\eta(a^{\prime\prime}_{1})=a_{1}$ and $\sigma(s^{\prime\prime}_{0}\cdot
a^{\prime\prime}_{1})=s^{\prime\prime}_{1}$. By Proposition 1.1, the morphism
$(\sigma,\eta):{\mathcal{A}}^{\prime\prime}(s^{\prime\prime}_{1})\to{\mathcal{A}}(s_{1})$
is open.
$\textstyle{s_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\textstyle{s^{\prime\prime}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{a^{\prime\prime}_{1}}$$\scriptstyle{\sigma^{\prime}}$$\textstyle{s^{\prime}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\prime}(a^{\prime\prime}_{1})}$$\textstyle{s_{1}}$$\textstyle{s^{\prime\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}~{}\sigma}$$\scriptstyle{\sigma^{\prime}}$$\textstyle{\sigma^{\prime}(s^{\prime\prime}_{1})}$
Figure 2: To the construction of open morphisms.
The map $\sigma^{\prime}$ of the morphism
$(\sigma^{\prime},\eta^{\prime}):{\mathcal{A}}^{\prime\prime}\to{\mathcal{A}}^{\prime}$
is total. It follows that $\sigma^{\prime}(s^{\prime\prime}_{1})\in
S^{\prime}$. By Proposition 1.1, the morphism
$(\sigma,\eta):{\mathcal{A}}^{\prime\prime}(s^{\prime\prime}_{1})\stackrel{{\scriptstyle(\sigma^{\prime},\eta^{\prime})}}{{\to}}{\mathcal{A}}^{\prime}(\sigma^{\prime}(s^{\prime\prime}_{1}))$
is open. The morphisms $(\sigma,\eta)$ and $(\sigma^{\prime},\eta^{\prime})$
preserve labels. By putting
$a^{\prime}_{1}=\eta^{\prime}(a^{\prime\prime}_{1})$ and
$s^{\prime}_{1}=\sigma^{\prime}(s^{\prime\prime}_{1})$, we obtain the desired.
$\Box$
###### Corollary 2.3
Let $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ be labelled asynchronous
systems. For every $w=a_{1}\cdots a_{k}\in E^{*}$ with $k\geqslant 0$
satisfying the condition $s_{0}\cdot w\in S$, there exists a word
$w^{\prime}=a^{\prime}_{1}\cdots a^{\prime}_{k}\in E^{\prime*}$ such that the
following two properies hold:
* •
$s^{\prime}_{0}\cdot w^{\prime}\in S^{\prime}$;
* •
the labelled asynchronous systems $({\mathcal{A}}(s_{0}\cdot w),\lambda,L)$
and $({\mathcal{A}}^{\prime}(s^{\prime}_{0}\cdot
w^{\prime}),\lambda^{\prime},L)$ are $Pom_{L}$-bisimilar.
Proof. For $k=0$, the word $w$ is empty, that is $w=1$. Taking $w^{\prime}=1$,
we get the $Pom_{L}$-bisimilar labelled asynchronous systems
$({\mathcal{A}},\lambda,L)$ and $({\mathcal{A}}^{\prime},\lambda^{\prime},L)$.
For $k=1$, the assertion follows from Proposition 2.2. Assuming that the
assertion is true for some $k>0$, we can prove by Proposition 2.2, that it
holds for $k+1$. So, it is true for all $k\geqslant 0$. $\Box$
### 2.2 Open maps and surjectivity
Let ${\mathcal{A}}$ be an asynchronous system. Denote by
$Q_{0}({\mathcal{A}})=S(s_{0})$ the set of all reachable states $s\in S$. For
every $n>0$, we consider sets
$Q_{n}({\mathcal{A}})=\\{(s,e_{1},\cdots,e_{n})\in S(s_{0})\times E^{n}\\\
s\cdot e_{1}\cdots e_{n}\in S~{}\&~{}(e_{i},e_{j})\in I\mbox{ for all
}1\leqslant i<j\leqslant n\\}$
Let $(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ be a morphism of
asynchronous system. If $\eta:E\to E^{\prime}$ is total, then for all
$n\geqslant 0$ the maps $Q_{n}(\sigma,\eta):Q_{n}({\mathcal{A}})\to
Q_{n}({\mathcal{A}}^{\prime})$ are defined by the formula
$Q_{n}(\sigma,\eta)(s,e_{1},\cdots,e_{n})=(\sigma(s),\eta(e_{1}),\cdots,\eta(e_{n})).$
###### Lemma 2.4
If $(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ is open, then for
every reachable $s^{\prime}\in S^{\prime}$ there exists $s\in S$ such that
$\sigma(s)=s^{\prime}$.
Proof. We have $\sigma(s_{0})=s^{\prime}_{0}$. If $s^{\prime}$ is reachable,
then there exists a path $\sigma(s_{0})=s^{\prime}_{0}\stackrel{{\scriptstyle
a^{\prime}_{1}}}{{\to}}s^{\prime}_{1}\to\cdots\to
s^{\prime}_{n-1}\stackrel{{\scriptstyle
a^{\prime}_{n}}}{{\to}}s^{\prime}_{n}$. The morphism $(\sigma,\eta)$ is open.
Hence for $a^{\prime}_{1}$ and $s^{\prime}_{1}$, there are $a_{1}$ and $s_{1}$
satisfying $\eta(a_{1})=a^{\prime}_{1}$ and $\sigma(s_{1})=s^{\prime}_{1}$.
Then we find $a_{2}\in E$ satisfying $\eta(a_{2})=a^{\prime}_{2}$. And so on
till we find $a_{n}\in E$ such that $\eta(a_{n})=a^{\prime}_{n}$ and
$\sigma(s_{n})=s^{\prime}$. Desired element $s$ will be equal to $s_{n}$.
$\Box$
###### Proposition 2.5
If a morphism $(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ is open,
then the maps $Q_{n}({\mathcal{A}})\to Q_{n}({\mathcal{A}}^{\prime})$ are
surjective.
Proof. Prove for $n=0$. If $s^{\prime}$ is reachable, then there exists a path
$\sigma(s_{0})=s^{\prime}_{0}\stackrel{{\scriptstyle
a^{\prime}_{1}}}{{\to}}s^{\prime}_{1}\stackrel{{\scriptstyle
a^{\prime}_{2}}}{{\to}}\ldots\stackrel{{\scriptstyle
a^{\prime}_{n}}}{{\to}}s^{\prime}_{k}=s^{\prime}.$
There are $a_{1}\in E$ and $s_{1}\in S$ for which $\eta(a_{1})=a^{\prime}_{1}$
and $(\sigma,\eta)(s_{0}\stackrel{{\scriptstyle
a_{1}}}{{\to}}s_{1})=(s^{\prime}_{0}\stackrel{{\scriptstyle
a^{\prime}_{1}}}{{\to}}s^{\prime}_{1})$:
$\textstyle{s_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\scriptstyle{\sigma}$$\textstyle{s^{\prime}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a^{\prime}_{1}}$$\textstyle{s_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{s^{\prime}_{1}}$
We have $\sigma(s_{1})=s^{\prime}_{1}$. There are $a_{2}\in E$ and $s_{2}\in
S$ satisfying $\sigma(s_{2})=s^{\prime}_{2}$ and $\eta(a_{2})=a^{\prime}_{2}$
and so on. By induction, we obtain $s_{k}\in S$ such that
$\sigma(s_{k})=s^{\prime}_{k}=s^{\prime}$. Therefore, $\sigma:S(s_{0})\to
S^{\prime}(s^{\prime}_{0})$ is surjective.
For $n=1$, the map $\\{(s,e_{1})|se_{1}\in
S\\}\to\\{(\sigma(s),e^{\prime}_{1})|\sigma(s)e^{\prime}_{1}\in S^{\prime}\\}$
is surjective by property (ii) of open morphisms.
Let $n\geqslant 2$. For each $s\in S(s_{0})$, consider the set
$Q_{n}({\mathcal{A}},s)=\\{(s,e_{1},\cdots,e_{n})\in\\{s\\}\times E^{n}~{}|\\\
s\cdot e_{1}\cdots e_{n}\in S~{}\&~{}(e_{i},e_{j})\in I\mbox{ for all
}1\leqslant i<j\leqslant n\\}$
and
$Q_{n}({\mathcal{A}}^{\prime},\sigma(s))=\\{(\sigma(s),e^{\prime}_{1},\cdots,e^{\prime}_{n})\in\\{\sigma(s)\\}\times{E^{\prime}}^{n}|\\\
\sigma(s)\cdot e^{\prime}_{1}\cdots e^{\prime}_{n}\in
S^{\prime}~{}\&~{}(e^{\prime}_{i},e^{\prime}_{j})\in I\mbox{ for all
}1\leqslant i<j\leqslant n.\\}$
For any $(\sigma(s),e^{\prime}_{1},\cdots,e^{\prime}_{n})\in
Q_{n}({\mathcal{A}}^{\prime},\sigma(s))$, there are $e_{1}$, $e_{2}$, …,
$e_{n}\in E$ for which $s_{1}=s\cdot e_{1}\in S$, $s_{2}=s\cdot e_{1}e_{2}\in
S$, …, $s_{n}=s\cdot e_{1}\cdots e_{n}\in S$, wherein
$\eta(e_{1})=e^{\prime}_{1}$, …, $\eta(e_{n})=e^{\prime}_{n}$.
By induction on $n$, we will prove that $(e_{i},e_{j})\in I$ for all
$1\leqslant i<j\leqslant n$. For this purpose, we assume that
$(e_{i},e_{j})\in I$ for all $1\leqslant i<j\leqslant n-1$. And we show that
$(e_{i},e_{n})\in I$ for all $1\leqslant i\leqslant n-1$. We have
$(s_{n-2},e_{n-1},s_{n-1})\in{\rm Tran}$, $(s_{n-1},e_{n},s_{n})\in{\rm
Tran}$, and $(\eta(e_{n-1}),\eta(e_{n}))\in I^{\prime}$. It follows by the
property (iii) that $(e_{n-1},e_{n})\in I$. By Axiom (ii) for a state space,
there is $t\in S$ such that $(s_{n-2},e_{n},t)\in{\rm Tran}$ and
$(t,e_{n-1},s_{n})\in{\rm Tran}$. It follows from
$(\eta(e_{n-2}),\eta(e_{n}))\in I^{\prime}$, that $(e_{n-2},e_{n})\in I$.
Again by Axiom (ii), there is $t_{1}\in S$ such that
$(s_{n-3},e_{n},t_{1})\in{\rm Tran}$ and $(t_{1},e_{n-2},s_{n})\in{\rm Tran}$.
It follows from $(\eta(e_{n-3}),\eta(e_{n}))\in I^{\prime}$, that
$(e_{n-3},e_{n})\in I$, and so on. In the end, we obtain $(e_{i},e_{n})\in I$
for all $1\leqslant i\leqslant n-1$. Consequently $(e_{i},e_{j})\in I$ for all
$1\leqslant i<j\leqslant n$. Thus, $(s,e_{1},\ldots,e_{n})\in
Q_{n}({\mathcal{A}},s)$. Therefore for every
$(s^{\prime},e^{\prime}_{1},\ldots,e^{\prime}_{n})\in Q_{n}({\mathcal{A}})$,
there is $(s,e_{1},\ldots,e_{n})\in Q_{n}({\mathcal{A}})$ mapped to
$(s^{\prime},e^{\prime}_{1},\ldots,e^{\prime}_{n})\in Q_{n}({\mathcal{A}})$.
$\Box$
###### Remark 2.2
The converse is not true. There are morphisms $(\sigma,\eta)$, for which the
map $Q_{n}(\sigma,\eta)$ is surjective for all $n\geqslant 0$, but the
$(\sigma,\eta)$ is not $Pom_{pt}$-open. For example, $S=\\{s_{0}\\}$,
$E=\\{a,b,c\\}$, $I=\\{(a,b),(b,a)\\}$, $S^{\prime}=\\{s^{\prime}_{0}\\}$,
$E^{\prime}=\\{a^{\prime},b^{\prime}\\}$,
$I^{\prime}=\\{(a^{\prime},b^{\prime}),(b^{\prime},a^{\prime})\\}$. Figure 3
shows the independence graphs and the map $\eta:E\to E^{\prime}$.
$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\eta(b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a}$$\textstyle{c}$$\textstyle{\eta(a)=\eta(c)}$
Figure 3: Example of surjection which is not $Pom_{pt}$
We have $(\eta(b),\eta(c))\in I^{\prime}$, but $(b,c)\notin I$. Hence, the
morphism $(\sigma,\eta)$ is not open.
For an reachable state $s\in S$ of asynchronous system
${\mathcal{A}}=(S,s,E,I,{\rm Tran})$, let ${\mathcal{A}}(s)=(S,s,E,I,{\rm
Tran})$ be the asynchronous system which differs only by the initial state.
###### Corollary 2.6
If $(\sigma,\eta):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ is open, then for
each reachable state $s\in S$, the maps $Q_{n}({\mathcal{A}}(s))\to
Q_{n}({\mathcal{A}}^{\prime}(\sigma(s)))$ are surjective for all $n\geqslant
0$.
## 3 Homology groups of asynchronous systems
We introduce the homology groups of labelled asynchronous systems. We will
prove that bisimulation equivalence is stronger than property to have
isomorphic homology groups.
### 3.1 Computing homology groups of simplicial schemes
Recall that a simplicial scheme $(A,{\mathfrak{M}})$ consists of a set $A$ of
vertices and a set ${\mathfrak{M}}$ of finite nonempty subsets $S\subseteq A$
satisfying the following conditions
* •
$(\forall a\in A)$ $\\{a\\}\in{\mathfrak{M}}$,
* •
$(\forall S,S^{\prime}\subseteq
A)~{}S\in{\mathfrak{M}}~{}\&~{}S^{\prime}\subseteq S\Rightarrow
S^{\prime}\in{\mathfrak{M}}$.
The elements of $\mathfrak{M}$ are called simplices. For $n\geqslant 0$, a
simplex $S$ is called $n$-dimensional or $n$-simplex if number $|S|$ of its
elements equals $n+1$.
Let $(A,\mathfrak{M})$ be a simplicial scheme. For the computing its homology
groups $H_{n}(A,\mathfrak{M})$, we define an arbitrary total order relation on
$A$. Consider the complex
$0\leftarrow{\,\mathbb{Z}}\mathfrak{M}_{0}\stackrel{{\scriptstyle
d_{1}}}{{\leftarrow}}{\,\mathbb{Z}}\mathfrak{M}_{1}\stackrel{{\scriptstyle
d_{2}}}{{\leftarrow}}{\,\mathbb{Z}}\mathfrak{M}_{2}\leftarrow\cdots\leftarrow{\,\mathbb{Z}}\mathfrak{M}_{n-1}\stackrel{{\scriptstyle
d_{n}}}{{\leftarrow}}{\,\mathbb{Z}}\mathfrak{M}_{n}\leftarrow\cdots$
where
$\mathfrak{M}_{n}=\\{(a_{0},a_{1},\ldots,a_{n})|a_{0}<a_{1}<\cdots<a_{n}~{}\&~{}\\{a_{0},a_{1},\ldots,a_{n}\\}\in\mathfrak{M}\\}$.
Elements of $\mathfrak{M}_{n}$ are called ordered $n$-simplices. Here
${\,\mathbb{Z}}\mathfrak{M}_{n}$ denotes the free Abelian group generated by
ordered $n$-simplices. The differentials $d_{n}$ are defined on ordered
$n$-simplices by the formula
$d_{n}(a_{0},a_{1},\ldots,a_{n})=\sum_{i=0}^{n}(-1)^{i}(a_{0},\ldots,\widehat{a_{i}},\ldots,a_{n})$
where $\widehat{a_{i}}$ denotes the operation of removing the symbol $a_{i}$
from the tuple. We will suppose that the sets of $n$-simplices are finite. In
this case, the differentials $d_{n}$ can be specified using integer matrices.
Each column of the matrix for $d_{n}$ corresponds to a tuple
$(a_{0},a_{1},\ldots,a_{n})\in\mathfrak{M}_{n}$. Each string corresponds to
$(a_{0},\ldots,a_{n-1})\in\mathfrak{M}_{n-1}$. For each column
$(a_{0},a_{1},\ldots,a_{n})$ and string
$(a_{0},\ldots,\widehat{a_{i}},\ldots,a_{n})$, at their intersection, the
entry equals $(-1)^{i}$. Other entries of the matrix equal $0$. For
calculating the homology groups, each matrix $d_{n}$ is reduced to the Smith
normal form. The homology groups $H_{n}=Ker(d_{n})/Im(d_{n+1})$ of this
complex is equal to
${\,\mathbb{Z}}^{|\mathfrak{M}_{n}|-rank(d_{n})-rank(d_{n+1})}\oplus{\,\mathbb{Z}}/\delta_{1}{\,\mathbb{Z}}\oplus\cdots\oplus{\,\mathbb{Z}}/\delta_{r}{\,\mathbb{Z}}$
where $r=rank(d_{n+1})$ and $\delta_{1},\cdots,\delta_{r}$ is the non-zero
diagonal entries of the Smith normal form for the matrix $d_{n+1}$.
### 3.2 Homology groups of labelled asynchronous systems
Let $({\mathcal{A}},\lambda,L)$ be a labelled asynchronous system.
Introduce homology groups of the labelled asynchronous systems. For this
purpose, consider the simplicial scheme $(\lambda^{+}E,{\mathfrak{M}})$ whose
vertices are the elements $\lambda(a)$, where $a\in E$ are elements for which
there are $s,s^{\prime}\in S(s_{0})$ satisfying $(s,a,s^{\prime})\in{\rm
Tran}$. Thus
$\lambda^{+}E=\\{\lambda(a)~{}|~{}(\exists s\in S(s_{0}))s\cdot a\in S\\}.$
Simplices are finite sets $\\{\lambda(a_{1}),\ldots,\lambda(a_{k})\\}$,
$k\geqslant 1$, for which the following two conditions hold:
* •
$(a_{i},a_{j})\in I$, for all $1\leqslant i<j\leqslant k$;
* •
there are $s\in S(s_{0})$ for which $s\cdot a_{1}\cdots a_{k}\in S$.
###### Remark 3.1
1. (i)
For every $(s,a_{1},\ldots,a_{k})\in Q_{k}({\mathcal{A}})$, we include the set
$\\{\lambda(a_{1}),\ldots,\lambda(a_{k})\\}\\}$ in ${\mathfrak{M}}$.
2. (ii)
If the elements are duplicated in
$\\{\lambda(a_{1}),\ldots,\lambda(a_{k})\\}$, then we remove them. For example
$\\{a,b,a,c,a,b\\}=\\{a,b,c\\}$.
###### Definition 3.2
Homology groups $H_{n}({\mathcal{A}},\lambda,L)$ of a labelled asynchronous
system is the homology groups $H_{n}(\lambda^{+}E,{\mathfrak{M}})$ of the
constructed simplicial scheme.
###### Example 3.3
Consider an asynchronous system ${\mathcal{A}}=(S,s_{0},E,I,{\rm Tran})$ where
$S=\\{000,001,010,011,100,101,110\\}$, $s_{0}=000$,
$E=\\{a_{1},a_{2},a_{3}\\}$,
$I=\\{(a_{1},a_{2}),(a_{2},a_{1}),(a_{1},a_{3}),(a_{3},a_{1}),(a_{2},a_{3}),(a_{3},a_{2})\\}$.
Transitions correspond to arrows of the diagram:
---
$\textstyle{001\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\textstyle{101}$$\textstyle{000\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{3}}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\textstyle{011}$$\textstyle{100\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{3}}$$\scriptstyle{a_{2}}$$\textstyle{010\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{3}}$$\scriptstyle{a_{1}}$$\textstyle{110}$
Let $L=E$ and let the label function $\lambda:E\to L$ is defined as
$\lambda(a)=a$ for all $a\in E$. The simplicial scheme consists of vertices
$E=\\{a_{1},a_{2},a_{3}\\}$ and simplices $\\{a_{1},a_{2}\\}$,
$\\{a_{1},a_{3}\\}$, $\\{a_{2},a_{3}\\}$. Define the order on vertices by
$a_{1}<a_{2}<a_{3}$. Homology groups is computed by the complex
$0\leftarrow{\,\mathbb{Z}}\\{a_{1},a_{2},a_{3}\\}\stackrel{{\scriptstyle
d_{1}}}{{\leftarrow}}{\,\mathbb{Z}}\\{(a_{1},a_{2}),(a_{1},a_{3}),(a_{2},a_{3})\\}\leftarrow
0$
Matrix for $d_{1}$ equals
$\displaystyle\quad\begin{array}[]{cccc}&~{}~{}~{}~{}(a_{1},a_{2})&~{}~{}~{}(a_{1},a_{3})&~{}~{}(a_{2},a_{3})\end{array}$
$\displaystyle\begin{array}[]{l}a_{1}\\\ a_{2}\\\
a_{3}\end{array}\quad\left(\begin{array}[]{ccc}~{}~{}~{}-1&~{}~{}~{}~{}~{}~{}~{}-1&~{}~{}~{}~{}~{}~{}~{}0\\\
~{}~{}~{}~{}1&~{}~{}~{}~{}~{}~{}~{}~{}0&~{}~{}~{}~{}~{}~{}-1\\\
~{}~{}~{}~{}0&~{}~{}~{}~{}~{}~{}~{}~{}1&~{}~{}~{}~{}~{}~{}~{}1\end{array}\quad\right)$
The Smith normal form for $d_{1}$ equals
$\displaystyle\quad\begin{array}[]{cccc}&~{}~{}~{}~{}(a_{1},a_{2})&~{}~{}~{}(a_{1},a_{3})&~{}~{}(a_{2},a_{3})\end{array}$
$\displaystyle\begin{array}[]{l}a_{1}\\\ a_{2}\\\
a_{3}\end{array}\quad\left(\begin{array}[]{ccc}~{}~{}~{}~{}1&~{}~{}~{}~{}~{}~{}~{}~{}0&~{}~{}~{}~{}~{}~{}~{}0\\\
~{}~{}~{}~{}0&~{}~{}~{}~{}~{}~{}~{}~{}1&~{}~{}~{}~{}~{}~{}~{}0\\\
~{}~{}~{}~{}0&~{}~{}~{}~{}~{}~{}~{}~{}0&~{}~{}~{}~{}~{}~{}~{}0\end{array}\quad\right)$
It follows that
$H_{0}({\mathcal{A}},\lambda,L)={\,\mathbb{Z}}^{3-0-2}\oplus{\,\mathbb{Z}}/1{\,\mathbb{Z}}\oplus{\,\mathbb{Z}}/1{\,\mathbb{Z}}\cong{\,\mathbb{Z}}$,
$H_{1}({\mathcal{A}},\lambda,L)={\,\mathbb{Z}}^{3-2-0}\cong{\,\mathbb{Z}}$.
Other homology groups equal $0$.
The complex for computing groups $H_{n}({\mathcal{A}}(s),\lambda,L)$ for
$s=001$ has unique non-zero term ${\,\mathbb{Z}}\\{a_{1},a_{2}\\}$. It follows
$H_{n}({\mathcal{A}}(s),\lambda,L)=\left\\{\begin{array}[]{cl}{\,\mathbb{Z}}\oplus{\,\mathbb{Z}},&\mbox{
if }n=0,\\\ 0,&\mbox{ if }n>0.\\\ \end{array}\right.$
The complex for computing $H_{n}({\mathcal{A}}(s),\lambda,L)$ for $s=011$
consists of zeros. Therefore $H_{n}({\mathcal{A}}(011),\lambda,L)=0$ for all
$n\geqslant 0$.
###### Theorem 3.1
If labelled asynchronous systems $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ are $Pom_{L}$-bisimilar, then
their homology groups are isomorphic.
Proof. Denote by $\mathfrak{M}$ and $\mathfrak{M^{\prime}}$ the simplicial
schemes corresponded to the labelled asynchronous systems. If the labelled
asynchronous systems are $Pom_{L}$-bisimilar, then there is a labelled
asynchronous system together with the morphisms
$({\mathcal{A}},\lambda,L)\stackrel{{\scriptstyle(\sigma,\eta)}}{{\longleftarrow}}({\mathcal{A}}^{\prime\prime},\lambda^{\prime\prime},L)\stackrel{{\scriptstyle(\sigma^{\prime},\eta^{\prime})}}{{\longrightarrow}}({\mathcal{A}}^{\prime},\lambda^{\prime},L)~{}.$
Let $P^{f}(L)$ be the set of all finite subsets of $L$. Consider a maps
$\lambda_{n}:Q_{n}({\mathcal{A}})\to P^{f}(L)$ acting as
$\lambda(s,a_{1},\ldots,a_{n})=\\{\lambda(a_{1}),\ldots,\lambda(a_{n})\\}$.
The function $\lambda$ can have equal values. Hence, the set
$\\{\lambda(a_{1}),\ldots,\lambda(a_{n})\\}$ can contain $<n$ elements
For $n=0$, we let $\lambda_{0}(s)=\emptyset$. By Proposition 2.5 the maps
$Q_{n}(\sigma,\eta)$ and $Q_{n}(\sigma^{\prime},\eta^{\prime})$ are
surjective. The pairs $(\sigma,\eta)$ and $(\sigma^{\prime},\eta^{\prime})$
are morphisms of asynchronous systems. Hence, the following diagram is
commutative
---
$\textstyle{Q_{n}({\mathcal{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{n}}$$\textstyle{Q_{n}({\mathcal{A}}^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{n}^{\prime\prime}}$$\scriptstyle{Q_{n}(\sigma,\eta)}$$\scriptstyle{Q_{n}(\sigma^{\prime},\eta^{\prime})}$$\textstyle{Q_{n}({\mathcal{A}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{\prime}_{n}}$$\textstyle{P^{f}(L)}$
We have the equalities
$Im(\lambda_{n})=Im(\lambda^{\prime\prime}_{n})=Im(\lambda^{\prime}_{n}).$
Consequently the simplicial sets $\mathfrak{M}$ and $\mathfrak{M^{\prime}}$
are equal. Therefore, the groups $H_{n}(\mathfrak{M})$ and
$H_{n}(\mathfrak{M^{\prime}})$ are isomorphic. $\Box$
###### Corollary 3.2
Let $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ be $Pom_{L}$-bisimilar
asynchronous systems. For each $w=a_{1}\cdots a_{k}\in E^{*}$, $k\geqslant 0$,
satifying $s_{0}\cdot w\in S$ there is a word $w^{\prime}=a^{\prime}_{1}\cdots
a^{\prime}_{k}\in E^{\prime*}$ such that $s^{\prime}_{0}\cdot w^{\prime}\in
S^{\prime}$ and
$(\forall n\geqslant 0)~{}H_{n}({\mathcal{A}}(s_{0}\cdot w),\lambda,L)\cong
H_{n}({\mathcal{A}}^{\prime}(s^{\prime}_{0}\cdot
w^{\prime}),\lambda^{\prime},L).$ (1)
Proof. By Proposition 2.3, in this case for the word $w$, there exists
$w^{\prime}$ for which $({\mathcal{A}}(s_{0}\cdot w),\lambda,L)$ and
$({\mathcal{A}}^{\prime}(s^{\prime}_{0}\cdot w^{\prime}),\lambda^{\prime},L)$
are $Pom_{L}$-bisimilar. Application of Theorem 3.1 to the obtained labelled
asynchronous systems leads us to desired isomorphism of the homology groups.
$\Box$
###### Example 3.4
Consider well known labelled asynchronous systems
$\textstyle{s_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\textstyle{s_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{s_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{s_{3}}$$\textstyle{s_{4}}$
$\textstyle{s_{0}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{s_{1}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{c}$$\textstyle{s_{2}^{\prime}}$$\textstyle{s_{3}^{\prime}}$
The first asynchronous system ${\mathcal{A}}$ consists of
$S=\\{s_{0},s_{1},s_{2},s_{3},s_{4}\\}$, $E=\\{a_{1},a_{2},b,c\\}$,
$I=\emptyset$, ${\rm
Tran}=\\{(s_{0},a_{1},s_{1}),(s_{0},a_{2},s_{2}),(s_{1},b,s_{3}),(s_{2},c,s_{4})\\}$.
The second asynchronous system ${\mathcal{A}}^{\prime}$ consists of
$S^{\prime}=\\{s^{\prime}_{0},s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3}\\}$,
$E^{\prime}=\\{a,b,c\\}$, $I^{\prime}=\emptyset$, ${\rm
Tran}=\\{(s^{\prime}_{0},a,s^{\prime}_{1}),(s^{\prime}_{1},b,s^{\prime}_{2}),(s^{\prime}_{1},c,s^{\prime}_{3})\\}$.
The label functions have values in $L=\\{a,b,c\\}$ and are defined by
$\lambda(a_{1})=\lambda(a_{2})=\lambda^{\prime}(a)=a,~{}\lambda(b)=\lambda^{\prime}(b)=b,~{}\lambda(c)=\lambda^{\prime}(c)=c.$
Compute $H_{n}({\mathcal{A}}(s_{1}),\lambda,L)$ by the complex
$0\leftarrow{\,\mathbb{Z}}\\{b\\}\leftarrow 0$. We have
$H_{n}({\mathcal{A}}(s_{1}),\lambda,L)=\left\\{\begin{array}[]{cl}{\,\mathbb{Z}},&\mbox{
if }n=0,\\\ 0,&\mbox{ if }n>0.\\\ \end{array}\right.$
The groups $H_{n}({\mathcal{A}}^{\prime}(s_{1}^{\prime}),\lambda^{\prime},L)$
are isomorphic to homology groups of the complex
$0\leftarrow{\,\mathbb{Z}}\\{b\\}\oplus{\,\mathbb{Z}}\\{c\\}\leftarrow 0$. We
have
$H_{n}({\mathcal{A}}^{\prime}(s_{1}^{\prime}),\lambda^{\prime},L)=\left\\{\begin{array}[]{cl}{\,\mathbb{Z}}\oplus{\,\mathbb{Z}},&\mbox{
if }n=0,\\\ 0,&\mbox{ if }n>0.\\\ \end{array}\right.$
The groups $H_{0}({\mathcal{A}}(s_{1}),\lambda,L)$ and
$H_{0}({\mathcal{A}}^{\prime}(s_{1}^{\prime}),\lambda^{\prime},L)$ are not
isomorphic. It follows from Corollary 3.2 that $({\mathcal{A}},\lambda,L)$ and
$({\mathcal{A}}^{\prime},\lambda^{\prime},L)$ are not $Pom_{L}$-bisimilar.
## 4 Homology groups of labelled Petri nets
Recall some definitions from theory of Petri nets. Then consider homology
groups of labelled Petri nets and prove that for each simplicial scheme, there
is a labelled Petri net homological equivalent to this simplicial scheme.
### 4.1 Petri nets
We view “display” and “function” as synonyms. For a finite set $P$, let
${\,\mathbb{N}}^{P}$ denotes a set of all functions $M:P\to{\,\mathbb{N}}$,
where ${\,\mathbb{N}}=\\{0,1,2,\ldots\\}$ is the set of non-neganbve integers.
For any $M_{1},M_{2}\in{\,\mathbb{N}}^{P}$, define a sum $M_{1}+M_{2}$ as a
function with values $(M_{1}+M_{2})(p)=M_{1}(p)+M_{2}(p)$ for all $p\in P$.
Let $M_{1}\geqslant M_{2}$ if $M_{1}(p)\geqslant M_{2}(p)$ for all $p\in P$.
If $M_{1}\geqslant M_{2}$, then we can define a difference $M_{1}-M_{2}$ as
the function with the values $M_{1}(p)-M_{2}(p)$. Define a scalar product by
$M_{1}\cdot M_{2}=\sum_{p\in P}M_{1}(p)M_{2}(p)$.
A Petri net ${\,\cal N}=(P,T,pre,post,M_{0})$ consists of finite sets $P$ and
$T$ with two maps $pre:T\to{\,\mathbb{N}}^{P}$, $post:T\to{\,\mathbb{N}}^{P}$
and a function $M_{0}:P\to{\,\mathbb{N}}$ called initial marking. Elements
$p\in P$ are called places, and $t\in T$ are events. A marking is an arbitrary
function $M:P\to{\,\mathbb{N}}$.
$t_{1}$$t_{2}$$t_{3}$$p_{2}$$p_{1}$ Figure 4: Example of Petri net
A Petri net can be given as a directed graph whose vertices are places
depicted by circles, and events depicted by rectangles. Every arrow goes from
an event to a place or from a place to an event. For any $t\in T$, the number
entering into it arrows equals $pre(t)(p)$ and the number of arrows outgoing
from $t$ equals $post(t)(p)$. The initial marking is given by drawing the
points in each place. These points are called tokens. The number of tokens in
a place $p$ is equal to $M_{0}(p)$. If $M_{0}(p)=0$, then the place is empty.
Fig. 4 shows a Petri net ${\,\cal N}=(P,T,pre,post,M_{0})$ where
$P=\\{p_{1},p_{2}\\}$, $T=\\{t_{1},t_{2},t_{3}\\}$. The values
$pre(t_{i})(p_{j})$ and $post(t_{i})(p_{j})$, $1\leqslant i\leqslant 3$,
$1\leqslant j\leqslant 2$, are equal to the entries of the matrices
$(pre(t_{i})(p_{j}))=\left(\begin{array}[]{cc}0&0\\\ 1&1\\\
0&1\end{array}\right)\qquad(post(t_{i})(p_{j}))=\left(\begin{array}[]{cc}2&1\\\
0&0\\\ 0&0\end{array}\right)$
### 4.2 Labelled asynchronous system for a Petri net and its homology groups
Let ${\,\cal N}=(P,T,pre,post,M_{0})$ be a Petri net. Consider a corresponding
asynchronous system ${\mathcal{A}}({\,\cal N})=(S,s_{0},E,I,{\rm Tran})$, with
$S={\,\mathbb{N}}^{P}$, $s_{0}=M_{0}$, $E=T$. The relation of independence $I$
consists of pairs $(e_{1},e_{2})\in T\times T$ for which the scalar product
$(pre(e_{1})+post(e_{1})\cdot(pre(e_{2})+post(e_{2}))$ equals $0$. This means
that $e_{1}$ and $e_{2}$ do not have common input or output places. The set
${\rm Tran}$ consists of triples $(M,e,M^{\prime})$ where $M$ and $M^{\prime}$
are markings and $e\in T$ satifies two following conditions
* •
$M\geqslant pre(e)$,
* •
$M-pre(e)+post(e)=M^{\prime}$.
If $(M,e,M^{\prime})\in{\rm Tran}$, then we say that the marking $M^{\prime}$
is obtained from $M$ by operation of event $e\in T$. For example, for Petri
net in Fig. 4, we have $pre(t_{2})\geqslant M_{0}$. The operation of the event
$t_{2}$ leads to the new magking $M_{1}=M_{0}-pre(t_{2})+post(t_{2})$ (Fig.
5).
$t_{1}$$t_{2}$$t_{3}$$p_{2}$$p_{1}$ Figure 5: The marking obtained by
operation of the event $t_{2}$
Let $L$ be an arbitrary nonempty set. A Petri net ${\,\cal N}$ with a function
$\lambda:T\to L$ is called labelled. The asynchronous system
${\mathcal{A}}({\,\cal N})$ corresponding ${\mathcal{A}}$ has the set of
events $E=T$. Hence, for any labelled Petri nets, it is defined the labelled
asynchronous system $({\mathcal{A}}({\,\cal N}),\lambda,L)$.
###### Definition 4.1
Let $({\,\cal N},\lambda,L)$ be a labelled Petri net. Its homology groups
$H_{n}({\,\cal N},\lambda,L)$ are defined as $H_{n}({\mathcal{A}}({\,\cal
N}),\lambda,L)$, $n\geqslant 0$.
###### Example 4.2
Consider the Petri net ${\,\cal N}=(P,T,pre,post,M_{0})$, in Fig. 6. Let
$L=E=\\{t_{1},t_{2},t_{3},t_{4}\\}$, $\lambda(t_{i})=i$, for all $1\leqslant
i\leqslant 4$.
$\textstyle{t_{1}}$$\textstyle{\bigodot\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}p_{1}}$$\textstyle{p_{3}~{}\bigodot\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{t_{3}}$$\textstyle{t_{2}}$$\textstyle{\bigodot\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces~{}p_{2}}$$\textstyle{p_{4}~{}\bigodot\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{t_{4}}$
Figure 6: Example of computing the homology groups of Petri net
The relation $I$ contains the pairs $(t_{1},t_{3})$, $(t_{1},t_{4})$,
$(t_{2},t_{3})$, $(t_{2},t_{4})\\}$, $(t_{3},t_{1})$, $(t_{3},t_{2})$,
$(t_{4},t_{1})$, $(t_{4},t_{2})$. The simplicial set $(E,{\mathfrak{M}})$ give
the following sets of simplices
${\mathfrak{M}}_{0}=\\{t_{1},t_{2},t_{3},t_{4}\\},\\\
{\mathfrak{M}}_{1}=\\{(t_{1},t_{3}),(t_{1},t_{4}),(t_{2},t_{3}),(t_{2},t_{4})\\},~{}$
and ${\mathfrak{M}}_{n}=\emptyset$ for $n\geqslant 2$. We get the following
complex for the computing the homology groups of the labelled Petri nets:
$0\leftarrow{\,\mathbb{Z}}^{4}\stackrel{{\scriptstyle
d_{1}}}{{\longleftarrow}}{\,\mathbb{Z}}^{4}\leftarrow 0.$
The differential $d_{1}$ is given by the matrix
$\displaystyle\quad\begin{array}[]{cccccc}&~{}~{}(t_{1},t_{3})&(t_{1},t_{4})&(t_{2},t_{3})&(t_{2},t_{4})\end{array}$
$\displaystyle\begin{array}[]{l}t_{1}\\\ t_{2}\\\ t_{3}\\\
t_{4}\end{array}\left(\begin{array}[]{cccc}-1&~{}~{}~{}~{}~{}-1&~{}~{}~{}~{}~{}~{}0&~{}~{}~{}~{}~{}~{}0\\\
~{}0&~{}~{}~{}~{}~{}~{}0&~{}~{}~{}~{}~{}-1&~{}~{}~{}~{}~{}-1\\\
+1&~{}~{}~{}~{}~{}~{}0&~{}~{}~{}~{}~{}+1&~{}~{}~{}~{}~{}~{}0\\\
~{}0&~{}~{}~{}~{}~{}+1&~{}~{}~{}~{}~{}~{}0&~{}~{}~{}~{}~{}+1\end{array}\right)$
Its Smith normal form has the diagonal entries $(1,1,1,0)$. Consequently
$H_{0}({\,\cal N},\lambda,L)\cong H_{1}({\,\cal
N},\lambda,L)={\,\mathbb{Z}}\mbox{ and }H_{n}({\,\cal N},\lambda,L)=0\mbox{
for all }n\geqslant 2.$
A sequence of Abelian groups $A_{k}$, $k\geqslant 0$, is called to be finite
if there is $n\geqslant 0$ such that $A_{k}=0$ for all $k>n$.
###### Theorem 4.1
For an arbitrary finite sequence of finitely generated Abelian groups $A_{0}$,
$A_{1}$, $A_{2}$, …where $A_{0}$ is free and is not equal to $0$, there exists
a labelled Petri net such that its $k$th homology groups are isomorphic to
$A_{k}$ for all $k\geqslant 0$.
Proof. In this case by [17, Chapter 4, Exercise C-7], there exists a compact
polyhedron with homology groups $A_{k}$ for all $k\geqslant 0$. Compact
polyhedra are precisely the topological spaces admitting triangulations [17,
Chapter 3, Corollary 20]. Hence, there exists a simplicial scheme
$(X,{\mathfrak{M}})$ the homology groups of which are isomorphic to $A_{k}$.
Let $(E,{\mathfrak{M}}^{\prime})$ be a barycentric subdivision of the
simplicial set $(E,{\mathfrak{M}})$. Vertices $e\in E$ of the barycentric
subdivision are simplices $\sigma\in{\mathfrak{M}}$. Simplices of
$(E,{\mathfrak{M}}^{\prime})$ are finite sets of simplices
$\\{\sigma_{0},\ldots,\sigma_{n}\\}$ totally ordered by the relation
$\subseteq$. It means that there is a permutation
$(\sigma_{i_{0}},\ldots,\sigma_{i_{n}})$ such that
$\sigma_{i_{0}}\subset\sigma_{i_{1}}\subset\ldots\subset\sigma_{i_{n}}$. It is
well known that homology groups of $(E,{\mathfrak{M}}^{\prime})$ are
isomorphic to homology groups of $(X,{\mathfrak{M}})$. Define a relation $I$
on $E$ by
$(\sigma,\sigma^{\prime})\in
I\Leftrightarrow\sigma\subset\sigma^{\prime}\vee\sigma^{\prime}\subset\sigma.$
$p_{1}$$p_{2}$$p_{m}$$e_{1}$$e_{2}$$e_{m}$ Figure 7: The constructing of a
Petri net
Building a Petri net is similar to the construction of the work [18]. Denote
the elements of $E$ by $e_{1}$, $e_{2}$, …, $e_{m}$ where $m=|E|$. Consider
the Petri net depicted in Fig. 7. It consists of places $p_{i}$, connected
with the events $e_{i}$ by the arrows where $i=1,2,\ldots,m$. The initial
marking is defined as $M_{0}(p_{i})=1$ for all $i=1,2,\ldots,e_{m}$. For every
$(e_{i},e_{j})\notin I$, we make the events $e_{i}$ and $e_{j}$ to be
dependent by adding two arrows as shown in Fig. 8.
$p_{i}$$p_{j}$$e_{i}$$e_{j}$ Figure 8: Adding arrows to the Petri net
Let $L=E$ and let the label function defined as $\lambda(e_{i})=e_{i}$ for all
$i=1,\ldots,m$. For every $e_{i}\in E$, we have $s_{0}\cdot e_{i}\in S$. It
follows that the set of vertices of a simplicial scheme corresponding to the
Petri net is equal to $E$. For each nonempty subset
$\\{e_{i_{0}},\ldots,e_{i_{n}}\\}\subseteq E$ consisting of mutually
independent elements, we have $s_{0}\cdot e_{i_{0}}\cdots e_{i_{n}}\in S$.
Consequently the simplicial set corresponding to the Petri net is equal to
$(E,{\mathfrak{M}}^{\prime})$. Thus, $H_{n}({\,\cal N},\lambda,L)=A_{n}$ for
all $n\geqslant 0$. $\Box$
###### Corollary 4.2
For any finite sequence of finitely generated Abelian groups $A_{0}$, $A_{1}$,
$A_{2}$, …where $A_{0}$ is free and non-zero, there is a labelled asynchronous
system the $k$th homology groups of which are isomorphic to $A_{k}$ for all
$k\geqslant 0$.
## References
* [1] A. Joyal, M. Nielsen and G. Winskel, Bisimulation from open maps, LICS93 BRICS Report RS-94-7, Aarhus Univ., 1994. 42 pp.
* [2] R. Milner, Communication and concurrency. International Series in Computer Science (Prentice Hall, New York, 1989).
* [3] M. Nielsen and G. Winskel, Petri nets and bisimulation, Theoret. Comput. Sci., 153:1-2 (1996) 211–244.
* [4] M. Herlihy and N. Shavit, The Topological Structure of Asynchronous Computability, Journal of ACM, 46:6 (1999) 858–923.
* [5] E. Goubault and T.P. Jensen, Homology of higher dimensional automata, Lecture Notes in Computer Science, Vol. 630, (Springer, Berlin, 1992) 254–268.
* [6] E. Goubault, The Geometry of Concurrency, Ph.D. Thesis, Ecole Normale Supérieure, 1995, 349 p.
* [7] L. Fajstrup, M. Raußen, E.Goubault. Algebraic topology and concurrency, Theoret. Comput. Sci. 357:1-3 (2006) 241–278.
* [8] E. Goubault, E. Haucourt and S. Krishnan, Covering space theory for directed topology, Theor. Appl. Categ. 22:9 (2009) 252–268.
* [9] U. Fahrenberg, A. Legay, History-Preserving Bisimilarity for Higher-Dimensional Automata via Open Maps, arXiv:1209.4927v2 [cs.LO], (Cornell Ubiversity, New York, 2012).
* [10] A. Husainov, On the homology of small categories and asynchronous transition systems, Homology Homotopy Appl., 6:1 (2004) 439–471.
* [11] A. A. Khusainov, V. E. Lopatkin, I. A. Treshchev, Studying a mathematical model of parallel computation by algebraic topology methods, Journal of Applied and Industrial Math. 3:3 (2009) 353-363.
* [12] A. A. Husainov, The cubical homology of trace monoids, Far Eastern Math. Journal 12:1 (2012) 108–122
http://mi.mathnet.ru/eng/dvmg/v12/i1/p108
* [13] A. A. Husainov, The Homology of Partial Monoid Actions and Petri Nets, Appl. Categor. Struct. (2012) DOI: 10.1007/s10485-012-9280-9.
* [14] G. Winskel and M. Nielsen, Models for Concurrency, in: Abramsky, Gabbay and Maibaum, eds., Handbook of Logic in Computer Science, Vol.4 (Oxford University Press, Oxford, 1995) 1–148.
* [15] M. A. Bednarczyk, Categories of Asynchronous Systems, Ph.D. thesis, University of Sussex, Report No. 1/88, 1988.
* [16] V. Diekert, Y. Métivier, Partial Commutation and Traces, in: Handbook of formal languages, Vol. 3, ( Springer, New York, 1997) 457–533.
* [17] E.H. Spanier, Algebraic topology, (McGraw-Hill Book Company, New York, 1966).
* [18] A. A. Khusainov, Homology groups of asynchronous systems, Petri nets, and trace languages, Sib. Electron. Mat. Izv., 2012. v. 9. P. 13-44. (Russian)
|
arxiv-papers
| 2013-07-20T06:34:50 |
2024-09-04T02:49:48.193801
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmet A. Husainov",
"submitter": "Ahmet Husainov A.",
"url": "https://arxiv.org/abs/1307.5377"
}
|
1307.5429
|
# Probing the anharmonicity of the potential well for magnetic vortex core in
a nanodot
O.V. Sukhostavets Departamento de Física de Materiales, Universidad del Pais
Vasco, 20018 San Sebastian, Spain B. Pigeau Service de Physique de l'État
Condensé (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France S. Sangiao
Service de Physique de l'État Condensé (CNRS URA 2464), CEA Saclay, 91191 Gif-
sur-Yvette, France G. de Loubens Service de Physique de l'État Condensé
(CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France V.V. Naletov
Service de Physique de l'État Condensé (CNRS URA 2464), CEA Saclay, 91191 Gif-
sur-Yvette, France Institute of Physics, Kazan Federal University, Kazan
420008, Russian Federation O. Klein [email protected] Service de Physique de
l'État Condensé (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France K.
Mitsuzuka S. Andrieu F. Montaigne Institut Jean Lamour, UMR CNRS 7198,
Université de Lorraine, 54 506 Nancy, France K.Y. Guslienko Departamento de
Física de Materiales, Universidad del Pais Vasco, 20018 San Sebastian, Spain
IKERBASQUE, The Basque Foundation for Science, 48011 Bilbao, Spain
###### Abstract
The anharmonicity of the potential well confining a magnetic vortex core in a
nanodot is measured dynamically with a Magnetic Resonance Force Microscope
(MRFM). The stray field of the MRFM tip is used to displace the equilibrium
core position away from the nanodot center. The anharmonicity is then inferred
from the relative frequency shift induced on the eigen-frequency of the vortex
core translational mode. An analytical framework is proposed to extract the
anharmonic coefficient from this variational approach. Traces of these shifts
are recorded while scanning the tip above an isolated nanodot, patterned out
of a single crystal FeV film. We observe $+10$% increase of the eigen-
frequency when the equilibrium position of the vortex core is displaced to
about one third of its radius. This calibrates the tunability of the
gyrotropic mode by external magnetic fields.
There has been recently a renewed interest, both theoretical and experimental,
in the problem of nonlinear (NL) magnetization dynamics inside confined
nanostructures Slavin2009 . NL phenomena are responsible for the creation of
novel dynamical objects Mohseni2013 , analogs of dynamical solitons. They also
set the figure of merit of spintronics devices, e.g. the spectral purity and
tuning sensitivity of spin transfer torque nano-oscillators Slavin2009 . On
the theoretical side, predictions on the amplitude of the NL coefficients have
been found to be extremely difficult to compute beyond the uniformly
magnetized ground state. The difficulty raises both from the magneto-dipolar
field, which introduces a non-local interaction, and from the kinetic part of
the effective field (or gauge field), which modifies the texture of the
magnetic configuration. On the experimental side, the most promising findings
have been discovered on non-uniform ground states, such as magnetic vortex
existing in ferromagnetic nanodot. Vortices have stimulated the emergence of
higher performance microwave oscillators using isolated Pribiag2007 or
dipolarly coupled Locatelli2011 nanodot, or for future magnetic memories by
allowing the resonant switching of the magnetic configuration Pigeau2010 .
Magnetic vortex corresponds to a curling in-plane magnetization spatial
distribution leaving a nanometric in size core region ($\sim$ the exchange
length), where the magnetization is pointing out-of-plane. The lowest energy
mode is a translational (or gyrotropic) mode of the vortex core position
$\bm{X}$, expressed here in reduced unit of the dot radius. Its properties are
governed by the magnetostatic potential well $W^{\text{(M)}}$ in which the
core evolves. For a circular nanodot, the magnetostatic energy is isotropic in
the dot plane and it can be written as a series expansion of even powers of
the dimensionless $\bm{X}$ Dussaux2012 :
$W^{\text{(M)}}=W^{\text{(M)}}_{0}+\frac{1}{2}\kappa|{\bm{X}}|^{2}+\frac{1}{4}\kappa^{\prime}|{\bm{X}}|^{4}+{\cal{O}}(|{\bm{X}}|^{6})\,,$
(1)
At the present, only the parabolicity of the confinement, $\kappa$, has been
well characterized experimentally and the measured value is in agreement with
theoretical predictions Novosad2005 . This is not the case for the higher
order terms and there is no consensus yet on the order of magnitude or the
sign of the anharmonic coefficient $\lambda\equiv\kappa^{\prime}/\kappa$
afferent to the depolarisation effect of a displaced vortex inside a large
planar circular nanodot. The asymptotic limit of large radius is the relevant
aspect ratio to test the dipole dominating limit and the circular symmetry is
necessary to avoid the additional complexity of non-isotropic potential found
for example in square shaped elements Drews2012 . Up to now, attempts to
measure $\lambda$ in circular nanodot using large rf excitation have so far
lead to inconsistent results between experiments Pigeau2011 (red shift) and
theory Gaididei2010 (blue shift). The measurement of $\lambda$ through a
variational approach, consisting in studying the small change of oscillation
period when a large static displacement of the vortex core equilibrium
position is produced, has so far failed too: this counterpart of large rf
oscillation has mostly revealed the potential well inhomogeneities leading to
pinning of the core Chen2012 ; Burgess2013 .
In this work, we report on an experimental measurement of $\lambda$ in a large
planar circular nanodot using a Magnetic Resonance Force Microscope (MRFM).
All the experimental measurements are performed on an individual nanodisk of
thickness $t=26.7$ nm thick and nominal radius $R=300$ nm, patterned out of a
single crystal FeV film. Only the perfect crystalline structure ensures an
unpinned displacement of the vortex core throughout the sample volume. We rely
here on the non-uniform stray field of the magnetic tip of the MRFM to
displace the vortex core away from the nanodot center. The anharmonic
coefficient is then inferred from the measurement of the relative variation of
the eigen-frequency of the gyrotropic mode as a function of the tip
displacement.
Figure 1: (Color online) a) Side and b) top views of the experimental setup:
the stray field of an MRFM tip is used to displace the static position
$\bm{X}_{0}$ of the vortex core. The magnetic vortex state is shown in a bi-
variate colormap of $\bm{m}=\bm{M}/M_{s}$ (amplitude-phase $\leftrightarrow$
luminance-hue). The insets are microscopy images of the magnetic tip and disk
sample.
We start first with a description of the experimental setup Klein2008
(FIG.1). The right inset shows an image of the sample: a circular nanodot,
which is patterned by standard lithography and ion-milling techniques from an
extended film of Fe-V (10% V)Mitsuzuka2012 with magnetization $4\pi M_{s}=17$
kG. A magnetic tip is brought in the vicinity of the sample (left image). The
tip consists of a soft Fe particle glued at the apex of micro-cantilever. The
MRFM is placed inside a superconducting coil magnet, which produces an
homogenous bias magnetic field $\bm{H}_{0}$ of 6 kOe along the $z$-direction
(parallel to the normal of the disk). The value of ${H}_{0}$ is chosen to be
strong enough to magnetize the MRFM tip close to its saturation value, while
remaining weak enough compared to the saturation field of the nanodot to
preserve the vortex ground state inside the sample. At ${H}_{0}$ the tip stray
field is
$\bm{H}^{\text{tip}}(\bm{r})=-\nabla(\bm{\mu}_{\text{tip}}\cdot\bm{r}/r^{3})$,
the dipolar field generated by a point magnetic moment
${\mu}_{\text{tip}}=4\times 10^{-10}$ emu oriented along ${\hat{\bm{z}}}$
Pigeau2012 . Effects of perpendicular magnetic field on magnetic vortex are
well established Ivanov2002 ; Loubens2009 : the in-plane spins are tilted
towards the applied field producing hereby a decrease 111At $H_{0}=6$ kOe, the
spins are tilted out-of-plane by about 20∘ generating a 7% decrease of the in-
plane component of the magnetization outside the vortex core. of the in-plane
component of the magnetization outside the vortex core (cone state Ivanov2002
).
We then proceed to the measurement of the variation of the excitation spectrum
of the gyrotropic mode when the tip is scanned by $\pm 0.85\mu$m along the
$x$-direction by steps of 50 nm (see FIG.2). The scan height is $0.9\pm
0.05\mu$m 222Although the use of piezo-actuators allows ultra-precise
displacement of the micro-cantilever, the value of $\delta_{z}$ has inherently
some uncertainty as it corresponds to the free axis of the cantilever. above
the nanodot or $\delta_{z}=3.0\pm 0.15$ in reduced units of $R$ (hereafter all
spatial displacements are expressed in units of the dot radius $R$.) Placing
the tip at the origin ($\delta_{x}=0$ or on the axis of the disk), attracts
the vortex core at the center of the nanodot. From there, lateral displacement
of the magnetic tip produces a vector shift ${\bm{X}}_{0}$ of the vortex core
equilibrium position from the dot center. The process is driven by the growth
of the in-plane domain parallel to the in-plane component of the tip stray
field. We observe in FIG.2 that the eigen-frequency _increases_ (blue shift)
upon increasing $\left|{\bm{X}}_{0}\right|$. Noting that the frequency shift
is symmetric and isotropic (the signature that the intrinsic potential is
being probed) we find that the vortex core dynamics can be tuned on a relative
large range ($\sim$ 10%), hereby demonstrating that the magnetostatic
potential must be anharmonic, since a purely parabolic shape would have lead
to a frequency independent behavior on ${\bm{X}}_{0}$.
Analysis of the amplitude of the MRFM signal gives a hint on the amount of
displacement $\left|\bm{X}_{0}\right|$ achieved during a scan. The MRFM signal
corresponds to the difference of vertical force $\Delta F_{z}$ acting on the
cantilever when the vortex motion is excited. Defining $\bm{m}=\bm{M}/M_{s}$
the reduced magnetization vector, the gyromotion produces a diminution of the
spontaneous magnetization along the local equilibrium direction $\Delta
m_{i}=\frac{1}{2}\left|\partial_{X}m_{i}+j\,\partial_{Y}m_{i}\right|^{2}_{\bm{X}=\bm{X}_{0}}$,
that mostly occurs outside the core region Guslienko2008a . The generated
force can be then calculated from the reaction force $-\Delta
F_{z}=VM_{s}\langle g_{zi}\Delta m_{i}\rangle$ acting on the nanodot due to
the gradient tensor of the tip:
${\widehat{\bm{g}}}=\nabla\bm{H}^{\text{tip}}$. Cartesian tensor notation is
used here, with repeated indices being assumed summed. The chevron bracket
indicates that the enclosed quantity is averaged over the volume $V$ of the
nanodot. The red-blue colormap in FIG.2 codes the amplitude MRFM signal: red
(blue) means attractive (repulsive) force. Small arrows at
$\delta_{x}\approx\pm 1.7$ indicate the compensation point of the force: where
the change of sign occurs. This distance is about twice smaller than the one
required to change the sign of the force in the saturated state (contribution
dominated by $\langle g_{zz}\Delta m_{z}\rangle$). The difference is
interpreted as due to the translation of the core position transversally to
the tip position (along the $y$-axis). The growth of this domain generates a
repulsive vertical force on the tip trough the cross gradient term $\langle
g_{zx}\Delta m_{x}\rangle$. We shall thus use the position of the compensation
point to calibrate precisely the amplitude of the displacement of the vortex
core.
Figure 2: (Color online) Density plot showing the experimentally measured
variation of the eigen-frequency of the gyrotropic mode upon a lateral
displacement, $\delta_{x}$, of the MRFM tip at fixed $\delta_{y}=0$ and
nominal $\delta_{z}=3.0$. A red-blue colormap shows the sign of the force
acting on the cantilever. The arrows indicate where the change of polarity
occurs.
Our next step is to develop an analytical framework allowing the extraction of
$\lambda$ from this variational study. The first stage of this analysis is to
calculate the equilibrium position $\bm{X}_{0}=(X_{0},Y_{0})$ (here $X_{0}$
and $Y_{0}$ are the two in-plane cartesian coordinates) by minimizing the
total energy $W=W^{\text{(M)}}+W^{\text{(H)}}$, the sum of $W^{\text{(M)}}$,
the magnetostatic self-energy of the vortex ground state which confines the
vortex core to the center of the nanodot and
$W^{\text{(H)}}=-VM_{s}\langle\bm{m}\cdot\bm{H}\rangle$, the Zeeman energy
which represents the interaction with the external magnetic field
$\bm{H}=\bm{H}_{0}+\bm{H}^{\text{tip}}$ and is responsible for the
displacement $\bm{X}_{0}$. The magnetic configuration inside the nanodot
$m_{x}+j\,m_{y}=2\mathit{w}/(1+\mathit{w}\mathit{w}^{\ast})$ is conveniently
described by a conformal mapping of the complex variable $\mathit{w}$ ($\ast$
indicating the complex conjugate), which is a piecewise function of the
complex position $\text{\raisebox{1.29167pt}{\footnotesize\cursive
z}}=(x+j\,y)/R$ with
$\mathit{w}=f(\text{\raisebox{1.29167pt}{\footnotesize\cursive
z}})/|f(\text{\raisebox{1.29167pt}{\footnotesize\cursive z}})|$ outside the
vortex core region and
$\mathit{w}=f(\text{\raisebox{1.29167pt}{\footnotesize\cursive z}})$ inside.
The function $f$ captures the texture of the spatial configuration. To
calculate the new equilibrium position $\mathcal{Z}_{0}=(X_{0}+j\,Y_{0})$, it
is appropriate to describe the dot magnetization, $\bm{m}$, by the rigid
vortex model (RVM) Guslienko2001 , written as
$f(\text{\raisebox{1.29167pt}{\footnotesize\cursive z}})=\pm
j\,(\text{\raisebox{1.29167pt}{\footnotesize\cursive
z}}-\mathcal{Z}_{0})/r_{c}$. Here $r_{c}=R_{c}/R$ denotes the core radius
$R_{c}$ in reduced unit of $R$ and the $\pm$ sign depends on the chirality of
the vortex. The static displacement $\bm{X}_{0}$ is then obtained by
minimizing the total energy 333All energies can be approximately calculated by
running the integrand solely on the nanodot volume outside the vortex core
region. with the analytical expression of $W^{\text{(M)}}$ obtained by the
RVM. In the RVM, the magnetostatic energy is generated by the surface magnetic
charges $\sigma$ located at the circumference of the disk (the volume charges
$\nabla\cdot\bm{M}$ are absent). The confinement potential follows from the
integral $W^{\text{(M)}}=\frac{1}{2}\int d\phi\int
d\phi^{\prime}\sigma(\phi)\sigma(\phi^{\prime})/\sqrt{2(1-\cos(\phi-\phi^{\prime}))}$
where the integration is taken over the disk periphery and $\sigma$ is given
by:
$\sigma(\phi)=+M_{s}\frac{-|\bm{X}_{0}|\sin(\phi-\phi_{0})}{\sqrt{1-2|\bm{X}_{0}|\cos(\phi-\phi_{0})+|\bm{X}_{0}|^{2}}}\,,$
(2)
$\phi_{0}$ is the azimuthal direction of the vortex equilibrium position
measured from the averaged in-plane bias field direction (here $x$-axis).
The implicit trajectory of $\bm{X}_{0}$ is shown in FIG.3a for three different
heights $\delta_{z}$ around the nominal value. As expected the in-plane
components of the tip magnetic field displace the vortex core mainly along the
$y$-axis. The displacement along $x$-axis is approximately twice smaller. The
resulting displacement distance $|\bm{X}_{0}|$ as a function of $\delta_{x}$
is shown in FIG.3b. We use here a skewed scale on the abcisse to show the
behavior when $\delta_{x}\gg 1$. We have also calculated the corresponding
dipolar force produced on the tip. The result is coded in the colormap using
the same convention as in FIG.2. We have placed small arrows at the
compensation points. Since decreasing the scan height increases the amplitude
of $|\bm{X}_{0}|$, we find that the position of the arrows sensitively depends
on $\delta_{z}$. Varying $\delta_{z}$ in the experimental error bars
$[2.6,3.0]$ displaces the compensation point by $\pm 0.3\cdot R$ (or $\pm 100$
nm) around the mean value $\delta_{x}=1.8$, in agreement with the experimental
data. We shall use this marker to evaluate the uncertainty window of
$|\bm{X}_{0}|$ in our experiment.
The second stage of this analysis is to perform a linearization of the vortex
equation of motion to a cyclic excitation field. The instantaneous response
$\bm{X}=(X,Y)$ is decomposed into the static component $\bm{X}_{0}$,
calculated previously, and a dynamic component $\bm{\xi}=\bm{X}-\bm{X}_{0}$
representing the small oscillating deviation of the vortex core position from
its equilibrium 444We estimate that $\xi=0.07$ in our experiment, where the
microwave field strength is $h_{\text{rf}}=0.6$ Oe.. In the dynamical case,
the dipolar pinning imposes a precession node at the dot circumference
Guslienko2008a . It implies that the dynamical magnetization comes from the
variation, $\partial_{X}{\bm{m}}+j\,\partial_{Y}{\bm{m}}$, of a magnetic
configuration that has no radial component at the dot border. Therefore, to
calculate the frequency of the small dynamic vortex displacement $\bm{\xi}$,
it is appropriate to use the surface charges free model or two vortex ansatz
(TVA) written as $f(\text{\raisebox{1.29167pt}{\footnotesize\cursive z}})=\mp
j\,\frac{1}{r_{c}}(\text{\raisebox{1.29167pt}{\footnotesize\cursive
z}}-\mathcal{Z})(\text{\raisebox{1.29167pt}{\footnotesize\cursive
z}}\mathcal{Z}^{\ast}-1)/(1+|\mathcal{Z}|^{2})$ Guslienko2002 with
$\mathcal{Z}=(X+j\,Y)$. In our notation, the dampingless Thiele equation
simply writes $\bm{G}\times\dot{\bm{\xi}}=\partial W/\partial\bm{\xi}$, where
$W$ is the total energy and $\left|\bm{G}\right|=2\pi M_{s}t/\gamma$ is the
gyrovector Guslienko2008 (the dot is the short hand notation for the time
derivative and $\gamma$ is the gyromagnetic ratio). Linearization around
$\bm{X}_{0}$ yields the gyrotropic angular frequency
$\omega^{2}=\frac{K_{xx}K_{yy}-K_{xy}^{2}}{G^{2}}\,,\ \text{with}\
K_{ij}\equiv\left.\frac{\partial^{2}W}{\partial{\xi}_{i}\partial{\xi}_{j}}\right|_{\bm{X}=\bm{X}_{0}}$
(3)
being the stiffness of the vortex core to small displacements in both the $i$
and $j$ directions. Distinction between different cartesian directions is
necessary once the trajectory becomes elliptical. This is precisely, what
occurs when $\bm{X}_{0}\gg\bm{\xi}$: the amplitude of the $\xi$-component
along $\bm{X}_{0}$ differs from the amplitude of the $\xi$-component
perpendicular to $\bm{X}_{0}$ (short axis of the ellips is along the radial
direction). The degree of ellipticity is determimed by the anharmonic
contribution $\lambda\left|\bm{X}_{0}\right|^{2}$. This is in contrast to the
opposite limit $\bm{X}_{0}\ll\bm{\xi}$, where the trajectory corresponds to a
large amplitude circular vortex core motion around the nanodot center
Dussaux2012 .
To calculate the different tensor elements of the stiffness
$K_{ij}=K^{\text{(M)}}_{ij}+K^{\text{(H)}}_{ij}$, one must decompose it in two
contributions corresponding respectively to the magnetostatic and Zeeman
energies. The first order value of the TVA magnetostatic stiffness, $\kappa$,
has been already expressed analytically Guslienko2006 . The analytical
expression of the anharmonic correction is obtained by inserting Eq.(1) in
Eq.(3) and it leads to a simplified expression
$K^{\text{(M)}}_{ij}=\kappa\left.\left(\delta_{ij}+\lambda|\bm{X}|^{2}\delta_{ij}+2\lambda
X_{i}X_{j}\right)\right|_{\bm{X}=\bm{X}_{0}}$. It turns out that the Zeeman
stiffness can be neglected. Indeed, it can be shown that the tip stray field
produces no Zeeman stiffness along the diagonal elements
($K^{\text{(H)}}_{ii}=0$). Only the cross-terms $K^{\text{(H)}}_{xy}\neq 0$
are non-vanishing but they represent a negligible correction ($<$ 3%). We thus
find that at $\bm{X}_{0}=0$ and $H_{z}=0$, Eq.(3) simplifies to the well known
expression $\omega(0,0)={\kappa}/{G}$ Guslienko2002 . At $\bm{X}_{0}=0$ and
$H_{z}\neq 0$, the stiffness of the magnetostatic potential is renormalized by
the in-plane magnetization projection of the cone state and one obtains
${\omega(0,H_{z})}/\omega(0,0)=1+H_{z}/(4\pi M_{s})$ Loubens2009 . In the
general case $\bm{X}_{0}\neq 0$ and $H_{z}\neq 0$, the relative frequency
shift reduces to the following analytical expression:
$\frac{\omega(\bm{X_{0}},H_{z})}{\omega(0,H_{z})}=1+2\lambda\left|\bm{X_{0}}\right|^{2}+{\cal{O}}(|{\bm{X}}|^{4})\,.$
(4)
Notice that the prefactor of 2 multiplying $\lambda$ is specific to the limit
$\bm{X}_{0}\gg\bm{\xi}$.
The next step is to plot in FIG.4a the experimental data extracted from FIG.2,
renormalized by the predicted dependence of ${\omega(0,H_{z})}$, as a function
of the calculated $|\bm{X}_{0}|$ during a lateral scan of the tip at fixed
$\delta_{z}=2.8$. Fitting the data of FIG.4a with a parabola (solid line)
yields an average curvature $\lambda=0.5$. We have plotted in FIG.4b the
experimentally measured relative frequency normalized by $\omega(0,H_{0})$.
The latter quantity is inferred experimentally by studying the decay of
$\omega$ upon increasing $\delta_{z}$, while keeping the tip on the symmetry
axis ($\delta_{x}=\delta_{y}=0$): a fit of the decay behavior yields the
asymptotic value $\omega(0,H_{0})$. In FIG.4b, the data point are colored
according to the colormap associated with the amplitude of the force. For
comparison, we have also plotted the predicted variation of $\omega$ by Eq.(4)
as a function of $\delta_{x}$ for two values of $\lambda$. Setting $\lambda=0$
in Eq.(4), would have produced the usual bell-shaped curve Pigeau2012 , which
corresponds to a diminution of $\omega(0,H_{z})$ when the tip moves away from
the nanodot axis. The behavior for $\lambda=0.5$ is in excellent agreement
with the experimental data, both in the amplitude of the NL frequency shift
and in the position of the compensation point of the force.
We have then repeated the analysis by varying $\delta_{z}$ in the experimental
error bar range: $\pm 0.2$ around the nominal value. Fit of the data by a
parabola would lead to larger (smaller) values of $\lambda$ depending if the
amplitude of the shift decreases (increases). This procedure yields an
uncertainty window of 30% for the determination of $\lambda$, shown as a
shaded area in FIG.4a. Our fitting analysis did not account for higher order
corrections in Eq.(1). In FIG.4a, the curvature increases with the
displacement distance. Inclusion in the fit of terms in
$\left|\bm{X}\right|^{4}$ would have decrease the value of $\lambda$ by about
one standard deviation. As an additional check, we have performed a simulation
of the expected $\omega(X_{0},H_{z})$ for our nanodot using a mesh-size of 2.3
nm and a GPU-accelerated micromagnetic code Vansteenkiste2011a . The result is
shown as crosses in FIG.4a, demonstrating that our determination of $\lambda$
is in quantitative agreement with numerical simulations. It is also in
agreement with the result obtained by Dussaux et al. from micromagnetic
simulations performed in the limit $\xi\gg X_{0}$ on a thinner dot with
approximately the same radius Dussaux2012 .
Figure 3: (Color online) a) Trajectory of the vortex core
${\bm{X}}_{0}=(X_{0},Y_{0})$ during an implicit lateral scan of the tip
$\delta_{x}\in[-8,+8]$ at 3 different heights, $\delta_{z}$. b) Norm of the
displacement vector, $\left|\bm{X}_{0}\right|$, as a function of the tip
position, $\delta_{x}$.
In summary, using an MRFM, we have measured quantitatively the anharmonicity
coefficient $\lambda=+0.5\pm 0.15$ produced by the depolarisation field of a
vortex in a planar nanodot 555A recent preprint posted during the review
process reports a new analytical expression for $\lambda$ compatible with our
measurement Metlov2013 . From a fundamental perspective, it is interesting to
note that the obtained value (dipole-dominated) is about twice smaller than
the $\lambda=1$ predicted by the local easy-plane model Ivanov1998 . Further
work is required to check if the value is independent of the out-of-plane
external magnetic field $\bm{H}_{0}$ or the dot aspect ratio, $R/t$, in
particular around the line of the vortex state stability $\kappa(R,t)=0$
Guslienko2008 . Finally, we mention that foldover experiments performed on the
same nanodot at $\bm{X}_{0}=0$ produce a red shift of the gyrotropic frequency
(regime $\bm{X}_{0}\ll\xi$), which is opposite with respect of the sign of
$\lambda>0$. This finding suggests that the NL frequency shift observed in the
foldover of the resonant curve is not dominated by the anharmonicity of the
magnetostatic potential, but perhaps by the NL damping Pigeau2011 .
Figure 4: (Color online) a) Plot of the relative variation of eigen-frequency
of the gyrotropic mode as a function of $|\bm{X}_{0}|$. A fit by a parabola
yields $\lambda=0.5\pm 0.15$. The cross are the results of micromagnetic
simulations b) Relative variation of the eigen-frequency as a function of
$\delta_{x}$. The lines show the analytically predicted behavior for a
vanishing (0) and finite (0.5) anharmonic coefficient.
###### Acknowledgements.
This research was partly supported by the EU grant MOSAIC (ICT-FP7-317950),
the French ANR Grant MARVEL (ANR-2010-JCJC-0410-01), the Spanish MEC Grants
PIB2010US-00153 and FIS2010-20979-C02-01. S.S., K.G. and O.S. acknowledge
support from the Marie Curie grant AtomicFMR (IEF-301656), from the IKERBASQUE
and from the UPV/EHU, respectively.
## References
* (1) A. Slavin and V. Tiberkevich, Ieee Transactions On Magnetics 45, 1875 (Apr. 2009)
* (2) S. M. Mohseni, S. R. Sani, J. Persson et al., , Science 339, 1295 (Mar. 2013)
* (3) V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nature Physics 3, 498 (Jul. 2007),
* (4) N. Locatelli, V. V. Naletov, J. Grollier, G. de Loubens, V. Cros, C. Deranlot, C. Ulysse, G. Faini, O. Klein, and A. Fert, Applied Physics Letters 98, 062501 (Feb. 2011)
* (5) B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, L. W. Molenkamp, V. S. Tiberkevich, and A. N. Slavin, Applied Physics Letters 96, 132506 (Mar. 2010)
* (6) A. Dussaux, A. V. Khvalkovskiy, P. Bortolotti, J. Grollier, V. Cros, and A. Fert, Physical Review B 86, 014402 (Jul. 2012)
* (7) V. Novosad, F. Y. Fradin, P. E. Roy, K. S. Buchanan, K. Y. Guslienko, and S. D. Bader, Phys. Rev. B 72, 024455 (Jul 2005),
* (8) A. Drews, B. Krüger, G. Selke, T. Kamionka, A. Vogel, M. Martens, U. Merkt, D. Möller, and G. Meier, Phys. Rev. B 85, 144417 (Apr 2012),
* (9) B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, and L. W. Molenkamp, Nature Physics 7, 26 (Jan. 2011)
* (10) Y. Gaididei, V. P. Kravchuk, and D. D. Sheka, International Journal of Quantum Chemistry 110, 83 (Jan. 2010)
* (11) T. Y. Chen, M. J. Erickson, P. A. Crowell, and C. Leighton, Phys. Rev. Lett. 109, 097202 (Aug 2012),
* (12) J. A. J. Burgess, A. E. Fraser, F. F. Sani, D. Vick, B. D. Hauer, J. P. Davis, and M. R. Freeman, Science 339, 1051 (2013),
* (13) O. Klein, G. de Loubens, V. V. Naletov, F. Boust, T. Guillet, H. Hurdequint, A. Leksikov, A. N. Slavin, V. S. Tiberkevich, and N. Vukadinovic, Physical Review B 78, 144410 (Oct. 2008)
* (14) K. Mitsuzuka, D. Lacour, M. Hehn, S. Andrieu, and F. Montaigne, Applied Physics Letters 100, 192406 (May 2012)
* (15) B. Pigeau, C. Hahn, G. de Loubens, V. V. Naletov, O. Klein, K. Mitsuzuka, D. Lacour, M. Hehn, S. Andrieu, and F. Montaigne, Phys. Rev. Lett. 109, 247602 (Dec 2012),
* (16) B. A. Ivanov and G. M. Wysin, Phys. Rev. B 65, 134434 (Mar 2002),
* (17) G. de Loubens, A. Riegler, B. Pigeau, F. Lochner, F. Boust, K. Y. Guslienko, H. Hurdequint, L. W. Molenkamp, G. Schmidt, A. N. Slavin, V. S. Tiberkevich, N. Vukadinovic, and O. Klein, Physical Review Letters 102, 177602 (May 2009)
* (18) At $H_{0}=6$ kOe, the spins are tilted out-of-plane by about 20∘ generating a 7% decrease of the in-plane component of the magnetization outside the vortex core.
* (19) Although the use of piezo-actuators allows ultra-precise displacement of the micro-cantilever, the value of $\delta_{z}$ has inherently some uncertainty as it corresponds to the free axis of the cantilever.
* (20) K. Y. Guslienko, A. N. Slavin, V. Tiberkevich, and S.-K. Kim, Phys. Rev. Lett. 101, 247203 (Dec 2008),
* (21) K. Y. Guslienko, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, Phys. Rev. B 65, 024414 (Dec 2001),
* (22) All energies can be approximately calculated by running the integrand solely on the nanodot volume outside the vortex core region.
* (23) We estimate that $\xi=0.07$ in our experiment, where the microwave field strength is $h_{\text{rf}}=0.6$ Oe.
* (24) K. Y. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, Journal of Applied Physics 91, 8037 (May 2002)
* (25) K. Y. Guslienko, Journal of Nanoscience and Nanotechnology 8, 2745 (Jun. 2008)
* (26) K. Y. Guslienko, X. F. Han, D. J. Keavney, R. Divan, and S. D. Bader, Physical Review Letters 96, 067205 (Feb. 2006)
* (27) A. Vansteenkiste and B. Van de Wiele, Journal of Magnetism and Magnetic Materials 323, 2585 (Nov. 2011)
* (28) A preprint posted on arXiv during the review process reports a new analytical expression for $\lambda$ compatible with our measurement Metlov2013
* (29) B. A. Ivanov, H. J. Schnitzer, F. G. Mertens, and G. M. Wysin, Phys. Rev. B 58, 8464 (Oct 1998),
* (30) K. L. Metlov, ArXiv e-prints(Aug. 2013), arXiv:1308.0240 [cond-mat.mes-hall]
|
arxiv-papers
| 2013-07-20T14:16:53 |
2024-09-04T02:49:48.205013
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "O.V. Sukhostavets, B. Pigeau, G. de Loubens, V.V. Naletov, O. Klein,\n K. Mitsuzuka, S. Andrieu, F. Montaigne, and K.Y. Guslienko",
"submitter": "Olivier Klein",
"url": "https://arxiv.org/abs/1307.5429"
}
|
1307.5448
|
# Software Carpentry: Lessons Learned
Greg Wilson Mozilla Foundation / [email protected]
###### Abstract
Over the last 15 years, Software Carpentry has evolved from a week-long
training course at the US national laboratories into a worldwide volunteer
effort to raise standards in scientific computing. This article explains what
we have learned along the way the challenges we now face, and our plans for
the future.
## Introduction
In January 2012, John Cook posted this to his widely-read blog [1]:
> In a review of linear programming solvers from 1987 to 2002, Bob Bixby says
> that solvers benefited as much from algorithm improvements as from Moore’s
> law: “Three orders of magnitude in machine speed and three orders of
> magnitude in algorithmic speed add up to six orders of magnitude in solving
> power. A model that might have taken a year to solve 10 years ago can now
> solve in less than 30 seconds.”
A million-fold speedup is impressive, but hardware and algorithms are only two
sides of the iron triangle of programming. The third is programming itself,
and while improvements to languages, tools, and practices have undoubtedly
made software developers more productive since 1987, the speedup is
percentages rather than orders of magnitude. Setting aside the minority who do
high-performance computing (HPC), the time it takes the “desktop majority” of
scientists to produce a new computational result is increasingly dominated by
how long it takes to write, test, debug, install, and maintain software.
The problem is, most scientists are never taught how to do this. While their
undergraduate programs may include a generic introduction to programming or a
statistics or numerical methods course (in which they’re often expected to
pick up programming on their own), they are almost never told that version
control exists, and rarely if ever shown how to design a maintainable program
in a systematic way, or how to turn the last twenty commands they typed into a
re-usable script. As a result, they routinely spend hours doing things that
could be done in minutes, or don’t do things at all because they don’t know
where to start [2, 3].
This is where Software Carpentry comes in. We ran 91 workshops for over 4300
scientists in 2013. In them, more than 100 volunteer instructors helped
attendees learn about program design, task automation, version control,
testing, and other unglamorous but time-tested skills [4]. Two independent
assessments in 2012 showed that attendees are actually learning and applying
at least some of what we taught; quoting [5]:
> The program increases participants’ computational understanding, as measured
> by more than a two-fold (130%) improvement in test scores after the
> workshop. The program also enhances their habits and routines, and leads
> them to adopt tools and techniques that are considered standard practice in
> the software industry. As a result, participants express extremely high
> levels of satisfaction with their involvement in Software Carpentry (85%
> learned what they hoped to learn; 95% would recommend the workshop to
> others).
Despite these generally positive results, many researchers still find it hard
to apply what we teach to their own work, and several of our experiments—most
notably our attempts to teach online—have been failures.
## From Red to Green
Some historical context will help explain where and why we have succeeded and
failed.
### Version 1: Red Light
In 1995-96, the author organized a series of articles in _IEEE Computational
Science & Engineering_ titled, “What Should Computer Scientists Teach to
Physical Scientists and Engineers?” [6]. The articles grew out of the
frustration he had working with scientists who wanted to run before they could
walk—i.e., to parallelize complex programs that weren’t broken down into self-
contained functions, that didn’t have any automated tests, and that weren’t
under version control [7].
In response, John Reynders (then director of the Advanced Computing Laboratory
at Los Alamos National Laboratory) invited the author and Brent Gorda (now at
Intel) to teach a week-long course on these topics to LANL staff. The course
ran for the first time in July 1998, and was repeated nine times over the next
four years. It eventually wound down as the principals moved on to other
projects, but two valuable lessons were learned:
1. 1.
Intensive week-long courses are easy to schedule (particularly if instructors
are travelling) but by the last two days, attendees’ brains are full and
learning drops off significantly.
2. 2.
Textbook software engineering is not the right thing to teach most scientists.
In particular, careful documentation of requirements and lots of up-front
design aren’t appropriate for people who (almost by definition) don’t yet know
what they’re trying to do. Agile development methods, which rose to prominence
during this period, are a less bad fit to researchers’ needs, but even they
are not well suited to the “solo grad student” model of working so common in
science.
### Versions 2 and 3: Another Red Light
The Software Carpentry course materials were updated and released in 2004-05
under a Creative Commons license thanks to support from the Python Software
Foundation [8]. They were used twice in a conventional term-long graduate
course at the University of Toronto aimed at a mix of students from Computer
Science and the physical and life sciences.
The materials attracted 1000-2000 unique visitors a month, with occasional
spikes correlated to courses and mentions in other sites. But while grad
students (and the occasional faculty member) found the course at Toronto
useful, it never found an institutional home. Most Computer Science faculty
believe this basic material is too easy to deserve a graduate credit (even
though a significant minority of their students, particularly those coming
from non-CS backgrounds, have no more experience of practical software
development than the average physicist). However, other departments believe
that courses like this ought to be offered by Computer Science, in the same
way that Mathematics and Statistics departments routinely offer service
courses. In the absence of an institutional mechanism to offer credit courses
at some inter-departmental level, this course, like many other
interdisciplinary courses, fell between two stools.
> It Works Too Well to be Interesting
>
> We have also found that what we teach simply isn’t interesting to most
> computer scientists. They are interested in doing research to advance our
> understanding of the science of computing; things like command-line history,
> tab completion, and “select * from table” have been around too long, and
> work too well, to be publishable any longer. As long as universities reward
> research first, and supply teaching last, it is simply not in most computer
> scientists own best interests to offer this kind of course.
Secondly, despite repeated invitations, other people did not contribute
updates or new material beyond an occasional bug report. Piecemeal improvement
may be normal in open source development, but Wikipedia aside, it is still
rare in other fields. In particular, people often use one another’s slide
decks as starting points for their own courses, but rarely offer their changes
back to the original author in order to improve it. This is partly because
educators’ preferred file formats (Word, PowerPoint, and PDF) can’t be handled
gracefully by existing version control systems, but more importantly, there
simply isn’t a “culture of contribution” in education for projects like
Software Carpentry to build on.
The most important lesson learned in this period was that while many faculty
in science, engineering, and medicine agree that their students should learn
more about computing, they _won’t_ agree on what to take out of the current
curriculum to make room for it. A typical undergraduate science degree has
roughly 1800 hours of class and laboratory time; anyone who wants to add more
programming, statistics, writing, or anything else must either lengthen the
program (which is financially and institutionally infeasible) or take
something out. However, everything in the program is there because it has a
passionate defender who thinks it’s vitally important, and who is likely
senior to those faculty advocating the change.
> It Adds Up
>
> Saying, “We’ll just add a little computing to every other course,” is a
> cheat: five minutes per hour equals four entire courses in a four-year
> program, which is unlikely to ever be implemented. Pushing computing down to
> the high school level is also a non-starter, since that curriculum is also
> full.
The sweet spot for this kind of training is therefore the first two or three
years of graduate school. At that point, students have time (at least, more
time than they’ll have once they’re faculty) and real problems of their own
that they want to solve.
### Version 4: Orange Light
The author rebooted Software Carpentry in May 2010 with support from Indiana
University, Michigan State University, Microsoft, MITACS, Queen Mary
University of London, Scimatic, SciNet, SHARCNet, and the UK Met Office. More
than 120 short video lessons were recorded during the subsequent 12 months,
and six more week-long classes were run for the backers. We also offered an
online class three times (a MOOC _avant la lettre_).
This was our most successful version to date, in part because the scientific
landscape itself had changed. Open access publishing, crowd sourcing, and
dozens of other innovations had convinced scientists that knowing how to
program was now as important to doing science as knowing how to do statistics.
Despite this, though, most still regarded it as a tax they had to pay in order
to get their science done. Those of us who teach programming may find it
interesting in its own right, but as one course participant said, “If I wanted
to be a programmer instead of a chemist, I would have chosen computer science
as my major instead of chemistry.”
Despite this round’s overall success, there were several disappointments:
1. 1.
Once again, we discovered that five eight-hour days are more wearying than
enlightening.
2. 2.
And once again, only a handful of other people contributed material, not least
because creating videos is significantly more challenging than creating
slides. Editing or modifying them is harder still: while a typo in a slide can
be fixed by opening PowerPoint, making the change, saving, and re-exporting
the PDF, inserting new slides into a video and updating the soundtrack seems
to take at least half an hour regardless of how small the change is.
3. 3.
Most importantly, the MOOC format didn’t work: only 5-10% of those who started
with us finished, and the majority were people who already knew most of the
material. Both figures are in line with completion rates and learner
demographics for other MOOCs [9], but are no less disappointing because of
that.
The biggest take-away from this round was the need come up with a scalable,
sustainable model. One instructor simply can’t reach enough people, and
cobbling together funding from half a dozen different sources every twelve to
eighteen months is a high-risk approach.
### Version 5: Green Light
Software Carpentry restarted once again in January 2012 with a new grant from
the Sloan Foundation, and backing from the Mozilla Foundation. This time, the
model was two-day intensive workshops like those pioneered by The Hacker
Within, a grassroots group of grad students helping grad students at the
University of Wisconsin – Madison.
Shortening the workshops made it possible for more people to attend, and
increased the proportion of material they retained. It also forced us to think
much harder about what skills scientists really needed. Out went object-
oriented programming, XML, Make, GUI construction, design patterns, and
software development lifecycles. Instead, we focused on a handful of tools
(discussed in the next section) that let us introduce higher-level concepts
without learners really noticing.
Reaching more people also allowed us to recruit more instructors from workshop
participants, which was essential for scaling. Switching to a “host site
covers costs” model was equally important: we still need funding for the
coordinator positions (the author and two part-time administrative assistants
at Mozilla, and part of one staff member’s time at the Software Sustainability
Institute in the UK), but our other costs now take care of themselves.
Our two-day workshops have been an unqualified success. Both the number of
workshops, and the number of people attending, have grown steadily:
Figure 1: Cumulative Number of Workshops Figure 2: Cumulative Enrolment
More importantly, feedback from participants is strongly positive. While there
are continuing problems with software setup and the speed of instruction
(discussed below), 80-90% of attendees typically report that they were glad
they attended and would recommend the workshops to colleagues.
## What We Do
So what does a typical workshop look like?
* •
_Day 1 a.m._ : The Unix shell. We only show participants a dozen basic
commands; the real aim is to introduce them to the idea of combining single-
purpose tools (via pipes and filters) to achieve desired effects, and to
getting the computer to repeat things (via command completion, history, and
loops) so that people don’t have to.
* •
_Day 1 p.m._ : Programming in Python (or sometimes R). The real goal is to
show them when, why, and how to grow programs step-by-step as a set of
comprehensible, reusable, and testable functions.
* •
_Day 2 a.m._ : Version control. We begin by emphasizing how it’s a better way
to back up files than creating directories with names like “final”,
“really_final”, “really_final_revised”, and so on, then show them that it’s
also a better way to collaborate than FTP or Dropbox.
* •
_Day 2 p.m._ : Using databases and SQL. The real goal is to show them what
structured data actually is—in particular, why atomic values and keys are
important—so that they will understand why it’s important to store information
this way.
As the comments on the bullets above suggest, our real aim isn’t to teach
Python, Git, or any other specific tool: it’s to teach _computational
competence_. We can’t do this in the abstract: people won’t show up for a
hand-waving talk, and even if they do, they won’t understand. If we show them
how to solve a specific problem with a specific tool, though, we can then lead
into a larger discussion of how scientists ought to develop, use, and curate
software.
We also try to show people how the pieces fit together: how to write a Python
script that fits into a Unix pipeline, how to automate unit tests, etc. Doing
this gives us a chance to reinforce ideas, and also increases the odds of them
being able to apply what they’ve learned once the workshop is over.
Of course, there are a lot of local variations around the template outlined
above. Some instructors still use the command-line Python interpreter, but a
growing number have adopted the IPython Notebook, which has proven to be an
excellent teaching and learning environment.
We have also now run several workshops using R instead of Python, and expect
this number to grow. While some people feel that using R instead of Python is
like using feet and pounds instead of the metric system, it is the _lingua
franca_ of statistical computing, particularly in the life sciences. A handful
of workshops also cover tools such as LaTeX, or domain-specific topics such as
audio file processing. We hope to do more of the latter going forward now that
we have enough instructors to specialize.
We aim for no more than 40 people per room at a workshop, so that every
learner can receive personal attention when needed. Where possible, we now run
two or more rooms side by side, and use a pre-assessment questionnaire as a
sorting hat to stream learners by prior experience, which simplifies teaching
and improves their experience. We do _not_ to shuffle people from one room to
another between the first and second day: with the best inter-instructor
coordination in the world, it still results in duplication, missed topics, and
jokes that make no sense.
Our workshops were initially free, but we now often have a small registration
fee (typically $20–40), primarily because it reduces the no-show rate from a
third to roughly 5%. When we do this, we must be very careful not to trip over
institutional rules about commercial use of their space: some universities
will charge us hundreds or thousands of dollars per day for using their
classrooms if any money changes hands at any point. We have also experimented
with refundable deposits, but the administrative overheads were unsustainable.
> Commercial Offerings
>
> Our material is all covered by the Creative Commons – Attribution license,
> so anyone who wants to use it for corporate training can do so without
> explicit permission from us. We encourage this: it would be great if
> graduate students could help pay their bills by sharing what they know, in
> the way that many programmers earn part or all of their living from working
> on open source software.
>
> What _does_ require permission is use of our name and logo, both of which
> are trademarked. We’re happy to give that permission if we’ve certified the
> instructor and have a chance to double-check the content, but we do want a
> chance to check: we have had instances of people calling something “Software
> Carpentry” when it had nothing to do with what we usually teach. We’ve
> worked hard to create material that actually helps scientists, and to build
> some name recognition around it, and we’d like to make sure our name
> continues to mean something.
As well as instructors, we rely local helpers to wander the room and answer
questions during practicals. These helpers may be participants in previous
workshops who are interested in becoming instructors, grad students who’ve
picked up some or all of this on their own, or members of the local open
source community; where possible, we aim to have at least one helper for every
eight learners.
We find workshops go a lot better if people come in groups (e.g., 4–5 people
from one lab) or have other pre-existing ties (e.g., the same disciplinary
background). They are less inhibited about asking questions, and can support
each other (morally and technically) when the time comes to put what they’ve
learned into practice after the workshop is over. Group signups also yield
much higher turnout from groups that are otherwise often under-represented,
such as women and minority students, since they know in advance that they will
be in a supportive environment.
## Small Things Add Up
As in chess, success in teaching often comes from the accumulation of
seemingly small advantages. Here are a few of the less significant things we
do that we believe have contributed to our success.
### Live Coding
We use live coding rather than slides: it’s more convincing, it enables
instructors to be more responsive to “what if?” questions, and it facilitates
lateral knowledge transfer (i.e., people learn more than we realized we were
teaching them by watching us work). This does put more of a burden on
instructors than a pre-packaged slide deck, but most find it more fun.
### Open Everything
Our grant proposals, mailing lists, feedback from workshops, and everything
else that isn’t personally sensitive is out in the open. While we can’t prove
it, we believe that the fact that people can see us actively succeeding,
failing, and learning buys us some credibility and respect.
### Open Lessons
This is an important special case of the previous point. Anyone who wants to
use our lessons can take what we have, make changes, and offer those back by
sending us a pull request on GitHub. As mentioned earlier, this workflow is
still foreign to most educators, but it is allowing us to scale and adapt more
quickly and more cheaply than the centralized approaches being taken by many
high-profile online education ventures.
### Use What We Teach
We also make a point of eating our own cooking, e.g., we use GitHub for our
web site and to plan workshops. Again, this buys us credibility, and gives
instructors a chance to do some hands-on practice with the things they’re
going to teach. The (considerable) downside is that it can be quite difficult
for newcomers to contribute material; we are therefore working to streamline
that process.
### Meet the Learners on Their Own Ground
Learners tell us that it’s important to them to leave the workshop with their
own working environment set up. We therefore continue to teach on all three
major platforms (Linux, Mac OS X, and Windows), even though it would be
simpler to require learners to use just one. We have experimented with virtual
machines on learners’ computers to reduce installation problems, but those
introduce problems of their own: older or smaller machines simply aren’t fast
enough. We have also tried using VMs in the cloud, but this makes us dependent
on university-quality WiFi…
### Collaborative Note-Taking
We often use Etherpad for collaborative note-taking and to share snippets of
code and small data files with learners. (If nothing else, it saves us from
having to ask students to copy long URLs from the presenter’s screen to their
computers.) It is almost always mentioned positively in post-workshop
feedback, and several workshop participants have started using it in their own
teaching.
We are still trying to come up with an equally good way to share larger files
dynamically as lessons progress. Version control does _not_ work, both because
our learners are new to it (and therefore likely to make mistakes that affect
classmates) and because classroom WiFi frequently can’t handle a flurry of
multi-megabyte downloads.
### Sticky Notes and Minute Cards
Giving each learner two sticky notes of different colors allows instructors to
do quick true/false questions as they’re teaching. It also allows real-time
feedback during hands-on work: learners can put a green sticky on their laptop
when they have something done, or a red sticky when they need help. We also
use them as minute cards: before each break, learners take a minute to write
one thing they’ve learned on the green sticky, and one thing they found
confusing (or too fast or too slow) on the red sticky. It only takes a couple
of minutes to collate these, and allows instructors to adjust to learners’
interests and speed.
### Pair Programming
Pairing is a good practice in real life, and an even better way to teach:
partners can not only help each other out during the practical, but clarify
each other’s misconceptions when the solution is presented, and discuss common
research interests during breaks. To facilitate it, we strongly prefer flat
seating to banked (theater-style) seating; this also makes it easier for
helpers to reach learners who need assistance.
### Keep Experimenting
We are constantly trying out new ideas (though not always on purpose). Among
our current experiments are:
_Partner and Adapt_
We have built a very fruitful partnership with the Software Sustainability
Institute (SSI), who now manage our activities in the UK, and are adapting our
general approach to meet particular local needs.
_A Driver’s License for HPC_
As another example of this collaboration, we are developing a “driver’s
license” for researchers who wish to use the DiRAC HPC facility. During
several rounds of beta testing, we have refined an hour-long exam to assess
people’s proficiency with the Unix shell, testing, Makefiles, and other
skills. This exam was deployed in the fall of 2013, and we hope to be able to
report on it by mid-2014.
_New Channels_
On June 24-25, 2013, we ran our first workshop for women in science,
engineering, and medicine. This event attracted 120 learners, 9 instructors, a
dozen helpers, and direct sponsorship from several companies, universities,
and non-profit organizations. Our second such workshop will run in March 2014,
and we are exploring ways to reach other groups that are underrepresented in
computing.
_Smuggling It Into the Curriculum_
Many of our instructors also teach regular university courses, and several of
them are now using part or all of our material as the first few lectures in
them. We strongly encourage this, and would welcome a chance to work with
anyone who wishes to explore this themselves.
## Instructor Training
To help people teach, we now run an online training course for would-be
instructors. It takes 2–4 hours/week of their time for 12–14 weeks (depending
on scheduling interruptions), and introduces them to the basics of educational
psychology, instructional design, and how these things apply to teaching
programming. It’s necessarily very shallow, but most participants report that
they find the material interesting as well as useful.
Why do people volunteer as instructors?
_To make the world a better place._
The two things we need to get through the next hundred years are more science
and more courage; by helping scientists do more in less time, we are helping
with the former.
_To make their own lives better._
Our instructors are often asked by their colleagues to help with computing
problems. The more those colleagues know, the more interesting those requests
are.
_To build a reputation._
Showing up to run a workshop is a great way for people to introduce themselves
to colleagues, and to make contact with potential collaborators. This is
probably the most important reason from Software Carpentry’s point of view,
since it’s what makes our model sustainable.
_To practice teaching._
This is also important to people contemplating academic careers.
_To help diversify the pipeline._
Computing is 12-15% female, and that figure has been _dropping_ since the
1980s. While figures on female participation in computational science are hard
to come by, a simple head count shows the same gender skew. Some of our
instructors are involved in part because they want to help break that cycle by
participating in activities like our workshop for women in science and
engineering in Boston in June 2013.
_To learn new things, or learn old things in more detail._
Working alongside an instructor with more experience is a great way to learn
more about the tools, as well as about teaching.
_It’s fun._
Our instructors get to work with smart people who actually want to be in the
room, and don’t have to mark anything afterward. It’s a refreshing change from
teaching undergraduate calculus…
## TODO
We’ve learned a lot, and we’re doing a much better job of reaching and
teaching people than we did eighteen months ago, but there are still many
things we need to improve.
### Too Slow _and_ Too Fast
The biggest challenge we face is the diversity of our learners’ backgrounds
and skill levels. No matter what we teach, and how fast or how slow we go, 20%
or more of the room will be lost, and there’s a good chance that a different
20% will be bored.
The obvious solution is to split people by level, but if we ask them how much
they know about particular things, they regularly under- or over-estimate
their knowledge. We have therefore developed a short pre-assessment
questionnaire (listed in the appendix) that asks them whether they could
accomplish specific tasks. While far from perfect, it seems to work well
enough for our purposes.
### Finances
Our second-biggest problem is financial sustainability. The “host site covers
costs” model allows us to offer more workshops, but does not cover the 2 full-
time equivalent coordinating positions at the center of it all. We do ask host
sites to donate toward these costs, but are still looking for a long-term
solution.
### Long-Term Assessment
Third, while we believe we’re helping scientists, we have not yet done the
long-term follow-up needed to prove this. This is partly because of a lack of
resources, but it is also a genuinely hard problem: no one knows how to
measure the productivity of programmers, or the productivity of scientists,
and putting the two together doesn’t make the unknowns cancel out.
What we’ve done so far is collect verbal feedback at the end of every workshop
(mostly by asking attendees what went well and what didn’t) and administer
surveys immediately before and afterwards. Neither has been done
systematically, though, which limits the insight we can actually glean. We are
taking steps to address that, but the larger question of what impact we’re
having on scientists’ productivity still needs to be addressed.
> Meeting Our Own Standards
>
> One of the reasons we need to do long-term follow-up is to find out for our
> own benefit whether we’re teaching the right things the right way. As just
> one example, some of us believe that Subversion is significantly easier for
> novices to understand than Git because there are fewer places data can
> reside and fewer steps in its normal workflow. Others believe just as
> strongly that there is no difference, or that Git is actually easier to
> learn. While learnability isn’t the only concern—the large social network
> centered around GitHub is a factor as well—we would obviously be able to
> make better decisions if we had more quantitative data to base them on.
### “Is It Supposed to Hurt This Much?”
Fourth, getting software installed is often harder than using it. This is a
hard enough problem for experienced users, but almost by definition our
audience is _inexperienced_ , and our learners don’t (yet) know about system
paths, environment variables, the half-dozen places configuration files can
lurk on a modern system, and so on. Combine that with two version of Mac OS X,
three of Windows, and two oddball Linux installations, and it’s almost
inevitable that every time we introduce a new tool, it won’t work as expected
(or at all) for at least one person in the room. Detailed documentation has
not proven effective: some learners won’t read it (despite repeated
prompting), and no matter how detailed it is, it will be incomprehensible to
some, and lacking for others.
> Edit This
>
> And while it may seem like a trivial thing, editing text is always harder
> than we expect. We don’t want to encourage people to use naive editors like
> Notepad, and the two most popular legacy editors on Unix (Vi and Emacs) are
> both usability nightmares. We now recommend a collection of open and almost-
> open GUI editors, but it remains a stumbling block.
### Teaching on the Web
Challenge #5 is to move more of our teaching and follow-up online. We have
tried several approaches, from MOOC-style online-only offerings to webcast
tutorials and one-to-one online office hours via VoIP and desktop sharing. In
all cases, turnout has been mediocre at the start and dropped off rapidly. The
fact that this is true of most high-profile MOOCs as well is little comfort…
### What vs. How
Sixth on our list is the tension between teaching the “what” and the “how” of
programming. When we teach a scripting language like Python, we have to spend
time up front on syntax, which leaves us only limited time for the development
practices that we really want to focus on, but which are hard to grasp in the
abstract. By comparison, version control and databases are straightforward:
what you see is what you do is what you get.
We also don’t as good a job as we would like teaching testing. The mechanics
of unit testing with an xUnit-style framework are straightforward, and it’s
easy to come up with representative test cases for things like reformatting
data files, but what should we tell scientists about testing the numerical
parts of their applications? Once we’ve covered floating-point roundoff and
the need to use “almost equal” instead of “exactly equal”, our learners quite
reasonably ask, “What should I use as a tolerance for my computation?” for
which nobody has a good answer.
### Standardization vs. Customization
What we _actually_ teach varies more widely than the content of most
university courses with prescribed curricula. We think this is a strength—one
of the reasons we recruit instructors from among scientists is so that they
can customize content and delivery for local needs—but we need to be more
systematic about varying on purpose rather than by accident.
### Watching vs. Doing
Finally, we try to make our teaching as interactive as possible, but we still
don’t give learners hands-on exercises as frequently as we should. We also
don’t give them as diverse a range of exercises as we should, and those that
we do give are often at the wrong level. This is partly due to a lack of time,
but disorganization is also a factor.
There is also a constant tension between having students do realistic
exercises drawn from actual scientific workflows, and giving them tasks that
are small and decoupled, so that failures are less likely and don’t have
knock-on effects when they occur. This is exacerbated by the diversity of
learners in the typical workshop, though we hope that will diminish as we
organize and recruit along disciplinary lines instead of geographically.
### Better Teaching Practices
Computing education researchers have learned a lot in the past two decades
about why people find it hard to learn how to program, and how to teach them
more effectively [10, 11, 12, 13, 14]. We do our best to cover these ideas in
our instructor training program, but are less good about actually applying
them in our workshops.
## Conclusions
To paraphrase William Gibson, the future is already here—it’s just that the
skills needed to implement it aren’t evenly distributed. A small number of
scientists can easily build an application that scours the web for recently-
published data, launch a cloud computing node to compare it to home-grown data
sets, and push the result to a GitHub account; others are still struggling to
free their data from Excel and figure out which of the nine backup versions of
their paper is the one they sent for publication.
The fact is, it’s hard for scientists to do the cool things their colleagues
are excited about without basic computing skills, and impossible for them to
know what other new things are possible. Our ambition is to change that: not
just to make scientists more productive today, but to allow them to be part of
the changes that are transforming science in front of our eyes. If you would
like to help, we’d like to hear from you.
### Competing Interests
The author is an employee of the Mozilla Foundation. Over the years, Software
Carpentry has received support from:
* •
The Sloan Foundation
* •
Microsoft
* •
NumFOCUS
* •
Continuum Analytics
* •
Enthought
* •
The Python Software Foundation
* •
Indiana University
* •
Michigan State University
* •
MITACS
* •
The Mozilla Foundation
* •
Queen Mary University London
* •
Scimatic Inc.
* •
SciNET
* •
SHARCNET
* •
The UK Met Office
* •
The MathWorks
* •
Los Alamos National Laboratory
* •
Lawrence Berkeley National Laboratory
### Grant Information
Software Carpentry is currently supported by a grant from the Sloan
Foundation.
### Acknowledgements
The author wishes to thank Brent Gorda, who helped create Software Carpentry
sixteen years ago; the hundreds of people who have helped organize and teach
workshops over the years; and the thousands of people who have taken a few
days to learn how to get more science done in less time, with less pain.
Particular thanks go to the following for their comments, corrections, and
inspiration:
* •
Azalee Bostroem (Space Telescope Science Institute)
* •
Chris Cannam (Queen Mary, University of London)
* •
Stephen Crouch (Software Sustainability Institute)
* •
Matt Davis (Datapad, Inc.)
* •
Luis Figueira (King’s College London)
* •
Richard “Tommy” Guy (Microsoft)
* •
Edmund Hart (University of British Columbia)
* •
Neil Chue Hong (Software Sustainability Institute)
* •
Katy Huff (University of Wisconsin)
* •
Michael Jackson (Edinburgh Parallel Computing Centre)
* •
W. Trevor King (Drexel University)
* •
Justin Kitzes (University of California, Berkeley)
* •
Stephen McGough (University of Newcastle)
* •
Lex Nederbragt (University of Oslo)
* •
Tracy Teal (Michigan State University)
* •
Ben Waugh (University College London)
* •
Lynne J. Williams (Rotman Research Institute)
* •
Ethan White (Utah State University)
## References
* [1] John D. Cook. Moore’s Law Squared, 2012. Viewed July 2013.
* [2] Jo Erskine Hannay, Hans Petter Langtangen, Carolyn MacLeod, Dietmar Pfahl, Janice Singer, and Greg Wilson. How do scientists develop and use scientific software? In Second International Workshop on Software Engineering for Computational Science and Engineering (SECSE09), 2009.
* [3] Prakash Prabhu, Thomas B. Jablin, Arun Raman, Yun Zhang, Jialu Huang, Hanjun Kim, Nick P. Johnson, Feng Liu, Soumyadeep Ghosh, Stephen Beard, Taewook Oh, Matthew Zoufaly, David Walker, and David I. August. A survey of the practice of computational science. In Proceedings of the 24th ACM/IEEE Conference on High Performance Computing, Networking, Storage and Analysis, 2011.
* [4] Greg Wilson, D. A. Aruliah, C. Titus Brown, Neil P. Chue Hong, Matt Davis, Richard T. Guy, Steven H.D. Haddock, Kathryn D. Huff, Ian M. Mitchell, Mark D. Plumbley, Ben Waugh, Ethan P. White, and Paul Wilson. Best practices for scientific computing. PLoS Biology, 12(1):e1001745, January 2014.
* [5] Jorge Aranda. Software Carpentry Assessment Report, 2012.
* [6] Gregory V. Wilson. What Should Computer Scientists Teach to Physical Scientists and Engineers? IEEE Computational Science and Engineering, Summer and Fall 1996\.
* [7] Greg Wilson. Where’s the Real Bottleneck in Scientific Computing? American Scientist, January-February 2006.
* [8] Greg Wilson. Software Carpentry: Getting Scientists to Write Better Code by Making Them More Productive. Computing in Science & Engineering, November-December 2006.
* [9] Katy Jordan. MOOC completion rates: The data, 2013. Viewed July 2013.
* [10] Mark Guzdial. Why is it so hard to learn to program? In Andy Oram and Greg Wilson, editors, Making Software: What Really Works, and Why We Believe It, pages 111–124. O’Reilly Media, 2010.
* [11] Mark Guzdial. Exploring hypotheses about media computation. In Proc. Ninth Annual International ACM Conference on International Computing Education Research, ICER’13, pages 19–26. ACM, 2013\.
* [12] Orit Hazzan, Tami Lapidot, and Noa Ragonis. Guide to Teaching Computer Science: An Activity-Based Approach. Springer, 2011.
* [13] Leo Porter, Mark Guzdial, Charlie McDowell, and Beth Simon. Success in introductory programming: What works? Communications of the ACM, 56(8), 2013.
* [14] Juha Sorva. Visual Program Simulation in Introductory Programming Education. PhD thesis, Aalto University, 2012.
## Appendix A Pre-Assessment Questionnaire
* •
What is your career stage?
* –
Undergraduate
* –
Graduate
* –
Post-doc
* –
Faculty
* –
Industry
* –
Support Staff
* –
Other:
* •
What is your discipline?
* –
Space sciences
* –
Physics
* –
Chemistry
* –
Earth sciences (geology, oceanography, meteorology)
* –
Life science (ecology, zoology, botany)
* –
Life science (biology, genetics)
* –
Brain and neurosciences
* –
Medicine
* –
Engineering (civil, mechanical, chemical)
* –
Computer science and electrical engineering
* –
Economics
* –
Humanities and social sciences
* –
Tech support, lab tech, or support programmer
* –
Administration
* –
Other:
* •
In three sentences or less, please describe your current field of work or your
research question.
* •
What OS will you use on the laptop you bring to the workshop?
* –
Linux
* –
Apple OS X
* –
Windows
* –
I do not know what operating system I use.
* •
With which programming languages, if any, could you write a program from
scratch which imports some data and calculates mean and standard deviation of
that data?
* –
C
* –
C++
* –
Perl
* –
MATLAB
* –
Python
* –
R
* –
Java
* –
Other:
* •
What best describes how often you currently program?
* –
I have never programmed.
* –
I program less than one a year.
* –
I program several times a year.
* –
I program once a month.
* –
I program once a week or more.
* •
What best describes the complexity of your programming? (Choose all that
apply.)
* –
I have never programmed.
* –
I write scripts to analyze data.
* –
I write tools to use and that others can use.
* –
I am part of a team which develops software.
* •
A tab-delimited file has two columns showing the date and the highest
temperature on that day. Write a program to produce a graph showing the
average highest temperature for each month.
* –
Could not complete.
* –
Could complete with documentation or search engine help.
* –
Could complete with little or no documentation or search engine help.
* •
How familiar are you with Git version control?
* –
Not familiar with Git.
* –
Only familiar with the name.
* –
Familiar with Git but have never used it.
* –
Familiar with Git because I have used or am using it.
* •
Consider this task: given the URL for a project on GitHub, check out a working
copy of that project, add a file called notes.txt, and commit the change.
* –
Could not complete.
* –
Could complete with documentation or search engine help.
* –
Could complete with little or no documentation or search engine help.
* •
How familiar are you with unit testing and code coverage?
* –
Not familiar with unit testing or code coverage.
* –
Only familiar with the terms.
* –
Familiar with unit testing or code coverage but have never used it.
* –
Familiar with unit testing or code coverage because I have used or am using
them.
* •
Consider this task: given a 200-line function to test, write half a dozen
tests using a unit testing framework and use code coverage to check that they
exercise every line of the function.
* –
Could not complete.
* –
Could complete with documentation or search engine help.
* –
Could complete with little or no documentation or search engine help.
* •
How familiar are you with SQL?
* –
Not familiar with SQL.
* –
Only familiar with the name.
* –
Familiar with SQL but have never used it.
* –
Familiar with SQL because I have used or am using them.
* •
Consider this task: a database has two tables: Scientist and Lab. Scientist’s
columns are the scientist’s user ID, name, and email address; Lab’s columns
are lab IDs, lab names, and scientist IDs. Write an SQL statement that outputs
the number of scientists in each lab.
* –
Could not complete.
* –
Could complete with documentation or search engine help.
* –
Could complete with little or no documentation or search engine help.
* •
How familiar do you think you are with the command line?
* –
Not familiar with the command line.
* –
Only familiar with the term.
* –
Familiar with the command line but have never used it.
* –
Familiar with the command line because I have or am using it.
* •
How would you solve this problem: A directory contains 1000 text files. Create
a list of all files that contain the word “Drosophila” and save the result to
a file called results.txt.
* –
Could not create this list.
* –
Would create this list using “Find in Files” and “copy and paste”.
* –
Would create this list using basic command line programs.
* –
Would create this list using a pipeline of command line programs.
|
arxiv-papers
| 2013-07-20T18:44:29 |
2024-09-04T02:49:48.214174
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Greg Wilson",
"submitter": "Greg Wilson",
"url": "https://arxiv.org/abs/1307.5448"
}
|
1307.5505
|
flat quasi-coherent sheaves of finite cotorsion dimension]
flat quasi-coherent sheaves of finite cotorsion dimension
Esmaeil Hosseini]Esmaeil Hosseini
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
Let $X$ be e quasi-compact and semi-separated scheme. If every flat
quasi-coherent sheaf has finite cotorsion dimension, we prove that
$X$ is $n$-perfect for some $n\geq 0$. If $X$ is coherent and
$n$-perfect(not necessarily of finite krull dimension), we prove
that every flat quasi-coherent sheaf has finite
pure injective dimension. Also, we show that there is an
equivalence $\KPIFX\lrt\DFX$ of homotopy categories, whenever
$\KPIFX$ is the homotopy category of pure injective flat
quasi-coherent sheaves and $\DFX$ is the pure derived category of
flat quasi-coherent sheaves.
§ INTRODUCTION
In this paper, $X$ denotes a quasi-compact and semi-separated
scheme, $\CO_X$-modules are quasi-coherent sheaves on $X$ and all
rings are commutative with identity.
Let $\mathfrak{Qco}X$ be the category of $\CO_X$-modules. An
$\CO_X$-module $\CF$ is called flat if for each $p\in X$,
is a flat $\CO_{X,p}$-module or equivalently the the functor
$\CF\otimes_{\CO_X}-:\mathfrak{Qco}X\lrt\mathfrak{Qco}X$ is exact.
In <cit.>, the authors proved that
the pair $(\mathrm{FlatX},\mathrm{CotX})$ is a complete cotorsion
theory in $\mathfrak{Qco}X$, whenever FlatX is the class of all flat
$\CO_X$-modules and
~~\Ext^1_X(\CF,\CG)=0\}$, is the class of all cotorsion
$\CO_X$-modules. So,
for a given $\CO_X$-module $\CG$, we can define cd$\CG$(the cotorsion
dimension of $\CG$).
In this paper we study those schemes $X$ such that over them every
flat $\CO_X$-module has finite cotorsion dimension. If $X$ is
affine, then every flat $\CO_X$-module has finite projective
dimension and so there exist an integer $n\geq 0$ such that for each
flat $\CO_X$-module $\CF$, $\pd\CF\leq n$(the projective dimension
of $\CF$). Unfortunately, this argument is not true when $X$ is
non-affine, since there is no non-zero projective $\CO_X$-modules.
In the following result we make a proof to such case.
The following conditions are equivalent:
(i) Every flat $\CO_X$-module has finite cotorsion dimension.
(ii) $X$ is $n$-perfect for some integer $n\geq 0$.
In the remainder of this paper we show that every flat
$\CO_X$-module has finite pure injective dimension. As an
application, we prove that if $X$ is coherent $n$-perfect then there
is an equivalence $\KPIFX\lrt\KPFX$ of homotopy categories, whenever
$\KFX$ be the homotopy category of complexes of flat
$\CO_X$-modules and $\KPIFX$ be the essential image of the homotopy
category of complexes of pure injective flat $\CO_X$-modules in
$\KFX$ in the sense of [3].
Setup. In this paper,
$\mathfrak{U}=\{\mathrm{Spec}A_i\}_{i=1}^m$ denotes a
semi-separating cover of $X$(i.e. each intersection of elements of
$\mathfrak{U}$ is also affine).
§ COTORSION ENVELOPE OF BOUNDED COMPLEXES
Let $\CBX$ be the category of bounded complexes of $\CO_X$-modules,
$\CBAFX$ be the category of bounded acyclic complexes of flat
$\CO_X$-modules and $\CBCOTX$ be the category of bounded complexes
of cotorsion $\CO_X$-modules. In this section, we prove that the
pair $(\CBAFX,\CBCOTX)$ is a complete cotorsion theory in $\CBX$,
i.e. $\CBAFX= {}^\perp\CBCOTX$, $\CBAFX^\perp=\CBCOTX$ and it has
enough projectives. For notations and definitions see [3] and [1].
Let $\[email protected]{\mathbf{G}:
0\ar[r]&\CG'\ar[r]&\CG\ar[r]&\CG''\ar[r]&0}$ be an exact sequence of
$\CO_X$-modules. Then there exists a morphism $\phi:\F\lrt
\mathbf{G}$ of complexes whenever $\F$ is a short exact sequence
of flat $\CO_X$-modules.
be the flat cover of $\CG''$. Consider the pullback diagram
\[\[email protected]@R-0.9pc{&&0\ar[d]&0\ar[d]\\&&\CC''\ar[d]\ar@{=}[r]&\CC''\ar[d]\\0\ar[r]&\CG'
\ar[r]\ar@{=}[d]&\CP\ar[r]\ar[d]&\CF''\ar[r]\ar[d]&0\\
and let
be the flat cover of $\CP$. Then the pullback of $i$ and $p$
completes the proof.
Recall that, a bounded complex $\F$ is called flat if
$\F\in\CBAFX$ and a bounded complex $\mathbf{C}$ is called
cotorsion if $\C\in\CBAFX^{\perp}$, where the orthogonal is
taken in the exact category
$\mathbf{C}^{\mathrm{b}}(\mathfrak{Qco}X)$. By similar argument that
used in <cit.>, we deduce the following
Let $\C$ be a bounded complex. Then $\C$ is cotorsion if and only
if it is a complex of cotorsion $\CO_X$-modules.
Let $\X$ be a bounded complex of $\CO_X$-modules. Then there exists
an exact sequence
$\[email protected]@R-0.9pc{0\ar[r]&\C\ar[r]&\F\ar[r]&\X\ar[r]&0}$ of
complexes, where $\F$ is flat and $\C$ is cotorsion.
By [8], there exists a quasi-isomorphism $f: \X[-1]\lrt\I$,
with $\I$ is a bounded complex of injective $\CO_X$-modules. By
Lemma <ref>, we construct the short exact sequence
with $\F$ is flat and $\C'$ is cotorsion complex. Then the
pullback of the morphisms
$\[email protected]{\F\ar[r]&\mathrm{cone}(f)}$ and
$\[email protected]{\I\ar[r]&\mathrm{cone}(f)}$ completes the proof.
The pair $(\CBAFX,\CBCOTX)$ is a complete cotorsion theory in
It suffices to show that $\CBAFX={}^\perp\CBCOTX$. Let
$\X\in{}^\perp\CBCOTX$. By Theorem <ref>, there exist an exact
of complexes, with $\F$ is flat and $\C$ is cotorsion. By
assumption this is split exact sequence. Then $\X\in\CBAFX$.
Therefore $(\CBAFX,\CBCOTX)$ is a cotorsion theory which is complete
by Theorem <ref>.
Every bounded complex of $\CO_X$-modules admits flat cover and
cotorsion envelope.
§ COTORSION DIMENSION OF FLAT $\CO_X$-MODULE
In this section we prove that if every flat $\CO_X$-module has
finite cotorsion dimension then $X$ is $n$-perfect for some $n\geq
Let $n\geq 0$ be an integer. $X$ is called $n$-perfect if
$n=\mathrm{sup}\{\mathrm{cd}\CF| \CF\in\mathrm{Flat}X\}$.
Let for each $1\leq i\leq m$, every flat $A_i$-module has finite
cotorsion dimension. Then $X$ is $n$-perfect for some $n$.
Let $\CF$ be a flat $\CO_X$-module and
\[\[email protected]@R-0.9pc{\mathbf{G}: 0\ar[r]&\CF\ar[r]&\mathfrak{C}^0(\mathfrak{U},\CF)\ar[r]
\CF)\ar[r]&\cdots\ar[r]&\mathfrak{C}^{m-2}(\mathfrak{U},\CF)\ar[r]&\mathfrak{C}^{m-1}(\mathfrak{U},\CF)\ar[r]&
be its $\check{\mathrm{C}}$heck resolution. By assumption, for
each $0\leq i\leq m-1$, $\mathfrak{C}^i(\mathfrak{U},\CF)$ has
finite cotorsion dimension, then by Theorem <ref> there exist a
of $\mathbf{G}$ by flat complexes of $\CO_X$-modules, where
$\mathbf{C}_0$, $\mathbf{C}_1$, ..., $\mathbf{C}_{n-1}$ are
cotorsion complexes and $\mathbf{C}_n$ is a flat complex such that
for each $i> 0$, $\mathbf{C}_n^i$ is cotorsion. Then the flat
\[\[email protected]@R-0.9pc{ 0\ar[r]&\CF\ar[r]&\mathbf{C}_0^0\ar[r]&\mathbf{C}_1^0\ar[r]&\cdots\ar[r]&\mathbf{C}_{n-1}^0\ar[r]&
\mathbf{C}_{n}^1\ar[r]&\mathbf{C}_{n}^2\ar[r]&\cdots\ar[r]&\mathbf{C}_n^m\ar[r]&
is a cotorsion resolution of $\CF$.
If every flat $\CO_X$-module has finite cotorsion dimension. Then
for each $1\leq i\leq m$, every flat $A_i$-module has finite
cotorsion dimension.
With out lose of generality we can assume that $i=1$. Let $F$ be a
flat $A_1$-module, $f:U_1\lrt X$ be the inclusion and
$\[email protected]@R-0.9pc{\mathbf{C}_F: 0\ar[r]&F\ar[r]^{\xi}&C^0
\ar[r]^{\delta^0}&C^1\ar[r]^{\delta^1}&C^2\ar[r]^{\delta^2}&\cdots,}$
be its minimal cotorsion resolution. By construction,
$\mathbf{C}_F$ is a pure acyclic complex of flat $A_1$-modules.
Apply the exact functor ${f}_{{}_*}$ and get the pure acyclic
\ar[r]^{{f}_{{}_*}\widetilde{\xi}}&{f}_{{}_*}\widetilde{C^0}
\ar[r]^{{f}_{{}_*}\widetilde{\delta^0}}&{f}_{{}_*}\widetilde{C^1}\ar[r]^{{f}_{{}_*}\widetilde{\delta^1}}
\ar[r]^{{f}_{{}_*}\widetilde{\delta^2}}&\cdots}$ of flat
$\CO_X$-modules. In fact, it is a cotorsion resolution of
${f}_{{}_*}\widetilde{F}$. The assumption implies that
$\im{f}_{{}_*}\widetilde{\delta^{n-1}}$ is cotorsion for some
integer $n$. So the exact sequence
of flat $\CO_X$-modules splits. Then
$C^n=\im\delta^{n-1}\oplus\im\delta^{n}$. It follows that $\cd F\leq
$\mathbf{Proof~ of ~Theorem ~1.1.}$ $(i)\lrt(ii)$ If every flat
$\CO_X$-module has finite cotorsion dimension. Then by Theorem
<ref>, for each $0\leq i \leq m$, every flat $A_i$-module has
finite cotorsion dimension. So, for each $i$ there exist an
integer $n_i\geq 0$ such that $A_i$ is $n_i$-perfect. Therefore the
proof of Theorem <ref> implies that $X$ is $n$-perfect for some
integer $n\geq 0$.
$(ii)\lrt(i)$ Clear.
A scheme $X$ is $n$-perfect if and only if for every $\CO_X$-module
$\CG$, $\cd~\CG\leq n$.
Let $X$ be $n$-perfect, $\CG$ be an $\CO_X$-module and
be the flat cover of $\CG$. Then for any flat $\CO_X$-module $\CF$
we have the following exact sequence
\[\[email protected]@R-0.9pc{0 = \Ext^{n+1}_X(\CF,\CC)\ar[r]&
\Ext^{n+1}_X(\CF,\CF')\ar[r]&\Ext^{n+1}_X(\CF,\CG)\ar[r]&
\Ext^{n+2}_X(\CF,\CC)=0.}\]
Then $\Ext^{n+1}_X(\CF,\CG)=0$ and hence
$\cd~\CG\leq n$.
The converse is trivial.
Now by using the main Theorem of [7] we give examples of
non-noetherian $n$-perfect schemes of infinite Krull dimension.
Let $R$ be a ring, $|R|\leq\aleph_n$ for some $n\geq 0$,
$A=R[x_1,x_2,...]$ be the polynomial ring of infinite indeterminate
and $X=\bigcup_{i=1}^{i=m}D(f_i)$ be an open subscheme of Spec$A$.
Then $X$ is a non-noetherian non-affine scheme of infinite krull
dimension(for definitions and notations see <cit.>). By the
same argument that used in the proof of Theorem <ref> we deduce
that $X$ is $k$-perfect for some $k$.
Let $\mathfrak{T}$ be a topological space of cardinality at most
$\aleph_n$ for some integer $n\geq 0$. If $\mathfrak{T}$ is not
$p$-space, then the commutative ring $\mathrm{C}(\mathfrak{T})$, the
ring of real valued continuous functions on $\mathfrak{T}$, is a
non-noetherian $(n+1)$-perfect ring of infinite krull dimension. For
example the metric space $\mathbb{R}$(real numbers) is not a
If $R$ is a noetherian ring of finite krull dimension $n$. Then it
is $n$-perfect.
If $R$ is $n$-perfect. Then $R[x]$ is also $(n+1)$-perfect.
The Nagata's example of a noetherian ring of infinite krull
dimension is $n$-perfect for some integer $n$, see[Appendix, Example
§.§ Pure injective dimension of flat $\CO_X$-modules
Recall that an exact sequence
$\[email protected]@R-0.9pc{0\ar[r]&\CK\ar[r]& \CG}$ of
$\CO_X$-modules is called pure if it remains exact after tensoring
with any $\CO_X$-module. An $\CO_X$-module $\CE$ is called pure
injective if it is injective with respect pure exact sequences of
$\CO_X$-modules. For a given $\CO_X$-module $\CF$, let $\CF^*=
\oplus_{i=1}^m{f_i}_*\widetilde{F_i^*}$, $\CF^{**}=
\oplus_{i=1}^m{f_i}_*\widetilde{F_i^{**}}$ such that for each $1\leq
i\leq m$, $F_i=\CF(U_i)$, $F_i^*=\mathrm{Hom}_{\Z}(F_i,{\Q}/{\Z})$,
and $\[email protected]@R-0.9pc{f_i:U_i\ar[r]&X}$ be the inclusion.
Then $\CF^*$ and $\CF^{**}$ are pure injective $\CO_X$-modules and
$\CF\lrt\CF^{**}$ is a pure monomorphism.
In this subsection we let $X$ be a coherent scheme. Recall
that $X$ is called coherent if $A_i$ is a coherent ring for each
$1\leq i\leq m$
Let $\CF$ be a flat $\CO_X$-module. Then $\CF$ is pure injective if
and only if it is cotorsion.
Let $\[email protected]@R-0.9pc{
0\ar[r]&\CF\ar[r]&\CC\ar[r]&\CG\ar[r]&0}$ be the cotorsion envelope
of $\CF$. Since $\CF$ is flat then this sequence is pure and hence
it is split.
Let $\CF$ be a cotorsion $\CO_X$-module and
0\ar[r]&\CF\ar[r]&\CF^{**}\ar[r]&\frac{\CF^{**}}{\CF}\ar[r]&0}$ be
its pure injective preenvelope. Since $\CF$ and $\CF^{**}$ are flat
$\CO_X$-module then $\frac{\CF^{**}}{\CF}$ is also flat and so
this sequence is split.
The pure injective dimension of an $\CO_X$-module $\CF$
can be defined in usual sense.
A scheme $X$ is $n$-perfect if and only if every $\CO_X$-module has
finite pure injective dimension.
Let $\CG$ be an $\CO_X$-module and
be its minimal cotorsion resolution. By Theorem <ref>,
$\im\delta^{n-1}$ is cotorsion flat and by Proposition <ref>, it is pure
injective. Therefore this pure exact sequence is a pure injective
resolution of $\CG$ of length $n$.
By Proposition <ref>, the converse is
§ APPLICATION
Let $\KFX$ be the homotopy category of complexes of flat
$\CO_X$-modules, $\KPFX$ be the full subcategory of $\KFX$
consisting of all pure acyclic complexes of flat $\CO_X$-modules and
$\KCOFX$ be the essential image of the homotopy category of
complexes of cotorsion flat $\CO_X$-modules. In [3], the authors proved that
there is an equivalence $\KCOFX\lrt\KFX/{\KPFX}=\DFX$ of homotopy
categories, whenever $\KPFX\cap\KCOFX=0$ and $\mathfrak{Qco}X$ have
enough flats. For instance such equivalenece of homotopy categories
exists, when $X$ is $n$-perfect (possibly non-noetherian of infinite
Krull dimension).
In this section we let $X$ be a coherent, $\CPFX$ be the category of
all flat complexes of $\CO_X$-modules and
$\mathbf{C}(\mathrm{Pinj}X)$ be the category of complexes of pure
injective $\CO_X$-modules.
Let $\C$ be a complex of $\CO_X$-modules. Then $\C\in\CPFX^\perp$ if
and only if it is a complex of pure injective $\CO_X$-modules .
Let $\C$ be a complex of pure injective $\CO_X$-modules. By <cit.>, there is a degree-wise split exact sequence
$\[email protected]@R-0.9pc{0\ar[r]&\C\ar[r]&\C'\ar[r]&\F'\ar[r]&0} $
where $\C'\in\CPFX^\perp$ and $\F'\in\CPFX$. Therefore we have a
canonical morphism $ u: \mathbf{F}' \rightarrow \Sigma \mathbf{C}$
such that $\[email protected]@R-0.9pc{
\mathbf{C}\ar[r]&\mathbf{C}'\ar[r]&\mathbf{F}'\ar[r]^u&\Sigma
\mathbf{C}}$is a triangle in $\KFX$. Moreover, $\F'$ is a complex
of cotorsion $\CO_X$-modules and hence it is contractible by
$n$-perfectness of $X$. It follows that for each flat complex $\F$,
$\Hom_{\K(X)}(\F,\C)\cong\Hom_{\K(X)}(\F,\C')=0$. Therefore by
<cit.>, $\C\in\CPFX^\perp$.
The converse follows from <cit.>.
The cotorsion theory $(\CPFX, \mathbf{C}(\mathrm{Pinj}X))$ is
The result follows from <cit.> and Theorem <ref>.
Let $\KPIFX$ be the essential image(in the sense of [3]) of
the homotopy category of complexes of pure injective flat
$\CO_X$-modules in $\KFX$.
There is an equivalence $\KPIFX\lrt \DFX$ of homotopy categories.
The pair $(\KPFX,\KPIFX)$ is a complete cotorsion theory in $\KFX$
in the sense of [3]. Then there is an equivalence $\KPIFX\lrt
\DFX$ of homotopy categories.
[EE] EE E. Enochs, S. Estrada, Relative homological algebra in the category of quasi-coherent sheaves ,
Adv.Math. 194 (2005), 284-295.
[1] E. Enochs , O. Jenda, Relative homological algebra,
Gordon and Breach S.Publishers, (2000).
[2] R. Hartshorne, Algebraic Geometry, Springer- Verlag, New York Inc. (1997).
[3] E. Hosseini, Sh. Salarian, A cotorsion theory in
the homotopy category of flat quasi-coherent sheaves, Proc. Amer. Math. Soc. 141 (3) (2013), 753-762.
[6] M. Nagata, Local rings,
R. E. Krieger Pub. Co. 234 (1975).
[7] D. Simson, A remark on projective dimension of flat
modules, Math. Ann. 209 (1974), 181-182.
[8]N. Spaltenstein, Resolutions of unbounded complexes,
Compositio Math. 65, no. 2 (1988), 121-154.
\[\[email protected]@R-0.9pc{0\ar[r]&\CF'\ar[r]\ar[d]&\CF\ar[r]\ar[d]&\CF''\ar[r]\ar[d]&0\\
§ PURE INJECTIVE DIMENSION OF FLAT COMPLEXES OF
This section is devoted to the study of purity in
$\C(\mathfrak{Qco}X)$. A morphism $f:\X\lrt\Y$ of complexes is
called a monomorphism if for each $i\in\Z$, $f^i$ is a monomorphism
of $\CO_X$-modules. Now we make our definition of purity in
Let $\CF$ be an $\CO_X$-module, $\CF^*=
\oplus_{i=1}^m{f_i}_*\widetilde{F_i^*}$, $\CF^{**}=
\oplus_{i=1}^m{f_i}_*\widetilde{F_i^{**}}$ and $\CF^{***}=
\oplus_{i=1}^m{f_i}_*\widetilde{F_i^{***}}$ such that for each
$1\leq i\leq m$, $F_i=\CF(U_i)$,
$F_i^*=\mathrm{Hom}_{\Z}(F_i,{\Q}/{\Z})$, and
$\[email protected]@R-0.9pc{f_i:U_i\ar[r]&X}$ is inclusion. Then
$\CF^*$, $\CF^{**}$ and $\CF^{***}$ are pure injective
$\CO_X$-modules and $\CF\lrt\CF^{**}$ is a pure monomorphism.
A monomorphism $f:\X\lrt \Y$ in $\C(\mathfrak{Qco}X)$ is a pure
monomorphism if $f^{*}:\Y^{*}\lrt \X^{*}$ is a split epimorphism in
the category $\C(\mathfrak{Qco}X)$.
Recall that a complex $\mathbf{P}$ of $\CO_X$-module is called
pure injective if it is injective with respect pure
exact sequence.
Let $\X$ be a complex of $\CO_X$-modules. Then the canonical
monomorphism $\X\lrt \X^{**}$ is pure and $\X^{**}$ is a pure
injective complex.
A complex $\X$ is pure injective if and only if it is a direct
summand of $\CX^{**}$. Moreover, for any complex $\CY$ of
$\CO_X$-modules, $\CY^*$ is pure injective. If $\X$ is pure
injective complex, then it possesses a pure injective $\CO_X$-module
in each degree. But the converse need not be true.
A complex $\F$ of $\CO_X$-modules is flat if and only if $\F^{**}$
is also a flat complex.
A complex $\F$ of flat $\CO_X$-modules is pure injective if and only
if for each $i\in\Z$, $\F^i$ is pure injective $\CO_X$-module.
Let $\X$ be a complex of $\CO_X$-modules, the pure injective
dimension of $\X$ can be defined as follows
$$\mathrm{pid}\X = \mathrm{min}\{n| \X ~~\emph{has a pure injective resolution of lenght}~~ n\}.$$
In the following theorem we show that, if $X$ is $n$-perfect,
sup$\{\textmd{cd}\CF| ~~\CF\in\textmd{Flat}(\mathfrak{Qco}X)\}$ =
sup$\{\textmd{pid}\CF| ~~\CF\in\textmd{Flat}(\mathfrak{Qco}X)\}$.
Let $X$ be an $n$-perfect scheme and $\F$ be a complex of flat
$\CO_X$-modules. Then $\mathrm{pid}\F \leq n$.
The pair $(\CPFX,\CPFX^\perp)$ is a complete cotorsion theory in
Let $\X$ be a complex of $\CO_X$-modules and for each $i\in\Z$,
$\[email protected]@R-0.9pc{\CG^i\ar[r]^{f^i}& \X^i}$ be the flat
cover of $\X^i$. Since the flat cover of an $\CO_X$-module is an
epimorphism, there is an epimorphism
\X}$ of complexes induced by
\X}\}_{i\in\Z}$. This implies that the pair $(\CPFX,\CPFX^\perp)$
is a cotorsion theory in $\C(\mathfrak{Qco}X)$.
Now, let $\kappa$ be a cardinal number such that $\kappa\geq
\mathrm{max}\{|\CO_X|, |\mathcal{V}|, \aleph_0\}$ and
$\CS=\{\F\in\CPFX|~~|\F|\leq\kappa\}$. By similar argument that
used in <cit.>, we deduce that the cotorsion theory
$(\CPFX,\CPFX^\perp)$ is cogenerated by $\CS$. Hence it is a
complete cotorsion theory in $\C(\mathfrak{Qco}X)$.
|
arxiv-papers
| 2013-07-21T09:01:39 |
2024-09-04T02:49:48.225906
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Esmaeil Hosseini",
"submitter": "Esmaeil Hosseini",
"url": "https://arxiv.org/abs/1307.5505"
}
|
1307.5523
|
# Orbital stability of standing waves of a class of fractional Schrödinger
equations with a general Hartree-type integrand
Y. Cho Department of Mathematics, and Institute of Pure and Applied
Mathematics, Chonbuk National University, Jeonju 561-756, South Korea. M.M.
Fall African Institute for Mathematical Sciences of Senegal, AIMS-Senegal, KM
2, Route de Joal, B.P. 14 18. Mbour, Sénégal. H. Hajaiej Department of
Mathematics, College of Science, King Saud University P.O. Box 2455, Riyadh
11451, Saudi Arabia. P.A. Markowich Division of Math & Computer Sc & Eng,
King Abdullah University of Science and Technology Thuwal 23955-6900, Saudi
Arabia. S. Trabelsi
###### Abstract
This article is concerned with the mathematical analysis of a class of a
nonlinear fractional Schrödinger equations with a general Hartree-type
integrand. We prove existence and uniqueness of global-in-time solutions to
the associated Cauchy problem. Under suitable assumptions, we also prove the
existence of standing waves using the method of concentration-compactness by
studying the associated constrained minimization problem. Finally we show the
orbital stability of standing waves which are the minimizers of the associate
variational problem.
###### keywords:
Fractional Schrödinger equation, Hartree type nonlinearity, standing waves,
orbital stability
## 1 Introduction
A partial differential equation is called fractional when it involves
derivatives or integrals of fractional order. Various physical phenomena and
applications require the use of fractional derivatives, for instance quantum
mechanics, pseudo-chaotic dynamics, dynamics in porous media, kinetic theories
of systems with chaotic dynamics. The latter application is based on the so
called fractional Schrödinger equation. This equation was derived using the
path integral over a kind of Lévy quantum mechanical path approach by Laskin
in Ref. [14, 15, 16]. The mathematical analysis of the fractional nonlinear
Schrödinger equation has been growing continually during the last few decades.
Many results have been obtained and we refer for instance to [5] and
references therein. This paper deals with the analysis of the following Cauchy
problem
$\displaystyle\mathscr{S}:\quad\left\\{\begin{array}[]{l}i\partial_{t}\phi+(-\Delta)^{s}\phi=\left(G(|\phi|)\star
V(|x|)\right)G^{\prime}(\phi),\\\ \\\ \phi(t=0,x)=\phi_{0}.\end{array}\right.$
In the system $\mathscr{S}$, $\phi(t,x)$ is a complex-valued function on
$\mathbb{R}\times{\mathbb{R}^{N}}$ and $\phi_{0}$ is a prescribed initial data
in $H^{s}(\mathbb{R}^{N})$. The operator $(-\Delta)^{s}$ denotes the
fractional Laplacian of power $0<s<1$. It is defined as a pseudo-differential
operator
$\mathcal{F}[(-\Delta)^{s}\,\phi](\xi)=|\xi|^{2s}\,\mathcal{F}[\phi](\xi)$
with $\mathcal{F}$ being the Fourier transform. The symbol $\star$ denotes the
convolution operator in ${\mathbb{R}^{N}}$ with the potential
$V(|x|)=|x|^{\beta-N}$ where $\beta>0$ is such that $\beta>N-2s$. The function
$G$ is a differentiable function from
$\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$,
$G^{\prime}(\phi):=\frac{dG}{d\phi}:=F(|\phi|)\phi$, where
$F:\mathbb{R}\rightarrow\mathbb{R}$. The above Cauchy problem reduces to the
massless boson Schrödinger equation in three dimensions when
$G(\phi)=|\phi|^{2}$, $V(|x|)=|x|^{-1}$ and $s=\frac{1}{2}$. In this case,
standing waves of the system $\mathscr{S}$, i.e. solutions of the form
$\phi(t,x)=u(x)e^{-i\kappa t}$, satisfy the following semilinear partial
differential equation
$(-\Delta)^{1/2}u-(|x|^{-1}\ast\;u^{2})u+\kappa u=0.$ (1)
The associated variational problem
$\displaystyle\mathcal{I}_{\lambda}$
$\displaystyle=\inf\left\\{{|\\!|}|\xi|^{\frac{1}{2}}\mathcal{F}[{u}](\xi)\|^{2}_{L^{2}({\mathbb{R}^{N}})}-\int_{\mathbb{R}^{N}\times{\mathbb{R}^{N}}}\frac{|u(x)|^{2}|u(y)|^{2}}{|x-y|}dxdy,\right.$
$\displaystyle\hskip 150.0pt\left.u\in
H^{\frac{1}{2}}({\mathbb{R}^{N}}),\>\int_{{\mathbb{R}^{N}}}|u(x)|^{2}\,dx=\lambda\right\\},$
(2)
has played a fundamental role in the mathematical theory of gravitational
collapse of boson stars, [18]. In Ref. [12], the authors studied the
associated variational problem
$\displaystyle\mathcal{I}^{G}_{\lambda}$
$\displaystyle=\inf\left\\{{|\\!|}|\xi|^{s}\mathcal{F}[{u}](\xi)\|^{2}_{L^{2}({\mathbb{R}^{N}})}-\int_{\mathbb{R}^{N}\times{\mathbb{R}^{N}}}G(u(x))V(|x-y|)G(u(y))dxdy,\right.$
$\displaystyle\hskip 170.0pt\left.u\in
H^{s}({\mathbb{R}^{N}}),\>\int_{{\mathbb{R}^{N}}}|u(x)|^{2}\,dx=\lambda\right\\},$
(3)
for a general nonlinearity $G$, a kernel $V(|x|)=|x|^{\beta-N}$ and dimension
$N$, where here and the following
$H^{s}(\mathbb{R}^{N}):=\\{u\in
L^{2}({\mathbb{R}^{N}}^{N}),\>\,{|\\!|}|\xi|^{s}\mathcal{F}[{u}](\xi)\|^{2}_{L^{2}({\mathbb{R}^{N}})}<\infty\\}.$
In the critical case $2s={N-\beta}{}$, they were able to extend the results of
[18]. Moreover, in the subcritical $2s>{N-\beta}$, they have also proved the
existence and symmetry of all minimizers of (3) by using rearrangement
techniques. More precisely, they showed that under suitable assumptions on
$G$, one can always take a radial and radial by decreasing minimizing sequence
of problem (3). Another very important issue related to the nonlinear
fractional Schrödinger equation $\mathscr{S}$ is the orbital stability of
standing waves. For such an issue, it is essential to show that all the
minimizing sequences are relatively compact in $H^{s}(\mathbb{R}^{N})$. This
is the gist of the breakthrough paper [4]. The line of attach consists of:
1. 1.
Prove the uniqueness of the solutions of $\mathscr{S}$.
2. 2.
Prove the conservation of energy and mass of the solutions.
3. 3.
Prove the relative compactness of all minimizing sequences of the problem (3).
Our first result concerns the well-posedness of the system $\mathscr{S}$.
Before stating it, we need to fix some conditions on $G$. We assume that $G$
is nonnegative and differentiable such that $G(0)=0$ and for all
$\psi\in\mathbb{R}_{+}$
$\mathcal{A}_{0}:\exists\,\mu\in\left[\left.2,1+\frac{2s+\beta}{N}\right)\right.\>\text{s.t.}\>\quad\left\\{\begin{array}[]{ll}&G(\psi)\leq\eta(|\psi|^{2}+|\psi|^{\mu}),\\\
&\\\ &|G^{\prime}(\psi)|\leq\eta(|\psi|+|\psi|^{\mu-1}).\end{array}\right.$
We have obtained the following
###### Theorem 1.1.
Let $N\geq 1,0<s<1,0<\beta<N,N-2s\leq\beta,\phi_{0}\in
H^{s}({\mathbb{R}^{N}})$ and $G$ such that $\mathcal{A}_{0}$ holds true. Then,
there exists a weak global-in-time solution $\phi(t,x)$ to the system
$\mathscr{S}$ such that
$\phi\in L^{\infty}(\mathbb{R}\,;\,H^{s}({\mathbb{R}^{N}}))\cap
W^{1,\infty}(\mathbb{R}\,;\,H^{-s}({\mathbb{R}^{N}})).$
Moreover, if $N=1$ and $\frac{1}{2}<s<1$ or if $N\geq 3$,
$\frac{N}{2(N-1)}<s<1$,
$N-s+\frac{1}{2}<\beta<\min(N,\frac{3N}{2}-s-\frac{N}{4s})$ and $\mu$ (in
$\mathcal{A}_{0}$) is such that
$\max\left(2,1+\frac{2\beta-N}{N-2s}\right)<\mu<2+\frac{N}{N-2s}\frac{2s-1-2N+2\beta}{2s-1+N},$
then the solution is unique.
The particular case $\mu=2$ and $2s=N-\beta$ was treated in Ref. [5] and for
lightness of the proofs, we shall sometimes omit it and focus on the case
$\mu\in\left(2,1+\frac{2s+\beta}{N}\right)$. The proof of the existence part
of Theorem 1.1 is based on a classical contraction argument and the
conservation laws associated to the dynamics of the system $\mathscr{S}$. The
uniqueness part for $N=1$ of Theorem 1.1 readily follows from the embedding
$H^{s}\hookrightarrow L^{\infty}$ for all $s>\frac{1}{2}$. The part for $N\geq
3$ is obtained using mixed norms to be defined later and weighted Strichartz
and convolution inequalities, which require $N\geq 3$. It would be very
interesting to find estimates to handle the uniqueness for $N=2$. Let us
mention that in Ref. [9] the authors showed the orbital stability of standing
waves in the case of power nonlinearities by assuming energy conservation and
time continuity without proving uniqueness, which is an inescapable and quite
hard step, especially in the fractional setting. As mentioned before, if
$\phi(t,x)=e^{i\kappa t}u(x)$ with $\kappa\in\mathbb{R}$ is a solution of the
system $\mathscr{S}$, then it is called a standing wave solution and $u(x)$
solves the following bifurcation problem
$\tilde{\mathscr{S}}:\quad(-\Delta)^{s}u-\kappa u=\left(G(|u|)\star
V(|x|)\right)G^{\prime}(u).$
In order to study the existence of a solution $(\kappa,u)$ to the stationary
equation $\tilde{\mathscr{S}}$, we use a variational method based on the
following minimization problem
$\mathcal{I}_{\lambda}=\inf\left\\{\mathcal{E}(u),\quad u\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|u(x)|^{2}\,dx=\lambda\right\\},$
(4)
where $\lambda$ is a positive prescribed number and
$\displaystyle\mathcal{E}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\int_{\mathbb{R}^{N}\times\mathbb{R}^{N}}G(|u(x)|)\,V(|x-y|)\,G(|u(y)|)\,dxdy,$
$\displaystyle:=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\,\mathcal{D}(G(|u|),G(|u|)).$
The kinetic energy is precisely expressed by the formula for all function $u$
in the Schwarz class
$\|\nabla_{s}u\|^{2}_{L^{2}({\mathbb{R}^{N}})}=C_{N,s}\int_{\mathbb{R}^{N}\times{\mathbb{R}^{N}}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}dxdy,$
(5)
with $C_{N,s}$ being a positive normalization constant. In order to prove the
existence of critical points to the functional $\mathcal{E}$ and thereby
solutions to the problem $\tilde{\mathscr{S}}$, we will need some extra grows
condition on $G$: for all $\psi\in\mathbb{R}_{+}$
$\mathcal{A}_{1}:\quad\left\\{\begin{array}[]{l}\exists
0<\alpha<1+\frac{2s+\beta}{N}\>\>s.t.\>\>\forall\psi,\>0<\psi\ll 1,\quad
G(\psi)\geq\eta\,\psi^{\alpha},\\\ \\\
G(\theta\,\psi)\geq\theta^{1+\frac{2s+\beta}{2N}}\,G(\psi).\end{array}\right.$
Our next main result is contained in the following
###### Theorem 1.2.
Let $0<s<1,0<\beta<N,N-\beta\leq 2s$ and $G$ such that $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ hold true. Then, for all $\lambda>0$, problem (4) has a
minimizer $u_{\lambda}\in H^{s}({\mathbb{R}^{N}})$ such that
$I_{\lambda}=\mathcal{E}(u_{\lambda})$.
In fact we will show that any minimizing sequence of problem 4 is –up to
suitable translations– relatively compact in $H^{s}({\mathbb{R}^{N}})$. The
proof of Theorem 1.2 is based on the concentration-compactness method of P-L.
Lions [17]. The last part of the paper deals with the stability of the
standing waves. For that purpose, we introduce the following problem
$\hat{\mathcal{I}}_{\lambda}=\inf\left\\{\mathcal{J}(z),\quad z\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|z|^{2}\,dx=\lambda\right\\},$
where $z=u+i\,v$ and
$\displaystyle\mathcal{J}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}z{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\,\mathcal{D}(G(|z(x)|),G(|z(x)|)),$
$\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}+\frac{1}{2}{|\\!|}\nabla_{s}v{|\\!|}^{2}_{L^{2}({\mathbb{R}^{N}})}-\frac{1}{2}\,\mathcal{D}(G((u^{2}+v^{2})^{\frac{1}{2}}),G((u^{2}+v^{2})^{\frac{1}{2}})),$
$\displaystyle:=$ $\displaystyle\mathcal{J}(u,v).$
We have obviously $\mathcal{E}(u)=\mathcal{J}(u,0)$. Following Ref. [4], we
introduce the following set
$\hat{\mathcal{O}}_{\lambda}=\left\\{z\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|z|^{2}\,dx=\lambda\>\>:\>\>\mathcal{J}(z)=\hat{\mathcal{I}}_{\lambda}\right\\}.$
The set $\hat{\mathcal{O}}_{\lambda}$ is the so called orbit of the standing
waves of $\mathscr{S}$ with mass $\sqrt{\lambda}$. We define the stability of
$\hat{\mathcal{O}}_{\lambda}$ as follows
###### Definition 1.3.
Let $\phi_{0}\in H^{s}({\mathbb{R}^{N}})$ be an initial data and $\phi(t,x)\in
H^{s}({\mathbb{R}^{N}})$ the associated solution of problem $\mathscr{S}$. We
say that $\hat{\mathcal{O}}_{\lambda}$ is $H^{s}({\mathbb{R}^{N}})-$stable
with respect to the system $\mathscr{S}$ if
* 1.
$\hat{\mathcal{O}}_{\lambda}\neq\varnothing$.
* 2.
For all $\varepsilon>0$, there exists $\delta>0$ such that for any
$\phi_{0}\in H^{s}({\mathbb{R}^{N}})$ satisfying
$\inf_{z\in\hat{\mathcal{O}}_{\lambda}}|\phi_{0}-z|<\delta$, we have
$\inf_{z\in\hat{\mathcal{O}}_{\lambda}}|\phi(t,x)-z|<\epsilon$ for all
$t\in\mathbb{R}$.
The notion of stability depends then intimately on the well-posedness of the
Cauchy problem $\mathscr{S}$ and the existence of standing waves. Therefore,
having in hand Theorems 1.1 and 1.2, we prove the following
###### Theorem 1.4.
Let $N\geq 3$, $\frac{N}{2(N-1)}<s<1$,
$N-s+\frac{1}{2}<\beta<\min(N,\frac{3N}{2}-s-\frac{N}{4s})$ and let $G$
satisfying $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ with $\mu$ (in
$\mathcal{A}_{0}$) such that
$\max\left(2,1+\frac{2\beta-N}{N-2s}\right)<\mu<2+\frac{N}{N-2s}\,\frac{2s-1-2N+2\beta}{2s-1+N}.$
Let $\phi_{0}\in H^{s}({\mathbb{R}^{N}})$ and $\phi(t,x)\in
H^{s}({\mathbb{R}^{N}})$ the associated solution to the problem $\mathscr{S}$.
Then $\hat{\mathcal{O}}_{\lambda}$ is $H^{s}({\mathbb{R}^{N}})-$stable with
respect to the system $\mathscr{S}$.
The paper is divided into three sections. The first one is dedicated to the
analysis of the dynamics of the system $\mathscr{S}$. More precisely, in this
section we prove Theorem 1.1. First of all, we prove a local-in-time existence
of solutions. Second we show that under extra assumptions, this solution is
actually unique. Eventually, we use the conservation laws to show the global-
in-time well-posedness. The second section is devoted to the proof of
existence of solution to the problem $\tilde{\mathscr{S}}$. For that purpose,
we use the classical concentration compactness method [17] to prove Theorem
1.2. The last section is dedicated to the proof of stability of standing
waves, namely Theorem 1.4. Here, we use ideas and techniques developed in
[13]. From this point onward, $\eta$ will denote variant universal constants
that may change from line to line of inequalities. When $\eta$ depends on some
parameter, we will write $\eta(\cdot)$ instead of $\eta$. In order to lighten
the notation and the calculation, we shall use $L^{p}$ and $H^{s}$ instead of
$L^{p}({\mathbb{R}^{N}})$ and $H^{s}({\mathbb{R}^{N}})$ respectively for real
or complex valued functions. Also, we shall use ${|\\!|}\cdot{|\\!|}_{p}$
instead of ${|\\!|}\cdot{|\\!|}_{L^{p}({\mathbb{R}^{N}})}$ for all
$p\in[1,\infty]$. The exponent $p^{\prime}$ will denotes the conjugate
exponent of $p$, that is $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. For a more
detailed account about the Sobolev spaces $H^{s}$, we refer the reader to any
textbook of functional analysis (see [3] for instance).
## 2 Well-posedness of the system $\mathscr{S}$
In this section we consider the local and global well-posedness of the problem
$\mathscr{S}$ and prove Theorem 1.1. Let us denote the nonlinear term
$[V(|x|)\star G(\phi)]G^{\prime}(\phi)$ by $\mathcal{N}(\phi)$. Since the
well-posedness of the case $\mu=2,2s=N-\beta$ was treated in [5], in this
paper we consider the initial value problem $\mathscr{S}$ with
$\mu\in\left(2,1+\frac{2s+\beta}{N}\right)$. Let $g=G^{\prime}$, that is,
$\int_{0}^{|z|}g(\alpha)\,d\alpha=G(z)$, and assume that
$g(z)=\frac{z}{|z|}g(|z|),z\neq 0$, $G(z)\geq 0$. Then, with
$\mathcal{A}_{0}$, the function $g$ satisfies obviously
$\displaystyle|g(z)|+|g^{\prime}(z)z|\leq C(|z|+|z|^{\mu-1})\;\;\mbox{for
all}\;\;z\in\mathbb{C}.$ (6)
### 2.1 Weak solutions
We first show existence of weak solutions to $\mathscr{S}$ in $H^{s}$. For
this purpose we prove that $\mathcal{N}$ is Lipschitz map from
$L^{p^{\prime}}$ to $L^{r}$ for some
$p,r\in\left.\left[2,\frac{2N}{N-2s}\right)\right.$. Then the rest of the
proof is quite straightforward from the Lipschitz map and well-known
regularizing arguments and we refer the readers to the book [3].
###### Proposition 2.1.
Let $N\geq 2$, $0<s<1$, $0<\beta<N$ and $2s\geq N-\beta$. If $g$ satisfies (6)
with $\mu\in\left(2,1+\frac{2s+\beta}{N}\right)$. Then there exists a weak
solution $\phi$ such that
$\displaystyle\phi\in L^{\infty}(-T_{min},T_{max};H^{s})\cap
W^{1,\infty}(-T_{min},T_{max};H^{-s}),$
$\displaystyle{|\\!|}\phi(t){|\\!|}_{2}={|\\!|}\phi_{0}{|\\!|}_{2},\;\;\mathcal{J}(\phi(t))\leq\mathcal{J}(\phi_{0}).$
for all $t\in(-T_{min},T_{max})$, where $(-T_{min},T_{max})$ is the maximal
existence time interval of $\phi$ for given initial data $\phi_{0}$.
###### Proof.
Let us introduce the following cut-off for the function $g$,
$g_{1}(\alpha)=\chi_{\\{0\leq\alpha<1\\}}g(\alpha)$ and
$g_{2}(\alpha)=\chi_{\\{\alpha\geq 1\\}}g(\alpha)$ and
$G_{i}(z)=\int_{0}^{|z|}g_{i}(\alpha)\,d\alpha$ with obvious definition of the
Euler function $\chi$. Then, one can writes
$\mathcal{N}(\phi)=\sum_{i,j=1,2}\mathcal{N}_{ij}(\phi)\>\>\text{where}\>\>\mathcal{N}_{ij}(\phi)=\int_{{\mathbb{R}^{N}}}|x-y|^{-(N-\beta)}G_{i}(|\phi|)\,dy\,g_{j}(\phi).$
We claim that there exist
$p_{ij},r_{ij}\in\left[\left.2,\frac{2N}{N-2s}\right)\right.$111If $N=1$ and
$\frac{1}{2}\leq s<1$, then $\frac{2N}{N-2s}$ is interpreted as $\infty$. such
that
$\displaystyle{|\\!|}\mathcal{N}_{ij}(\phi)-\mathcal{N}_{ij}(\psi){|\\!|}_{p^{\prime}_{ij}}\leq\eta(K){|\\!|}\phi-\psi{|\\!|}_{r_{ij}},$
(7)
for some constant $\eta(K)$ with $\eta(K)\leq\eta\,K^{a_{i\\!j}}$,
$a_{i\\!j}>0$ for all $1\leq i,j\leq 2$, provided
${|\\!|}\phi{|\\!|}_{H^{s}}+{|\\!|}\psi{|\\!|}_{H^{s}}\leq K$. This implies
that $\mathcal{N}:H^{s}\to H^{-s}$ is a Lipschitz map on a bounded sets of
$H^{s}$. Indeed, let $\mu_{1}=2$ and $\mu_{2}=\mu$. Then we have
$\displaystyle|\mathcal{N}_{ij}(\phi)-\mathcal{N}_{ij}(\psi)|$
$\displaystyle\leq\eta\int_{{\mathbb{R}^{N}}}|x-y|^{-(N-\beta)}(|\phi|^{\mu_{i}-1}+|\psi|^{\mu_{i}-1})|\phi-\psi|\,dy|\phi|^{\mu_{j}-1}$
$\displaystyle+\eta\int_{{\mathbb{R}^{N}}}|x-y|^{-(N-\beta)}|\psi|^{\mu_{i}}\,dy(|\phi|^{\mu_{j}-2}+|\psi|^{\mu_{j}-2})|\phi-\psi|.$
By Hölder’s and Hardy-Littlewood-Sobolev inequalities with indices
$p_{ij},r_{ij}$ such that
$\displaystyle
1-\frac{1}{p_{ij}}=\frac{\mu_{i}}{r_{ij}}-\frac{\beta}{N}+\frac{\mu_{j}-1}{r_{ij}},\;\;\frac{\mu_{i}}{r_{ij}}>\frac{\beta}{N},$
(8)
we obtain
$\displaystyle{|\\!|}\mathcal{N}_{ij}(\phi)-\mathcal{N}_{ij}(\psi){|\\!|}_{p^{\prime}_{ij}}$
$\displaystyle\leq\eta\,\left[({|\\!|}\phi{|\\!|}_{r_{ij}}^{\mu_{i}-1}+{|\\!|}\psi{|\\!|}_{r_{ij}}^{\mu_{i}-1}){|\\!|}\phi{|\\!|}_{r_{ij}}^{\mu_{j}-1}\right.$
$\displaystyle\left.+{|\\!|}\psi{|\\!|}_{r_{ij}}^{\mu_{i}}({|\\!|}\phi{|\\!|}_{r_{ij}}^{\mu_{j}-2}+{|\\!|}\psi{|\\!|}_{r_{ij}}^{\mu_{j}-2})\right]{|\\!|}\phi-\psi{|\\!|}_{r_{ij}}.$
Thus if $p_{ij},r_{ij}\in\left.\left[2,\frac{2N}{N-2s}\right)\right.$, then
Sobolev inequality shows (7). Now we show that there exist
${p_{ij}},{r_{ij}}\in[2,\frac{2N}{N-2s})$ such that the combinations (8) hold
true. If ${p_{ij}},{r_{ij}}$ satisfy (8), then they are on the line
$\displaystyle\frac{1}{{r_{ij}}}=\frac{1}{\mu_{i}+\mu_{j}-1}(1+\frac{\beta}{N}-\frac{1}{{p_{ij}}}).$
(9)
Since $\frac{1}{\mu_{i}+\mu_{j}-1}(1+\frac{\beta}{N}-\frac{1}{2})<\frac{1}{2}$
and
$\frac{N-2s}{2N}<\frac{1}{\mu_{i}+\mu_{j}-1}(1+\frac{\beta}{N}-\frac{N-2s}{2N})$,
the line (9) of $(\frac{1}{{p_{ij}}},\frac{1}{{r_{ij}}})$ always passes
through the open square
$(\frac{N-2s}{2N},\frac{1}{2})\times(\frac{N-2s}{2N},\frac{1}{2})$. We have
only to find a pair $(\frac{1}{{p_{ij}}},\frac{1}{{r_{ij}}})$ of line (9) such
that $\frac{\mu_{i}}{{r_{ij}}}>\frac{\beta}{N}$. If
$\frac{\mu_{i}}{{r_{ij}}}>\frac{\beta}{N}$, then
$\frac{1}{{p_{ij}}}<1-\frac{\mu_{j}-1}{\mu_{i}}\frac{\beta}{N}.$
So, it suffices to show that
$\displaystyle\max\left(\frac{1}{p_{0}},\frac{N-2s}{2N}\right)<1-\frac{\mu_{j}-1}{\mu_{i}}\frac{\beta}{N},$
(10)
where $\frac{1}{p_{0}}$ is the point of line (9) when
$\frac{1}{{r_{ij}}}=\frac{1}{2}$, that is,
$\frac{1}{p_{0}}=1+\frac{\beta}{N}-\frac{\mu_{i}+\mu_{j}-1}{2}$. In fact, it
is an easy matter to show (10) from the condition
$\mu\in\left(2,1+\frac{\beta+2s}{N}\right)$ and we leave the proof to the
reader. The proof of Proposition 2.1 follows now by a straightforward
application of a contraction argument. ∎
### 2.2 Uniqueness
Since the case $N=1$ can be treated as in [3], we omit the details. When
$N\geq 3$, the uniqueness of weak solutions can be shown by a weighted
Strichartz and convolution estimates. For that purpose, we introduce the
following mixed norm for all $1\leq m,\widetilde{m}<\infty$
${|\\!|}h{|\\!|}_{L_{\rho}^{m}L_{\sigma}^{\widetilde{m}}}:=(\int_{0}^{\infty}(\int_{S^{N-1}}|h(\rho\sigma)|^{\widetilde{m}}\,d\sigma)^{\frac{m}{\widetilde{m}}}\,\rho^{n-1}d\rho)^{\frac{1}{m}}.$
The case $m=\infty$ or $\widetilde{m}=\infty$ can be defined is a usual way.
Then we have the following.
###### Proposition 2.2.
Let $N\geq 3$, $\frac{N}{2(N-1)}<s<1$,
$N-s+\frac{1}{2}<\beta<\min(N,\frac{3N}{2}-s-\frac{N}{4s})$, and $g$ such that
the condition (6) holds true with
$\max\left(2,1+\frac{2\beta-N}{N-2s}\right)<\mu<2+\frac{N}{N-2s}\,\frac{2s-1-2N+2\beta}{2s-1+N}.$
Then the $H^{s}$-weak solution to the problem $\mathscr{S}$ constructed in
proposition 2.1 is unique.
The dimension restriction $N\geq 3$ is necessary for $\frac{N}{2(N-1)}<s<1$
and $N-s+\frac{1}{2}<\beta<\frac{3N}{2}-s-\frac{N}{4s}$, which are needed for
the exponents appearing in (14).
###### Proof.
Let $U(t)=e^{it(-\Delta)^{s}}$, then the solution ${\phi}$ constructed in
Proposition 2.1 satisfies the integral equation
$\displaystyle{\phi}(t)=U(t)\varphi-i\int_{0}^{t}U(t-t^{\prime})\mathcal{N}({\phi}(t^{\prime}))\,dt^{\prime}\;\;\mbox{a.e.}\;t\in(-T_{min},T_{max}).$
(11)
Before going further, let us recall the following weighted Strichartz estimate
(see for instance Lemma 6.2 of [5] and Lemma 2 of [6]).
###### Lemma 2.3.
Let $N\geq 2$ and $2\leq q<4s$. Then, for all $\psi\in L^{2}$, we have
${|\\!|}|x|^{-\delta}U(t)\psi{|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\leq\eta\,{|\\!|}\psi{|\\!|}_{2},$
where $\delta=\frac{N+2s}{q}-\frac{N}{2}$,
$\frac{1}{\widetilde{q}}=\frac{1}{2}-\frac{1}{N-1}\left(\frac{2s}{q}-\frac{1}{2}\right)$
and $\eta$ is independent of $t_{1},t_{2}$.
In [5] it was shown that
${|\\!|}|x|^{-\delta}D_{\sigma}^{\frac{2s}{q}-\frac{1}{2}}U(t)\psi{|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{2})}\leq\eta{|\\!|}\psi{|\\!|}_{2}.$
Lemma 2.3 can be derived by Sobolev embedding on the unit sphere. Here
$D_{\sigma}=\sqrt{1-\Delta_{\sigma}}$ where $\Delta_{\sigma}$ is the Laplace-
Beltrami operator on the unit sphere. Now, let us recall the following
weighted convolution inequality we shall use in the sequel
###### Lemma 2.4 (Lemma 4.3 of [7]).
Let $r\in[1,\infty]$ and $0\leq\delta\leq\gamma<N-1$. If
$\frac{1}{r}>\frac{\gamma}{N-1}$, then for all $f$ such that
$|x|^{-(\gamma-\delta)}f\in L^{1}$, we have
${|\\!|}|x|^{\delta}(|x|^{-\gamma}\ast
f){|\\!|}_{L_{\rho}^{\infty}L_{\sigma}^{r}}\leq\eta{|\\!|}|x|^{-(\gamma-\delta)}f{|\\!|}_{1}.$
Therefore, using Lemma 2.3 one can readily deduce that
$\displaystyle{|\\!|}|x|^{-\delta}\int_{0}^{t}U(t-t^{\prime})f(t^{\prime}){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\leq\eta{|\\!|}f{|\\!|}_{L^{1}(-t_{1},t_{2};L^{2})}.$
(12)
Thus, if we set $f=\mathcal{N}({\phi})-\mathcal{N}({\psi})$ and
$\gamma=N-\beta$. Then from (11) we infer
$\displaystyle{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}$
$\displaystyle\leq$
$\displaystyle\eta\sum_{i,j=1}^{2}\int_{-t_{1}}^{t_{2}}{|\\!|}\mathcal{N}_{ij}({\phi})-\mathcal{N}_{ij}({\psi}){|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq$
$\displaystyle\eta\sum_{i,j=1}^{2}\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{{\mathbb{R}^{N}}}|x-y|^{-\gamma}(|{\phi}|^{\mu_{i}-1}+|{\psi}|^{\mu_{i}-1})|{\phi}-{\psi}|\,dy|{\phi}|^{\mu_{j}-1}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle+$
$\displaystyle\eta\sum_{i,j=1}^{2}\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{{\mathbb{R}^{N}}}|x-y|^{-\gamma}|{\psi}|^{\mu_{i}}\,dy(|{\phi}|^{\mu_{j}-2}+|{\psi}|^{\mu_{j}-2})|{\phi}-{\psi}|{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\equiv$
$\displaystyle\sum_{i,j=1}^{2}(\mathcal{T}^{1}_{ij}+\mathcal{T}^{2}_{ij}).$
We first estimate $\mathcal{T}^{1}_{ij}$ using Hölder’s and Hardy-Littlewood-
Sobolev inequalities. On the one side if $(i,j)=(1,2)$, since
$\mu\in\left(1+\frac{2\beta-N}{N-2s},1+\frac{\beta+2s}{N}\right)$, $0<\beta<N$
and $2s>\gamma=N-\beta$, we can find $r\in\left[2,\frac{2N}{N-2s}\right]$ such
that
$\frac{\beta}{N}=\frac{1}{r}+\frac{(\mu-1)(N-2s)}{2N},\;\;\frac{1}{r}+\frac{1}{2}>\frac{\beta}{N}.$
Thus, we can write
$\displaystyle\mathcal{T}^{1}_{12}$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}({|\\!|}{\phi}{|\\!|}_{r}+{|\\!|}{\psi}{|\\!|}_{r}){|\\!|}{\phi}-{\psi}{|\\!|}_{2}|{\phi}|_{\frac{2N}{N-2s}}^{\mu-1}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2})({|\\!|}{\phi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu}+{|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu}){|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$
On the opposite side, if $(i,j)\neq(1,2)$, then we can choose
$r\in\left[2,\frac{2N}{N-2s}\right]$ such that
$\frac{\beta}{N}=\frac{\mu_{i}-1}{r}+\frac{(\mu_{j}-1)}{r},\;\;\frac{\mu_{i}-1}{r}+\frac{1}{2}>\frac{\beta}{N}.$
Such a combination is always possible thanks to our conditions on $\mu,\beta$
and $s$. Therefore, we get as above
$\displaystyle\mathcal{T}^{1}_{ij}$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}({|\\!|}{\phi}{|\\!|}_{r}^{\mu_{i}-1}+{|\\!|}{\psi}{|\\!|}_{r}^{\mu_{i}-1}){|\\!|}{\phi}-{\psi}{|\\!|}_{2}{|\\!|}{\phi}{|\\!|}_{\frac{r}{\mu_{j}-1}}^{\mu_{j}-1}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2})({|\\!|}{\phi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu_{j}-2}+{|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu_{j}-2}){|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$
We are kept with the estimates of $\mathcal{T}^{2}_{ij}$. If $j=1$, then we
can use Hardy-Sobolev inequality such that for $0<q<N$ and $2\leq p<\infty$
$\displaystyle{|\\!|}|x|^{-\frac{q}{p}}f{|\\!|}_{p}\leq\eta{|\\!|}f{|\\!|}_{\dot{H}^{\frac{N}{2}-\frac{N-q}{p}}}.$
(13)
In fact, we have
$\displaystyle\mathcal{T}^{2}_{11}$
$\displaystyle\leq\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{\mathbb{R}^{N}}|x-y|^{-\gamma}|{\psi}|^{2}\,dy{|\\!|}_{L_{x}^{\infty}}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}{|\\!|}{\psi}{|\\!|}_{\dot{H}^{\frac{\gamma}{2}}}^{2}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2}){|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{2}{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$
Since $\frac{N}{2}-\frac{\beta}{\mu}\leq s$ we also have
$\displaystyle\mathcal{T}^{2}_{21}$
$\displaystyle\leq\int_{-t_{1}}^{t_{2}}{|\\!|}\int_{\mathbb{R}^{N}}|x-y|^{-\gamma}|{\psi}|^{\mu}\,dy{|\\!|}_{L_{x}^{\infty}}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq
C\int_{-t_{1}}^{t_{2}}{|\\!|}{\psi}{|\\!|}_{\dot{H}^{\frac{N}{2}-\frac{\beta}{\mu}}}^{\mu}{|\\!|}{\phi}-{\psi}{|\\!|}_{2}\,dt^{\prime},$
$\displaystyle\leq
C(t_{1}+t_{2}){|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{2}{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}.$
When $j=2$, we use the weighted convolution inequality (Lemma 2.4). The
hypothesis on $\beta,\mu$ guarantees the existence of exponents
$q,\widetilde{q}$ and $r$ satisfying the conditions of Lemmas 2.3, 2.4 and
also the following combination
$\displaystyle\frac{1}{2}=\frac{(\mu-2)(N-2s)}{2N}+\frac{1}{q}=\frac{1}{r}+\frac{(\mu-2)(N-2s)}{2N}+\frac{1}{\widetilde{q}}.$
(14)
Hence, using the Hardy-Sobolev inequality (13) we write
$\displaystyle\mathcal{T}^{2}_{i,2}$
$\displaystyle\leq\int_{-t_{1}}^{t_{2}}{|\\!|}|x|^{\delta}\int_{\mathbb{R}^{N}}|x-y|^{-\gamma}|{\psi}|^{\mu_{i}}\,dy{|\\!|}_{L_{\rho}^{\infty}L_{\sigma}^{r}}\left({|\\!|}{\phi}{|\\!|}_{\frac{2N}{N-2s}}^{\mu-2}+{|\\!|}{\psi}{|\\!|}_{\frac{2N}{N-2s}}^{\mu-2}\right)\times$
$\displaystyle\hskip
200.0pt\times{|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L_{\rho}^{q}L_{\sigma}^{\widetilde{q}}}\,dt^{\prime},$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}{|\\!|}|x|^{(-\gamma-\delta)}|{\psi}|^{\mu_{i}}{|\\!|}_{1}\left({|\\!|}|{\phi}{|\\!|}_{H^{s}}^{\mu-2}+{|\\!|}{\psi}{|\\!|}_{H^{s}}^{\mu-2}\right){|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L_{\rho}^{q}L_{\sigma}^{\widetilde{q}}}\,dt^{\prime},$
$\displaystyle\leq\eta\int_{-t_{1}}^{t_{2}}{|\\!|}{\psi}{|\\!|}_{\dot{H}^{\frac{N}{2}-\frac{\beta+\delta}{\mu_{i}}}}^{\mu_{i}}\left({|\\!|}{\phi}{|\\!|}_{H^{s}}^{\mu-2}+{|\\!|}{\psi}{|\\!|}_{H^{s}}^{\mu-2}\right){|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L_{\rho}^{q}L_{\sigma}^{\widetilde{q}}}\,dt^{\prime},$
$\displaystyle\leq\eta(t_{1}+t_{2})^{1-\frac{1}{q}}\left({|\\!|}{\phi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu-2}+{|\\!|}{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};H^{s})}^{\mu_{i}+\mu-2}\right)\times$
$\displaystyle\hskip
200.0pt\times{|\\!|}|x|^{-\delta}({\phi}-{\psi}){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}.$
Now, if $(-t_{1},t_{2})\subset[-T_{1},T_{2}]$ and
${|\\!|}{\phi}{|\\!|}_{L^{\infty}(-T_{1},T_{2};H^{s})}+{|\\!|}\psi{|\\!|}_{L^{\infty}(-T_{1},T_{2};H^{s})}\leq
K$, then by combining all the estimates above we infer
$\displaystyle{|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\leq\eta(K^{2}+K^{2\mu-2})\times$
$\displaystyle\hskip
40.0pt\times(t_{1}+t_{2})^{1-\frac{1}{q}}\left({|\\!|}{\phi}-{\psi}{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\right).$
Thus, ${\phi}=\psi$ on $[-t_{1},t_{2}]$ for sufficiently small $t_{1},t_{2}$.
Let $I=(-a,b)$ be the maximal interval of $[-T_{1},T_{2}]$ with
${|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-c,d;L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-c,d;L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}=0,\>c<a,d<b.$
Assume that $a<T_{1}$ or $b<T_{2}$. Without loss of generality, we may also
assume that $a<T_{1}$ and $b<T_{2}$. Then for a small $\varepsilon>0$ we can
find $a<t_{1}<T_{1},b<t_{2}<T_{2}$ such that
$\displaystyle{|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}$
$\displaystyle\leq(K^{2}+K^{2\mu-2})(t_{1}+t_{2}-a-b)^{1-\frac{1}{q}}\times$
$\displaystyle\hskip
50.0pt\times\left({|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\right),$
$\displaystyle\leq(1-\varepsilon)\left({|\\!|}{\phi}-\psi{|\\!|}_{L^{\infty}(-t_{1},t_{2};L^{2})}+{|\\!|}|x|^{-\delta}({\phi}-\psi){|\\!|}_{L^{q}(-t_{1},t_{2};L_{\rho}^{q}L_{\sigma}^{\widetilde{q}})}\right).$
This contradicts the maximality of $I$. Thus $I=[-T_{1},T_{2}]$. Since
$[-T_{1},T_{2}]$ is arbitrarily taken in $(-T_{min},T_{max})$, we finally get
the whole uniqueness and the Proposition 2.2 is now proved. ∎
### 2.3 Global well-posedness
Using the argument of [3], one can show that the uniqueness implies actually
well-posedness and conservation laws:
$\displaystyle\bullet\;{\phi}\in C(-T_{min},T_{max};H^{s})\cap
C^{1}(-T_{min},T_{max};H^{-s}),$
$\displaystyle\bullet\;{\phi}\;\;\mbox{depends continuously
on}\;\;\phi_{0}\;\;\mbox{in}\;\;H^{s},$
$\displaystyle\bullet\;{|\\!|}{\phi}(t){|\\!|}_{2}={|\\!|}\phi_{0}{|\\!|}_{2}\;\;\mbox{
and}\;\;\mathcal{J}({\phi}(t))=\mathcal{J}(\phi_{0})\;\;\forall\;t\in(-T_{min},T_{max}).$
The proofs of these points are standard, we omit them and refer to [3]. Now we
remark that the well-posedness is actually global by establishing a uniform
bound on the $H^{s}$ norm of $\phi(t)$ for all $t\in(-T_{min},T_{max})$. We
first consider the global existence of weak solutions. Suppose ${\phi}$ is a
weak solution on $(-T_{min},T_{max})$ as in Proposition 2.1. We show that
${|\\!|}{\phi}(t){|\\!|}_{H^{s}}$ is bounded for all $t\in(-T_{min},T_{max})$.
For this purpose let us introduce the following notation
$\mathcal{D}(G(|\phi|),G(|\phi|))=\sum_{i,j=1}^{2}\mathcal{D}_{i,j}(|\phi|),\quad\mathcal{D}_{i,j}(|\phi|):=\mathcal{D}(G_{i}(|\phi|),G_{j}(|\phi|)),$
(15)
where obviously we set $G_{i}:=\int_{0}^{|z|}g_{i}(\alpha)\,d\alpha$ and
recall that the $g_{i}$ are defined as
$g_{1}(\alpha)=\chi_{\\{0\leq\alpha<1\\}}g(\alpha)$ and
$g_{2}(\alpha)=\chi_{\\{\alpha\geq 1\\}}g(\alpha)$. Using Hardy-Littlewood-
Sobolev and the fractional Gagliardo-Nirenberg inequalities and the assumption
$\mathcal{A}_{0}$ we can write the following estimates
$\displaystyle\mathcal{D}_{1,1}\leq\eta\,{|\\!|}u{|\\!|}^{4}_{\frac{4N}{N+\beta}}\leq\eta{|\\!|}u{|\\!|}^{4-\frac{N-\beta}{s}}_{2}\,{|\\!|}u{|\\!|}^{\frac{N-\beta}{s}}_{\dot{H}^{s}},$
(16)
$\displaystyle\mathcal{D}_{2,2}\leq\eta\,{|\\!|}u{|\\!|}^{2\mu}_{\frac{2N\mu}{N+\beta}}\leq\eta{|\\!|}u{|\\!|}^{2\mu-\frac{N(\mu-1)-\beta}{s}}_{2}\,{|\\!|}u{|\\!|}^{\frac{N(\mu-1)-\beta}{s}}_{\dot{H}^{s}},$
(17)
$\displaystyle\mathcal{D}_{1,2},\,\mathcal{D}_{2,1}\leq\eta\,{|\\!|}u{|\\!|}^{\mu+2-\frac{N\mu-2\beta}{2s}}_{2}\,{|\\!|}u{|\\!|}^{\frac{N\mu-2\beta}{2s}}_{\dot{H}^{s}}.$
(18)
Since $N-\beta<2s$, then $0<\frac{N-\beta}{s}<2$ and $4-\frac{N-\beta}{s}>2$.
As well since $2\leq\mu<1+\frac{2s+\beta}{N}$, then
$0<\frac{N-\beta}{s}\leq\frac{N(\mu-1)-\beta}{s}<2$ and
$2<\mu-\frac{N(\mu-1)-\beta}{s}$. Eventually, we have
$0<\frac{N-\beta}{s}\leq\frac{N\mu-2\beta}{2s}$ and
$2\leq\mu<\mu+1+\frac{\beta-N}{2s}<\mu+2-\frac{N\mu-2\beta}{2s}$. The
estimates above can be summarized as follows with $\mu_{1}=2$ and
$\mu_{2}=\mu$.
$\displaystyle\mathcal{D}_{i,j}(|u|)$ $\displaystyle\leq$
$\displaystyle\eta\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|u(x)|^{\mu_{i}}|u(y)|^{\mu_{j}}}{|x-y|^{N-\beta}}\,dxdy,$
(19) $\displaystyle\leq$
$\displaystyle\eta\,{|\\!|}u{|\\!|}^{\mu_{i}+\mu_{j}-\gamma_{i,j}}_{2}\,{|\\!|}u{|\\!|}^{\gamma_{i,j}}_{\dot{H}^{s}}$
where
$\gamma_{i,j}=\frac{N}{s}\left(1+\frac{\beta}{N}\right)-\left(\frac{N}{2s}-1\right)(\mu_{i}+\mu_{j}).$
Thus, we have clearly
$\displaystyle\frac{1}{2}{|\\!|}{\phi}{|\\!|}_{H^{s}}^{2}$
$\displaystyle=\frac{1}{2}{|\\!|}{\phi}{|\\!|}_{2}^{2}+\mathcal{J}({\phi})+\mathcal{D}(G(|\phi|),G(|\phi|)),$
$\displaystyle\leq\frac{1}{2}{|\\!|}\phi_{0}{|\\!|}_{2}^{2}+\mathcal{J}(\phi_{0})+\eta\sum_{i,j=1,2}{|\\!|}\phi_{0}{|\\!|}_{2}^{\frac{2\gamma_{ij}}{2-\mu_{i}-\mu_{j}+\gamma_{ij}}}+\frac{1}{4}{|\\!|}{\phi}{|\\!|}_{H^{s}}^{2}.$
Thus
${|\\!|}{\phi}{|\\!|}_{H^{s}}\leq\eta\left({|\\!|}\phi_{0}{|\\!|}_{H^{s}}\right),\quad\text{for
all}\>t\in(-T_{min},T_{max}).$
Therefore $T_{min}=T_{max}=\infty$. If $s,\beta,\mu$ satisfy the hypothesis of
Proposition 2.2, then we get the global well-posedness. Eventually, combining
this fact with the Propositions 2.1 and 2.2 prove Theorem 1.1.
## 3 Existence of standing waves
In this section we study the minimization problem $\tilde{\mathscr{S}}$. We
prove the existence of a solution to $\tilde{\mathscr{S}}$ using a variational
approach via the concentration-compactness method of P-L. Lions [17]. Indeed,
we aim to prove the existence of critical points to the energy functional
$\displaystyle\mathcal{E}(u)$ $\displaystyle=$
$\displaystyle\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{2}-\frac{1}{2}\,\mathcal{D}(G(|u|),G(|u|)).$
In other words, we look for a function $u_{\lambda}$ such that
$\mathcal{E}(u_{\lambda})=\mathcal{I}_{\lambda}=\inf\left\\{\mathcal{E}(u),\quad
u\in
H^{s}(\mathbb{R}^{N}),\quad\int_{\mathbb{R}^{N}}|u(x)|^{2}\,dx=\lambda\right\\}.$
As noticed in the introduction of this paper, this problem has been studied in
various situation depending on the value of $s$ and the conditions on $\beta$
and the integrand $G$ in Ref. [5, 12, 18]. In order to prove the existence of
critical points to the functional $\mathcal{E}$, we start with the following
claim
###### Proposition 3.1.
For all $\lambda>0$ and $G$ such that $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$
hold true, we have
* 1.
The functional $\mathcal{E}\in C^{1}(H^{s},\mathbb{R})$ and there exists a
constant $\eta>0$ such that
${|\\!|}\mathcal{E}^{\prime}(u){|\\!|}_{H^{-s}}\leq\eta\left({|\\!|}u{|\\!|}_{H^{s}}+{|\\!|}u{|\\!|}_{H^{s}}^{\frac{2s+\beta}{N}}\right).$
* 2.
$-\infty<\mathcal{I}_{\lambda}<0$.
* 3.
Each minimizing sequence for the problem $\mathcal{I}_{\lambda}$ is bounded in
$H^{s}$.
###### Proof.
Let us mention that only assumption $\mathcal{A}_{0}$ is needed to prove the
$C^{1}$ property of the energy functional $\mathcal{E}$. The proof of this
claim is standard and we refer the reader to Ref. [11] for details. Now, we
prove the second assertion. Let $u\in H^{s}(\mathbb{R}^{N})$ such that
${|\\!|}u{|\\!|}_{2}=\sqrt{\lambda}$ and assume $\mathcal{A}_{0}$. Then, on
the one hand, thanks to (16 –18), it is rather easy to show using Young’s
inequality that for all $\epsilon_{1},\epsilon_{2}$ and $\epsilon_{3}$, there
exist $C_{\epsilon_{1}},C_{\epsilon_{2}},C_{\epsilon_{3}}>0$ such that
$\displaystyle\mathcal{D}_{1,1}$ $\displaystyle\leq$
$\displaystyle\eta\left(\epsilon_{1}\,{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+C_{\epsilon_{1}}\lambda^{e_{1}}\right),\quad
e_{1}:=\frac{4s+\beta-N}{2s+\beta-N}.$ (20) $\displaystyle\mathcal{D}_{1,2}$
$\displaystyle\leq$
$\displaystyle\eta\left(\epsilon_{2}\,{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+C_{\epsilon_{2}}\lambda^{e_{2}}\right),\quad
e_{2}:=\frac{2s\mu+\beta-N(\mu-1)}{2s+\beta-N(\mu-1)}.$ (21)
$\displaystyle\mathcal{D}_{1,2},\,\mathcal{D}_{2,1}$ $\displaystyle\leq$
$\displaystyle\eta\left(\epsilon_{1}{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right),\quad
e_{3}:=1+\frac{2s\mu}{4s-N\mu+2\beta}.$ (22)
Observe that $0<2s+\beta-N<4s+\beta-N$ so that $e_{1}>1$. Also,
$0<2s+\beta-N(\mu-1)<2s\mu+\beta-N(\mu-1)$ so that $e_{2}>1$. Eventually,
$4s-N\mu+2\beta>2s+\beta-N>0$ so that $\frac{2s\mu}{4s-N\mu+2\beta}>0$ and
$e_{3}>1$. Therefore, for sufficiently small $\epsilon_{1},\epsilon_{2}$ and
$\epsilon_{3}$, one has
$\displaystyle\mathcal{E}(u)$ $\displaystyle\geq$
$\displaystyle\left(\frac{1}{2}-\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})\right)\,{|\\!|}u{|\\!|}^{2}_{H^{s}}-\frac{1}{2}-\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right),$
$\displaystyle\geq$
$\displaystyle-\frac{1}{2}\lambda-\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right).$
Thus, we obtain $\mathcal{I}_{\lambda}>-\infty$. On the other hand, let us
introduce for all $\kappa\in\mathbb{R}$, the rescaled function
$u_{\kappa}=\kappa^{\frac{1}{2}}u(\kappa^{\frac{1}{N}}\cdot)$. Obviously, one
has $\int_{\mathbb{R}^{N}}|u_{\kappa}|^{2}=\lambda$ and using
$\mathcal{A}_{1}$
$\displaystyle\mathcal{E}(u_{\kappa})\leq\frac{1}{2}\kappa^{\frac{2s}{N}}\int_{\mathbb{R}^{N}}|(-\Delta)^{s}u(x)|^{2}dx-\frac{\kappa^{\alpha-\left(1+\frac{\beta}{N}\right)}}{2}\,\mathcal{D}(|u(x)|^{\alpha},|u(y)|^{\alpha}).$
We have $0<\alpha-\left(1+\frac{\beta}{N}\right)<\frac{2s}{N}$, therefore we
can take $\kappa$ small enough to get $\mathcal{E}(u_{\kappa})<0$. Thus,
$\mathcal{I}_{\lambda}\leq\mathcal{E}(u_{\kappa})<0$. We are kept with the
proof of the third assertion. Let $(u_{n})_{n\in\mathbb{N}}$ be a minimizing
sequence for the problem $\mathcal{I_{\lambda}}$. Therefore, thanks to
(20–22), we have for all $u\in H^{s}$
$\displaystyle\mathcal{D}(G(|u|),G(|u|))$ $\displaystyle\leq$
$\displaystyle\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})\,{|\\!|}u{|\\!|}^{2}_{\dot{H}^{s}}+\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right).$
Hence
$\displaystyle{|\\!|}u_{n}{|\\!|}^{2}_{H^{s}}$ $\displaystyle=$ $\displaystyle
2\,\mathcal{E}(u_{n})+{|\\!|}u_{n}{|\\!|}^{2}_{2}+\mathcal{D}(G(|u_{n}|),G(|u_{n}|)),$
$\displaystyle\leq$ $\displaystyle
2\,\mathcal{I}_{\lambda}+\lambda+\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})\,{|\\!|}u_{n}{|\\!|}^{2}_{H^{s}}+\eta\left(C_{\epsilon_{1}}\lambda^{e_{1}}+C_{\epsilon_{2}}\lambda^{e_{2}}+C_{\epsilon_{3}}\lambda^{e_{3}}\right).$
Eventually, we pick $\epsilon_{1},\epsilon_{2}$ and $\epsilon_{3}$ such that
$\eta(\epsilon_{1}+\epsilon_{2}+\epsilon_{3})<1$, we get immediately that the
minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ is bounded in $H^{s}$. ∎
Before going further, let us introduce the so called Lévy concentration
function
$\mathcal{Q}_{n}(r)=\sup_{y\in{\mathbb{R}^{N}}}\int_{B(y,r)}\,|u_{n}(x)|^{2}dx.$
It is known that each $\mathcal{Q}_{n}$ is nondecreasing on $(0,+\infty)$.
Also, with the Helly’s selection Theorem, the sequence
$(\mathcal{Q}_{n})_{n\in\mathbb{N}}$ has a subsequence that we still denote
$(\mathcal{Q}_{n})_{n\in\mathbb{N}}$ by abuse of notation, such that there is
a nondecreasing function $\mathcal{Q}(r)$ satisfying
$\mathcal{Q}_{n}(r)\xrightarrow[n\to+\infty]{}\mathcal{Q}(r),\quad\text{for
all}\quad r>0.$
Since $0\leq\mathcal{Q}_{n}(r)\leq\lambda$, there exists $\beta\in\mathbb{R}$
such that $0\leq\beta\leq\lambda$ such that
$\mathcal{Q}(r)\xrightarrow[r\to+\infty]{}\gamma.$
Briefly speaking, a minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ for the
problem $\mathcal{I_{\lambda}}$ can only be in one of the following
situations:
* 1.
Vanishing, i.e. $\gamma=0$.
* 2.
Dichotomy, i.e. $0<\gamma<\lambda$.
* 3.
Compactness, i.e. $\gamma=\lambda$.
In the sequel we shall proceed by elimination and show that vanishing and
dichotomy do not occur. Therefore, compactness holds true and we are done. We
start with the following
###### Proposition 3.2.
Let $\lambda>0$ and $(u_{n})_{n\in\mathbb{N}}$ be a minimizing sequence of
problem $\mathcal{I}_{\lambda}$ with $G$ such that $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ hold true. Then $\gamma>0$.
The proposition claims then that the situation of vanishing does not occurs.
In the proof of Proposition 3.2, we shall use, for all subset of
$A\subset{\mathbb{R}^{N}}$, the notation
$\mathcal{D}|_{A}(G(|u|),G(|u|)):=\int_{A\times
A}G(|u(x)|)\,V(|x-y|)\,G(|u(y)|)\,dxdy.$
###### Proof.
Let us first prove that $\mathcal{D}(G(|u_{n}|),G(|u_{n}|))$ is lower bounded.
In other words, we show that for $n\in\mathbb{N}$ large enough there exists
$\delta>0$ such that
$\delta<\mathcal{D}(G(|u_{n}|),G(|u_{n}|)).$ (23)
We argue by contradiction and assume that there exist no such $\delta$,
therefore $\liminf_{n\rightarrow+\infty}\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\leq
0$, thus
$\displaystyle\mathcal{I}_{\lambda}=\lim_{n\rightarrow+\infty}\mathcal{E}(u_{n})$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow+\infty}\left(\frac{1}{2}{|\\!|}\nabla_{s}u_{n}{|\\!|}^{2}_{2}-\frac{1}{2}\,\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\right)$
$\displaystyle\geq$
$\displaystyle-\frac{1}{2}\lim_{n\rightarrow+\infty}\,\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\geq
0.$
The inequality above is in contradiction with the fact that
$\mathcal{I}_{\lambda}<0$. On the other hand, arguing by contradiction and
assuming that the minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ vanishes,
i.e. assume that $\gamma=0$. Then there exists a subsequence
$(u_{n_{k}})_{k\in\mathbb{N}}$ of $(u_{n})_{n\in\mathbb{N}}$ and a radius
$\tilde{r}>0$ such that
$\sup_{y\in{\mathbb{R}^{N}}}\int_{B(y,\tilde{r})}\,|u_{n_{k}}(x)|^{2}dx\xrightarrow[k\to+\infty]{}0.$
Next, since the sequence $(u_{n_{k}})_{k\in\mathbb{N}}$ is bounded in $H^{s}$,
then one can find $r_{\epsilon}>0$ such that
$\mathcal{D}|_{|x-y|\geq
r_{\epsilon}}({G(|u_{n_{k}}|),G(|u_{n_{k}}|))}\leq\frac{\epsilon}{2}.$
Now, we cover ${\mathbb{R}^{N}}$ by balls of radius $r$ and centers $c_{i}$
for $i=1,2,\ldots$ such that each point of ${\mathbb{R}^{N}}$ is contained in
at most $N+1$ ball. Therefore, there exists $N_{\epsilon}$ ball and a
subsequence $(c_{i_{l}})_{l=1,\ldots,N_{\epsilon}}$ such that
$\displaystyle\mathcal{D}|_{|x-y|\geq
r_{\epsilon}}({G(|u_{n_{k}}|),G(|u_{n_{k}}|))}\leq\eta\sum_{p,q=1}^{2}{\mathcal{D}_{p,q}|_{|x-y|\geq
r_{\epsilon}}}(|u_{n_{k}}|),$ $\displaystyle\leq$
$\displaystyle\eta\sum_{p,q=1}^{2}\sum_{l=1}^{\infty}\sum_{i=1}^{N_{\epsilon}}\int_{B_{x}(c_{l},r)}\int_{B_{y}(c_{i_{l}},r)}\frac{|u_{n_{k}}(x)|^{\mu_{p}}|u_{n_{k}}(y)|^{\mu_{q}}}{|x-y|^{N-\beta}}dxdy,$
$\displaystyle\leq$
$\displaystyle\eta\sum_{p,q=1}^{2}\sum_{l=1}^{\infty}\sum_{i=1}^{N_{\epsilon}}{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}{|\\!|}\int_{B_{y}(c_{l_{i}},r)}\frac{|u_{n_{k}}(y)|^{\mu_{q}}}{|x-y|^{N-\beta}}dy\,|u_{n_{k}}|^{\mu_{p}-1}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}\,,$
$\displaystyle\leq$ $\displaystyle
N_{\epsilon}\,\eta\left(\sum_{l=1}^{\infty}{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}\right){|\\!|}u_{n_{k}}{|\\!|}_{r}\,{|\\!|}u_{n_{k}}{|\\!|}^{\mu-1}_{\frac{2N}{N-2s}}\sup_{y\in\mathbb{R}^{N}}\,{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B(y,r))}$
$\displaystyle+$ $\displaystyle
N_{\epsilon}\,\eta\sum_{(p,q)\neq(1,2),p,q=1}^{2}\left(\sum_{l=1}^{\infty}{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B_{x}(c_{l},r))}\right)\,{|\\!|}u_{n_{k}}{|\\!|}^{\mu_{q}-1}_{{r_{pq}}}\,{|\\!|}u_{n_{k}}{|\\!|}^{\mu_{p}-1}_{{\frac{r_{pq}}{\mu_{p}-1}}}\times$
$\displaystyle\hskip
250.0pt\times\sup_{y\in\mathbb{R}^{N}}\,{|\\!|}u_{n_{k}}{|\\!|}_{L^{2}(B(y,r))},$
where $r$ and $r_{pq}$ are such that
$\displaystyle\frac{\beta}{N}=\frac{1}{r}+(\mu-1)\left(\frac{1}{2}-\frac{s}{N}\right),\quad\frac{1}{r}+\frac{1}{2}>\frac{\beta}{N},$
$\displaystyle\frac{\mu_{p}-1}{r_{pq}}+\frac{\mu_{q}-1}{r_{pq}}=\frac{\beta}{N},\quad\frac{\mu_{p}-1}{r_{pq}}+\frac{1}{2}>\frac{\beta}{N},\quad(p,q)\neq(1,2).$
Since $1+\frac{2\beta-N}{N-2s}<\mu<1+\frac{\beta+2s}{N},0<\beta<N$ and
$s>\frac{N-\beta}{2}$, it is rather clear that one can find (as in section 2)
$r,r_{p,q}\in\left[2,\frac{2N}{N-2s}\right]$. Consequently, we have obviously
$\displaystyle\mathcal{D}|_{|x-y|\geq
r_{\epsilon}}({G(|u_{n_{k}}|),G(|u_{n_{k}}|))}\leq(N+1)\,N_{\epsilon}\,\eta{|\\!|}u_{n_{k}}{|\\!|}_{{2}}\times$
$\displaystyle\hskip
50.0pt\times\left({|\\!|}u_{n_{k}}{|\\!|}^{\mu}_{H^{s}}+{|\\!|}u_{n_{k}}{|\\!|}^{2}_{H^{s}}+{|\\!|}u_{n_{k}}{|\\!|}^{2(\mu-1)}_{H^{s}}\right)\left(\sup_{y\in\mathbb{R}^{N}}\,\int_{B(y,r)}|u_{n_{k}}|^{2}\right)^{\frac{1}{2}}.$
$\displaystyle\hskip 300.0pt\xrightarrow[n\to+\infty]{}0.$
This shows that if the minimizing sequence $(u_{n})_{n\in\mathbb{N}}$
vanishes, then
$\mathcal{D}(G(|u_{n}|),G(|u_{n}|))\xrightarrow[n\to+\infty]{}0.$
This is in contradiction with the property (23), namely for $n\in\mathbb{N}$
large enough there exists $\gamma>0$ such that
$\mathcal{D}(G(|u_{n}|),G(|u_{n}|))>\gamma$. Thus, vanishing does not occurs.
∎
Now, we show the following
###### Proposition 3.3.
Let $0<\pi<\lambda$ and $G$ such that $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$
hold true. Then the mapping $\lambda\mapsto\mathcal{I}_{\lambda}$ is
continuous and
$\mathcal{I}_{\lambda}<\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}$.
###### Proof.
Let $\lambda>0$ and $(\lambda_{k})_{k\in\mathbb{N}}$ be a sequence of positive
numbers such that $\lambda_{k}\xrightarrow[k\to+\infty]{}\lambda$. Let
$\epsilon>0$ and $u\in H^{s}(\mathbb{R}^{N})$ such that
${|\\!|}u{|\\!|}_{2}=\sqrt{\lambda}$ and
$\mathcal{I}_{\lambda}\leq\mathcal{E}(u)\leq\mathcal{I}_{\lambda}+\frac{\epsilon}{2}.$
For all $k\in\mathbb{N}$, let $u_{k}=\sqrt{\frac{\lambda_{k}}{\lambda}}u$.
Obviously $u_{k}\in H^{s}(\mathbb{R}^{N})$ and
${|\\!|}u_{k}{|\\!|}^{2}_{2}=\lambda_{k}$ so that for all
$k\in\mathbb{N},\,\mathcal{I}_{\lambda_{k}}\leq\mathcal{E}(u_{k})$. Now, we
show that $\mathcal{E}(u_{k})\xrightarrow[k\to+\infty]{}\mathcal{E}(u)$.
First, for all $k\in\mathbb{N}$
${|\\!|}u_{k}-u{|\\!|}_{\dot{H}^{s}}\leq{|\\!|}u_{k}{|\\!|}_{\dot{H}^{s}}\,\left|1-\sqrt{\frac{\lambda_{k}}{\lambda}}\right|.$
Since any sequence of $\mathcal{I}_{\lambda}$ is bounded in
$H^{s}(\mathbb{R}^{N})$ and $\lambda_{k}\xrightarrow[k\to+\infty]{}\lambda$,
then we have obviously
$\frac{1}{2}{|\\!|}\nabla_{s}u_{k}{|\\!|}^{2}_{2}\xrightarrow[k\to+\infty]{}\frac{1}{2}{|\\!|}\nabla_{s}u{|\\!|}^{2}_{2}$.
Next, following the first assertion of Proposition 3.1, we have
$\mathcal{E}(u)\in C^{1}(H^{s}(\mathbb{R}^{N}),\mathbb{R})$. In particular,
one can easily see from the proof of this point that
$D(u):=\mathcal{D}(G(|u|),G(|u|))\in C^{1}(H^{s}(\mathbb{R}^{N}),\mathbb{R})$
and
$\left|\mathcal{D}^{\prime}(u)\right|\leq\eta\left({|\\!|}u{|\\!|}_{H^{s}}+{|\\!|}u{|\\!|}^{\frac{2s+\beta}{N}}_{H^{s}}\right).$
(24)
We refer to Ref. [11] for details. Therefore, we have
$\displaystyle\left|\mathcal{D}(u_{k})-\mathcal{D}(u)\right|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{t}\frac{d}{dt}\mathcal{D}(tu_{k}+(1-t)u)dt\right|,$
$\displaystyle\leq$
$\displaystyle\eta\sup_{u\in{H^{s}},{|\\!|}u{|\\!|}_{H^{s}}\leq\eta}{|\\!|}\mathcal{D}^{\prime}(u){|\\!|}_{H^{-s}}\,{|\\!|}u_{k}-u{|\\!|}_{H^{s}},$
$\displaystyle\leq$
$\displaystyle\eta\,{|\\!|}u_{k}{|\\!|}_{H^{s}}\,\left|1-\sqrt{\frac{\lambda_{k}}{\lambda}}\right|\,\xrightarrow[k\to+\infty]{}0.$
Thus, we have $\mathcal{E}(u_{k})\xrightarrow[k\to+\infty]{}\mathcal{E}(u)$.
Consequently, we have
$\mathcal{I}_{\lambda_{k}}\leq\mathcal{I}_{\lambda}+\epsilon$ for $k$ large
enough. Next, for all $k\in\mathbb{N}$, let us choose $\tilde{u}_{k}\in
H^{s}(\mathbb{R}^{N})$ such that
${|\\!|}\tilde{u}_{k}{|\\!|}_{2}=\sqrt{\lambda}_{k}$ and
$\mathcal{E}(\tilde{u}_{k})\leq\mathcal{I}_{\lambda_{k}}+\frac{1}{k}$.
Moreover, for all $k\in\mathbb{N}$, we set
$\bar{u}_{k}=\sqrt{\frac{\lambda}{\lambda_{k}}}\tilde{u}_{k}$. Obviously,
since $\bar{u}_{k}\in H^{s}(\mathbb{R}^{N})$ and
${|\\!|}\bar{u}_{k}{|\\!|}_{2}^{2}=\lambda$, we have
$\mathcal{I}_{\lambda}\leq\mathcal{E}(\bar{u}_{k})$. Exactly the same argument
as above shows that
$\mathcal{E}(\tilde{u}_{k})\xrightarrow[k\to+\infty]{}\mathcal{E}(\bar{u})$ so
that for $k$ large enough, we have
$\mathcal{I}_{\lambda}\leq\mathcal{I}_{\lambda_{k}}+\epsilon$. Whence,
$\lambda\mapsto\mathcal{I}_{\lambda}$ is continuous on
$\mathbb{R}_{+}^{\star}$. Eventually, using the energy estimates (16-18) or
(19), it is rather easy to show that $I_{\lambda}\xrightarrow[\lambda\to
0^{+}]{}0$. This shows that the mapping $\lambda\mapsto\mathcal{I}_{\lambda}$
is continuous.
Let us now prove the strict sub–additivity inequality. For that purpose, we
introduce $u_{\theta}=\theta^{\kappa}u(\theta^{\frac{\kappa}{N}})$ for all
$\kappa>\frac{N}{N+2s}$. Obviously $u_{\kappa}\in H^{s}(\mathbb{R}^{N})$ and
${|\\!|}u_{\theta}{|\\!|}_{{L^{2}(\mathbb{R}^{N})}}=\sqrt{\theta\lambda}$.
Moreover, using $\mathcal{A}_{1}$, we have
$\displaystyle\mathcal{E}(u_{\theta})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u_{\theta}|^{2}dx-\frac{1}{2}\mathcal{D}(G(|u_{\theta}|),G(|u_{\theta}|)),$
$\displaystyle\leq$
$\displaystyle\frac{\theta^{\kappa\left(1+\,\frac{2s}{N}\right)}}{2}\left(\int_{{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx-\mathcal{D}(G(|u|),G(|u|))\right)={\theta^{\kappa\left(1+\,\frac{2s}{N}\right)}}\mathcal{E}(u).$
Thus, we deduce that
$\mathcal{I}_{\theta\lambda}\leq{\theta^{\kappa\left(1+\,\frac{2s}{N}\right)}}\mathcal{I}_{\lambda}$
for all $\theta>0$. Now we let $0<\pi<\lambda$, therefore since
$\kappa\left(1+\,\frac{2s}{N}\right)>1$ we have
$\displaystyle\mathcal{I}_{\lambda}\leq\lambda^{\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{1}$
$\displaystyle<$
$\displaystyle\pi^{\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{1}+(\lambda-\pi)^{\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{1},$
$\displaystyle\leq$
$\displaystyle\pi^{\kappa\left(1+\,\frac{2s}{N}\right)}\pi^{-\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{\pi}+(\lambda-\pi)^{\kappa\left(1+\,\frac{2s}{N}\right)}(\lambda-\pi)^{-\kappa\left(1+\,\frac{2s}{N}\right)}\mathcal{I}_{\lambda-\pi},$
$\displaystyle=$ $\displaystyle\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}.$
In summary, for all $0<\pi<\lambda$, we have
$\mathcal{I}_{\lambda}<\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}$. ∎
Now, we are able to claim the following
###### Proposition 3.4.
Let $\lambda>0$ and $(u_{n})_{n\in\mathbb{N}}$ be a minimizing sequence of
problem $\mathcal{I}_{\lambda}$ with $G$ such that $\mathcal{A}_{0}$ and
$\mathcal{A}_{1}$ hold true. Then dichotomy does not occur for
$(u_{n})_{n\in\mathbb{N}}$.
###### Proof.
Let us introduce $\xi$ and $\chi$ in $C^{\infty}$ such that $0\leq\xi,\chi\leq
1$ and
$\xi(x)=\left\\{\begin{array}[]{lcl}1&\text{if}&|x|\leq 1\\\ &&\\\
0&\text{if}&|x|\geq
2\end{array}\right.,\>\chi(x)=1-\xi(x),\>{|\\!|}\nabla\xi{|\\!|}_{\infty},{|\\!|}\nabla\chi{|\\!|}_{\infty}\leq
2.$
For all $r>0$, let $\xi_{r}(\cdot)=\xi(\frac{\cdot}{R})$ and
$\chi_{r}(\cdot)=\chi(\frac{\cdot}{R})$. we will show that dichotomy does not
occur by contradicting the fact that for all $0<\pi<\lambda$, we have
$\mathcal{I}_{\lambda}<\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}$ proved in
Proposition 3.3. Indeed, let $(u_{n})_{n\in\mathbb{N}}$ be a minimizing
sequence of problem $\mathcal{I}_{\lambda}$ and assume that dichotomy holds.
Then, using the construction of [17], there exist
* 1.
$0<\pi<\lambda$,
* 2.
a sequence $(y_{n})_{n\in\mathbb{N}}$ of points in ${\mathbb{R}^{N}}$,
* 3.
two increasing sequences of positive real number $(r_{1,n})_{n\in\mathbb{N}}$
and $(r_{2,n})_{n\in\mathbb{N}}$ such that
$r_{1,n}\xrightarrow[n\to+\infty]{}+\infty\quad\text{and}\quad\frac{r_{2,n}}{2}-r_{1,n}\xrightarrow[n\to+\infty]{}+\infty,$
such that the sequences $u_{1,n}=\xi_{r_{1,n}}(\cdot-y_{n})u_{n}$ and
$u_{2,n}=\chi_{r_{2,n}}(\cdot-y_{n})u_{n}$ satisfy
$\left\\{\begin{array}[]{lll}&u_{n}=u_{1,n}\>\text{on}\>B(y_{n},{r_{1,n}}),\\\
&\\\
&u_{n}=u_{2,n}\>\text{on}\>B^{c}(y_{n},{r_{2,n}})={\mathbb{R}^{N}}\setminus
B(y_{n},{r_{2,n}}),\\\ &\\\
&\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\pi,\>\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\lambda-\pi,\\\
&\\\
&{|\\!|}u_{n}-(u_{1,n}+u_{2,n}){|\\!|}_{p}\xrightarrow[n\to+\infty]{}0,\>\text{for
all}\>2\leq p<\frac{2N}{N-2s},\\\ &\\\
&{|\\!|}u_{n}{|\\!|}_{L^{p}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\xrightarrow[n\to+\infty]{}0,\>\text{for all}\>2\leq
p<\frac{2N}{N-2s},\\\ &\\\
&\mathrm{dist}(\mathrm{Supp}(u_{1,n}),\mathrm{Supp}(u_{2,n}))\xrightarrow[n\to+\infty]{}+\infty.\end{array}\right.$
We have obviously
$\displaystyle\mathcal{E}(u_{n})$ $\displaystyle=$
$\displaystyle\mathcal{E}(u_{1,n})+\mathcal{E}(u_{2,n})+\frac{1}{2}\int_{\mathbb{R}^{N}}|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-\frac{1}{2}\mathcal{D}(G(|u_{n}|),G(|u_{n}|))dx$
$\displaystyle-$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}+|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(\mathcal{D}(G(|u_{1,n}|),G(|u_{1,n}|))+\mathcal{D}(G(|u_{2,n}|),G(|u_{2,n}|))\right).$
Now we show the existence of $\epsilon>0$ such that for sufficiently large
radius $r_{1,n}$ and $r_{1,n}$ we have
$\displaystyle\frac{1}{2}\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx\geq-\eta\epsilon.$
(25)
Firs of all, it is rather easy to show that by construction of the sequences
$u_{i,n}$ for $i=1,2$, we have
$\displaystyle\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx$
$\displaystyle\hskip
70.0pt\geq-\,\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$
$\displaystyle\hskip
85.0pt-\,\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy.$
Indeed, the estimate above is justified using the definition (5) combined with
the following basic fact for $u_{1,n}$
$\displaystyle|u_{1,n}(x)-u_{1,n}(y)|^{2}$
$\displaystyle=|\xi_{r_{1,n}}(x-y_{n})u_{n}(x)-\xi_{r_{1,n}}(y-y_{n})u_{n}(y)|^{2}$
$\displaystyle\leq\frac{1}{2}|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}\left(|u_{1,n}(x)|^{2}+|u_{1,n}(y)|^{2}\right)$
$\displaystyle+\frac{1}{2}\left(|\xi_{r_{1,n}}(x-y_{n})|^{2}+|\xi_{r_{1,n}}(y-y_{n})|^{2}\right)|u_{1,n}(x)-u_{1,n}(y)|^{2}.$
and equivalently for $u_{2,n}$
$\displaystyle|u_{2,n}(x)-u_{2,n}(y)|^{2}$
$\displaystyle=|\chi_{r_{2,n}}(x-y_{n})u_{n}(x)-\chi_{r_{2,n}}(y-y_{n})u_{n}(y)|^{2}$
$\displaystyle\leq\frac{1}{2}|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}\left(|u_{2,n}(x)|^{2}+|u_{2,n}(y)|^{2}\right)$
$\displaystyle+\frac{1}{2}\left(|\chi_{r_{2,n}}(x-y_{n})|^{2}+|\chi_{r_{2,n}}(y-y_{n})|^{2}\right)|u_{2,n}(x)-u_{2,n}(y)|^{2}.$
In order to show (25), it suffices to show that there exist $\epsilon>0$ such
that for large radius $r_{1,n}$ and $r_{2,n}$, we have
$\displaystyle\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy\leq\eta\epsilon,$
$\displaystyle\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy\leq\eta\epsilon.$
We prove the first assertion and the second one follows equivalently. Indeed,
we split the sum in two part as follows
$\displaystyle\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$
$\displaystyle\hskip 30.0pt=\int_{|x-y|\leq
r_{1,n}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$
$\displaystyle\hskip
30.0pt+\int_{|x-y|>r_{1,n}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy:=\mathcal{T}_{1}+\mathcal{T}_{2}$
Now, we write
$\displaystyle\mathcal{T}_{1}$ $\displaystyle\leq{r^{-2}_{1,n}}\int_{|x-y|\leq
r_{1,n}}\frac{|u_{n}(x)|^{2}}{|x-y|^{N+2s-2}}dxdy$
$\displaystyle\leq{r^{-2}_{1,n}}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx\int_{|x|\leq
r_{1,n}}\frac{1}{|x|^{N+2s-2}}dx\leq\eta\,r^{-2s}_{1,n}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx.$
Moreover,
$\displaystyle\mathcal{T}_{2}$ $\displaystyle\leq
r^{-s}_{1,n}\int_{|x-y|>r_{1,n}}\frac{|\xi_{r_{1,n}}(x-y_{n})-\xi_{r_{1,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+s}}dxdy$
$\displaystyle\leq\eta\,r^{-s}_{1,n}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx\int_{|x-y|>r_{1,n}}\frac{1}{|x-y|^{N+s}}dy\leq\eta\,r^{-s}_{1,n}\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx.$
Eventually summing up $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ and use the same
argument in order to handle the term
$\int_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}\frac{|\chi_{r_{2,n}}(x-y_{n})-\chi_{r_{2,n}}(y-y_{n})|^{2}|u_{n}(x)|^{2}}{|x-y|^{N+2s}}dxdy$,
one ends with
$\displaystyle\int_{\mathbb{R}^{N}}\left(|(-\Delta)^{\frac{s}{2}}\,u_{n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{1,n}|^{2}-|(-\Delta)^{\frac{s}{2}}\,u_{2,n}|^{2}\right)dx\,$
$\displaystyle\hskip
100.0pt\geq-\eta\,(r^{-2s}_{1,n}+\,r^{-s}_{1,n}+r^{-2s}_{2,n}+\,r^{-s}_{2,n})\,\int_{{\mathbb{R}^{N}}}|u_{n}(x)|^{2}dx.$
The estimate (25) follows for $r_{1,n}$ and $r_{2,n}$ large enough. Next,
observe that $|u_{n}-u_{1,n}-u_{2,n}|\leq
3\,\mathds{1}_{(B(y_{n},r_{2,n})\setminus B(y_{n},r_{1,n}))}$ where
$\mathds{1}_{(B(y_{n},r_{2,n})\setminus B(y_{n},r_{1,n}))}$ denotes the
characteristic function of $B(y_{n},r_{2,n})\setminus B(y_{n},r_{1,n})$. Now,
we have
$\displaystyle\left|\mathcal{D}(G(|u_{n}|),G(|u_{n}|))-\mathcal{D}(G(|v_{n}|),G(|v_{n}|))-\mathcal{D}(G(|w_{n}|),G(|w_{n}|))\right|$
$\displaystyle\leq$
$\displaystyle\int_{B(y_{n},2r)\setminus\bar{B}(y_{n},2r)}\left(\left|\frac{G(|u_{n}|)G(|u_{n}|)}{|x-y|^{N-\beta}}\right|+\left|\frac{G(|v_{n}|)G(|v_{n}|)}{|x-y|^{N-\beta}}\right|\right.$
$\displaystyle\hskip
200.0pt+\left.\left|\frac{G(|w_{n}|)G(|w_{n}|)}{|x-y|^{N-\beta}}\right|\right)dxdy,$
$\displaystyle\leq$
$\displaystyle\eta\left({|\\!|}u{|\\!|}^{4-\frac{N-\beta}{s}}_{L^{2}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\,{|\\!|}u{|\\!|}^{\frac{N-\beta}{s}}_{H^{s}}+{|\\!|}u{|\\!|}^{2\mu-\frac{N(\mu-1)-\beta}{s}}_{L^{2}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\,{|\\!|}u{|\\!|}^{\frac{N(\mu-1)-\beta}{s}}_{H^{s}}\right)$
$\displaystyle+$
$\displaystyle\eta\,{|\\!|}u{|\\!|}^{\mu+2-\frac{N\mu-2\beta}{2s}}_{L^{2}(B(y_{n},r_{2,n})\setminus
B(y_{n},r_{1,n}))}\,{|\\!|}u{|\\!|}^{\frac{N\mu-2\beta}{2s}}_{H^{s}}\xrightarrow[n\to+\infty]{}0.$
where we used the estimates (16–18). Thus, for $r_{2,n}$ and $r_{1,n}$ large
enough we have
$-\frac{1}{2}\left(\mathcal{D}(G(|u_{n}|),G(|u_{n}|))-\mathcal{D}(G(|v_{n}|),G(|v_{n}|))-\mathcal{D}(G(|w_{n}|),G(|w_{n}|))\right)\geq-\eta\epsilon.$
(26)
Summing up (25) and (26), we end up for large $r_{1,n}$ and $r_{2,n}$ with
$\mathcal{E}(u_{n})-\mathcal{E}(u_{1,n})-\mathcal{E}(u_{2,n})\geq-\eta\epsilon.$
(27)
Since we have
$\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\pi$ and
$\int_{\mathbb{R}^{N}}|u_{1,n}|^{2}dx\xrightarrow[n\to+\infty]{}\lambda-\pi$,
there exist two positive real sequences $(\mu_{1,n})_{n\in\mathbb{N}}$ and
$(\mu_{2,n})_{n\in\mathbb{N}}$ such that
$|\mu_{1,n}-1|,|\mu_{2,n}-1|<\epsilon$ and
$\int_{\mathbb{R}^{N}}|\mu_{1,n}u_{1,n}|^{2}dx=\pi,\quad\int_{\mathbb{R}^{N}}|\mu_{2,n}u_{2,n}|^{2}dx=\lambda-\pi,$
so that
$\displaystyle\mathcal{I}_{\pi}\leq\mathcal{E}(\mu_{1,n}u_{1,n})\leq\mathcal{E}(u_{1,n})+\frac{\eta\epsilon}{2},$
$\displaystyle\mathcal{I}_{\lambda-\pi}\leq\mathcal{E}(\mu_{2,n}u_{2,n})\leq\mathcal{E}(u_{2,n})+\frac{\eta\epsilon}{2}.$
Thus, with (27), we have and the continuity of the mapping
$\lambda\mapsto\mathcal{I}_{\lambda}$ for all $\lambda>0$, we have
$\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}-3\eta\epsilon\leq\mathcal{E}(u_{1,n})+\mathcal{E}(u_{2,n})-\eta\epsilon\leq\mathcal{E}(u_{n})\xrightarrow[n\to+\infty]{}\mathcal{I}_{\lambda}.$
In summary, we proved that for all $0<\pi<\lambda$, we have
$\mathcal{I}_{\pi}+\mathcal{I}_{\lambda-\pi}\leq\mathcal{I}_{\lambda}$
contradicting the strict sub–additivity inequality proved above. Then, the
dichotomy does not occur. ∎
Now, we finish the proof of Theorem 1.2. Since vanishing and dichotomy do not
occur for any minimizing sequence $(u_{n})_{n\in\mathbb{N}}$ for the problem
$\mathcal{I}_{\lambda}$, then the compactness certainly occurs. Following the
concentration-compactness principle [17], we know that every minimizing
sequence $(u_{n})_{n\in\mathbb{N}}$ of $\mathcal{I}_{\lambda}$ satisfies (up
to extraction if necessary)
$\lim_{r\to+\infty}\lim_{n\to+\infty}\sup_{y\in{\mathbb{R}^{N}}}\int_{B(y,r)}|u_{n}(x)|^{2}dx=\lambda.$
That is, for all $\epsilon>0$, there exist $r_{\epsilon}>0$ and
$n_{\epsilon}\in\mathbb{N}^{\star}$ and $\\{y_{n}\\}\subset{\mathbb{R}^{N}}$
such that for all $r>r_{\epsilon}$ and $n\geq n_{\epsilon}$, we have
$\int_{B(y_{n},r)}|u_{n}(x)|^{2}dx=\lambda-\epsilon$
Now, let $w_{n}=u_{n}(x+y_{n})$, we have obviously that
${|\\!|}w_{n}{|\\!|}_{H^{s}}={|\\!|}u_{n}{|\\!|}_{H^{s}}$ is bounded in
$H^{s}(\mathbb{R}^{N})$, therefore $(w_{n})_{n\in\mathbb{N}}$ (up to
extraction if necessary) converges weakly to $w$ in $H^{s}(\mathbb{R}^{N})$.
In particular $(w_{n})_{n\in\mathbb{N}}$ converges weakly to $w$ in
${L^{2}(\mathbb{R}^{N})}$ and ${|\\!|}w_{n}{|\\!|}_{2}=\sqrt{\lambda}$. Now,
let $\tilde{r}_{\epsilon}>r_{\epsilon}$ such that
${|\\!|}w{|\\!|}_{L^{2}(B^{c}(0,\tilde{r}_{\epsilon}))}<\frac{\epsilon}{2}$.
Thus, there exists
$\tilde{n}_{\epsilon}\in\mathbb{N}^{\star},\>\tilde{n}_{\epsilon}>n_{\epsilon}$
such that for all $n\geq\tilde{n}_{\epsilon}$, we have
${|\\!|}w_{n}-w{|\\!|}_{L^{2}(B(0,\tilde{r}_{\epsilon}))}<\frac{\epsilon}{2}$.
Therefore, with the triangle inequality, we have
$\displaystyle{|\\!|}w{|\\!|}_{2}$ $\displaystyle\geq$
$\displaystyle{|\\!|}u_{n}{|\\!|}_{2}-{|\\!|}w_{n}-w{|\\!|}_{L^{2}(B(0,\tilde{r}_{\epsilon}))}-{|\\!|}w_{n}-w{|\\!|}_{L^{2}(B^{c}(0,\tilde{r}_{\epsilon}))},$
$\displaystyle\geq$
$\displaystyle{|\\!|}u_{n}{|\\!|}_{L^{2}(B(y_{n},\tilde{r}_{\epsilon}))}-{|\\!|}w_{n}-w{|\\!|}_{L^{2}(B(0,\tilde{r}_{\epsilon}))}-{|\\!|}w{|\\!|}_{L^{2}(B^{c}(0,\tilde{r}_{\epsilon}))}\geq\sqrt{\lambda-\epsilon}-\epsilon.$
Passing to the limit we get ${|\\!|}w{|\\!|}_{2}\geq\sqrt{\lambda}$. Since the
$L^{2}$ is lower semi continuous, we obtain that
${|\\!|}w{|\\!|}_{2}\leq\liminf_{n\to+\infty}{|\\!|}w_{n}{|\\!|}_{2}=\sqrt{\lambda}$.
Eventually, we get ${|\\!|}w{|\\!|}_{2}=\sqrt{\lambda}$, therefore the
sequence $(w_{n})_{n\in\mathbb{N}}$ converges strongly in
${L^{2}(\mathbb{R}^{N})}$ to $w$. Also, we have
$\displaystyle\left|\mathcal{D}(G(|w_{n}|),G(|w_{n}|))-D(G(|w|),G(|w|))\right|$
$\displaystyle\hskip
70.0pt\leq\left|\int_{0}^{t}\frac{d}{dt}\mathcal{D}(tG(|w_{n}|)+(1-t)G(|w|))\,dt\right|,$
$\displaystyle\hskip
70.0pt\leq\eta\sup_{u\in{H^{s}},{|\\!|}u{|\\!|}_{H^{s}}\leq\eta}{|\\!|}\mathcal{D}^{\prime}(u){|\\!|}_{H^{-s}}\,{|\\!|}w_{n}-w{|\\!|}_{H^{s}},$
$\displaystyle\hskip
70.0pt\leq\eta\,{|\\!|}w_{n}-w{|\\!|}_{2}+\eta{|\\!|}w_{n}-w{|\\!|}_{\frac{2s+\beta}{N}}\xrightarrow[n\to+\infty]{}0.$
In the last line we used 24-kind inequality and again we refer to [11] for a
proof. Using the lower semi-continuity of the $-s$ norm, we have
${|\\!|}w{|\\!|}_{H^{s}}\leq\liminf_{n\to+\infty}{|\\!|}w_{n}{|\\!|}_{H^{s}}$.
Summing up, we get clearly
$\mathcal{I}_{\lambda}\leq\mathcal{E}(w)\leq\liminf_{n\to+\infty}\mathcal{E}(w_{n})=\mathcal{I}_{\lambda}.$
This shows that $w$ is a minimizer of $\mathcal{I}_{\lambda}$ and
$w_{n}\xrightarrow[n\to+\infty]{}w$ in $H^{s}(\mathbb{R}^{N})$. Theorem 1.1 is
now proved.
## 4 Stability of standing waves
In this section, we prove the orbital stability of standing waves in the sense
of Definition 1.3. That is we prove Theorem 1.4.
We argue par contradiction. Assume that $\hat{\mathcal{O}}_{\lambda}$ is not
stable, then either $\hat{\mathcal{O}}_{\lambda}$ is empty or there exist
$w\in\hat{\mathcal{O}}_{\lambda}$ and a sequence $\phi^{n}_{0}\in H^{s}$ such
that ${|\\!|}\phi^{n}_{0}-w{|\\!|}_{H^{s}}\xrightarrow[n\to+\infty]{}0$ as
$n\rightarrow\infty$ but
$\displaystyle{\inf_{z\in\hat{\mathcal{O}}_{\lambda}}}{|\\!|}\phi^{n}(t_{n},.)-z{|\\!|}_{H^{s}}\geq\varepsilon,$
(28)
for some sequence $t_{n}\subset\mathbb{R}$, where $\phi^{n}(t_{n},.)$ is the
solution of the Cauchy problem $\mathscr{S}$ corresponding to the initial
condition $\phi^{n}_{0}$. Now let $w_{n}=\phi^{n}(t_{n},.)$, since
${\mathcal{J}}(w)=\hat{\mathcal{I}}_{\lambda}$, it follows from the continuity
of the $L^{2}$ norm and $\mathcal{J}$ in $H^{s}$ that
${|\\!|}\phi^{n}_{0}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}$ and
$\mathcal{J}(w_{n})=\mathcal{J}(\phi^{n}_{0})=\hat{\mathcal{I}}_{\lambda}$.
With the conservation of mass and energy associated with the dynamics of the
system $\mathscr{S}$, we deduce that
$\displaystyle{|\\!|}w_{n}{|\\!|}_{2}={|\\!|}\phi^{n}_{0}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}\quad\text{and}\quad\mathcal{J}(w_{n})=\mathcal{J}(\phi_{0}^{n})\xrightarrow[n\to+\infty]{}\hat{\mathcal{I}}_{\lambda}.$
Therefore if $(w_{n})_{n\in\mathbb{N}}$ has a subsequence converging to an
element $w\in H^{s}$: ${|\\!|}w{|\\!|}_{2}=\sqrt{\lambda}$ and
$\mathcal{J}(w)=\hat{\mathcal{I}}_{c}$. This shows that
$w\in\hat{\mathcal{O}}_{\lambda}$, but
$\inf_{z\in\hat{\mathcal{O}}_{\lambda}}{|\\!|}\phi^{n}(t_{n},.)-z{|\\!|}_{H^{s}}\leq{|\\!|}w_{n}-w{|\\!|}_{H^{s}}$
contradicting (28).
In summary, to show the orbital stability of $\hat{\mathcal{O}}_{\lambda}$,
one has to prove that $\hat{\mathcal{O}}_{\lambda}$ is not empty and that any
sequence $(w_{n})_{n\in\mathbb{N}}\subset H^{s}$ such that
${|\\!|}w_{n}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}\quad\mbox{
and
}\quad\mathcal{J}(w_{n})\xrightarrow[n\to+\infty]{}\hat{\mathcal{I}}_{\lambda},$
(29)
is relatively compact in $H^{s}$ (up to a translation).
From now on, we consider a sequence $(w_{n})_{n\in\mathbb{N}}$ satisfying
(29). Our aim is to prove that it admits a convergent subsequence to an
element $w\in H^{s}$. If $(w_{n})_{n\in\mathbb{N}}\subset H^{s}$, it is easy
to see that
$(|w_{n}|)_{n\in\mathbb{N}}\subset H^{s}\,;\quad w_{n}=(u_{n},v_{n}).$
Thanks to $\mathcal{A}_{0}$, we have that $(w_{n})_{n\in\mathbb{N}}$ is
bounded in $H^{s}$ and hence by passing to a subsequence, there exists
$w=(u,v)\in H^{s}$ such that
$\left\\{\begin{array}[]{l}u_{n}\mbox{ converges weakly to }u\mbox{ in
}H^{s},\\\ \\\ v_{n}\mbox{ converges weakly to }v\mbox{ in }H^{s},\\\ \\\
\mbox{ the limit when}\>n\>\mbox{goes to
}+\infty\>\mbox{of}\>{|\\!|}\nabla_{s}u_{n}{|\\!|}_{2}+{|\\!|}\nabla_{s}v_{n}{|\\!|}_{2}\>\mbox{
exists }.\end{array}\right.$ (30)
Now, a straightforward calculation shows that
$\mathcal{J}(w_{n})-\mathcal{E}(|w_{n}|)=\frac{1}{2}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}_{2}-\frac{1}{2}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}\geq
0.$ (31)
Thus we have
$\hat{\mathcal{I}}=\lim_{n\to+\infty}\mathcal{J}(w_{n})\geq\limsup_{n\to+\infty}\mathcal{E}(|w_{n}|).$
(32)
But
${|\\!|}|w_{n}|{|\\!|}^{2}_{2}={|\\!|}w_{n}{|\\!|}^{2}_{2}=\lambda_{n}\xrightarrow[n\to+\infty]{}\lambda.$
(33)
By the continuity of the mapping $\lambda\mapsto\mathcal{I}_{\lambda}$ (see
Proposition 3.3), we obtain
$\lim_{n\to+\infty}\mathcal{J}(w_{n})\geq\liminf_{n\to+\infty}\mathcal{I}_{\lambda_{n}}=\mathcal{I}_{\lambda}\geq\hat{\mathcal{I}}_{\lambda}.$
(34)
Hence
$\lim_{n\rightarrow+\infty}\mathcal{J}(w_{n})=\lim_{n\rightarrow+\infty}\mathcal{E}(|w_{n}|)=\mathcal{I}_{\lambda}=\hat{\mathcal{I}}_{\lambda}.$
The properties (30) and the inequalities (31) and (34) imply that
$\lim_{n\rightarrow+\infty}{|\\!|}\nabla_{s}u_{n}{|\\!|}^{2}_{2}-{|\\!|}\nabla_{s}v_{n}{|\\!|}^{2}_{2}-{|\\!|}\nabla_{s}(u^{2}_{n}+v_{n}^{2})^{1/2}{|\\!|}^{2}_{2}=0,$
(35)
which is equivalent to say that
$\lim_{n\rightarrow+\infty}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}=\lim_{n\rightarrow+\infty}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}.$
(36)
The convergence (33), the inequality (34) and Theorem 1.2 imply that $|w_{n}|$
is relatively compact in $H^{s}$ (up to a translation). Therefore, there
exists $\varphi\in H^{s}$ such that
$(u^{2}_{n}+v^{2}_{n})^{1/2}\rightarrow\varphi\mbox{ in}\>\>H^{s}\>\>\mbox{
and
}\>\>{|\\!|}\varphi{|\\!|}_{2}=\sqrt{\lambda}\>\>\mbox{with}\>\>\mathcal{E}(\varphi)=I_{\lambda}.$
Let us prove that $\varphi=|w|=(u^{2}+v^{2})^{1/2}$. Using (30), it follows
that $u_{n}\xrightarrow[n\to+\infty]{}u$ and
$v_{n}\xrightarrow[n\to+\infty]{}v$ in $L^{2}(B(0,R))$
$\displaystyle|(u^{2}_{n}+v^{2}_{n})^{1/2}-(u^{2}+v^{2})^{1/2}|\leq|u_{n}-u|^{2}+|v_{n}-v|^{2},$
$\displaystyle(u^{2}_{n}+v^{2}_{n})^{1/2}\xrightarrow[n\to+\infty]{}(u^{2}+v^{2})^{1/2}\quad\mbox{
in }L^{2}(B(0,R)).$
Thus we certainly have that $(u^{2}+v^{2})^{1/2}=|w|=\varphi$. On the other
hand
${|\\!|}|w_{n}|{|\\!|}_{2}={|\\!|}w_{n}{|\\!|}_{2}\xrightarrow[n\to+\infty]{}\sqrt{\lambda}={|\\!|}w{|\\!|}_{2}={|\\!|}|w|{|\\!|}_{2}$.
Therefore, we are done if we prove that
$\lim_{n\to\infty}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}_{2}={|\\!|}\nabla_{s}w{|\\!|}^{2}_{2}$.
From (36), we have that
$\lim_{n\to+\infty}{|\\!|}\nabla_{s}w_{n}{|\\!|}^{2}_{2}=\lim_{n\to+\infty}|\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}$
and
$\lim_{n\to+\infty}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}={|\\!|}\nabla_{s}|w|{|\\!|}^{2}_{2}$.
Hence by the lower semi-continuity of ${|\\!|}\nabla_{s}\cdot{|\\!|}_{2}$, we
obtain
${|\\!|}\nabla_{s}w{|\\!|}^{2}_{2}\leq\lim_{n\to+\infty}{|\\!|}\nabla_{s}|w_{n}|{|\\!|}^{2}_{2}={|\\!|}\nabla_{s}|w|{|\\!|}^{2}_{2}.$
(37)
Eventually, using (31), it follows that
${|\\!|}\nabla_{s}w{|\\!|}^{2}_{2}\geq{|\\!|}\nabla_{s}|w|{|\\!|}^{2}_{2}.$
Since by (30), we know that $w_{n}$ converges weakly to $w$ in $H^{s}$, it
follows that $w_{n}\xrightarrow[n\to+\infty]{}w$ in $H^{s}$, which completes
the proof.
Now, we turn to the characterization of the Orbit
$\hat{\mathcal{O}}_{\lambda}$. We show the following
###### Proposition 4.1.
With the same assumptions of Theorem 1.4, we have
$\hat{\mathcal{O}}_{\lambda}=\left\\{e^{i\sigma}w(.+y),\quad\sigma\in\mathbb{R},y\in\mathbb{R}^{N}\right\\},$
$w$ is a minimizer of (4).
###### Proof.
Let $z=(u,v)\in\hat{\mathcal{O}}_{\lambda}$ and set
$\varphi=(u^{2}+v^{2})^{1/2}$. By the previous section, we know that
$\mathcal{E}(\varphi)=I_{\lambda}$, thus $\varphi$ satisfies the partial
differential equation :
$(-\Delta)^{s}\varphi+\kappa\varphi=V\star G(|\varphi|)G^{\prime}(\varphi),$
(38)
where $\kappa$ is a Lagrange multiplier. Furthermore the equality
$\|\nabla_{s}w\|_{2}=\|\nabla_{s}|w|\|_{2}$ implies that
$u(x)v(y)-v(x)u(y)=0.$ (39)
By Proposition 4.2, it is plain that $\varphi\in C(\mathbb{R}^{N})$ and
$V\star G(|\varphi|)\in C(\mathbb{R}^{N})$. We can write
$(-\Delta)^{s}\varphi+\kappa\varphi=V\star
G(|\varphi|)\frac{G^{\prime}(\varphi)}{\varphi}\chi_{\\{\varphi\neq
0\\}}\varphi$, with $\chi_{A}$ being the characteristic function of the set
$A$. Since $\varphi$ is nontrivial and $V\star
G(|\varphi|)\frac{G^{\prime}(\varphi)}{\varphi}\chi_{\\{\varphi\neq 0\\}}\in
L^{\infty}_{loc}({\mathbb{R}^{N}})$, we conclude that $\varphi>0$ in
${\mathbb{R}^{N}}$ by the Harnack inequality (see Lemma 4.9 in [2]) and a
standard argument of intersecting balls.
Case 1 : $u\equiv 0$
Case 2 : $v\equiv 0$
Case 3 : $u\neq 0$ and $v\neq 0$ everywhere.
Then (39) implies that
$\displaystyle\frac{u(x)}{v(x)}=\frac{u(y)}{v(y)}\;\quad\forall\;x,y\in\mathbb{N}^{N},$
$\displaystyle\Rightarrow\frac{u(x)}{v(x)}=\alpha\Rightarrow u(x)=\alpha
v(x)\quad\forall\;x\in\mathbb{R}^{N},$ $\displaystyle z=(\alpha+i)v\Rightarrow
z=e^{i\sigma}w,w=|z|.$
Let us now prove (39). By the fact that
$\mathcal{J}(z)=\hat{\mathcal{I}}_{\lambda}$, we can find a Lagrange
multiplier $\alpha\in\mathbb{C}$ such that
$\mathcal{J}^{\prime}(z)(\xi)=\displaystyle{\frac{\alpha}{2}\int_{\mathbb{R}^{N}}}z\bar{\xi}+\xi\bar{z}$
for all $\xi\in H^{s}$. Putting $\xi=z$, it follows immediately that
$\alpha\in\mathbb{R}$ and
$\left\\{\begin{array}[]{l}\displaystyle{\int}_{{\mathbb{R}^{N}}}\nabla_{s}u\nabla_{s}f-\displaystyle{\int}_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dyG^{\prime}(f(x))dx\\\
\\\ \hskip 215.0pt=\alpha\displaystyle{\int}_{{\mathbb{R}^{N}}}u(x)f(x)dx,\\\
\\\
\displaystyle{\int}_{{\mathbb{R}^{N}}}\nabla_{s}v\nabla_{s}f-\displaystyle{\int}_{{\mathbb{R}^{N}}\times{\mathbb{R}^{N}}}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dyG^{\prime}(f(x))dx\\\
\\\ \hskip
215.0pt=\alpha\displaystyle{\int}_{{\mathbb{R}^{N}}}v(x)f(x)dx,\end{array}\right.$
$\nabla_{s}$ denotes the fractional gradient, for all $f\in H^{s}$. It follows
that $u$ and $v$ solve the following system
$\left\\{\begin{array}[]{ll}(-\Delta)^{s}\,u+\displaystyle{\int}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dy\,G^{\prime}(u(x))+\alpha
u(x)=0,\\\
(-\Delta)^{s}\,v+\displaystyle{\int}G(u^{2}+v^{2})^{1/2}(y)V(|x-y|)dy\,G^{\prime}(v(x))+\alpha
v(x)=0.\end{array}\right.$
By Proposition 4.2, we have that $u$ and $v\in C(\mathbb{R}^{N})$ because
$(u^{2}+v^{2})^{1/2}\in H^{s}(\mathbb{R}^{N})$. Let
$\Omega=\\{x\in\mathbb{R}^{N}:u(x)=0\\}$, obviously $\Omega$ is closed since
$u$ is continuous. Let us prove that it is also open. Suppose that
$x_{0}\in\Omega$. Knowing that $\varphi(x_{0})>0$, we can find a ball $B$
centered in $x_{0}$ such that $v(x)\neq 0$ for any $x\in B$. Replacing $u$ and
$v$ in (35), we certainly have that
$u(x)v(y)-v(x)u(y)=0\quad\forall\;x,y\in B.$
This proves the result. ∎
## Appendix
In this appendix, we prove the following
###### Proposition 4.2.
Let $s\in(0,1),N-2s\leq\beta<N,\beta>0,u,\varphi\in H^{s}(\mathbb{R}^{N})$,
$G$ such that $\mathcal{A}_{0}$ holds and $\kappa$ is a real number such that
$(-\Delta)^{s}u-\kappa u=[V\star G(\varphi)]G^{\prime}(u).$ (40)
Then, there exists $\alpha\in(0,1)$ depending only on $N,\kappa,s,\beta$ such
that $u\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$. Moreover, if $\varphi\in
L^{\infty}_{loc}({\mathbb{R}^{N}})$, then $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ if $\beta\leq 1$ and $u\in
C^{1,\alpha}_{loc}({\mathbb{R}^{N}})$ if $\beta>1$ and in addition $V\star
G(\varphi)\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$.
###### Proof.
We start by recalling the Gagliardo-Nirenberg inequality
${|\\!|}\varphi{|\\!|}_{L^{p}({\mathbb{R}^{N}})}\leq
c_{N,s,p}{|\\!|}\varphi{|\\!|}_{H^{s}({\mathbb{R}^{N}})}\quad\textrm{ for all
$\varphi\in H^{s}(\mathbb{R}^{N})$ },$
for $p\in\left[2,\frac{2N}{N-2s}\right]$ if $N>2s$ and for all
$p\in\left[\left.2,\frac{2N}{N-2s}\right)\right.$ and $2s\geq N$ (here we put
$\frac{2N}{N-2s}\equiv+\infty$). Also we recall the Hardy-Littlewood-Sobolev
inequality:
$\|V\star g\|_{L^{\frac{qN}{N-q\beta}}({\mathbb{R}^{N}})}\leq
C_{N,\beta,q}\|g\|_{L^{q}({\mathbb{R}^{N}})}\quad\textrm{ for every $g\in
L^{q}({\mathbb{R}^{N}})$,}$
for $N-q\beta>0$. First of all we focus on the case $N>2s$. Thus, we have
${|\\!|}G(\varphi){|\\!|}_{L^{q}({\mathbb{R}^{N}})}\leq{|\\!|}\varphi^{2}{|\\!|}_{L^{q}({\mathbb{R}^{N}})}+{|\\!|}|\varphi|^{\mu}{|\\!|}_{L^{q}({\mathbb{R}^{N}})}={|\\!|}\varphi{|\\!|}_{L^{2q}({\mathbb{R}^{N}})}^{2}+{|\\!|}\varphi{|\\!|}_{L^{\mu
q}({\mathbb{R}^{N}})}^{\mu}.$
Hence, since $\varphi\in H^{s}(\mathbb{R}^{N})$, we infer that $G(\varphi)\in
L^{q}({\mathbb{R}^{N}})$ provided that $1\leq q\leq\frac{N}{N-2s}$ and
$\frac{2}{\mu}\leq q\leq\frac{1}{\mu}\frac{2N}{N-2s}$, that is $1\leq
q\leq\frac{N}{N-2s}$ and $1\leq q\leq\frac{2N^{2}}{(N-2s)(N+2s+\beta)}$. Now,
thanks to the fact that $N-2s\leq\beta<N$, we get
$1<\frac{N}{\beta}\leq\frac{N}{N-2s}$ and
$1<\frac{N}{\beta}\leq\frac{2N^{2}}{(N-2s)(N+2s+\beta)}$. In particular, we
deduce that $G(\varphi)\in L^{q}({\mathbb{R}^{N}})$ for all
$q\in\left[1,\frac{N}{\beta}\right]$. Now, for all $\epsilon>0$ we let
$q_{\epsilon}=\frac{N}{\beta}-\epsilon>1$. Using the Hardy-Littelwood-Sobolev
inequality, we get $V\star G(\varphi)\in
L^{\frac{Nq_{\epsilon}}{\epsilon\beta}}({\mathbb{R}^{N}})$ which in turns with
the fact that $\beta\geq N-2s$ shows that $V\star G(\varphi)\in
L^{r}({\mathbb{R}^{N}})$ for all $r>\frac{N}{N-\beta}\geq\frac{N}{2s}$. Now,
using the notation $b(x)=\frac{G^{\prime}(u)}{1+|u|}$ and
$sign(u)=\frac{u}{|u|}$, we reformulate the equation (40) as follows
$\displaystyle(-\Delta)^{s}u(x)-\kappa u(x)$ $\displaystyle=$
$\displaystyle[V\star G(\varphi)]\,b(x)\,(1+|u|)),$ $\displaystyle=$
$\displaystyle\int_{{\mathbb{R}^{N}}}V(|x-y|)G(\varphi(y))dy\,b(x)\,(1+sign(u)\,u).$
Observing that $\mu-2<\frac{2N}{N-2s}-2=\frac{4s}{N-2s}$, then for all
$r>\frac{N}{2s}$, we can write
$\displaystyle{|\\!|}[V\star G(\varphi)]b{|\\!|}_{L^{r}({\mathbb{R}^{N}})}$
$\displaystyle={|\\!|}[V\star
G(\varphi)]\,\frac{G^{\prime}(u)}{1+|u|}{|\\!|}_{L^{r}({\mathbb{R}^{N}})},$
$\displaystyle\leq c\,{|\\!|}[V\star
G(\varphi)]\frac{|u|+|u|^{\mu-1}}{1+|u|}{|\\!|}_{L^{r}({\mathbb{R}^{N}})},$
$\displaystyle\leq c\,{|\\!|}V\star
G(\varphi){|\\!|}_{L^{r}({\mathbb{R}^{N}})}+c\,{|\\!|}[V\star
G(\varphi)]|u|^{\mu-2}{|\\!|}_{L^{r}({\mathbb{R}^{N}})}.$
In order to deduce that the right hand side of this estimate is finite, we use
Hölder’s inequality to get
$\displaystyle{|\\!|}[V\star
G(\varphi)]\,|u|^{\mu-2}{|\\!|}^{r}_{L^{r}({\mathbb{R}^{N}})}$
$\displaystyle\leq{|\\!|}V\star
G(\varphi)]{|\\!|}_{L^{\frac{r\,\theta}{\theta-1}}({\mathbb{R}^{N}})}\,{|\\!|}|u|{|\\!|}^{\mu-2}_{L^{r\,(\mu-2)\,\theta}({\mathbb{R}^{N}})},$
for all $\theta>1$. Therefore, we can choose $r>\frac{N}{2s}$ and $\theta>1$
respectively close to $\frac{N}{2s}$ and $1$ so that
$1<r\,(\mu-2)\,\theta<\frac{2N}{N-2s}$. Hence using the Gagliardo-Nirenberg
inequality and the fact that $V\star G(\varphi)\in L^{r}({\mathbb{R}^{N}})$
for all $r>\frac{N}{N-\beta}\geq\frac{N}{2s}$ and $u\in
H^{s}(\mathbb{R}^{N})$, we end up with $[V\star G(\varphi)]b\in L^{r}$ for
some $r>\frac{N}{2s}$, hence $u\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$. Now,
we write the equation (40) as follows
$\displaystyle(-\Delta)^{s}u(x)=c(x)u(x)+d(x):$ $\displaystyle
c(x)=\kappa+[V\star G(\varphi)]\,b(x)\,sign(u)\in L^{r}({\mathbb{R}^{N}}),$
$\displaystyle d(x)=[V\star G(u)]\,b(x)\in L^{r}({\mathbb{R}^{N}}),$
for some $r>\frac{N}{2s}$. Thus, using the regularity result of Ref. [20], we
conclude that $u\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for some
$\alpha\in(0,1)$ provided $\frac{N}{2s}>1$. If $N=1$ and $s>\frac{1}{2}$, then
it is well-known that $H^{s}(\mathbb{R}^{N})$ is embedded in
$C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ with
$\alpha=s-\frac{1}{2}-\left[s-\frac{1}{2}\right]$ so that $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$. Moreover, if $N=1$ and $s=\frac{1}{2}$,
we have obviously $u\in L^{p}({\mathbb{R}^{N}})$ for every $p\geq 2$ and
classical elliptic regularity yields $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for some $\alpha\in(0,1)$. In the
following, $[\cdot]$ stands for the integer part of $\cdot$. Let us introduce
a cutoff function $\eta\in C_{c}^{\infty}({\mathbb{R}^{N}})$ such that
$\eta\equiv 1$ in the closed ball $B_{R}$ of center $0$ and radius $R>0$ and
$\eta\equiv 0$ in ${\mathbb{R}^{N}}\setminus B_{2R}$. To alleviate the
notation, we denote $f=G(\varphi)$ which belongs to
$L^{\infty}_{loc}({\mathbb{R}^{N}})\cap L^{q}({\mathbb{R}^{N}})$ with
$1<q\leq\frac{N}{\beta}$. We define $V_{1}(\varphi):=V\star(\eta f)$ and
$V_{2}(\varphi):=V\star((1-\eta)f)$. Then using Fourier transform, we get
$(-\Delta)^{\frac{\beta}{2}}V_{1}(\varphi)=f$ in the sense of distributions.
Now, if $\frac{\beta}{2}\in\mathbb{N}^{\star}$, then it is rather easy to show
using classical regularity theory that $V\star G(\varphi)\in
C^{\beta}({\mathbb{R}^{N}})$. Next, if $0<\frac{\beta}{2}<1$, then we apply
Proposition 2.1.9 of Ref. [19] to show that $V_{1}(\varphi)\in
C^{0,\alpha}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and $V_{1}(\varphi)\in
C^{[\beta],\alpha}({\mathbb{R}^{N}})$ for $\beta>1$ and some $\alpha\in(0,1)$.
Now, $V_{2}(\varphi)$ is smooth on $B_{R}$ since it is
$\frac{\beta}{2}-$harmonic in such a ball, see Ref. [1]. Hence, $V\star
G(\varphi)\in C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and
$V\star G(\varphi)\in C^{[\beta],\alpha}_{loc}({\mathbb{R}^{N}})$ for
$\beta>1$ and some $\alpha\in(0,1)$. Let us now turn to the case of
$\frac{\beta}{2}>1$ and $\frac{\beta}{2}\not\in\mathbb{N}$. we let
$\sigma=\frac{\beta}{2}-\left[\frac{\beta}{2}\right]$. Using Fourier
transform, we have
$(-\Delta)^{\left[\frac{\beta}{2}\right]}V_{1}(\varphi)=(-\Delta)^{\left[\frac{\beta}{2}\right]}\left((-\Delta)^{\sigma}V_{1}(\varphi)\right)=\eta\,f$
in the sense of distributions. Again, classical regularity theory arguments
implies that $(-\Delta)^{\sigma}V_{1}(\varphi)\in
C^{[\beta]}({\mathbb{R}^{N}})$ and so $V_{1}(\varphi)\in
C^{[\beta]}({\mathbb{R}^{N}})$. Similarly, we have
$(-\Delta)^{\frac{\beta}{2}}V_{2}(\varphi)=(-\Delta)^{\sigma}\left((-\Delta)^{\left[\frac{\beta}{2}\right]}V_{2}(\varphi)\right)=(1-\eta)\,f$
in the sense of distributions. Therefore the function
$g:=(-\Delta)^{\left[\frac{\beta}{2}\right]}V_{2}(\varphi)$ is given by
$(-\Delta)^{\left[\frac{\beta}{2}\right]}V_{2}(\varphi)(x)=\int_{{\mathbb{R}^{N}}}\frac{(1-\eta(y))\,f(y)}{|x-y|^{N-\sigma}}\,dy.$
Also, using the Hardy-Littelwood-Sobolev inequality, it is rather
straightforward to see that $g\in L^{p}({\mathbb{R}^{N}})$ for some $p>1$.
Thus, $g$ belongs to the set
$\left\\{u,\>\int_{{\mathbb{R}^{N}}}\frac{|u(x)|}{1+|x|^{N+2\sigma}}\,dx<+\infty\right\\}$.
Again, since $g$ is $\sigma-$harmonic in $B_{R}$, we deduce that $g$ is smooth
on $B_{R}$ by Ref. [1]. The radius $R$ being arbitrary, it follows that
$V_{2}(\varphi)$ is smooth on ${\mathbb{R}^{N}}$. In particular, we have
$V_{2}(\varphi)\in C^{[\beta]}({\mathbb{R}^{N}})$ because
$\left[\frac{\beta}{2}\right]$ is a positive integer. Recalling that we showed
$V_{1}(\varphi)\in C^{[\beta]}({\mathbb{R}^{N}})$, we conclude $V\star
G(\varphi)\in C^{[\beta]}({\mathbb{R}^{N}})$. Let us now summarize and
conclude the proof. We considered the partial differential equation (40) and
proved that for some $\alpha\in(0,1)$, we have $V\star G(\varphi)\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and $V\star
G(\varphi)\in C^{[\beta],\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta>1$. Since
$G^{\prime}$ is locally Lipschitz, we deduce that $u\in
C^{0,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta\leq 1$ and $u\in
C^{1,\alpha}_{loc}({\mathbb{R}^{N}})$ for $\beta>1$ by adapting the proof of
Lemma 3.3 of Ref. [8] for $N>2s$. If $N=1$ and $2s\geq 1$, we have that
$[V\star G(\varphi)]\,G^{\prime}(u)\in C^{0,\gamma}_{loc}({\mathbb{R}^{N}})$
for some $\gamma\in(0,1)$, thus using Proposition 2.1.8 of Ref. [19], we get
$u\in C^{1,\alpha}_{loc}({\mathbb{R}^{N}})$. ∎
## Acknowledgments
Y. Cho was supported by NRF grant 2010-0007550 (Republic of Korea). M. M. Fall
is supported by the Alexander von Humboldt foundation.
## References
* [1] K. Bogdan, T. Byczkowski, Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domains, Studia Math. 133 (1999), no. 1, 53-92.
* [2] X. Cabre, Y Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates. Annales de l’Institut Henri Poincaré (C) Non Linear Analysis, to appear.
* [3] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
* [4] T. Cazenave, P.-L. Lions, Orbital stability of standing waves for some nonlinear Schr dinger equations, Comm. Math. Phys. Volume 85, Number 4 (1982), 549-561.
* [5] Y. Cho, H. Hajaiej, G. Hwang, T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity to appear in Funkcialaj Ekvacioj (arXiv:1209.5899).
* [6] Y. Cho, G. Hwang, T. Ozawa, Global well-posedness of critical nonlinear Schrödinger equations below $L^{2}$, DCDS-A 33 (2013), 1389-1405.
* [7] Y. Cho, K. Nakanishi, On the global existence of semirelativistic Hartree equations, RIMS Kokyuroku Bessatsu, B22 (2010), 145-166.
* [8] M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, http://arxiv.org/abs/1301.5119. Comm. Partial Differential Equations, to appear.
* [9] B. Guo, D. Huang, Existence and stability of standing waves for nonlinear fractional Schrödinger equations, J. Math. Phys. 53 (2012). 083702
* [10] X. Guo, M. Xu, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrodinger equation, J. Math. Phys. 47, 082104 (2006).
* [11] H. Hajaiej, Existence of minimizers of functionals involving the fractional gradient in the absence of compactness, symmetry and monotonicity, J. Math. Anal. Appl. 399 (2013) 17-26.
* [12] H Hajaiej, L. Molinet, T. Ozawa, B. Wan,, Necessary and Sufficient Conditions for the Fractional Gagliardo-Nirenberg Inequalities and Applications to Navier-Stokes and Generalized Boson Equations, RIMS Kokyuroku Bessatsu RIMS Kokyuroku Bessatsu B26, 159-175, 2011-05-Kyoto University
* [13] H. Hajaiej, C. A. Stuart, On the Variational Approach to the Stability of Standing Waves for the Nonlinear Schrödinger Equation, Advanced Nonlinear Studies 4 (2004), 469-501.
* [14] N. Laskin, Fractional quantum mechanics and Lévy integral, Phys. Lett. A 268, 298305 (2000).
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* [17] P.-L. Lions, The concentration-compactness method in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. pincaré Anal. Non Linéaire 1 (1984) 109–145; Part II, Ann. Inst. H. pincaré Anal. Non Linéaire 1 (1984) 223–283
* [18] E.H. Lieb, H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.
* [19] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60(1):67-112, 2007.
* [20] J. Tan; J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst. 31 (2011), no. 3, 975 983.
* [21] D. Wu, Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity, arXiv:1210.3887
|
arxiv-papers
| 2013-07-21T12:08:21 |
2024-09-04T02:49:48.234757
|
{
"license": "Public Domain",
"authors": "Y. Cho, M.M. Fall, H. Hajaiej, P.A. Markowich, S. Trabelsi",
"submitter": "Hichem Hajaiej",
"url": "https://arxiv.org/abs/1307.5523"
}
|
1307.5598
|
# New discrete method for investigating the response properties in finite
electric field
Myong-Chol Pak Nam-Hyok Kim Hak-Chol Pak Song-Jin Im Department of
Physics, Kim Il Sung university, Pyongyang , Democratic People’s Republic of
Korea
###### Abstract
In this paper we develop a new discrete method for calculating the dielectric
tensor and Born effective charge tensor in finite electric field by using
Berry’s phase and the gauge invariance. We present a new method to overcome
non-periodicity of the potential in finite electric field due to the gauge
invariance, and construct the dielectric tensor and Born effective charge
tensor that satisfy translational symmetry in finite electric field. In order
to demonstrate the correctness of this method, we also perform calculations
for the semiconductors AlAs and GaAs under the finite electric field to
compare with the preceding method and the experiment.
###### keywords:
Berry’s phase , Gauge invariance , Dielectric tensor , Born effective charge
## 1 Introduction
The investigation for calculating the dielectric tensor and Born effective
charge tensor in finite electric field is very important in studying of bulk
ferroelectrics, ferroelectric films, superlattices, lattice vibrations in
polar crystals, and so on[1,2,3].
Recently, the investigation of response properties to the external electric
field is becoming interested theoretically as well as practically. In
particular, the dielectric tensor and Born effective charge tensor in finite
electric field are important physical quantities for analyzing and modeling
the response of material to the electric field. In case of zero electric
field, these response properties have already been studied by using DFPT
(Density Functional Perturbation theory), and excellent results have been
obtained [1].
DFPT[4]provides a powerful tool for calculating the $2^{nd}$-order derivatives
of the total energy of a periodic solid with respect to external
perturbations, such as strains, atomic sublattice displacements, a homogeneous
electric field etc. In contrast to the case of strains and sublattice
displacements for which the perturbing potential remains periodic, treatment
of homogeneous electric fields is subtle, because the corresponding potential
requires a term that is linear in real space, thereby breaking the
translational symmetry and violating the conditions of Bloch’s theorem.
Therefore, electric field perturbations have already been studied using the
long-wave method, in which the linear potential caused by applied electric
field is obtained by considering a sinusoidal potential in the limit that its
wave vector goes to zero[5]. In this approach, however, the response tensor
can be evaluated only at zero electric field.
In nonzero electric field, the investigation of the response properties can’t
be performed using method based on Bloch’s theorem, for nonperiodicity of the
potential with respect to electric field. Therefore, several methods for
overcoming it have been developed [2,3].
Ref.[2] introduces the electric field-dependent energy functional by Berry’s
phase, and suggests the methodology for calculating by using finite-difference
scheme. Ref.[3] discusses the proposal for calculating it by the discretized
form of Berry’s phase term and response theory with respect to perturbation of
the finite electric field. However, in these methods, the nonperiodicity of
the potential due to electric field is resolved by introducing polarized WFs
(Wannier Functions) due to finite electric field. This requires much cost in
calculating its inverse matrix in the perturbation expansion of Berry’s phase
and yields instability of results.
In this paper, we developed a new discrete method for calculating the
dielectric tensor and Born effective charge tensor in finite electric field by
using Berry’s phase and the gauge invariance. We present a new method for
overcoming non-periodicity of the potential in finite electric field due to
the gauge invariance, and calculate the dielectric tensor and Born effective
charge tensor in a discrete different way than ever before.
This paper is organized as follows. In Sec. 2, instead of preceding
investigation in which the total field-dependent energy functional is divided
into Kohn-Sham energy, Berry’s phase and Lagrange multiplier term, we discuss
the method for studying the response properties with a new discrete way by
using the polarization written with Berry’s phase and unit cell periodic
function polarized by field. In Sec. 3, we calculate the dielectric tensor and
Born effective charge tensor in finite electric field by constructing the
polarized Bloch wave Function and evaluating linear response of the wave
function with Sternheimer equation. We also calculate the $2^{nd}$-order
nonlinear dielectric tensor indicating nonlinear response property with
respect to electric field. In order to demonstrate the correctness of the
method, we also perform calculations for the semiconductors AlAs and GaAs
under the finite electric field. In Sec. 4, summary and conclusion are
presented.
## 2 New discrete method by using Berry’s phase and the gauge invariance
The response tensors with respect to electric field in finite electric field
are presented by the $2^{nd}$-order derivatives of the field-dependent total
energy functional with respect to the atomic sublattice displacements and the
homogeneous electric field. Here, the field-dependent energy functional[6] is
$E[\\{u^{(\vec{\varepsilon})}\\},\vec{\varepsilon}]=E_{KS}[\\{u^{(\vec{\varepsilon})}\\}]-\Omega\vec{\varepsilon}\cdot{\bf{P}}[\\{u^{(\vec{\varepsilon})}\\}$
(1)
where $E_{KS},\vec{\varepsilon},{\bf{P}}$ are the Kohn-Sham energy functional,
the finite electric field, and the cell volume, respectively. In addition,
$u^{(\vec{\varepsilon})}$ is a set of unit cell periodic function polarized by
field, and polarization ${\bf{P}}$ written through Berry’s phase is
${\bf{P}}=-{{ife}\over{(2\pi)^{3}}}\sum\limits_{n=1}^{M}{\int\limits_{BZ}{d^{3}k\left\langle{u_{n{\bf{k}}}^{(\vec{\varepsilon})}}\right|}}\nabla_{\bf{k}}\left|{u_{n{\bf{k}}}^{(\vec{\varepsilon})}}\right\rangle$
(2)
where $f$ is the spin degeneracy, and $f=2$.
In fact, the polarization is calculated by the following discretized form that
is suggested by King-Smith and Vanderbilt[7].
${\bf{P}}={{ef}\over{2\pi\Omega}}\sum\limits_{i=1}^{3}{{{{\bf{a}}_{i}}\over{N_{\bot}^{(i)}}}}\sum\limits_{l=1}^{N_{\bot}^{(i)}}{{\mathop{\rm
Im}\nolimits}\ln\prod\limits_{j=1}^{N_{i}}{\det
S_{{\bf{k}}_{lj},{\bf{k}}_{lj+1}}}},$ (3)
(Ref.[6] and [7] point out the meaning of every parameter in Eq. 2 and Eq. 3.)
Next, if we consider the orthonormality constraints of the unit cell periodic
function polarized by field
$\left\langle{{u_{m{\bf{k}}}^{(\vec{\varepsilon})}}}\mathrel{\left|{\vphantom{{u_{m{\bf{k}}}^{(\vec{\varepsilon})}}{u_{n{\bf{k}}}^{(\vec{\varepsilon})}}}}\right.\kern-1.2pt}{{u_{n{\bf{k}}}^{(\vec{\varepsilon})}}}\right\rangle=\delta_{mn}$
(4)
,the total energy functional is divided into 3 parts as follows.
$F=F_{KS}+F_{BP}+F_{LM}$ (5)
where $F_{KS}=E_{KS}$ is Kohn-Sham energy and
$F_{BP}=-\Omega\vec{\varepsilon}\cdot{\bf{P}}$ is the coupling between the
electric field and the polarization by Berry’s phase, and the constraints are
given by Lagrange multiplier term, $F_{LM}$. Next, the set of unit cell
periodic functions polarized by field, $\\{u^{(\vec{\varepsilon})}\\}$ is
determined with variational method. The set of its function is different from
a set of unit cell periodic function in zero field. Although, strictly
speaking, calculated ground state is not exact ground state, this method is a
way to overcome nonperiodicity of the potential caused by electric field[3].
Therefore, it does not include explicitly the gauge invariant property and
requires big cost in calculating its inverse matrix in the perturbation
expansion of Berry’s phase, yielding instability of results.
We apply the perturbation expansion by using DFPT, and investigate the
response property with a new discrete method by using Eq. 2 and unit cell
periodic functions polarized by field. Since the general perturbation
expansion methods were mentioned in Refs.[1,2,3], we consider the response
tensors, dielectric tensor and Born effective charge tensor in case of
perturbation with respect to the atomic sublattice displacements and the
homogeneous electric field.
In Gaussian system the dielectric tensor is
$\in_{\alpha\beta}=\delta_{\alpha\beta}+4\pi\chi_{\alpha\beta}$ (6)
and then electric susceptibility tensor can be written by perturbation
expansion.
$\begin{split}\chi_{\alpha\beta}&=-{1\over\Omega}{{\partial^{2}F}\over{\partial\varepsilon_{\alpha}\partial\varepsilon_{\beta}}}=-{f\over{2(2\pi)^{3}}}\int\limits_{BZ}{d^{3}k\sum\limits_{n=1}^{M}{[\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right|T+v_{ext}\left|{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right\rangle+}}\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right|T+v_{ext}\left|{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right\rangle]\\\
&+\sum\limits_{n=1}^{M}{[\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right|(ie{\partial\over{\partial
k_{\beta}}})\left|{u_{n{\bf{k}}}^{(0)}}\right\rangle+\left\langle{u_{n{\bf{k}}}^{(0)}}\right|(ie{\partial\over{\partial
k_{\beta}}})\left|{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right\rangle+\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right|(ie{\partial\over{\partial
k_{\alpha}}})\left|{u_{n{\bf{k}}}^{(0)}}\right\rangle}\\\
&+\left\langle{u_{n{\bf{k}}}^{(0)}}\right|(ie{\partial\over{\partial
k_{\alpha}}})\left|{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right\rangle]+{f\over{2(2\pi)^{3}}}\int\limits_{BZ}{d^{3}k\sum\limits_{m,n=1}^{M}{\Lambda_{mn}^{(0)}({\bf{k}})[\left\langle{{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}}\mathrel{\left|{\vphantom{{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}{u_{m{\bf{k}}}^{\varepsilon_{\beta}}}}}\right.\kern-1.2pt}{{u_{m{\bf{k}}}^{\varepsilon_{\beta}}}}\right\rangle+\left\langle{{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}}\mathrel{\left|{\vphantom{{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}{u_{m{\bf{k}}}^{\varepsilon_{\alpha}}}}}\right.\kern-1.2pt}{{u_{m{\bf{k}}}^{\varepsilon_{\alpha}}}}\right\rangle]}}-{1\over{2\Omega}}{{\partial^{2}E_{XC}}\over{\partial\varepsilon_{\alpha}\partial\varepsilon_{\beta}}}\end{split}$
(7)
Though Eq. 7 reflects successfully the response properties with respect to
perturbation in finite electric field, it does not describe sufficiently the
periodic effect of crystal. Because the operator,$ie\nabla_{\bf{k}}$ hidden
Berry’s phase must be applied to gauge invariant quantity in order to overcome
nonperiodicity of potential caused by field[7]. Therefore, using the gauge
invariant form,$ie{\partial\over{\partial
k_{\alpha}}}\sum\limits_{m=1}^{M}{\left|{u_{m{\bf{k}}}^{(0)}}\right\rangle}\left\langle{u_{m{\bf{k}}}^{(0)}}\right|$
and considering
$0^{th}$-order,$\Lambda_{mn}^{(0)}({\bf{k}})=\varepsilon_{n{\bf{k}}}^{(0)}\delta_{mn}$,
dielectric tensor is
$\begin{split}\chi_{\alpha\beta}&=\left.{-{1\over\Omega}{{\partial^{2}F}\over{\partial\varepsilon_{\alpha}\partial\varepsilon_{\beta}}}}\right|_{\varepsilon=\varepsilon^{(0)}}\\\
&=-{f\over{2(2\pi)^{3}}}\int\limits_{BZ}{d^{3}k\sum\limits_{n=1}^{M}{[\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right|T+v_{ext}-\varepsilon_{n{\bf{k}}}^{(0)}\left|{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right\rangle+}}\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right|T+v_{ext}-\varepsilon_{n{\bf{k}}}^{(0)}\left|{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right\rangle\\\
&+\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right|(ie{\partial\over{\partial
k_{\beta}}}\sum\limits_{m=1}^{M}{\left|{u_{m{\bf{k}}}^{(0)}}\right\rangle}\left\langle{u_{m{\bf{k}}}^{(0)}}\right|)\left|{u_{n{\bf{k}}}^{(0)}}\right\rangle+\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right|(ie{\partial\over{\partial
k_{\alpha}}}\sum\limits_{m=1}^{M}{\left|{u_{m{\bf{k}}}^{(0)}}\right\rangle}\left\langle{u_{m{\bf{k}}}^{(0)}}\right|)\left|{u_{n{\bf{k}}}^{(0)}}\right\rangle\\\
&-\left\langle{u_{n{\bf{k}}}^{(0)}}\right|(ie{\partial\over{\partial
k_{\beta}}}\sum\limits_{m=1}^{M}{\left|{u_{m{\bf{k}}}^{(0)}}\right\rangle}\left\langle{u_{m{\bf{k}}}^{(0)}}\right|)\left|{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right\rangle-\left\langle{u_{n{\bf{k}}}^{(0)}}\right|(ie{\partial\over{\partial
k_{\alpha}}}\sum\limits_{m=1}^{M}{\left|{u_{m{\bf{k}}}^{(0)}}\right\rangle}\left\langle{u_{m{\bf{k}}}^{(0)}}\right|)\left|{u_{n{\bf{k}}}^{\varepsilon_{\beta}}}\right\rangle]-\left.{{1\over{2\Omega}}{{\partial^{2}E_{XC}}\over{\partial\varepsilon_{\alpha}\partial\varepsilon_{\beta}}}}\right|_{\varepsilon=\varepsilon^{(0)}}\end{split}$
(8)
where BZ(Brillouin Zone) integration is performed by Monkhorst-Pack special
point method. Meanwhile, the partial derivative is calculated with following
discretized method.
${\partial\over{\partial
k_{x}}}\left|{u_{m,i,j,k}}\right\rangle\left\langle{u_{m,i,j,k}}\right|={1\over{2\Delta
k_{x}}}(\left|{u_{m,i+1,j,k}}\right\rangle\left\langle{u_{m,i+1,j,k}}\right|-\left|{u_{m,i-1,j,k}}\right\rangle\left\langle{u_{m,i-1,j,k}}\right|)$
(9) ${\partial\over{\partial
k_{y}}}\left|{u_{m,i,j,k}}\right\rangle\left\langle{u_{m,i,j,k}}\right|={1\over{2\Delta
k_{y}}}(\left|{u_{m,i,j+1,k}}\right\rangle\left\langle{u_{m,i,j+1,k}}\right|-\left|{u_{m,i,j-1,k}}\right\rangle\left\langle{u_{m,i,j-1,k}}\right|)$
(10) ${\partial\over{\partial
k_{z}}}\left|{u_{m,i,j,k}}\right\rangle\left\langle{u_{m,i,j,k}}\right|={1\over{2\Delta
k_{z}}}(\left|{u_{m,i,j,k+1}}\right\rangle\left\langle{u_{m,i,j,k+1}}\right|-\left|{u_{m,i,j,k-1}}\right\rangle\left\langle{u_{m,i,j,k-1}}\right|)$
(11)
Additionally, the $1^{st}$-order wave function response with respect to finite
electric field is calculated with the following Sternheimer equation
$P_{c{\bf{k}}}(T+v_{ext}-\varepsilon_{n{\bf{k}}}^{(0)})P_{c{\bf{k}}}\left|{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right\rangle=-P_{c{\bf{k}}}(ie{\partial\over{\partial
k_{\alpha}}}\sum\limits_{m=1}^{M}{\left|{u_{m{\bf{k}}}^{(0)}}\right\rangle}\left\langle{u_{m{\bf{k}}}^{(0)}}\right|)\left|{u_{n{\bf{k}}}^{(0)}}\right\rangle$
(12)
Generally, investigation of the $2^{nd}$-order energy response requires up to
$1^{st}$-order wave function response with respect to perturbation by using
$"$2n+1 $"$theorem. Therefore, every result can be calculated with only the
$1^{st}$-order wave function response to finite electric field.
In this way, Born effective charge tensor is
$\begin{split}Z_{\kappa,\alpha\beta}^{*}&=\left.{-{{\partial^{2}F}\over{\partial\varepsilon_{\alpha}\partial\tau_{\kappa,\beta}}}}\right|_{\varepsilon=\varepsilon^{(0)}}\\\
&={{f\Omega}\over{(2\pi)^{3}}}\int\limits_{BZ}{d^{3}k\sum\limits_{n=1}^{M}{[\left\langle{u_{n{\bf{k}}}^{(0)}}\right|(T+v_{ext})^{\tau_{\kappa,\beta}}\left|{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right\rangle+\left\langle{u_{n{\bf{k}}}^{\varepsilon_{\alpha}}}\right|(T+v_{ext})^{\tau_{\kappa,\beta}}\left|{u_{n{\bf{k}}}^{(0)}}\right\rangle+\left.{{{\partial^{2}E_{XC}}\over{\partial\tau_{\kappa,\beta}\partial\varepsilon_{\alpha}}}}\right|_{\varepsilon=\varepsilon^{(0)}}}}\end{split}$
(13)
Eq. 13 also calculate with DFPT and the wave function polarized by field.
## 3 Results and Analysis
The calculation of the dielectric permittivity tensor and the Born effective
charge tensor is performed in three steps. First, a ground state calculation
in finite electric field is performed using the Berry’s phase method
implemented in the ABINIT code, and the field-polarized Bloch functions are
stored for the later linear-response calculation. Second, the linear-response
calculation is performed to obtain the first order response of Bloch
functions. Third, the matrix elements of the dielectric and Born effective
charge tensors are computed using these $1^{st}$-order responses.
To verify the correctness of our method, we have performed test calculation on
two prototypical semiconductors, AlAs and GaAs. In this calculation, we have
used the HSC norm-conserving pseudopotential method based on Density
Functional Theory with LDA (Local Density Approximation). The cutting energy,
$E_{cut}=20Ry$ and $6\times 6\times 6$ Monkhorst-Pack mesh for $k$-point
sampling were used.
In Table 1, we present the calculated values of dielectric tensor and Born
effective charge tensor of AlAs and GaAs, when such finite electric field as
in Ref. [3] is applied along the [100] direction. In order to compare our
method with the preceding one, we present the calculated values in our method
and preceding one (Ref.[3]), and the experimental values. As you see in Table
1, the calculated value of dielectric tensor in our method goes to the
experiment one[2] more closely than the calculated one in preceding method
(Ref.[3]).However, in case of Born effective charge tensor, the difference
between our method and preceding one does not almost occur. It shows that in
calculating Born effective charge tensor, there exist the $1^{st}$-order
contribution of the potential with respect to atomic sublattice displacements
and one of the polarized wave function with respect to the finite electric
field, the latter playing the essential role.
Table 1: Calculated and experimental values of dielectric tensor and Born effective charge tensor in finite electric field Material | Method | $\in$ | $Z^{*}$
---|---|---|---
| Our Method | 9.48 | 2.05
AlAs | Preceding Method[3] | 9.72 | 2.03
| Experiment[2] | 8.2 | 2.18
| Our Method | 12.56 | 2.20
GaAs | Preceding Method[3] | 13.32 | 2.18
| Experiment[2] | 10.9 | 2.07
We also calculated the $2^{nd}-order\ nonlinear\ dielectric\ tensor$,
nonlinear response property with respect to electric field. The $2^{nd}$-order
nonlinear dielectric tensor is
$\chi_{123}^{(2)}={1\over
2}{{\partial^{2}P_{2}}\over{\partial\varepsilon_{1}\partial\varepsilon_{3}}}={1\over
2}{{\partial\chi_{23}}\over{\partial\varepsilon_{1}}}$ (14)
Table 2 shows calculated value of the $2^{nd}$-order nonlinear dielectric
tensor on AlAs.
Table 2: Calculated value of the $2^{nd}$-order nonlinear dielectric tensor on AlAs Method | $\chi_{123}^{(2)}(pm/V)$
---|---
Our Method | 67.32
Preceding Method[3] | 60.05
Experiment[8] | 78 $\pm$ 20
As shown in Table 2, the calculated value of $2^{nd}$-order nonlinear
dielectric tensor in our method coincides with the experimental value[8] more
closely than the calculated one in the preceding method (Ref.[3]).
## 4 Summary
We suggested a new method for calculating the dielectric tensor and Born
effective charge tensor in finite electric field. In particular, in order to
overcome nonperiodicity of potential caused by electric field, a new
transformation conserving gauge invariant property is introduced. In future,
this methodology can be expanded not only to perturbation with respect to
field and atomic replacement but also to the other cases, such as strain and
chemical composition of solid solution.
## Acknowledgments
It is pleasure to thank Jin-U Kang, Chol-Jun Yu, Kum-Song Song, Kuk-Chol Ri
and Song-Bok Kim for useful discussions. This work was supported by the
Physics faculty in Kim Il Sung university of Democratic People’s Republic of
Korea.
## References
[1] C.-J. Yu and H. Emmerich. J. Phys.:Condens. Matter, 19:306203, 2007.
[2] I. Souza, J. Iniguez, and D. Vanderbilt. Phys. Rev. Lett., 89:117602,
2002.
[3] X. Wang and D. Vanderbilt. Phys. Rev. B, 75:115116, 2007.
[4] S. Baroni, Stefano de Gironcoli, and Andrea Dal Corso. Rev.Mod.Phys.,
73:515, 2001.
[5] X. Gonze and C. Lee. Phys. Rev. B, 55:10355, 1997.
[6] R. W. Nunes and X. Gonze. Phys. Rev. B, 63:155107, 2001.
[7] R. D. King-Smith and D.Vanderbilt. Phys. Rev. B, 48:4442, 1993.
[8] I. Shoji, T. Kondo, and R. Ito. Opt. Quantum Electron, 34:797, 2002
|
arxiv-papers
| 2013-07-22T06:39:50 |
2024-09-04T02:49:48.249998
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Myong Chol Pak, Nam-Hyok Kim, Hak-Chol Pak and Song-Jin Im",
"submitter": "Myong Chol Pak",
"url": "https://arxiv.org/abs/1307.5598"
}
|
1307.5619
|
# On the mailbox problem
Uri Abraham and Gal Amram
Departments of Mathematics and Computer Science,
Ben-Gurion University, Beer-Sheva, Israel
###### Abstract
The Mailbox Problem was described and solved by Aguilera, Gafni, and Lamport
in [4] with an algorithm that uses two flag registers that carry 14 values
each. An interesting problem that they ask is whether there is a mailbox
algorithm with smaller flag values. We give a positive answer by describing a
mailbox algorithm with 6 and 4 values in the two flag registers.
## 1 Introduction: the mailbox problem
The Mailbox Problem is a theoretical synchronization problem that arises from
analyzing the situation in which a processor must cater to occasional requests
from some device. The problem, as presented (and solved) in [4] requires the
implementation of three operations: deliver, check, and remove. The device
executes a deliver operation whenever it wants to get the processor’s
attention, and the processor executes from time to time check operations to
find out if there are any unhandled device requests. After receiving a
positive answer for its check operation the processor executes a remove
operation to find-out the nature of the request and to clear the interrupt
controller. It is required that a check operation $C$ returns a positive
answer if and only if the number of deliver occurrences that precede $C$ is
strictly greater than the number of remove operations executed before $C$. The
Mailbox Problem is to design a deliver/check/remove algorithm in which the
check operation is as efficient as possible, namely that it employs bounded
registers (called “flags”) that are as small as possible.
In [4] the problem is presented first informally by means of a story involving
two processes, a postman (which is the device) and a home owner (the
processor), in which the postman delivers its letters, and the owner removes
them one by one every time she approaches the mailbox. The problem is to find
an algorithm that ensures that the home owner approaches her mailbox if and
only if it is nonempty. The check function tells the home-owner whether the
mailbox is empty or not, and she approaches her mailbox only after receiving a
“nonempty” response from a check execution. As noted in [4], depending on the
assumptions made on the communication between the device and processor the
mailbox problem can be extremely easy or surprisingly difficult. The following
very easy solution (figure 1) shows that if the homeowner process can read an
unbounded register then the mailbox problem becomes trivial. In this unbounded
algorithm the postman adds its letter to $Q$ (the queue of requests), and then
it writes on its D_num register the number of letters so far added. The home-
owner, in executing her check operation, reads register D_num to know how many
letters were deposited, and determines the number of messages removed so far
by consulting her remove-number local variable $rn$, and then she concludes
that the mailbox is nonempty if the number of letters deposited exceeds the
number of letters removed.
deliver $(letter)$: 1 add letter to $Q[\mbox{\it dn}]$; 2 $\mbox{\it
dn}:=\mbox{\it dn}+1$; 3 $\mbox{D\\_num}:=\mbox{\it dn}$; check$()$: 1
$dn:=\mbox{D\\_num}$; 2 if $dn>rn$ then $rn:=rn+1$; return true else return
false; ————————————— remove( ) 1 remove the letter from $Q[rn]$;
Figure 1: The unbounded Mailbox Algorithm. Local variable $dn$ of the postman,
and local variable $rn$ of the home-owner are initially 0.
Another easy solution to the mailbox problem can be obtained with stronger
communication objects. For example, a simple algorithm is suggested in [4] in
which the postman and home-owner employ a flag at the mailbox. The postman can
atomically (in a single step) deliver mail to the box and raise the flag, and
the owner atomically removes mail from the box and lowers the flag. The
mailbox problem becomes highly non-trivial when limitations are imposed on the
communication devices. Specifically, Aguilera et al. require in [4], for
efficiency reasons, that the mailbox solutions use only the simplest possible
means, and the check operation (which is possibly invoked at higher frequency)
should access only a bounded register. As formulated in [4], the mailbox
problem asks for solutions that satisfy the following requirements111In an
interesting note in his list of publications home-page, Lamport tells that
when he first thought about this problem he believed it has no solution under
these requirements.:
1. 1.
Only registers with read/write actions can be employed.
2. 2.
Whereas the deliver and remove operations are allowed unbounded registers, the
home-owner can only read bounded value registers in check operation
executions.
3. 3.
Moreover, in her check operations the home-owner cannot use persistent local
variables, that is variables that retain their values from one invocation of
the operation to the following one.
4. 4.
The algorithms for the three operations (deliver, check, and remove) are
bounded wait-free.
A solution is presented in [4] in which each of the two processes uses
unbounded and bounded registers (the bounded registers are called ‘flags’) and
the check operation (as required) decides on the value to return by reading
only the bounded flag registers. The algorithm of [4] needs 14 values in each
of the two flag registers, and a question is posed there if leaner solutions
exist. We give a positive answer here by describing an algorithm in which the
flag registers of the postman and the home owner carry 6 and 4 values in each
of the flag registers; that is 10 values in total as opposed to 28 values in
[4].
We shall describe now in more details and greater formality the mailbox
problem of [4]. The mailbox problem assumes two serial processes, a postman
process and a home-owner process, and their mission is to implement three
operations: deliver(), check(), and remove(). deliver() takes a letter as
parameter, check returns a boolean value, and remove() returns a letter. It is
required that the algorithm is bounded wait-free, which means that each
operation completes before the process executing it has taken $k$ (atomic)
steps, for some fixed constant $k$, irrespectively of what steps the other
processes take.
The postman and the home-owner are serial processes which operate
concurrently. The postman executes forever the following routine: he gets a
letter $\ell$ and (if the letter is addressed to the home owner) he executes
the deliver$(\ell)$ operation which adds the letter to the owner’s mailbox. So
it is quite possible that the total number of deliver operations is finite.
The home-owner process executes forever the following routine:
$\begin{array}[]{l}{\sf repeat}\\\ \ v:=\mbox{\it check}()\\\ {\sf until}\
v=\mbox{\sf true};\\\ \ {\it remove}().\end{array}$ (1)
Thus the check operations are executed ad infinitum, although it is possible
that only a finite number of them are positive (return the value true).
The safety property is expressed in [4] by first stating its sequential
specification, and then requiring that a linearization exists which satisfies
this sequential specification. This is the well-known approach to
linearizability as defined by Herlihy and Wing in [6]. The following is the
formulation in [4] for the sequential specification:
> If the owner and postman never execute concurrently, then the value returned
> by an execution of check is true if and only if there are more deliver than
> remove executions before this execution of check.
To this specification we add the obvious requirement that a queue is
implemented, namely that the letters removed are those delivered, and that the
letters are removed in the order of delivery. The original mailbox paper [4]
mentions no queues in its algorithms because its authors decided to
concentrate on the coordination problem222In section 2.1 of [4] we read: “The
remove and deliver procedures are used only for synchronization; the actual
addition and removal of letters to/from the mailbox are performed by code
inserted in place of the comments. Since it is only the correctness of the
synchronization that concerns us, we largely ignore those comments and the
code they represent”.. We prefer however to put the queue in the foreground,
since it seems that the requirement that the home-owner receives the messages
of the postman (the device) and receives them in order is important for the
functionality of the system.
We sum-up the requirements of a linear mailbox in Figure 2.
1. The events are partitioned into deliver, check, and remove events, and are totally ordered by $\prec$ in the order-type of the natural numbers (if not finite). 2. For every check event $C$, $\mbox{\it Val}(C)\in\\{\mbox{\sf true},\mbox{\sf false}\\}$. For every remove event $R$ there is a check event $C$ such that $\mbox{\it Val}(C)=\mbox{\sf true}$, $C\prec R$ and there is no check or remove event $X$ with $C\prec X\prec R$. 3. For every check event $C$ let the removal number, $\mbox{\it removal\\_num}(C)$, be the number of remove events $R$ with $R\prec C$, and let $\mbox{\it deliver\\_num}(C)$ be the number of deliver events $D$ such that $D\prec C$. Then $\mbox{\it Val}(C)=``\mbox{\it removal\\_num}(C)<\mbox{\it deliver\\_num}(C)",$ that is to say the boolean value of $C$ is true iff the number of deliver events that precede $C$ exceeds the number of letters that were removed by remove events that precede $C$. 4. If $D_{1}\prec D_{2}\cdots$ and $R_{1}\prec R_{2}\cdots$ are the enumerations in increasing order of the deliver and of the remove events, then for every $i$ the letter removed by $R_{i}$ is the letter delivered by $D_{i}$.
Figure 2: Linear mailbox specification.
As for the liveness requirements, [4] requires that the algorithm is bounded
wait-free, which means (see [7] under the term loop-free, or [5]) that each
operation completes before the process executing it has taken $k$ steps, for
some fixed constant $k$.
For communication, the Mailbox Problem as formulated in [4] requires atomic
single-writer registers (shared variables). Recall that a register is serial
if its read/write events are totally ordered (by the precedence relation) and
the value of any read action is equal to the value of the last write action
that precedes it. A register is atomic if its read/write actions are
linearizable into a serial register. That is, the partially ordered precedence
relation has an extension into a total ordering so that the resulting register
is serial. In this paper we assume that all registers are serial. This
simplifies somewhat the presentation of the correctness proof because we do
not have to speak about extending the partial order into a linear one, but it
evidently does not limit the applicability of our algorithm which works as
well with atomic registers.
For any serial register R we define a function $\omega$ over the read actions
of register R, such that for any read $r$, $\omega(r)$ is the last write
action on R that precedes $r$. That is, $\omega(r)<r$ and there is no write
action $w$ on R with $\omega(r)<w<r$. Then $r$ and $\omega(r)$ have the same
value: $\mbox{\it Val}(r)=\mbox{\it Val}(\omega(r))$. (To ensure that
$\omega(r)$ is defined on all read actions, we have to assume an initial write
event that precedes all read events.)
As we have said, the mailbox algorithm uses both unbounded and bounded
registers, but the check operation can access only the bounded registers.
Following [4] the bounded registers are called “flags”, and so we have the
postman flag which we call $F_{P}$ and the home-owner flag which we call
$F_{H}$. The check operation only reads these flag registers (and contains no
write on any register).
An additional “access restriction” is made in [4] for efficiency’s sake which
requires that the check operation uses no persistent private variables in a
check operation. Namely, the owner’s decision on whether to approach the
mailbox or not should depend just on her readings of the $F_{P}$ and $F_{H}$
values and not on any internal information sustained from some previous
operation. While one may argue that a small persistent variable would not harm
the efficiency of the check operations, keeping the access restriction allows
a comparison of the different mailbox algorithms (which obey the same
restrictions). In fact, if we allow a persistent variables into our check
algorithm, then the algorithm would need just one postman flag register of 6
values and a boolean flag for the home-owner.
## 2 The 6/4 mailbox algorithm
In this section we define in Figure 4 a mailbox algorithm with 6 and 4 values
in its two bounded flag registers $F_{P}$ and $F_{H}$. The algorithm uses only
serial registers. Registers D_num, $T_{P}$ and $F_{P}$ are written by the
postman process, and registers R_num, $T_{H}$ and $F_{H}$ are written by the
home-owner. Both processes can read these registers, but the check procedure
only reads the bounded registers: $F_{H}$, $T_{H}$, $F_{P}$ and $T_{P}$.
Registers D_num and R_num are unbounded (they carry natural numbers). The
bounded registers of the postman process, namely $T_{P}$ and $F_{P}$, are
collectively its flag register. Since register $T_{P}$ carries two values and
register $F_{P}$ three values, the combined flag of the postman carries six
values. The bounded registers of the homeowner process, $T_{H}$ and $F_{H}$,
are both boolean, so that there are four values in these two registers which
are the flag of the homeowner.
registers | type | initially
---|---|---
of the postman: | |
D_num | natural number | $0$
$T_{P}$ | $\\{0,1\\}$ | $0$
$F_{P}$ | $\\{0,1,2\\}$ | $2$
of the homeowner: | |
R_num | natural number | $0$
$T_{H}$ | $\\{0,1\\}$ | $0$
$F_{H}$ | Boolean | false
Figure 3: Registers, their types and initial values.
We describe the data structures of the different registers in figure 3.
$F_{P}$ values for example are in $\\{0,1,2\\}$. The initial values of the
registers is also defined in this figure. The initial value of the $F_{P}$
register for example is $2$.
In addition to the registers, we have the FIFO queue $Q$ which supports two
operations: addition of a letter (executed by the postman process), and
removal of a letter (executed by the home owner when $Q$ is nonempty). $Q$ is
initially empty.
The local variables of the algorithm are as follows. (Variables with
unspecified initial values can take any initial value.)
Local variables of postman:
$dn$ is a natural number, initially $0$. $rn$ is a natural number, and $t$ is
in $\\{0,1\\}$.
Local variables of home-owner:
Procedure check uses variable $fh$ (Boolean), $th$ and $tp$ (in $\\{0,1\\}$),
and fp (in $\\{0,1,2\\}$). The remove procedure uses $rn$ and $dn$ that are
natural numbers, and $t\in\\{0,1\\}$. Initially $rn=0$. Local variables of the
postman process are obviously different from those of the home-owner even when
they have the same name.
deliver $(letter)$: 1 add letter to $Q$; 2 $\mbox{\it dn}:=\mbox{\it dn}+1$;
$\mbox{D\\_num}:=\mbox{\it dn}$; 3 $t:=\mbox{$T_{H}$}$; 4
$\mbox{$T_{P}$}:=1-t$; 5 $rn:=\mbox{R\\_num}$; 6 if $rn<dn$ then
$\mbox{$F_{P}$}:=1-t$ else $\mbox{$F_{P}$}:=2$; check$()$: 1 $\mbox{\it
fh}:=\mbox{$F_{H}$}$; if fh return true; 2 $\mbox{\it th}:=\mbox{$T_{H}$}$ ; 3
$\mbox{\it tp}:=\mbox{$T_{P}$}$; 4 $\mbox{\it fp}:=\mbox{$F_{P}$}$; 5 return
$\mbox{\it tp}\not=\mbox{\it th}\ \wedge\ \mbox{\it fp}=\mbox{\it tp}$;
remove( ) 1 remove one letter from $Q$; 2 $\mbox{\it rn}:=\mbox{\it rn}+1$ ;
$\mbox{R\\_num}:=\mbox{\it rn}$; 3 $t:=\mbox{$T_{P}$}$ ; 4
$\mbox{$T_{H}$}:=t$; 5 $dn:=\mbox{D\\_num}$ ; 6 $\mbox{$F_{H}$}:=``\mbox{\it
rn}<dn"$;
Figure 4: The 6/4 Mailbox Algorithm.
In order to ensure that the pseudocode of figure 4 is well-understood, we
shall go over some of its instructions, make some simple definitions (that
will be used later), and then we shall explain intuitively some of the main
ideas of the algorithm.
A deliver operation execution $D$ is an execution of lines 1–6 of that code.
It is a high-level event, namely the set of lower-level actions which are the
executions of the code instructions. Any deliver execution is invoked with
some letter parameter, and the first line of the code is an enqueue operation
in which this letter is added to $Q$ (the mailbox queue).
Variable $dn$ (the delivery number) is initially $0$, so that if $D$ is the
$i$’th deliver operation execution ($i=1,2,\ldots$) and $dn(D)$ denotes the
value of $dn$ after line 2 is executed in $D$, then $dn(D)=i$. Register D_num
thus contains the current delivery number.
We shall use this sort of notation $dn(D)$ for other variables as well. We
note that in our algorithms any local variable is assigned a value in a unique
instruction. So if $v$ is a local variable and $E$ some operation execution
that assigns a value to $v$, then the notation $v(E)$ for that value that $E$
assigns to $v$ is meaningful and well defined. Likewise, if $G$ is any
register such that $E$ contains a write into $G$ then we denote with $G(E)$
the value of that write. Again, since any operation execution contains at most
one write action into any register, this notation is well defined.
In line 3, register $T_{H}$ is read into variable $t$ and then the opposite
value is written onto register $T_{P}$. So, the postman is always changing the
color obtained from the homeowner process, while the homeowner always copy the
value obtained (see lines 3 and 4 in the remove code).
In executing line 5, register R_num is read into local variable $rn$, and in
line 6 condition $rn<\mbox{\it dn}$ is checked. If it holds then $1-t$ is
written in $F_{P}$, but otherwise the value $2$ is written. So $2$ is an
indication that the mailbox is empty.
There are two sorts of check operations. A “short” check $C$ is one that
returns true immediately after line 1 is executed. In this line, the homeowner
process reads her own register $F_{H}$ and returns true if that register’s
value is true. Note that line 1 is the only place in the algorithm where this
register is read, and hence the register is in fact dispensable and a local
homeowner variable could replace it. The access restriction however prohibits
persistent variables, and hence the need for this register which does nothing
more than replacing a persistent local variable.
A “longer” check $C$ is one in which all lines 1 to 5 are executed. In lines
2, 3, 4 registers $T_{H}$, $T_{P}$ and $F_{P}$ are read, and the value that
$C$ returns is a conjunction of two statements that involve tp, th, and fp.
Note that $T_{P}$ and $F_{P}$ are registers of the postman process, but th is
the value of register $T_{H}$ that the previous remove operation determined or
else is the initial value of that register (which is $0$) in case $C$ has no
previous remove operation.
A remove operation is an execution $R$ of lines 1–6 of the remove code. First
a letter is dequeued (and we have to prove that the queue is nonempty when
this instruction is executed) and then the current removal number $rn(R)$ is
written on register R_num. For any remove operation execution $R$, $rn(R)$ is
the value of variable $rn$ after line 2 is executed in $R$. We have already
noted that this notation is well defined since $rn$ is assigned a value in $R$
only at the execution of line 2. It follows that $rn(R)$ is equal to $i$ where
$R$ is the $i$-th remove operation execution. In lines 3 and 4 the homeowner
copies the value read in register $T_{P}$ into register $T_{H}$. Register
D_num is then read into dn (line 5) and the boolean value $``rn<\mbox{\it
dn}"$ is written in register $F_{H}$.
The differentiation between a short and longer check operations reflects a
main idea of the algorithm, namely that if the homeowner realizes in executing
remove operation $R$ that $``rn<\mbox{\it dn}"$ (namely that the queue is
nonempty), then no subsequent postman operations can change this fact, and
hence the first check operation that comes after $R$ can rely on this
information and return true in a short execution.
There are two or three main ideas that shape our mailbox algorithm. The first
one (very roughly speaking) is that the inequality of registers $T_{P}$ and
$T_{H}$ indicates a nonempty queue. Initially both registers are $0$, and in
any deliver operation the postman reads $T_{H}$ and writes in $T_{P}$ a
different value, thus indicating that the mailbox is nonempty. The homeowner
cancels this indication in any remove operation, but the equality of the
values of registers $T_{P}$ and $T_{H}$ is not an assurance that the queue is
empty. For example, after several letters were deposited, the homeowner
removes a single letter, leaving the two registers with equal value, and yet
the queue is still nonempty. Of course, registers R_num and D_num give an
exact estimation of the number of letters in the mailbox (namely
$\mbox{D\\_num}-\mbox{R\\_num}$), but since the check operation is not allowed
to access these unbounded registers it has to rely on the bounded registers.
The homeowner also checks the boolean value $F_{H}$ and if it is true then the
queue must be nonempty and the check operation is short in this case. (The
queue is nonempty in this case because if the previous remove operation has
established that $\mbox{D\\_num}-\mbox{R\\_num}>0$ then the mailbox is
nonempty since no remove operations were executed between the previous remove
and the present check.) If, however, $F_{H}$ is false, the homeowner needs a
more complex evidence in order to deduce that the mailbox is nonempty: the
inequality of colors $\mbox{\it tp}\not=\mbox{\it th}$, and the accordance
$\mbox{\it fp}=\mbox{\it tp}$ (which also indicates that $fp\not=2$).
An example can be useful here to explain why this condition $\mbox{\it
tp}\not=\mbox{\it th}\ \wedge\ \mbox{\it fp}=\mbox{\it tp}$ cannot be replaced
with the simpler condition $\mbox{\it tp}\not=\mbox{\it th}$. We see in figure
5 the following course of events.
1. 1.
postman execute a deliver operation $D_{1}$.
2. 2.
home-owner execute a check operation $C_{1}$, since postmanhas just delivered
a letter, $C_{1}$ is longer and positive.
3. 3.
postman starts to execute a second deliver operation D2 and execute the
commands in lines 1 and 2. It sends the letter, writes 2 into register D_num
and stops for awhile.
4. 4.
home-owner execute a remove operation $R_{1}$. This is the first remove
operation and home-owner reads in this operation 2 from $D_{n}um$ (the value
that postman wrote to $D_{n}um$ in $D_{2}$). Hence, $R_{1}$ is positive.
5. 5.
home-owner execute a check operation. Since $R_{1}$ is positive, $C_{2}$ is
short and positive.
6. 6.
home-owner execute a remove execution $R_{2}$. $rn(R_{2})=2$ and $dn(R_{2})=2$
(the value that postman wrote to D_num in $D_{2}$). Thus, $R_{2}$ is negative.
7. 7.
$P_{1}$ completes the execution of $D_{2}$, and execute the commands in line
3-6. It reads a value $c$ from $T_{H}$ (this value has been written to $T_{H}$
during the execution of R2) and writes to register $T_{P}$, $1-c$.
8. 8.
home-owner execute a check operation $C_{3}$. home-owner reads the value $c$
from $T_{H}$ (written in the execution of $R_{2}$) and reads the value $1-c$
from register $T_{P}$ (written in the execution of $D_{2}$). Since only
condition $tp\not=th$ is checked in $C_{3}$, $C_{3}$ is positive. Since there
are only two deliver events and only two remove events in this execution, and
since all of these executions precedes $C_{3}$, $C_{3}$ should be negative.
Thus, this is an incorrect execution.
$D_{1}$$C_{1}$$D_{2}(1-2)$0.2(4,3.5)(12,3.5)
$R_{1}$$C_{2}$$R_{2}$$D_{2}(3-6)$$C_{3}$
Figure 5: An example for an incorrect execution where a long check event only
checks condition $tp\not=th$.
## 3 Correctness of the algorithm
In order to prove that our algorithm implements a mailbox (as specified in
Figure 2) we need to define some functions and predicates that will serve us
in this proof. An action is an execution of an atomic instruction of the
algorithm such as a read or a write of a register or a queue action. Since we
assume that the registers are serial, and as the queue operations (to add or
remove a letter) are also instantaneous, we have a total ordering $<$ on these
actions. We write $a<b$ to say that $a$ precedes $b$ in this ordering. (A
relation $<$ is a total ordering when it is a transitive and irreflexive
relation such that for any two different members $a$ and $b$ in its domain we
have $a<b$ or $b<a$.)
An operation execution is an execution of the deliver, check, or remove
algorithm. Every operation execution is a high-level event, namely a set of
lower-level actions (also called lower-level events, as in [8]). The total
ordering $<$ on the lower-level actions induces a partial ordering on the
operation executions: for operation executions $A$ and $B$ we define that
$A<B$ if $a<b$ for every $a\in A$ and $b\in B$. It is also very convenient to
relate high-level events and lower-level actions: $A<x$ for a high-level event
$A$ and a lower-level event $x$ means that $a<x$ for every $a$ in $A$. And
similarly $x<A$ is defined when $x<a$ for every $a$ in $A$. The fact that we
use the same symbol $<$ to denote both the total ordering relation on the
actions and the resulting partial ordering relation on the high-level events
should not be a source of confusion.
The aim of the correctness proof is to define a total ordering $\prec$ on the
operation executions that extends the partial ordering $<$, and then to prove
that the specifications of Figure 2 hold.
We assume two initial high-level events $I_{p}$ and $I_{h}$ by the postman and
home-owner processes that determine the initial values of the registers
(defined in Figure 3) and the initial values of the variables. $I_{p}$
contains the initial write actions on registers D_num, $T_{P}$, and $F_{P}$,
and $I_{h}$ contains the initial write actions on registers R_num, $T_{H}$ and
$F_{H}$. These initial high-level events are concurrent. That is, it is
neither the case that $I_{p}<I_{h}$ nor that $I_{h}<I_{p}$.
If $a$ is any read/write action, then $[a]$ denotes that high level event to
which $a$ belongs. (Every low level action belongs to some operation
execution, except for the assumed initial write actions which belong to the
initial events $I_{h}$ and $I_{p}$.)
We shall name the different actions that compose the three operations.
1. 1.
Let $D$ be a deliver operation execution (which completed execution of lines
1–6 of the deliver code of Figure 4). We shall name the different actions of
$D$. First, the addition of the letter to the queue $Q$ is denoted $\mbox{\it
enq}(D)$. $D$ contains three write actions denoted $w1(D)$ $w2(D)$ and $w3(D)$
(corresponding to lines 2, 4, and 6 respectively, namely the writes on
registers $\mbox{D\\_num},\mbox{$T_{P}$}$ and $F_{P}$). $D$ contains two read
actions $r1(D)$ and $r2(D)$ (which correspond to lines 3 and 5, namely to the
reads of registers $T_{H}$ and R_num).
2. 2.
There are two sorts of check executions. A short operation $C$ is an execution
of line 1 that returns the value true. It contains a single read, denoted
$r0(C)$, of register $F_{H}$. A longer check operation is one that contains
executions of lines 1–5, and so it contains three additional read actions
denoted $r1(C)$, $r2(C)$ and $r3(C)$. $r1(C)$ is the read of register $T_{H}$,
$r2(C)$ is the read of register $T_{P}$, and $r3(C)$ is a read of register
$F_{P}$. A check operation contains no write actions.
3. 3.
A remove operation execution $R$ begins with a dequeue action on the mailbox
queue $Q$ which is denoted $\mbox{\it deq}(R)$. An important part of the
correctness proof is to prove that whenever $\mbox{\it deq}(R)$ is executed,
$Q$ is nonempty. There are two read actions in $R$, $r1(R)$ and $r2(R)$ which
correspond to lines 3 and 5. These are the reads of registers $T_{P}$ and
D_num. Then we notate the three write actions: $w1(R)$ is the write on
register R_num, $w2(R)$ is the write on register $T_{H}$, and $w3(R)$ is the
write on $F_{H}$.
If $X$ is a deliver (remove) operation execution, then $X$ contains a read
action of the R_num (respectively D_num) register. Specifically, $r=r2(X)$ is
the read of the R_num (respectively D_num) register, and then $\omega(r)$ is
the write action of that register which affected $r$. That is, $\omega(r)$ is
the last write action on register R_num (respectively D_num) that precedes $r$
(see section 1). Any action belongs to a unique higher level event, and if $Y$
is that higher level event that contains the write $\omega(r)$, then we define
$Y=\alpha(X)$.
A succinct definition of the function $\alpha$ can be given by the following
equation. For any deliver or remove operation $X$ we define
$\alpha(X)=[\omega(r2(X))].$ (2)
Recall that $[a]$ denotes the higher level event that contains action $a$. In
case $X=D$ is a deliver operation execution, $[\omega(r2(D))]$ is that high-
level event that contains $\omega(r)$, and so $\alpha(D)$ can either be an
operation execution that contains $\omega(r)$, or else the initial event
$I_{h}$ of the home-owner process in case $\omega(r)$ is the assumed initial
write.
In case $R$ is a remove operation execution, we have that
$\alpha(R)=[\omega(r2(R)]$. So if $r=r2(R)$ is the read of register D_num in
$R$, then $\omega(r)$ is the corresponding write action on that register. We
shall prove in Proposition 3.8 that $\omega(r)$ is not the initial write in
$I_{p}$, and so $D=\alpha(R)$ is a deliver operation execution and thus
$\omega(r)=w1(D)$.
The following lemma is an easy consequence of the fact that the registers (and
specifically the D_num register) are serial.
###### Lemma 3.1
If $R_{1}<R_{2}$ are two remove operations, then
$\alpha(R_{1})\leq\alpha(R_{2})$.
We say that a check operation $C$ is “positive” in case it returns the value
true. We say that it is “negative” when it returns false. Likewise, a remove
operation $R$ is positive when it writes true on its $F_{H}$ register (in
executing line 6), and it is negative when it writes false. And, again, a
deliver operation $D$ is positive if condition $``\mbox{\it rn}<dn"$ holds at
line 6 of $D$, and it is negative otherwise.
Now we define two functions, pre_rem and $\rho$, on the check events.
###### Definition 3.2
Let $C$ be any check operation execution. Define $\mbox{pre\\_rem}(C)$ as the
last remove operation execution $R$ such that $R<C$ if there is such a remove
execution that precedes $C$, and $\mbox{pre\\_rem}(C)=I_{h}$ as the assumed
initial home-owner event otherwise.
We note that a short check operation is positive, and hence a check operation
$C$ is short if and only if $\mbox{pre\\_rem}(C)$ is positive. Since the
assumed initial homeowner event is negative (as the initial value of $F_{H}$
is false), if $C$ is short then $\mbox{pre\\_rem}(C)$ is not the initial
event– it is necessarily a positive remove operation execution.
The following is a key definition in our correctness proof. It relates every
check operation $C$ to $\rho(C)$ which is the deliver operation (or initial
$I_{p}$ event) that $C$ considers in order to calculate the value (true or
false) to return.
###### Definition 3.3
For any check operation execution $C$ we define $\rho(C)$ as follows. In case
$C$ is a short check operation let $R=\mbox{pre\\_rem}(C)$ (which is a remove
operation execution as we noted) and then define $\rho(C)=[\omega(r2(R))]$. So
$\rho(C)=\alpha(\mbox{pre\\_rem}(C))$ when $C$ is short. In case $C$ is a
longer operation, define $\rho(C)=[\omega(r3(C))]$. ($r3(C)$ is the read of
$F_{P}$ in $C$.)
We note that $C<\rho(C)$ is impossible, by properties of the $\omega$ function
(namely by the fact that $\omega(r)<r$ for any read action $r$). The following
is therefore established.
###### Proposition 3.4
If $C<D$ (where $C$ is a check and $D$ a deliver operation) then $\rho(C)<D$.
###### Lemma 3.5
Suppose that $C<R$ are a check and remove operation executions. Then
$\rho(C)\leq\alpha(R)$.
Proof. If $C$ is short then $\rho(C)=\alpha(\mbox{pre\\_rem}(C))$, and since
$R^{\prime}=\mbox{pre\\_rem}(C)<C<R$, $R^{\prime}<R$ follows and so the proof
is concluded in this case with Lemma 3.1. Suppose next that $C$ is a longer
check operation and $\rho(C)=D$. Then $D=[\omega(r3(C))]$ by definition of
$\rho$. This implies that $D<r3(C)$. (Because if $D=I_{p}$ is the initial
event then $D<C$, and if $D$ is a deliver operation then the fact that the
write on $F_{P}$ is the last action in $D$ implies that $D<r3(C)$.) So $D<R$
and hence $D\leq\alpha(R)$.
We remind the reader that if $E$ is any operation execution and $x$ a variable
(or a register) whose value is assigned in $E$, then $x(E)$ denotes this
value.
For any remove operation $R$, $rn(R)$ is the value of variable $rn$ that is
determined in executing line 2 and is written on register R_num. We also set
$rn(I_{h})=0$ (and the initial value of variable $rn$ is $0$).
$rn(R)$ is called the “removal number”; it is the number of remove operations
$R^{\prime}$ such that $R^{\prime}\leq R$. Clearly, if $R_{1}<R_{2}<\cdots$ is
the sequence of remove operations in increasing order, then $rn(R_{i})=i$.
The check code does not contain a variable named $rn$, and so the number
$rn(C)$ for a check operation execution $C$ is defined directly as the number
of remove operations $R$ such that $R<C$. In other words,
$rn(C)=\\#\\{R\mid R\ \text{is a {\it remove}\ operation and }R<C\\}.$ (3)
Where $\\#A$ denotes the cardinality of the set $A$.
The “delivery number”, $dn(D)$, of a deliver operation $D$ is equal to the
number of deliver operations $D^{\prime}$ such that $D^{\prime}\leq D$. Thus
if $D_{1}<D_{2}<\cdots$ is the enumeration of the deliver operations in
increasing order, then $dn(D_{i})=i$. We also define $dn(I_{p})=0$.
It is convenient to define the “color” of operations. If $D$ is a deliver
operation, then $color(D)=\mbox{$T_{P}$}(D)=1-t(D)$. That is, $color(D)$ is
that value $c=0,1$ that is written into register $T_{P}$ when line 4 is
executed in $D$. (If condition $rn<dn$ holds in line 6, then $color(D)$ is
also the value that is written in register $F_{P}$.)
The color of any remove operation $R$ is defined by
$color(R)=t(R)=\mbox{$T_{H}$}(R)$. That is, the color of $R$ is the value read
from register $T_{P}$ and written into $T_{H}$. The color of the initial event
$I_{p}$ is $0$ which is the initial value of $T_{H}$.
If $C$ is a long check operation, then we define $color(C)=tp(C)$. That is,
the color of a long check operation is the value read from register $T_{P}$.
Note that if $C$ is a long check operation and $D=\rho(C)$, if $C$ is positive
then $D$ is positive and $color(D)=color(C)$. Indeed, $D=\rho(C)$ implies that
the value read in register $F_{P}$ in $C$ (namely $fp(C)$) is the value
written by $D$. Hence this value is not 2 (because $C$ is positive and
condition $tp=fp$ implies that $fp=0,1$). Hence $rn<dn$ holds in $D$, and
therefore $D$ is positive, and $color(D)=color(C)$ follows.
Note also that if $C$ is a long check operation and $S=\mbox{pre\\_rem}(C)$ (a
remove operation or $I_{p}$), if $C$ is positive then $color(C)\not=color(S)$.
This follows from equality $tp\not=th$ which holds at line 5 if $C$ is
positive.
We gather these observations into the following.
###### Lemma 3.6
If $C$ is a long check operation and $S=\mbox{pre\\_rem}(C)$, then $C$ is
positive iff $\rho(C)$ is positive, $color(C)\not=color(S)$, and
$color(\rho(C))=color(C)$.
Our aim now is to prove some properties of the functions and predicates that
we have defined above. These properties will be used to define a linear
ordering (total ordering) $\prec$ on the operation executions and to prove
that the properties of Figure 2 hold.
###### Lemma 3.7
Suppose that $C$ is a long check operation and $D=\rho(C)$. If
$S=\mbox{pre\\_rem}(C)$ is a remove operation such that $w2(D)<r1(S)$, then
$C$ is negative.
Proof. Let $c=color(D)$ be, as we have defined above, the value written into
register $T_{P}$, and assume for a contradiction that $C$ is positive. So
$color(C)=color(D)$ by Lemma 3.6. Since $w2(D)<r1(S)$ are a write and read
actions on register $T_{P}$, $w2(D)\leq\omega(r1(S))$.
1. Case 1.
$w2(D)=\omega(r1(S))$. This entails that $color(D)=color(S)$, and hence that
$color(C)=color(S)$ which implies by Lemma 3.6 that $C$ is negative.
2. Case 2.
$w2(D)<\omega(r1(S))$. This implies that $D<D^{\prime}$ where
$D^{\prime}=[\omega(r1(S))]$, and $\omega(r1(S))=w2(D^{\prime})$. So
$w2(D^{\prime})<r1(S)$ (as $\omega(r)<r$ for every read action $r$). Since it
is not the case that $w3(D^{\prime})<r3(C)$ (as $\rho(C)=D$), we get that
$r3(C)<w3(D^{\prime})$. Hence $w2(D^{\prime})<r1(S)<r2(C)<w3(D^{\prime})$.
This implies that $r1(S)$ and $r2(C)$ (which are both reads of register
$T_{P}$) get the same value of the write $w2(D^{\prime})$. Hence
$color(S)=color(C)$ which implies, again by Lemma 3.6, that $C$ is negative.
###### Proposition 3.8
If $C$ is a positive check operation and $\rho(C)=D$, then $D$ is a deliver
operation execution and
$rn(C)<dn(D).$ (4)
Proof. Assume first that $C$ is a short check operation and let
$R=\mbox{pre\\_rem}(C)$ be the previous remove operation, which necessarily
has set its register $F_{H}$ to be true at line 6. So $rn(R)=rn(C)$, and
inequality
$rn(R)<dn(R)$ (5)
holds. Let $r=r2(R)$ be the read of register D_num which obtained the value
$dn(R)$. By definition of $\rho(C)$ when $C$ is short,
$D=\rho(C)=[\omega(r)]$, and $dn(D)=dn(R)$ follows. Since $dn(R)>0$ follows
from (5) and as $dn(I_{p})=0$, $D\not=I_{p}$ is concluded and necessarily $D$
is a deliver operation execution and (4) follows.
Now suppose that $C$ is a longer check operation, and let $r2=r2(C)$ and
$r3=r3(C)$ be its reads of registers $T_{P}$ and $F_{P}$ (respectively). By
definition of $D=\rho(C)$, $D=[\omega(r3)]$. Then $D$ is either a deliver
operation execution (in which case $\omega(r3)=w3(D)$ is the write in register
$F_{P}$) or else is the initial event $I_{p}$ in case $\omega(r3)\in I_{p}$.
We claim that $D$ is not the initial event $I_{p}$. Indeed, the initial value
of $F_{p}$ is $2$, but as $C$ is positive condition $fp=tp$ holds in $C$,
which excludes the possibility that $fp(C)=2$ (as $tp(C)\in\\{0,1\\}$). Hence
$fp(C)=fp(D)$ is not $2$ and so $\mbox{\it rn}<dn$ is evaluated to true when
line 6 is executed in $D$. So
$rn(D)<dn(D)$
holds.
Define $R=\alpha(D)$; that is $R=[\omega(r2(D))]$. Then $rn(D)=rn(R)$. We
shall prove that $R=\mbox{pre\\_rem}(C)$. This will show that $rn(C)=rn(R)$,
and hence that $rn(C)<dn(D)$ as required. It thus remain to prove that
$R=\mbox{pre\\_rem}(C)$.
Suppose on the contrary that $R\not=\mbox{pre\\_rem}(C)$, and then
$R<\mbox{pre\\_rem}(C)$ follows (from the fact that $w1(R)<r2(D)<w3(D)<r3(C)$
which implies that $R<C$). Say $S=\mbox{pre\\_rem}(C)$. Since $\omega(r2(D))$
is in $R$, $r2(D)<w1(S)$. But $w2(D)<r2(D)$. Hence $w2(D)<w1(S)<r1(S)$ and
this implies by Lemma 3.7 that $C$ is not positive, which yields a
contradiction.
###### Proposition 3.9
If $D<C$ are a deliver and check operations such that $rn(C)<dn(D)$, then $C$
is positive.
Proof. A short check operation is always positive (it returns true), and hence
we may assume that $C$ is a longer check. Say $R=\mbox{pre\\_rem}(C)$ and then
$rn(R)=rn(C).$ (6)
Suppose first that $R=I_{h}$ is the initial event. In reading $T_{H}$, $C$
obtains the initial value $0$. We shall prove that $fp(C)=tp(C)=1$, and hence
that $C$ returns true at line 5, as required.
Define $E=\rho(C)=[\omega(r3(C))]$. Then $w3(E)<r3(C)$ (the write on $F_{P}$
in $E$ precedes the read of this register in $C$), and $D<C$ implies that
$D\leq E$. Now $\alpha(E)=I_{h}$ follows from the assumption that
$\mbox{pre\\_rem}(C)=I_{h}$. $rn(E)$ is $0$ (the initial value of R_num), but
$dn(E)>0$. So $``rn<dn"$ holds in $E$ when line 6 is executed in $E$, and
hence the value of $w3(E)$ is $1-t(E)$. But $t(E)=0$ because the initial value
of $T_{H}$ is $0$, and hence the value of $w3(E)$ is $1$. So $fp(C)=1$. The
proof that $tp(C)=1$ is very similarly obtained by taking $[\omega(r2(C))]$.
So now we assume that $R$ is a remove execution. In case $w1(D)<r2(R)$,
$w1(D)\leq\omega(r2(R))$ follows, and hence the read of D_num in $R$ obtains
the write in $D$ or a later write. Hence $dn(D)\leq dn(R)$. The fact that
$rn(R)=rn(C)$ and our assumption that $rn(C)<dn(D)$ imply that $rn(R)<dn(R)$.
So $R$ is positive and $C$ is a short positive check operation.
So we may assume that $r2(R)<w1(D)$. It follows from this assumption that
$w2(R)<r1(D)<C$. Say $c=color(R)$ (that is, by definition, the value of
$w2(R)$, which is the write in $T_{H}$).
Claim. If $E$ is any deliver operation such that $w2(R)<r1(E)<r3(C)$ (the
write on $T_{H}$ in $R$ precedes the read of $T_{H}$ in $E$ which itself
precedes the end of $C$) then $\omega(r1(E))=w2(R)$ and $color(E)=1-c$.
Proof of claim. Since $r1(E)$ is before the end of $C$ there is no write
action on register $T_{H}$ between $w2(R)$ and $r1(E)$. Hence
$w2(R)=\omega(r1(E))$. So $color(E)=1-c$.
In particular, if $E_{0}=\rho(C)$, then $D\leq E_{0}$ and the conditions of
the claim hold. (Recall that $r3(C)$ is the read of register $F_{P}$ in $C$,
and $\rho(C)=[\omega(r3(C))]$. Since $D<C$, $D\leq E_{0}$. And as the write on
$F_{P}$ is the last action in $E_{0}$, $E_{0}<r3(C)$.) Thus
$color(E_{0})=1-c$. Moreover, $R=\alpha(E_{0})$. To prove this fact note that
$w1(R)<w2(R)<r1(E_{0})<r2(E_{0})$ and $r2(E_{0})$ is before the end of $C$;
this implies that $w1(R)=\omega(r2(E_{0}))$ and hence that $rn(E_{0})=rn(R)$.
But (6) and the lemma’s assumption give $rn(R)=rn(C)<dn(D)$, and since $D\leq
E_{0}$ yields $dn(D)\leq dn(E_{0})$, condition $rn(E_{0})<dn(E_{0})$ holds.
Hence the value of $F_{P}$ that is written by $E_{0}$ is the color of $E_{0}$
which is $1-c$. Since $E_{0}=\rho(C)$, this implies that
$fp(C)=color(E_{0})=1-c$, and thus
$fp(C)=1-c.$
Condition $fp=tp$ holds in $C$ by the following argument. $tp(C)$ is the value
of the read of $T_{P}$, namely the value of $r2(C)$. Say $E=[\omega(r2(C))]$,
that is $w2(E)=\omega(r2(C))$. Since $D<C$, $D\leq E$. Also, $r1(E)$ is before
the end of $C$. As we noted in the above claim, this implies that
$color(E)=1-c$, and hence
$tp(C)=1-c.$
In view of the formula displayed above, this yields that $fp=tp$ holds in $C$.
Since $color(R)=c$, $R$ writes $c$ on $T_{H}$. But $R=\mbox{pre\\_rem}(C)$,
and so $th(C)=c$ follows. Hence condition $tp\not=th$ holds in $C$ because
$tp(C)=1-c$ but $th(C)=c$. So $C$ is indeed positive.
We are now ready to define the linear ordering $\prec$ on the deliver, check
and remove operations. We shall define first a relation $<^{*}$ that extends
$<$ on the operation executions, and then prove that $<^{\ast}$ has no cycles,
and that any linear ordering $\prec$ that extends $<^{*}$ satisfies the linear
mailbox specifications of Figure 2. This will complete the proof. We define
the relation $<^{*}$ as a union of $<$ with the relation $\lhd$ that relates
some check operations $C$ and deliver operations $D$ as follows.
$\begin{array}[]{rcl}\lhd&=&\\{\langle C,D\rangle\mid C\ \text{is negative and
}rn(C)<dn(D)\\}\\\ &&\hskip 17.07164pt\cup\\{\langle D,C\rangle\mid C\
\text{is positive and }dn(D)=rn(C)+1\\}.\end{array}$
Before we proceed we want to explain the intuition behind this definition of
$\lhd$. If $C$ is a negative check operation and $rn(C)<k$, if $D$ is the
$k$th deliver operation or a later deliver, then we surely want to have $D$
after $C$ in the linear ordering $\prec$ that we look for. (Otherwise, if $D$
is before $C$, then $C$ is required to be positive.) If, on the other hand,
$C$ is positive then among the operations that are before $C$ in the $\prec$
ordering we must have more deliver than remove operations and hence the $k+1$
deliver operation must be before $C$.
###### Lemma 3.10
If $X\lhd Y$ then it is not the case that $Y<X$.
Proof. We have to check two cases as in the definition of $X\lhd Y$.
1. 1.
Suppose first that $\langle X,Y\rangle=\langle C,D\rangle$ where $C$ is a
negative check operation and $D$ is a deliver operation such that
$rn(C)<dn(D)$. We have to prove that it is not the case that $D<C$. But if
$D<C$ then Proposition 3.9 implies that $C$ is positive.
2. 2.
Suppose next that $\langle X,Y\rangle=\langle D,C\rangle$ where $C$ is a
positive check operation and $D$ is a deliver operation such that
$dn(D)=rn(C)+1$. We have to prove that it is not the case that $C<D$. But if
$C<D$, then $D^{\prime}=\rho(C)<D$ (by Proposition 3.4) and hence
$dn(D^{\prime})<dn(D)$. So $dn(D^{\prime})\leq rn(C)$, in contradiction to
Proposition 3.8.
###### Lemma 3.11
If $C$ and $C^{\prime}$ are check operations and $D$ is a deliver operation
such that $C\lhd D\lhd C^{\prime}$, then $C<C^{\prime}$.
Proof. Since $C\lhd D$, the definition of $\lhd$ implies that $C$ is negative,
and
$rn(C)<dn(D).$
Now from $D\lhd C^{\prime}$ we get that $C^{\prime}$ is positive and
$dn(D)=rn(C^{\prime})+1$. So, firstly, we infer that $C\not=C^{\prime}$ (one
is negative and the other positive). If it is not the case that
$C<C^{\prime}$, then $C^{\prime}<C$ holds. In this case, since $C^{\prime}$ is
positive, there is a remove operation between $C^{\prime}$ and $C$, and hence
$rn(C^{\prime})<rn(C)$. So, $rn(C^{\prime})<rn(C)<dn(D)$ which is in
contradiction to $dn(D)=rn(C^{\prime})+1$.
###### Lemma 3.12
If $D$ and $D^{\prime}$ are deliver operations and $C$ a check operation, then
$D\lhd C\lhd D^{\prime}$ is impossible.
Proof. $D\lhd C$ implies that $C$ is positive but $C\lhd D^{\prime}$ implies
that it is negative.
A cycle of length $k\geq 1$ in a relation $T$ is a sequence
$X_{1},\ldots,X_{k+1}$ so that $X_{i}TX_{i+1}$ for $1\leq i\leq k$, and
$X_{k+1}=X_{1}$. We say that $X_{i+1}$ is the successor of $X_{i}$ in this
cycle.
###### Lemma 3.13
Relation $<^{*}\;=(<\;\cup\;\lhd)$ has no cycles, and hence can be extended to
a linear ordering of the operation executions.
Proof. By the definition of the union of two relations, $X<^{\ast}Y$ if $X<Y$
or $X\lhd Y$. Suppose on the contrary that there is a cycle
$X_{1}<^{*}X_{2}<^{*}\cdots<^{*}X_{n}$ of length $n\geq 1$ in the $<^{\ast}$
relation. Take such a cycle of minimal length. Since $<$ is transitive, there
are no two successive occurrences of the $<$ relation in this minimal cycle.
But it is also impossible to have two successive occurrences of the $\lhd$
relation (by lemmas 3.11 and 3.12). The cycle is not of length one, since both
$<$ and $\lhd$ are irreflexive. The cycle is not of length two (use Lemma 3.10
to see that it is not of the form $X\lhd Y<X$ or $X<Y\lhd X$).
We may assume that the cycle begins with the $<$ relation, and so it begins
$X_{1}<X_{2}\lhd X_{3}<X_{4}\cdots$. But $X_{2}\lhd X_{3}$ implies (by Lemma
3.10) that it is not the case that $X_{3}<X_{2}$. So ${\it begin}(X_{2})<{\it
end}(X_{3})$, where ${\it begin}(X)$ and ${\it end}(X)$ are the first and last
actions in $X$. Hence $X_{1}<X_{4}$ follows in contradiction to the minimality
of the cycle.
As $<^{\ast}$ has no cycles it can be extended to a linear ordering.
###### Theorem 3.14
Let $\prec$ be any linear ordering (total ordering) that extends $<^{*}$. Then
the specifications of Figure 2 hold.
Proof. For any check operation $C$ we define $\mbox{\it Val}(C)=\mbox{\sf
true}$ if $C$ is a positive, and $\mbox{\it Val}(C)=\mbox{\sf false}$ when $C$
is negative. We now check the four items of Figure 2.
1. 1.
$\prec$ is chosen to be a linear ordering that extends $<^{\ast}$, and hence
it also extends the $<$ ordering on the operation executions. We want to show
that for every operation execution $X$ the set $\\{Y\mid Y\prec X\\}$ is
finite. This is a consequence of the finiteness property of the $<$ relation
which says that for every event $X$ there is only a finite number of events
$Y$ such that $X<Y$ does not hold. Hence for all but a finite number of events
$X\prec Y$ holds.
2. 2.
If $R$ is any remove operation, then $R$ is preceded by a positive check
operation $C$. This is a requirement on how the operations are invoked, and
since the home-owner process is a serial process the two ordering $<$ and its
extension $\prec$ agree on the operations of that process, and so there is no
check or remove operation execution $X$ with $C\prec X\prec R$.
3. 3.
Recall that for every check operation execution $X$, $\mbox{\it
deliver\\_num}(X)$ and $\mbox{\it removal\\_num}(X)$ are the number of deliver
operations $D$ such that $D\prec X$, and (respectively) the number of remove
operations $R$ such that $R\prec X$. We have defined (in (3)) the number
$rn(C)$ as the number of remove operations $R$ such that $R<C$. Since the
homeowner process is serial, relations $<$ and $\prec$ coincide on the
homeowner events, and hence
$rn(C)=\mbox{\it removal\\_num}(C).$ (7)
And similarly, for any deliver $D$
$dn(D)=\mbox{\it deliver\\_num}(D).$ (8)
We have to show that
$\mbox{\it Val}(C)=``\mbox{\it removal\\_num}(C)<\mbox{\it
deliver\\_num}(C)".$ (9)
(Where $``\varphi"$ is the truth value of $\varphi$.) Consider first the case
that $C$ is negative, and assume that in contradiction to (9) $\mbox{\it
removal\\_num}(C)<\mbox{\it deliver\\_num}(C)$. Say $\mbox{\it
removal\\_num}(C)=k$. So
$k<\mbox{\it deliver\\_num}(C).$ (10)
If $D_{1}<D_{2}\cdots$ is an enumeration in increasing $<$ order of the
deliver operations, then $D_{k+1}\prec C$ (for otherwise, as $\prec$ is a
linear ordering, $C\prec D_{k+1}$ and hence
$\\{D\mid D\ \text{is a \mbox{\it deliver}\ operation and }D\prec
C\\}\subseteq\\{D_{1},\ldots,D_{k}\\}$
which implies that $\mbox{\it deliver\\_num}(C)\leq k$ in contradiction to
(10)). Yet, as $C$ is negative, $rn(C)=k$ and $dn(D_{k+1})=k+1$, the
definition of $\lhd$ dictates that $C\lhd D_{k+1}$, which is in contradiction
to $D_{k+1}\prec C$.
Consider now the case that $C$ is positive. Say $D=\rho(C)$. By Proposition
3.8, $rn(C)<dn(D)$. Hence we do have a deliver operation $D$ with
$dn(D)=rn(C)+1$. Then $D\lhd C$ and hence $D\prec C$. This shows that
$\mbox{\it deliver\\_num}(D)\leq\mbox{\it deliver\\_num}(C)$. But
$rn(C)<dn(D)$ and equations (7) and (8) show that $\mbox{\it
removal\\_num}(C)<\mbox{\it deliver\\_num}(D)$ and hence that (9) holds.
4. 4.
The fourth property of Figure 2 is that $R_{i}$ obtains the letter of $D_{i}$.
Let $C$ be that positive check operation that precedes $R_{i}$. Then
$rn(C)=i-1$. Define $D=\rho(C)$. By Proposition 3.8, $rn(C)<dn(D)$. Hence
$dn(D)\geq i$. So
$D_{i}\leq D.$
This implies that $\mbox{\it enq}(D_{i})<\mbox{\it deq}(R_{i})$ (see below)
and since this relation holds for every $i$ and as we assume that the queue
$Q$ that the algorithm employs is a fifo queue, it follows that the value
dequeued by $R_{i}$ is the value enqueued by $D_{i}$. Why $\mbox{\it
enq}(D_{i})<\mbox{\it deq}(R_{i})$? If this is not the case and $\mbox{\it
deq}(R_{i})<\mbox{\it enq}(D_{i})$, then the fact that the enqueue action is
the first in any deliver operation yields (together with $C<R_{i}$) that
$C<D_{i}\leq D$. But $C<D$ is in contradiction to $D=\rho(C)$.
## 4 A note on the proof
Our correctness proof of the linearizability of the mailbox algorithm that was
given in the previous section is clearly divided into two parts. The first
part consists in defining relations and functions such as $\alpha$ and $\rho$,
and in proving properties of the operation executions that are expressed by
means of these relations and functions. This part of the proof is extended
from Lemma 3.1 to Proposition 3.9 and it relies on the text of the algorithm.
The proof in the second part defines the linearization ordering $\prec$ and
shows that it possesses the required properties (those that are displayed in
Figure 2). In this part, the algorithm is not mentioned and only properties
established in the first part are used in an abstract way. Although the proof
of both parts was quite detailed and (we hope) convincing, we cannot claim
that it is a formal proof because something very definite is lacking which we
want to explicate. The correctness condition (linearizability) is about
executions of the algorithm, but we never defined what these executions are;
we never defined mathematical objects that represent executions and so we did
not explicate in a precise way how to formulate and formally prove theorems
about executions.
The standard way to define executions of a distributed algorithm is the
following which is based on the notions of states, steps and runs. A state is,
informally speaking, a description of the system as if frozen at a certain
moment. Formally, a state is a function that assigns values to the state
variables. Variables of our system are, for example, $PC_{p}$ (the postman
program counter) which can take any of the values in $\\{1,\ldots,6\\}$,
$PC_{h}$ (which is the homeowner program counter), $T_{H}$ (which is the
register with values in $\\{0,1\\}$) etc. If $S$ is a state and $x$ is any of
the state variables, then $S(x)$ denotes the value of $x$ in state $S$. An
initial state is a state $S$ such that $S(PC_{p})=1$, $S(\mbox{D\\_num})=0$,
and so on as in Figure 3.
A step is a pair of states $(S,T)$ that represents an execution of an (atomic)
instruction by one of the processes. So, for example, a “read of register
$T_{H}$” by the postman process is a step $(S,T)$ such that $S(PC_{p})=3$,
$T(PC_{p})=4$, $T(t_{p})=S(\mbox{$T_{H}$})$ and for any variable $x$ different
from $PC_{p}$ and $t_{p}$ $T(x)=S(x)$.
A run is defined to be a sequence of states $S_{0},\ldots$ such that $S_{0}$
is an initial state and for every $i$ $(S_{i},S_{i+1})$ is a step by one of
the processes. Runs represent executions of the algorithm.
These runs cannot support the lemmas and propositions of the first part of our
linearization proof and certainly they do not suffice for its second part,
simply because the high level events, namely the operation executions, are not
an integral part of these runs. Proposition 3.8 for example, requires the
notion of check and deliver operations, as well as the functions $\alpha$,
$dn$ and $rn$. Now, incorporating these higher level events and functions is
nothing very deep. We can simply take a run with its actions (formed by the
steps) and define sets of actions that form the operation executions. This
yields a structure that contains both actions and higher level events, and the
functions $\alpha$, $\rho$, etc. can be defined in this resulting structure as
we did in the previous section. A detailed description of this process by
which the extended run structure is obtained may be quite long, but it is
quite straightforward. In fact, there are possibly more then one reasonable
way to achieve this construction and a particular one can be found in [2] and
[1].
If we denote with $H$ some run of the system, that is some sequence of states
$H=(S_{0},\ldots)$ so that every pair $(S_{i},S_{i+1})$ is a step, and if we
let $\overline{H}$ be the resulting extended structure that contains both the
actions, the higher level operation executions and the required functions,
then all the lemmas and propositions of the first part of our proof refer to
the structure $\overline{H}$ (or more correctly to the set of all structures
$\overline{H}$ obtained from runs $H$ of the system).
Now for the second part of the proof we no longer need the actions and
references to the algorithm instructions. The structures that interest us are
those obtained by forgetting all references to lower level actions and keeping
only the higher level operation executions and the required functions and
relations that are defined over them. Let $\overline{H}$ be the extended
structure that results from a run $H$. Then we can form a structure $M$ by
keeping only the operation executions (as members of the universe of $M$), the
precedence relation $<$ over these members and all functions and predicates
that are defined over them. The resulting structure $M$ is the one on which
the second part of our linearization proof is about. $M$ is a structure in the
standard sense that is given in mathematical logic books. It is an
interpretation of some definite relational language. Any structure $M$
obtained in this way satisfies the properties that were established in
Propositions 3.8 to 3.9 and some additional obvious properties, and the second
part of the correctness proof establishes that any structure that satisfies
these properties possesses a linearization as required by the linear mailbox
specification of Figure 2. We refer to structures such as $M$ as Tarskian
system executions333This term was chosen in order to indicate that we
incorporate here the notion of system execution defined by Lamport [8] with
the work and ideas of Alfred Tarski..
A careful reader would surely not be happy with our “additional obvious
properties”, and she would rightly request a more detailed definition. What is
needed (for a careful correctness proof) is a definition of a first-order
language $L$ and a list of properties $PL$ that include not only those
enunciated by the propositions but also all those additional properties that
are required for the proof. Then the fact that the structures $M$ are detached
from the algorithm help us to check that indeed only the assumptions made in
the list $PL$ (and all of these properties) are used in the second part of the
proof. In our experience, this separation of the correctness proof into two
parts with the corresponding separation of the modeling structures helps to
improve the algorithms whose correctness we try to prove. What often happens
is that when the second part of the proof is established and it is evident
that only the properties listed in $PL$ are needed, then the algorithm itself
can be changed and improved by the designer who knows that if only these
properties of $PL$ still hold then the algorithm is correct.
To give an idea of what we have in mind for the list $PL$ we spell out in
details such a list, but we first describe the language to which the
statements of this list belong. The $L$ language is a multi-sorted language
that contains the following elements.
1. 1.
There are two sorts: Event and Number. (The role of sort Event is to represent
the operation executions, and the role of Number is to represent the set of
natural numbers.)
2. 2.
The following unary predicates are defined over Event. deliver, check, remove,
positive and negative.
3. 3.
A binary relation $<$ is defined on the Event sort. (This is called the
precedence relation.) The same symbol $<$ is also used for the “smaller than”
relation on the Number sort. The successor function $x+1$ is also assumed
here.
4. 4.
The functions $rn$ and $dn$ are defined over the Event sort and they take
Number values.
5. 5.
The function $\rho$ is defined on the Event sort and with values in this sort.
(In fact, we are only interested in $\rho(C)$ when $C$ is a positive check
event, and in this case $\rho(C)$ is a deliver event.)
The $PL$ properties are defined to be the following “axioms”. (For simplicity
we did not introduce queue events and did not relate the deliver and remove
events to the queue events.)
1. 1.
Relation $<$ is irreflexive and transitive on the Event sort, and it satisfies
the following property444This is the Russell–Wiener property which
characterizes interval orderings..
1. (a)
For every Event members $X_{1},X_{2},X_{3},X_{4}$:
$\text{if }X_{1}<X_{2},\ X_{3}<X_{4}\ \text{and }X_{2},X_{3}\text{ are
incomparable in }<,\text{ then }X_{1}<X_{4}.$
2. (b)
For every event $A$ there is a finite set of events $F$ such that if $Y$ is
any event not in $F$ then $X<Y$.
2. 2.
The deliver, check, and remove predicates are disjoint. We write $\mbox{\it
home-owner}(x)$ for $\mbox{\it check}(x)\vee{\it remove}(x)$.
3. 3.
The deliver events are linearly ordered. That is, if $\mbox{\it
deliver}(e_{1})$ and $\mbox{\it deliver}(e_{2})$, if $e_{1}\not=e_{2}$, then
$e_{1}<e_{2}$ or $e_{2}<e_{1}$.
The function $dn$ is an enumeration of the deliver events in their ordering.
That is, for every deliver event $d$, $dn(d)$ is the number of deliver events
$d^{\prime}$ such that $d^{\prime}\leq d$. (So $dn$ is one-to-one, into Number
and with values $>0$, so that for every deliver events $d_{1}$ and $d_{2}$
$dn(d_{1})<dn(d_{2})$ iff $d_{1}<d_{2}$, and if $dn(d)=k$ then for every
$1\leq j<k$ there exists some deliver $d^{\prime}$ with $dn(d^{\prime})=j$.)
4. 4.
The home-owner set of events is linearly ordered, and if $\mbox{\it home-
owner}(x)$ then $rn(x)$ is the number of remove events $r$ such that $r\leq
x$.
5. 5.
We assume an initial event $I$ and $I<e$ for any other event $e$.
6. 6.
Any check event is either positive or else negative. If $C$ is a positive
check event then there exists some remove event $R$ such that $C<R$ and there
is no home-owner event $X$ with $C<X<R$.
If $R$ is a remove event then there is some positive check $C$ such that $C<R$
and there is no home-owner event $X$ with $C<X<R$.
7. 7.
If $C$ is a positive check event and $\rho(C)=D$, then $D$ is a deliver
operation and $rn(C)<dn(D)$.
8. 8.
If $D<C$ are a deliver and (respectively) a check events such that
$rn(C)<dn(D)$ then $C$ is positive.
9. 9.
If $C<D$ are a positive check and (respectively) a deliver events, then
$\rho(C)<D$.
The last three items, 7,8 and 9, are the main properties and they were
established in propositions 3.8, 3.9 and 3.4. The reader can return now to
section 3 and re-read the second part of the proof, but now as if it were an
abstract proof about arbitrary structures that posses the nine properties
listed above. The reader can check that indeed only these properties are used
in the proof and each one serves at some point. (The argument that involves
the begin and end functions can be adapted to one the employs the
Russell–Wiener property.)
The role of the function $\rho$ is intuitively evident. If $C$ is a positive
check operation then it must be the case that $C$ relies on some deliver
operation execution $D$ that ensured $C$ that it may return true. The function
$D=\rho(C)$ gives this assurance, based on the inequality $dn(D)>rn(C)$. And
of course, we cannot expect that $C$ relies on some future event: hence
$C<\rho(C)$ is ruled out. It is not difficult to check that $\rho$ is not only
intuitively appropriate, but it is in fact necessary in the sense that if we
do have a mailbox algorithm for which a linear ordering $\prec$ exists that
satisfies the condition of Figure 2 then a function $\rho$ can be defined that
satisfies items 7 and 9.
## 5 Conclusion
In [4], Aguilera, Gafni, and Lamport define the Mailbox problem, and present a
solution in which the check operation reads two registers (the “flag”
registers) that can carry 14 values each. Moreover, they prove that there is
no solution to the Mailbox problem with two binary flags. We have presented
here a much simpler solution to the Mailbox problem with two flags that can
carry 6 and 4 values each. The gap between the impossibility of solving the
Mailbox problem with binary flags and our solution with flags that have 10
values in total is meaningful and it poses interesting theoretical questions:
to improve on the lower bound of [4], and to find a better solution to the
Mailbox problem than the one presented here.
Another problem from [4] is whether the space efficiency of the mailbox
algorithm presented in that paper can be improved. The algorithm of [4] uses
$\Theta(n\log n)$ bits of shared memory, where n is the number of executions
of deliver and remove. The authors of [4] conjecture that there is a solution
using logarithmic space, and indeed our algorithm uses two registers D_num and
R_num of width exactly $\log n$ for $n$ executions.
An interesting problem (connected with the Mailbox problem) is posed in [4]:
the bounded, wait-free Signaling problem for which [4] gives only a non-
blocking solution and leaves the wait-free problem open. The ideas developed
in this paper have contributed to a solution of the wait-free Signaling
problem which was obtained by the second author.
There are other problems around the Mailbox problem that seem to be quite
interesting. Are there solutions to the mailbox problem in which all registers
(not only the flag registers) are bounded? What solutions to the mailbox
problem can be obtained in which the flags are simple registers but the other
registers and queues can be more complex shared memory devices (for example
queues that have consensus number 2).
The last section of our paper discusses the structure of the correctness proof
and outlines a more abstract, two-stage proof in which the first stage
investigates the algorithm and the resulting behavior of the higher level
operation executions, and the second stage deals with abstract properties that
are detached from the algorithm’s text. In our experience, this division of
the correctness proof into two distinct parts has some marked benefit that
justifies further investigation. Not only that the correction proof seems
clearer in our eyes when its two parts are thus formally delineated, but the
method helps to fashion better algorithms. In developing the algorithm there
is a stage when the second part of the proof (its higher level, abstract part)
is established but the algorithm itself is not yet completely determined;
there are some features in the algorithm that can still be changed, some
actions that can be omitted, and some data structures that can be reduced.
When the designer of the algorithm has a clear and accessible aim in mind,
namely when the higher level properties that the algorithm has to ensure are
written down, then this process of improving the design of the algorithm
follows a sure path. For example, in the process of designing the mailbox
algorithm, once we understood that it suffices for the algorithm to satisfy
the nine properties listed above in order to solve the mailbox problem we
could play with changes and improvements knowing that as long as propositions
remain correct we are on the right path.
## References
* [1] U. Abraham. Models for Concurrency. Gordon and Breach, 1999\.
* [2] U. Abraham. Logical Classification of Distributed Algorithms (Bakery Algorithms as an example). Theor. Comput. Sci. 412(25): 2724-2745 (2011)
* [3] M. K. Aguilera, E. Gafni, L. Lamport. The Mailbox Problem (Extended Abstract). Disk 2008
* [4] M. K. Aguilera, E. Gafni, L. Lamport. The Mailbox Problem. Distributed Computing, 23(2), pp. 113-134, October 2010.
* [5] M. P. Herlihy. Wait-free synchronization. ACM Transactions on Programming Languages and Systems, 13(1):124–149, Jan. 1991.
* [6] M. Herlihy and J. Wing. Linearizability: A correctness condition for concurrent objects. ACM Transactions on Programming Languages and Systems, 12(3):463-492, 1990.
* [7] L. Lamport. A new solution of Dijkstra’s concurrent programming problem. Communications of the ACM, 17(8):453 - 455, Aug. 1974.
* [8] L. Lamport. On Interprocess Communication, Part I: Basic formalism, Part II: Algorithms. Distributed Computing, Vol. 1, pp. 77 - 101. 1986.
|
arxiv-papers
| 2013-07-22T08:27:31 |
2024-09-04T02:49:48.257816
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Uri Abraham, Gal Amram",
"submitter": "Uri Abraham",
"url": "https://arxiv.org/abs/1307.5619"
}
|
1307.5773
|
# Bulk Majorana mass terms and Dirac neutrinos in Randall Sundrum Model
Abhishek M Iyer [email protected] Centre for High Energy Physics,
Indian Institute of Science, Bangalore 560012 Sudhir K Vempati
[email protected] Centre for High Energy Physics, Indian Institute of
Science, Bangalore 560012
###### Abstract
We present a novel scheme where Dirac neutrinos are realized even if lepton
number violating Majorana mass terms are present. The setup is the Randall-
Sundrum framework with bulk right handed neutrinos. Bulk mass terms of both
Majorana and Dirac type are considered. It is shown that massless zero mode
solutions exist when the bulk Dirac mass term is set to zero. In this limit,
it is found that the effective 4D small neutrino mass is primarily of Dirac
nature with the Majorana type contributions being negligible. Interestingly,
this scenario is very similar to the one known with flat extra dimensions.
Neutrino phenomenology is discussed by fitting both charged lepton masses and
neutrino masses simultaneously. A single Higgs localised on the IR brane is
highly constrained as unnaturally large Yukawa couplings are required to fit
charged lepton masses. A simple extension with two Higgs doublets is presented
which facilitates a proper fit for the lepton masses.
###### pacs:
73.21.Hb, 73.21.La, 73.50.Bk
1. As of today, we have no experimental indication whether neutrinos are of Majorana type or the Dirac type. On the theoretical side, models of both Dirac and Majorana type have been considered. The popular seesaw mechanism with lepton number violation in the right handed neutrino sector Mohapatra leads to small Majorana type masses for the left handed neutrinos. Dirac type neutrinos on the other hand traditionally require lepton number conservation. In most models either this conservation is imposed by hand/construction or is a residue of some larger flavour symmetry 10 ; 11 ; 12 ; 13 ; gross ; 15 ; Dienes ; 17 ; 19 ; 20 ; 21 ; 22 ; 23 ; Memenga . Conservation of global quantum numbers like lepton number is typically disfavored theoretically due to arguments based on quantum gravity and worm holes Witten . For this reason, Dirac neutrinos are considered to be some what unnatural.
One possibility could be that lepton number is violated only by Planck scale
operators. If these operators are then some how suppressed, this would
naturally pave way for Dirac neutrino masses 111It should be noted that an
alternate approach would be to consider discrete flavour symmetries which are
imposed to avoid Majorana mass terms and thus leading to Dirac neutrino masses
Aranda:2013gga .. In four dimensions the impact of the lepton number violation
at the Planck scale is characterized by the effective operator $LH.LH/M_{Pl}$
at the weak scale. This leads to corrections to the neutrino mass matrix
$\sim~{}\mathcal{O}(10^{-3})~{}\text{eV}$ if one assumes $\mathcal{O}(1)$
coefficients. In higher dimensions explicit constructions can be done with
specific Planck scale lepton number violating operators and their impact on
weak scale physics can be studied. In fact in a particular example presented
in planckgher it has been shown that lepton number violation at the Planck
scale can almost be hidden from weak scale physics.
In this case, a Randall-Sundrum (RS) RS setup with two branes located at the
two orbifold fixed points is considered. The two fixed points, located at the
$y=0$ and $y=\pi R$ are identified with the $UV\sim M_{Pl}$ and the
$IR\sim\text{TeV}$ scales respectively. The line-element for the RS background
is given as
$ds^{2}=G_{MN}dx^{M}dx^{N}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$
(1)
where $\sigma(y)=k|y|$. Fermions and gauge bosons are allowed to propagate in
the bulk while the Higgs is localized on the IR brane It has been shown that
in the limit when the right handed neutrinos are highly IR localized, whereas
the Majorana mass terms are localised on the Planck brane, lepton number
violating effects in effective neutrino mass matrix in four dimensions are
highly suppressed. Neutrino masses can be of Dirac type by localised operators
on the IR brane planckgher . The main idea here being that the geometric
‘separation’ of the fields and the lepton number violating operators leads to
suppressed effects of the latter in the effective neutrino mass matrix. In the
present letter we present a novel way of realizing Dirac masses in the same RS
setup. We will consider dimensionful Majorana mass terms as lepton number
violating operators. In particular we show that the lepton number violating
operators need not be localized on the UV brane but instead can be present in
the bulk. The magnitude of the operators can be as large as the Planck scale.
The 4D neutrinos are almost Dirac like, in the limit when the bulk Dirac mass
terms for the right handed neutrinos are set to zero. Interestingly this case
is very similar to the case discussed in flat extradimensions.
2. Extra space dimensions offer a new way of looking at the fermion mass hierarchy in the SM. Fermion bulk wave-functions are ‘split’ and are localized at different points in the extra-dimension ArkaniHamed . The point of localization is determined by the bulk Dirac mass parameters introduced separately for the left and right components of the 5D matter fields. The overlap of the zero mode wave-functions with the Higgs field determines the effective four dimensional Yukawa coupling of the fermion. In the Randall-Sundrum setup, where the bulk geometry is warped, localization of the fermions is natural; the point of localization is again determined by the bulk Dirac mass terms. Neutrinos are however different from other matter fields as they allow both Dirac and Majorana mass terms. The profile of the zero mode crucially depends on the interplay between these two terms and the boundary conditions one chooses.
We now give a brief review of bulk (right handed) neutrino fields in the RS
set up. This case has been discussed in several papers warpedseesaw ; a ; b ;
c ; d ; planckgher ; perez ; Watanabe ; iyer . The leptonic and the quark part
of the action is given by
$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle S_{Kinetic}+\int
d^{4}x\int
dy\sqrt{-g}\left[\frac{1}{2}\left(m_{M}\bar{N}N^{c}+\text{h.c.}\right)+\ldots\right.$
$\displaystyle+$
$\displaystyle\left.\left(Y_{N}\bar{L}\tilde{H}N+Y_{E}\bar{L}EH+\ldots\right)~{}\delta(y-\pi
R)\right]$ $\displaystyle S_{Kinetic}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\int
dy~{}\sqrt{-g}~{}\left(~{}\bar{Q}(i\not{D}-m_{Q})Q+\bar{u}(i\not{D}-m_{u})u+\bar{d}(i\not{D}-m_{d})d\right.$
(2) $\displaystyle+$
$\displaystyle\left.\bar{N}(i\not{D}-m_{N})N+\bar{L}(i\not{D}-m_{L})L+\bar{E}(i\not{D}-m_{E})E~{}\right).$
where the covariant derivative is defined ass
$D_{M}=\partial_{M}+\Omega_{M}+\frac{ig_{5}}{2}\tau^{a}W_{M}^{a}(x,y)+\frac{ig^{\prime}}{2}Q_{Y}B_{M}(x,y)$
(3)
with $\Omega_{M}=(-k/2e^{-ky}\gamma_{\mu}\gamma^{5},0)$ being the spin
connection and $Q_{Y}$ is the hypercharge. $M$ is the five dimensional Lorentz
index. In the above, $N$ ( $E$ ) are the neutrino (charged lepton ) singlet
fields, $L$ are the lepton doublet fields and the Higgs field, is denoted by
$H$ with $\tilde{H}=i\sigma_{2}H^{*}$. Generation indices have been
suppressed. The bulk mass parameters for the $N$ fields are $m_{M}$ ($m_{N}$)
for the Majorana (Dirac) type. Here $N^{c}=C_{5}\bar{N}^{T}$ with $C_{5}$
being the five-dimensional charge conjugation matrix222$C_{5}$ is taken to be
$C_{4}$.. $m_{L}$($m_{E}$) stands for the bulk (Dirac type) mass terms of the
doublet (singlet) fields. All the bulk mass parameters are expressed in terms
of the so called $c$ parameters, for example, $m_{M}=c_{M}k$, with $k$ being
the reduced Planck scale. Similarly, $m_{N}$ = $c_{N}k$ etc333In the
following, we will consider all the mass parameters to be real.. $Y_{N,E}$ are
the Yukawa parameters with mass dimensions, $[Y]=-1$. Finally, let us note
that in the above action we assumed the Higgs field to be localized on the IR
brane.
The bulk fields can be Kaluza-Klein (KK) expanded in terms of their four
dimensional fields and profiles in the fifth direction. For the discussion
relevant here, we consider the expansions of $N$ and $L$ fields as:
$\displaystyle N(x,y)=\sum_{n=0}^{\infty}{e^{2\sigma(y)}\over\sqrt{\pi
R}}\left(N^{(n)}_{1}(x)g_{1}^{(n)}(y)+N^{(n)}_{2}(x)g_{2}^{(n)}(y)\right)$
$\displaystyle L(x,y)=\sum_{n=0}^{\infty}{e^{2\sigma(y)}\over\sqrt{\pi
R}}\left(L^{(n)}_{L}(x)f_{L}^{(n)}(y)+L^{(n)}_{R}(x)f_{R}^{(n)}(y)\right)$ (4)
where $N_{1}^{(n)}(x)$ and $N_{2}^{(n)}(x)$ are the two Weyl components of the
neutrino singlet field with $g_{1}^{(n)}(y)$ and $g_{2}^{(n)}(y)$ representing
their profiles in the $y$ direction444We will specify the $Z_{2}$ properties
of these components separately for each case we consider.. Similarly, for the
$L$ field, $L_{L}^{(n)}(x)$ and $L_{R}^{(n)}(x)$ represent the Weyl components
along with $f_{L}^{(n)}(y)$ and $f_{R}^{(n)}(y)$ represent the respective
profiles. The profiles can be derived from the action after imposing the
ortho-normality conditions. For the KK modes of the $N$ field, the profiles
are the solutions of the following couple differential equations warpedseesaw
$\displaystyle(\partial_{y}+m_{N})g_{1}^{(n)}(y)=m_{n}e^{\sigma}g_{2}^{(n)}(y)-m_{M}g_{2}^{(n)}(y)$
$\displaystyle(-\partial_{y}+m_{N})g_{2}^{(n)}(y)=m_{n}e^{\sigma}g_{1}^{(n)}(y)-m_{M}g_{1}^{(n)}(y)$
(5)
A crucial point to note is that zero mode solutions, $m_{n}=0$ in the set of
equations in Eq.(5), are not consistent with the boundary conditions at the
orbifold fixed pointswarpedseesaw . Solutions however do exist for higher
modes and they can be obtained numerically. A detailed phenomenological
analysis can be found in iyer .
3. Let us now revisit the result of Ref.planckgher where Dirac neutrinos are realized in the above RS setup with Majorana operators. Zero mode solutions for Eq.(5) are however possible if the Majorana mass terms are localized on the UV or IR boundary. For the case where they are confined to the UV boundary, the bulk eigenvalue equations for the $N$ fields in Eq.(5) simply reduce to
$\displaystyle(\partial_{y}+m_{N})g_{1}^{(n)}(y)=m_{n}e^{\sigma}g_{2}^{(n)}(y)$
$\displaystyle(-\partial_{y}+m_{N})g_{2}^{(n)}(y)=m_{n}e^{\sigma}g_{1}^{(n)}(y)$
(6)
Analytical solutions can easily be derived for $m_{n}=0$. Lets consider
$N_{1}$ component to be even under the $Z_{2}$ symmetry and $N_{2}$ component
to be odd. The profile of $N_{1}^{(0)}=N_{1}$ is given by
$g_{1}^{(0)}(y)=g_{1}(y)=\mathcal{N}_{N}e^{-c_{N}ky}$, where as the $Z_{2}$
odd field, $N_{2}$, has no zero modes. The normalization factor
$\mathcal{N}_{N}$ is given as
$\mathcal{N}_{N}=\sqrt{\frac{0.5-c_{N}}{\epsilon^{2c_{N}-1}-1}}$ (7)
where $\epsilon=e^{-kR\pi}$. For a regular RS setup $kR\sim 11.4$, which
implies $\epsilon\sim 10^{-16}$. It should be noted that profiles of the zero
modes of $L,E$ fields also carry the same form. The zero mode of $N_{1}$ field
picks up a Majorana mass due to the localised term at the UV boundary given
by:
$\displaystyle m_{N^{(0)}}$ $\displaystyle\sim$ $\displaystyle
m_{M}g_{1}(0)^{2}$ (8)
Assuming $c_{N}<0.5$ and $m_{M}\sim k$, the above equation becomes
$\displaystyle m_{N^{(0)}}$ $\displaystyle\sim$ $\displaystyle
k~{}(0.5-c_{N})e^{-(1-2c_{N})kR\pi}$ (9) $\displaystyle\sim$ $\displaystyle
1~{}\text{TeV}~{}\epsilon^{-2c_{N}}$
where $k\epsilon\sim 1~{}\text{TeV}$. To analyze the neutrino mass matrix, we
should also consider the IR brane localised terms, the second line of Eq.(Bulk
Majorana mass terms and Dirac neutrinos in Randall Sundrum Model), generated
from the Yukawa interaction. These are Dirac mass terms and are given by
$m_{D_{\nu}}^{(0,0)}=\frac{v}{\sqrt{2}}g_{1}(\pi R)Y_{N}^{\prime}f_{L}(\pi R)$
(10)
where the $\mathcal{O}$(1) parameter $Y_{N}^{\prime}=2kY_{N}$ and
$f_{L}^{(0)}=\mathcal{N}_{L}e^{-c_{L}ky}$ denotes the zero mode profile of the
doublet. The resultant neutrino mass matrix (with one KK mode ) has Type-I
seesaw structure. In the basis $\eta^{T}=\\{{\nu_{L}^{(0)},N_{1}^{(0)}}\\}$,
the Majorana mass to the lowest order is given as
$\mathcal{L}_{m}=-{1\over
2}\eta^{T}\mathcal{M}_{\nu}\eta\;\;\;;\;\;\;\;\mathcal{M}_{\nu}=\begin{pmatrix}0&m_{D_{\nu}}\\\
m_{D_{\nu}}&m_{N^{(0)}}\end{pmatrix}$ (11)
where we have assumed one flavour for simplicity. From Eq.(9) we see that as
$c_{N}\rightarrow-\infty$, $m_{N^{(0)}}\rightarrow 0$ 555Note that
$c_{N}\sim-1$ is sufficient to make $m_{N^{(0)}}$ insignificant.. Note that
this limit holds while $c_{M}$ is taken to be $\mathcal{O}$(1) and the
Majorana mass terms can be $\mathcal{O}(M_{PL})$. As a result the Majorana
mass for the right handed neutrino almost vanishes. In this limit, the
eigenvalues of the neutrino mass matrix in Eq.(11) are $\pm
m_{D_{\nu}}^{(0,0)}$. This implies that the localization of the zero mode of
the neutrino singlet very close to the IR brane results in its negligible
overlap with the lepton number violating operator situated on the UV brane.
The small neutrino masses are determined entirely by the brane localized
Yukawa coupling in Eq.(Bulk Majorana mass terms and Dirac neutrinos in Randall
Sundrum Model) thus attributing a Dirac nature to the neutrinos.
4. We now discuss the alternative possibility of realizing Dirac neutrinos in the presence of lepton number violating terms. However we will assume $m_{N}=0$ in Eq.(Bulk Majorana mass terms and Dirac neutrinos in Randall Sundrum Model). Bulk flavour symmetry groups can be imposed to achieve this limit. For example consider the following bulk flavour symmetry group for leptons
$G_{lepton}\equiv SU(3)_{L}\times SU(3)_{E}\times O(3)_{N_{R}}$ (12)
The transformation of the leptonic fields under $G_{lepton}$ is given as
$\displaystyle L\sim(3,1,1)\;\;\;E\sim(1,3,1)\;\;\;\
N_{R}\sim(1,1,3)\;\;\;\;N_{L_{i}}\sim(1,1,1)~{}(i=1,2,3)$ (13)
Note that we have given different representations for the left and right
chiralities of the $N$ field. The $Z_{2}$ odd field $N_{L}$ is considered to
transform as a singlet under $O(3)$. This choice leads to a vanishing bulk
Dirac mass ($m_{N}=0$) for the singlet field. In this case, the Majorana mass
terms are no longer localised on the UV brane, but are present in the bulk. In
this limit where the bulk Dirac mass for $N$ vanishes, the solutions to Eq.(5)
are simple to obtain and are given as
$\displaystyle g_{1}^{(n)}(y)$ $\displaystyle=$
$\displaystyle\xi\sin(\frac{m_{n}e^{\sigma}}{k}-m_{M}y),$ $\displaystyle
g_{2}^{(n)}(y)$ $\displaystyle=$
$\displaystyle\xi\cos(\frac{m_{n}e^{\sigma}}{k}-m_{M}y)$ (14)
where $\xi\sim\sqrt{\pi Rk}e^{-0.5\sigma(\pi R)}$. Imposing proper boundary
conditions we can project out the $Z_{2}$ odd components on the boundary. For
example, we choose as before that the $N_{2}$ component is $Z_{2}$ odd and
$N_{1}$ is the $Z_{2}$ even component. The boundary conditions for which the
$Z_{2}$ odd part say $g_{2}$ vanishes on the boundary are given as
$\frac{m_{n}e^{\sigma}}{k}-m_{M}\pi R=(2n+1){\pi\over 2}$ (15)
where n=0,1,2…. The zero mode666In principle massless modes are also possible
by choosing $m_{M}=\frac{-p}{R}$ where $p\in\mathcal{Z}^{+}$.($n=0$, massless
solutions) can exist if $m_{M}$ takes values $m_{M}=\frac{-1}{2R}$. In this
case, we have
$\displaystyle g_{1}^{(0)}(y)$ $\displaystyle=$
$\displaystyle\xi\sin(-m_{M}y)$ $\displaystyle g_{2}^{(0)}(y)$
$\displaystyle=$ $\displaystyle\xi\cos(m_{M}y)$ (16)
This corresponds to $c_{M}=\frac{-1}{2kR}$. The masses of the higher KK modes
are determined from the boundary conditions in Eq.(15) and are given as
$m_{n}\sim nk\pi\epsilon\;\;\;;\;n=1,2,3\ldots$ (17)
Unlike the other RS fields, the KK modes for the $N$ field are regularly
spaced at $1\pi$ TeV, $2\pi$ TeV etc. This reminds us of the KK bulk fields in
Arkani-Hamed, Dimopouplos, Dvali (ADD) models ADD ; Antoniadis:1998ig . Thus
if one considers bulk Majorana mass terms instead of Dirac mass terms, the
profiles in the bulk for the zero and higher modes are periodic.
Let us consider the total neutrino mass matrix in this case. In the basis,
$\chi^{T}=\\{\nu_{L}^{(0)},N^{(0)}_{1},N^{(1)}_{1}\ldots\\}$ the neutrino mass
matrix takes the form
$\mathcal{L}_{m}=-{1\over
2}\chi^{T}\mathcal{M}\chi\;\;\;;\;\;\;\;\mathcal{M}=\begin{pmatrix}0&m_{D_{\nu}}^{(0,0)}&-m_{D_{\nu}}^{(0,1)}&m_{D_{\nu}}^{(0,1)}&\ldots\\\
m_{D_{\nu}}^{(0,0)}&0&0&0&\ldots\\\ -m_{D_{\nu}}^{(0,1)}&0&m_{1}&0&\ldots\\\
m_{D_{\nu}}^{(0,1)}&0&0&m_{2}&\ldots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\
\end{pmatrix}$ (18)
where the 4D Dirac mass for the neutrino induced on the IR brane is given as
$m_{D_{\nu}}^{(0,0)}=\frac{v}{\sqrt{2}}Y^{\prime}_{N}\sqrt{\frac{0.5-c_{L}}{\epsilon^{2c_{L}-1}-1}}\epsilon^{c_{L}-0.5}$
(19)
where $Y^{\prime}_{N}=2kY_{N}$ and
$|m_{D_{\nu}}^{(0,0)}|=|m_{D_{\nu}}^{(0,i)}|$ $\forall$ $i\geq 1$. The form of
this mass matrix is very similar to the one with bulk right singlet neutrinos
and brane localised left handed doublet lepton fields in flat extra dimensions
considered in Ref.Dienes . The analysis of Ref. Dienes can be used to find
the eigenvalues of the mass matrix (18).
If there are $n_{0}$ KK modes in the theory, the effective neutrino mass
matrix in the basis $(\nu_{L}^{(0)},N^{(0)}_{1})$, after integrating our the
KK modes is given as Dienes
$\mathcal{M}_{eff}=\begin{pmatrix}a_{0}&m_{D_{\nu}}^{(0,0)}\\\
m_{D_{\nu}}^{(0,0)}&0\end{pmatrix}$ (20)
where $a_{0}\sim-\frac{m^{2}R}{\epsilon}\ln{n_{0}}$. The neutrino mass
eigenvalues can be written as
$m_{\nu}\sim\pm m_{D_{\nu}}^{(0,0)}-\frac{m^{2}R}{\epsilon}\ln{n_{0}}$ (21)
In the limit where $a_{0}~{}\ll~{}m_{D_{\nu}^{(0,0)}}$, neutrinos are
automatically Dirac-like. The second term in Eq.(21) is the Majorana “seesaw”
like terms which is a result of integrating out heavy KK modes. In contrast
with the ADD case, this limit is natural in the RS case. In the RS case, $R$
is small, $\sim{1\over k}$, which makes the contribution from the Majorana
term negligible. This result holds even for a large $n_{0}\sim 10^{18}$. On
the other hand, in the ADD case, a very large radius, $R\sim\text{eV}^{-1}$ is
required to have a large contribution from the “seesaw” term in the limit
$n_{0}$ becomes very large. $m_{D_{\nu}}^{(0,0)}$ can be made small $\sim
m_{atm}$ by choosing $c_{L}$ values appropriately. For example, this can be
achieved by localizing the leptonic doublets close to the UV brane.
Finally let us note that the above situation can be easily generalized to
three generations with three bulk right handed neutrinos. In the next section
we discuss neutrino phenomenology in detail.
5. We simultaneously fit neutrino mass differences and charged lepton masses along with the mixing angles to the $c$ parameters and the $\mathcal{O}(1)$ Yukawa couplings. More details of the fit can be found in iyer . For the standard RS set up, $\epsilon\sim 10^{-16}$, we find that to reproduce the atmosphere neutrino mass scale, $\mathcal{O}$(0.03) eV we need a $c_{L}\sim 1.3$. We have chosen the $\mathcal{O}(1)$ Yukawa coupling $Y^{\prime}$ to be 1. Such $c$ values for the lepton doublets make it difficult to fit simultaneously the charged lepton masses777Such a situation was also encountered in the case neutrinos get their masses through higher dimensional lepton number violating operator iyer .. The charged lepton mass matrix is given as
$m_{ij}=(Y^{\prime}_{E})_{ij}\mathcal{N}_{L_{i}}\mathcal{N}_{E_{j}}\epsilon^{c_{L_{i}}+c_{E_{j}}-1}$
(22)
where $\mathcal{N}_{L,E}$ have the same form as Eq.(7) and the dimensionless
$\mathcal{O}(1)$ Yukawa coupling for a brane localized Higgs is defined as
$Y^{\prime}_{E}=2kY_{E}$. The required bulk mass parameters for the charged
lepton fields, $c_{E}$ turn out to be large and negative. This introduces a
host of other problems like non-perturbative Yukawa couplings etc. On the
other hand, changing the warp factor will not have much impact on the results.
For example, for a $\epsilon\sim 10^{-2}$, we find that $c_{L}$ values
required are even larger.
A simple solution would be to disentangle the Higgs fields responsible for
neutrino masses and charged leptons by introducing an additional Higgs doublet
as in a two Higgs doublet model. We denote the Higgs doublet generating the
Dirac neutrino masses by $H_{u}$ and the other as $H_{d}$. $H_{u}$ is
localised on the IR brane where as the $H_{d}$ is a bulk Higgs field, whose
localisation is fixed by the charged lepton masses. The Yukawa part of the
lagrangian is now given as
$L_{Yuk}\subset\int d^{4}xdy\left[\left(\delta(y-\pi
R)Y_{N}\bar{L}H_{u}N+Y_{E}\bar{L}H_{d}E\ldots\right)\right]$ (23)
As before, the neutrino masses are given by Eq.(19) with $H$ replaced by
$H_{u}$. To determine the charged lepton mass matrices with a bulk Higgs, we
briefly review the the derivation for the profile equation as well as the
Yukawa couplings for a bulk Higgs.
For a bulk scalar field in a warped background, the presence of zero modes
requires the addition of brane localized mass terms. For the bulk field
$H_{d}$, the action is given as gherghetta ; Gherghetta1
$S=\int
d^{4}xdy\sqrt{-g}\left[\partial_{M}H_{d}^{*}\partial^{M}H_{d}+\left[m_{H_{d}}^{2}+2bk\left(\delta(y)-\delta(y-\pi
R)\right)\right]|H_{d}|^{2}\right]$ (24)
where we parametrize the bulk mass as $m_{H_{d}}^{2}=ak^{2}$ with $a,b$ being
dimensionless quantities. Ideally one would expect them to be
$\mathcal{O}$(1). The zero mode profile for a bulk scalar is given as
$f_{H_{d}^{(0)}}=\sqrt{k\pi R}\zeta_{H_{d}}e^{(b-1)ky}$ (25)
where the normalization factor $\zeta_{\phi}$ is given as
$\zeta_{H_{d}}=\sqrt{\frac{2(b-1)}{\epsilon^{2(1-b)}-1}}$ (26)
The brane parameter $b$ must be tuned to be $b=2\pm\sqrt{4+a}$ to satisfy the
boundary conditions for the zero modes. $b>1(b<1)$ implies the zero mode of
the Higgs is localized towards the IR(UV) brane.
For a bulk Higgs the fundamental Yukawa couplings $Y_{E}$ have mass dimension
-1/2. After performing the KK expansion and integrating over the extra-
dimension the zero mode mass matrix for all charged leptons in general is
given as 888The down type quarks have the same form of the mass matrix as the
charged leptons while the up type quark mass matrix is similar to Eq.(22).
$m_{ij}=v_{d}(Y^{\prime}_{E})_{ij}\zeta_{H_{d}}\mathcal{N}_{{L_{i}}}\mathcal{N}_{{E_{j}}}\left(\frac{\epsilon^{(c_{L_{i}}+c_{E_{j}}-b)}-1}{b-c_{L_{i}}-c_{E_{j}}}\right)$
(27)
where we have defined the dimensionless $\mathcal{O}$(1) Yukawa coupling as
$Y^{\prime}_{E}=2\sqrt{k}Y_{E}$ and the normalization factor $\mathcal{N}_{i}$
is defined in Eq.(7). The corresponding $\mathcal{O}$(1) parameters for the up
and the down sector quarks are denoted as $Y^{\prime}_{U}$ and
$Y^{\prime}_{d}$ respectively.
The ratio of the vev of the two Higgs doublet is defined as
$tan\beta=\frac{v_{u}}{v_{d}}$. We choose $tan\beta=10$ for illustration.
While $H_{u}$ is localized on the IR brane, $H_{d}$ is localized near the UV
brane ($b<1$). Fitting Eq.(19) for small neutrino masses require $c$ value for
the doublet to be $\sim 1.3$. Corresponding to this and fitting Eq.(27) for
the charged leptons we choose $b=0.3$. We find that the electron mass can be
conveniently fit by choosing $c_{E_{R}}\sim 0.3$ while the remaining charged
leptons can be fit by choosing a range $0.4<c<1$ for the corresponding bulk
mass parameters of the singlets. Table[1] shows the range of $c$ parameters
obtained which fit the lepton masses and mixing angles. For the leptonic case
we assume normal hierarchy of neutrino mass eigenvalues. The magnitude of
$\mathcal{O}(1)$ Yukawa parameters in the leptonic sector i.e.
$Y^{\prime}_{N,E}$ were chosen to be in the range $[0.08,4]$ to fit the data.
This configuration of Higges can also accommodate quark masses by the
introduction of 9 bulk mass parameters i.e $c_{Q},c_{U},c_{d}$ in Eq.(Bulk
Majorana mass terms and Dirac neutrinos in Randall Sundrum Model). To fit the
top quark mass, the third generation top singlet require $c_{t_{R}}\leq-3.0$
while the lighter generations including the third generation doublet can be
fit by choosing the corresponding $c$ values to be $0.3<c<1$. Table[2] shows
the range of $c$ parameters for the hadronic sector which fit the quark masses
and CKM angles. The magnitude of $\mathcal{O}$(1) Yukawa parameters for the
quark sector were also chosen to lie between $0.08<|Y^{\prime}_{a}|<4$ where
$a=d,U$.
parameter | range | parameter | range
---|---|---|---
$c_{L_{1}}$ | 1.27-2.4 | $c_{E_{1}}$ | 0.19-0.36
$c_{L_{2}}$ | 1.26-1.4 | $c_{E_{2}}$ | 0.35-0.40
$c_{L_{3}}$ | 1.25-1.38 | $c_{E_{3}}$ | 0.44-0.52
Table 1: Range of $c$ parameters which fit the lepton masses and mixing angles in the bulk two Higgs doublet model with bulk Majorana masses. $\mathcal{O}$(1) Yukawa parameters are chosen to lie between $0.08<|Y^{\prime}|<4$. Normal hierarchy is assumed for the neutrino masses. parameter | range | parameter | range | parameter | range
---|---|---|---|---|---
$c_{Q_{1}}$ | 0.39-0.57 | $c_{D_{1}}$ | 0.30-0.49 | $c_{U_{1}}$ | 0.67-0.79
$c_{Q_{2}}$ | 0.47-0.55 | $c_{D_{2}}$ | 0.37-0.93 | $c_{U_{2}}$ | 0.53-0.58
$c_{Q_{3}}$ | 0.502-0.507 | $c_{D_{3}}$ | 0.68-0.97 | $c_{U_{3}}$ | $-6.8$ \- $-3.0$
Table 2: Range of $c$ parameters which fit the quark masses and mixing angles
in the bulk two Higgs doublet model with bulk Majorana masses.
$\mathcal{O}$(1) Yukawa parameters are chosen to lie between
$0.08<|Y^{\prime}|<4$.
6. Nature has not yet spoken whether neutrinos are Dirac or Majorana. While theoretically we are prejudiced to consider that Majorana neutrinos are more natural as quantum gravity does not conserve global symmetries, it is not uncommon to find examples where Dirac neutrinos can exist even with lepton number violation. In the present work, using the RS set up, we presented a scenario in which Dirac neutrinos can be obtained in the presence of lepton number violating terms in the bulk.
We have considered lepton localisation purely with bulk Majorana mass terms.
Bulk Dirac mass terms are set to zero. This case leads to periodic KK modes
similar to ADD models. Zero modes can exist for particular values of the
Majorana mass terms. In this case, the Majorana contribution can be shown to
be negligible leading to Dirac neutrinos. Phenomenologically, this model
requires large $c_{E}$ values for the bulk mass parameters which is
problematic to fit charged lepton masses. A simple extension in terms of two
Higgs doublet model is presented where a good fit to neutrino masses and
charged lepton masses is simultaneously obtained. We have not commented about
electroweak precision constraints nor flavour constraints in this model as our
focus has been purely on fermion masses. Electroweak precision constraints are
expected to be strong and one possible way out is to consider a much larger
gauge group and particle spectrum towards a custodially symmetric RS model
Agashe:2003zs . We expect that inclusion of both the Higgses should be
straightforward. On the other hand flavour is expected to be strong due to
presence of new Higgs contributions. Fortunately, one can utilise Minimal
Flavour Violation (MFV) techniques to reduce flavour violation. For the
flavour symmetry group presented in Eq.(12), the Yukawa couplings transform as
$Y_{E}\sim(3,1,\bar{3})\;\;\;\;\;Y_{N}\sim(3,1,3)$ (28)
The bulk mass parameters can be expressed in terms of the Yukawa as
Fitzpatrick
$c_{L}=I+\alpha
Y_{E}Y_{E}^{\dagger}+\alpha^{\prime}Y_{N}Y_{N}^{\dagger}\;\;\;\;\;c_{E}=\beta
Y_{E}^{\dagger}Y_{E}\;\;\;\;\;c_{N}=0$ (29)
where $\alpha,\alpha^{\prime},\beta\in\mathcal{R}$. This leads to a strong
suppression of flavour violation. A detailed analysis of flavour violation in
RS models with two Higgs doublets will be presented elsewhere ourrs2 . The
investigation of the fermion mass hierarchy in two Higgs doublets models and
the resulting implications is interesting enough to be considered in greater
detail.
Acknowledgement
SKV acknowledges support from DST Ramanujam fellowship SR/S2/RJN-25/2008 of
Govt. of India.
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|
arxiv-papers
| 2013-07-22T16:51:34 |
2024-09-04T02:49:48.278209
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Abhishek M Iyer and Sudhir K Vempati",
"submitter": "Abhishek Iyer M",
"url": "https://arxiv.org/abs/1307.5773"
}
|
1307.5782
|
11institutetext: Department of Physics and Astronomy, University of Tennessee
Knoxville, Tennessee 37996, USA 22institutetext: Oak Ridge National
Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831, USA 33institutetext:
Faculty of Physics, University of Warsaw, ul. Hoża 69, 00-681 Warsaw, Poland
44institutetext: Institut für Theoretische Physik, Staudtstr. 7 D-90158
Universität Erlangen/Nürnberg, Erlangen, Germany 55institutetext: Physics
Department, Faculty of Science, University of Zagreb, Zagreb, Croatia
# Symmetry energy in nuclear density functional theory
W. Nazarewicz 1-31-3 P.-G. Reinhard 44 W. Satuła 33 D. Vretenar 55
(Received: date / Revised version: date)
###### Abstract
The nuclear symmetry energy represents a response to the neutron-proton
asymmetry. In this survey we discuss various aspects of symmetry energy in the
framework of nuclear density functional theory, considering both non-
relativistic and relativistic self-consistent mean-field realizations side-by-
side. Key observables pertaining to bulk nucleonic matter and finite nuclei
are reviewed. Constraints on the symmetry energy and correlations between
observables and symmetry-energy parameters, using statistical covariance
analysis, are investigated. Perspectives for future work are outlined in the
context of ongoing experimental efforts.
###### pacs:
21.65.EfSymmetry energy and 21.60.JzNuclear Density Functional Theory and
21.65.CdAsymmetric matter, neutron matter and 21.10.-kProperties of nuclei
## 1 Introduction
Density Functional Theory (DFT) is a universal approach used to describe
properties of complex, strongly correlated many body systems. Originally
developed in the context of many-electron systems in condensed matter physics
and quantum chemistry (Hoh64) ; (Koh65) (also known under the name of Kohn-
Sham DFT), it is also a tool of choice in microscopic studies of complex heavy
nuclei. The basic implementation of this framework is in terms of self-
consistent mean-field (SCMF) models (Vau72) ; (Neg72) ; (Ben03) .
Extending the DFT to atomic nuclei, the nuclear DFT, is not straightforward as
nuclei are self-bound, small, superfluid aggregations of two kinds of
fermions, governed by strong surface effects. Their smallness leads to
appreciable quantal fluctuations (finite-size effects) which are difficult to
incorporate into the energy density functional (EDF). The lack of external
binding potential implies that the nuclear DFT must be necessarily formulated
in terms of intrinsic normal and anomalous (pairing) densities (Mes09) . A
density matrix expansion of the effective interaction suggests that, in
addition to the standard local nucleon density, superior EDFs should also
include more involved nucleon aggregates such as the kinetic-energy density
and spin-orbit density (Vau72) ; (Neg72) ; (Ben03)
The commonly-used single-reference SCMF methods include the local (Skyrme),
non-local (Gogny) and covariant (relativistic) approaches (Ben03) ; (Lal04) ;
(Vre05) . All these approaches are thought to be different realizations of an
underlying effective field theory (Wei99) with the ultraviolet physics hidden
in free parameters adjusted to observations. For that reason, predictions for
low-energy (infrared) physics should be fairly independent of the particular
variant used in calculations (Pug03) ; (Car08) ; (Dru10) ; (Dob12) . The
underlying EDFs are constructed in phenomenological way, with coupling
constants optimized to selected nuclear data and expected properties of
homogeneous nuclear matter.
In practice, nuclear EDFs differ in their functional form and are subject to
different optimization strategies causing that their predictions vary even
within a single family of EDFs. In particular, large uncertainties remain in
the isovector channel, which is poorly constrained by experiment. A key
quantity characterizing the interaction in the isovector channel is the
nuclear symmetry energy (NSE) describing the static response of the nucleus to
the neutron-proton asymmetry.
As discussed in this Topical Issue, the NSE influences a broad spectrum of
phenomena, ranging from subtle isospin mixing effects in $N\sim Z$ nuclei to
particle stability of neutron-rich nuclei, to nuclear collective modes, and to
radii and masses of neutron stars. Various nuclear observables are sensitive
probes of NSE, and numerous phenomenological indicators can be constructed to
probe its various aspects.
It is the aim of this contribution to analyze the relations between NSE and
measurable observables in finite nuclei. The most promising observables for
isovector properties that have stimulated vigorous experimental and
theoretical activity include neutron radii, neutron skins, dipole
polarizability, and neutron star radii. The ongoing efforts are focused on
better constraining the uncertainties concerning the equation of state (EOS)
of the symmetric and asymmetric nucleonic matter (NM) and, in particular, the
symmetry energy and its density dependence. Parameters that characterize the
NSE are not entirely independent. They are affected by key nuclear observables
in different ways. Thus it is the sine qua non of a further progress in this
area to understand the correlation pattern between NSE parameters and finite-
nuclei observables, and to provide uncertainty quantification on theoretical
predictions using the powerful methods of statistical analysis Rei10 .
A second aim is to understand the dependences from a formal perspective and to
explore the impact of configuration mixing. Within the independent particle
picture the isovector response can be described in terms of a charge-dependent
symmetry potential that shifts the neutron well with respect to the proton
average potential. The effect can be estimated quantitatively within the
Fermi-gas model (FGM) augmented by a schematic isospin-isospin interaction
(Boh69)
$V_{TT}=\frac{1}{2}\kappa\hat{\vec{T}}\cdot\hat{\vec{T}}.$ (1)
In the Hartree approximation this model gives rise to a quadratic dependence
of the NSE on the neutron excess $I=(N-Z)/A$:
$E_{\rm sym}/A=a_{\rm sym}I^{2}=(a_{\rm sym,kin}+a_{\rm sym,int})I^{2},$ (2)
as $T=|T_{z}|=|N-Z|/2$ in the ground-states of almost all nuclei. The FGM, in
spite of its simplicity, has played an important role in our understanding of
the NSE. In particular, it separates the NSE strength into kinetic and
interaction (potential) contributions, and predicts a near-equality $a_{\rm
sym,kin}\approx a_{\rm sym,int}$ of these contributions. It also provides an
estimate $a_{\rm sym}\approx 25$ MeV for the NSE coefficient (see Ref. (Mek12)
for a recent discussion).
Furthermore, we note that the SCMF approach can lead to spontaneous breaking
of symmetries. This apparent drawback can be turned into an advantage, as the
symmetry breaking mechanism allows to incorporate many inter-nucleon
correlations within a single product state or, alternatively, within a single-
reference DFT sacrificing good quantum numbers; broken symmetries have to be
restored a posteriori. We will address this topic using the example of isospin
mixing which naturally has an impact on isovector properties.
This survey is organized as follows. Section 2 outlines the SCMF approaches
and details various theoretical ingredients of the models employed in this
work. Observables pertaining to bulk NM and finite nuclei that are essential
for NSE are discussed in Sec. 3. Constraints on NSE and correlations between
observables and NSE parameters, using the statistical covariance technique,
are presented in Sec. 4. Section 5 summarizes the current status of NSE
parameters. The planned extensions of the current DFT work are laid out in
Sec. 6.1. Finally, Sec. 7 contains the conclusions of this survey.
## 2 Nuclear DFT
The nuclear EDF constitutes a crucial ingredient for a set of DFT-based
theoretical tools that enable an accurate description of ground-state
properties, collective excitations, and large-amplitude dynamics over the
entire chart of nuclides, from relatively light systems to superheavy nuclei,
and from the valley of $\beta$-stability to the nucleon drip-lines. In general
EDFs are not directly related to any specific microscopic inter-nucleon
interaction, but rather represent universal functionals of nucleon densities
and currents. With a small set of global parameters adjusted to empirical
properties of nucleonic matter and to selected data on finite nuclei (Klu09) ;
(Kor10) , models based on EDFs enable a consistent description of a variety of
nuclear structure phenomena.
The unknown exact and universal nuclear EDF is approximated by simple, mostly
analytical, functionals built from powers and gradients of nucleonic densities
and currents, representing distributions of matter, spins, momentum and
kinetic energy. When pairing correlations are included, they are represented
by pair (anomalous) densities. In the field of nuclear structure this method
is analogous to Kohn-Sham DFT. SCMF models effectively map the nuclear many-
body problem onto a one-body problem using auxilliary Kohn-Sham single-
particle orbitals. By including many-body correlations in EDF, the Kohn-Sham
method in principle goes beyond the Hartree-Fock (HF) or Hartree-Fock-
Bogolyubov (HFB) approximations and, in addition, it has the advantage of
using local potentials. A broad range of nuclear properties have been very
successfully described using SCMF models based on Skyrme EDFs, relativistic
EDFs, and the Gogny interaction (Ben03) ; Sto07aR ; (Erl11) ; (Lal04) ;
(Vre05) ; (Men06) ; (Nik11) . (Note that the Gogny model is not strictly local
as the other EDFs.) In the remainder of this section we briefly outline the
Skyrme-Hartree-Fock (SHF) method and the relativistic mean-field (RMF)
approach. As both methods are widely used and extensively described in the
literature, we keep the presentation short and concentrate on a side-by-side
comparison of the models.
The basis of any mean-field approach is a set of single-nucleon canonical
(Kohn-Sham) orbitals $\psi_{\alpha}(\mathbf{r})$, with occupations amplitudes
$v_{\alpha}$. The $\psi_{\alpha}$ denote Dirac four-spinor wave functions in
the RMF framework, and two-component-spinor wave functions in the SHF which is
a classical mean-field model. The canonical occupation amplitudes $v_{\alpha}$
are determined by the pairing interaction. The starting point of a particular
model is an EDF expressed in terms of $\psi_{\alpha},v_{\alpha}$ and the local
densities derived therefrom. The energy functional for the SHF method reads
$\displaystyle E$ $\displaystyle=$
$\displaystyle\\!\int\\!d^{3}r\,\left({\mathcal{E}}_{\rm
kin}+{\mathcal{E}}_{\rm pot}\right)+E_{\rm Coul}+E_{\rm pair}+E_{\rm cm},$ (3)
$\displaystyle{\mathcal{E}}_{\rm kin}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{\mathrm{p}}}\tau_{\mathrm{p}}+\frac{\hbar^{2}}{2m_{\mathrm{n}}}\tau_{\mathrm{n}}$
$\displaystyle E_{\rm cm}^{\mbox{}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2mA}\langle\big{(}\hat{P}_{\mathrm{cm}}\big{)}^{2}\rangle.$
The kinetic energy ${\mathcal{E}}_{\rm kin}$ is expressed in terms of single-
nucleon wave functions. The Skyrme functional is contained in the interaction
part with the potential-energy density $\mathcal{E}_{\rm pot}$. The Coulomb
energy $E_{\rm Coul}$ consists of the direct Coulomb term, and the Coulomb
exchange that is usually taken into account at the level of the Slater
approximation. In most applications the center-of-mass correction $E_{\rm
cm}^{\mbox{}}$ is applied a posteriori because its variation would
considerably complicate the mean-field equations. The pairing functional
$E_{\rm pair}$ will be detailed later. The RMF approach is usually formulated
in terms of a Lagrangian:
$\displaystyle{L}$ $\displaystyle=\\!\int\\!d^{3}r\,\left(\mathcal{L}_{\rm
kin}-\mathcal{E}_{\rm pot}\right)-E_{\rm Coul}-E_{\rm pair}-E_{\rm cm},$ (4)
$\displaystyle\mathcal{L}_{\rm kin}$
$\displaystyle=\sum_{\alpha}v_{\alpha}^{2}\psi^{\dagger}_{\alpha}\hat{\gamma}_{0}(i\hat{\bm{\gamma}}\cdot\bm{\partial}-m)\psi^{\mbox{}}_{\alpha},$
(5)
where $\hat{\gamma}$ is the Dirac matrix. Again, the kinetic part is expressed
explicitly in terms of Dirac spinor wave functions, whereas interaction terms
are included in the potential energy density $\mathcal{E}_{\rm pot}$. Further
contributions from Coulomb, pairing and center-of-mass motion are treated
similarly as in the SHF approach.
The basic building blocks of an EDF are local densities and currents built
from single-nucleon wave functions Eng75a ; (Ben03) . These are summarized in
the upper part of Table 1.
densities
---
SHF | RMF
$T=0$ | $T=1$ | $T=0$ | $T=1$
$\displaystyle\rho_{0}({\bf r})=\sum_{\alpha}v_{\alpha}^{2}\,\psi_{\alpha}^{\dagger}\psi_{\alpha}^{\mbox{}}$ | $\displaystyle\rho_{1}({\bf r})=\sum_{\alpha}v_{\alpha}^{2}\,\psi_{\alpha}^{\dagger}\hat{\tau}_{3}\psi_{\alpha}^{\mbox{}}$ | $\displaystyle\rho_{0}({\bf r})=\sum_{\alpha}v_{\alpha}^{2}\,\psi_{\alpha}^{\dagger}\psi_{\alpha}^{\mbox{}}$ | $\displaystyle\rho_{1}({\bf r})=\sum_{\alpha}v_{\alpha}^{2}\,\psi_{\alpha}^{\dagger}\hat{\tau}_{3}\psi_{\alpha}^{\mbox{}}$
$\displaystyle\tau_{0}({\bf r})=\sum_{\alpha}v_{\alpha}^{2}\,{\bm{\nabla}}\psi_{\alpha}^{\dagger}{\bm{\nabla}}\psi_{\alpha}^{\mbox{}}$ | $\displaystyle\tau_{1}({\bf r})=\sum_{\alpha}v_{\alpha}^{2}\,{\bm{\nabla}}\psi_{\alpha}^{\dagger}\hat{\tau}_{3}{\bm{\nabla}}\psi_{\alpha}^{\mbox{}}$ | $\displaystyle\rho_{\mathrm{S}}({\bf r})=\sum_{\alpha}v_{\alpha}^{2}\,\psi_{\alpha}^{\dagger}\hat{\gamma}_{0}\psi_{\alpha}^{\mbox{}}$ |
$\displaystyle\mathbf{J}_{0}({\bf r})=-\mathrm{i}\sum_{\alpha}v_{\alpha}^{2}\,\psi_{\alpha}^{\dagger}{\bm{\nabla}}\\!\times\\!\hat{\mbox{\boldmath$\sigma$}}\psi_{\alpha}^{\mbox{}}$ | $\displaystyle\mathbf{J}_{1}({\bf r})=-\mathrm{i}\sum_{\alpha}v_{\alpha}^{2}\,\psi_{\alpha}^{\dagger}\hat{\tau}_{3}{\bm{\nabla}}\\!\times\\!\hat{\mbox{\boldmath$\sigma$}}\psi_{\alpha}^{\mbox{}}$ | |
potential-energy density
---
| SHF | RMF-PC | RMF-ME
| $T=0$ | $T=1$ | $T=0$ | $T=1$ | $T=0$ | $T=1$
$\rho\rho$ | $C_{0}^{\rho}\rho_{0}^{2}$ | $C_{1}^{\rho}\rho_{1}^{2}$ | $G_{\omega}\rho_{0}^{2}$ | $G_{\rho}\rho_{1}^{2}$ | $G_{\omega}\rho_{0}\frac{1}{-\Delta+m_{\omega}^{2}}\rho_{0}$ | $G_{\rho}{\rho_{1}}\frac{1}{-\Delta+m_{\rho}^{2}}{\rho_{1}}$
mass | $C_{0}^{\tau}\rho_{0}\tau_{0}$ | $C_{1}^{\tau}{\rho_{1}}{\tau_{1}}$ | $G_{\sigma}\rho_{\mathrm{S}}^{2}$ | | $G_{\sigma}\rho_{\mathrm{S}}\frac{1}{-\Delta+m_{\sigma}^{2}}\rho_{\mathrm{S}}$ |
${\bm{\ell}}\cdot{\bm{s}}$ | $C_{0}^{{\bm{\nabla}}\mathbf{J}}\rho_{0}{\bm{\nabla}}\\!\cdot\\!\mathbf{J}_{0}$ | $C_{1}^{{\bm{\nabla}}\mathbf{J}}{\rho}_{1}{\bm{\nabla}}\\!\cdot\\!{\mathbf{J}_{1}}$ | “ | | “ |
gradient | $C_{0}^{\Delta\rho}({\bm{\nabla}}\rho_{0})^{2}$ | $C_{1}^{\Delta\rho}({\bm{\nabla}}{\rho_{1}})^{2}$ | $f_{\mathrm{S}}({\bm{\nabla}}\rho_{\mathrm{S}})^{2}$ | | |
dens.dep. | $C_{0}^{\rho}=c_{0}^{\rho}+d_{0}^{\rho}\rho_{0}^{a}$ | $C_{1}^{\rho}=c_{1}^{\rho}+d_{1}^{\rho}\rho_{0}^{a}$ | $\displaystyle G_{i}=a_{i}+(b_{i}+c_{i}x)e^{-d_{i}x}$ | $\displaystyle G_{i}=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}}$ | $\displaystyle G_{\rho}=g_{\rho}e^{-a_{\rho}(x-1)}$
| $C_{T}^{\tau}=c_{T}^{\tau}\;,\;C_{T}^{\Delta\rho}=c_{T}^{\Delta\rho}\;,\;C_{T}^{{\bm{\nabla}}\mathbf{J}}=c_{T}^{{\bm{\nabla}}\mathbf{J}}$ | $i\in\\{\sigma,\omega,\rho\\}\,$, $\,\displaystyle x=\frac{\rho_{0}}{\rho_{\mathrm{sat}}}$ | $i\in\\{\sigma,\omega\\}\,$, $\,\displaystyle x=\frac{\rho_{0}}{\rho_{\mathrm{sat}}}$ |
Table 1: Upper: The basic isoscalar ($T=0$) and isovector ($T=1$) local
densities of SHF (left) and RMF (right). Lower: The potential-energy densities
in the three considered SCMF models. Model parameters (third row) defining the
coupling constants are indicated by lowercase latin letters. For further
explanation see text.
All densities appear in two flavors (Per04) ; (Roh10) : isoscalar ($T=0$), or
total density (sum of proton and neutron densities), and isovector ($T=1$)
density (difference between neutron and proton densities). Both can be
conveniently expressed using the isospin operator $\hat{\tau}_{3}$. The basic
ingredients of an EDF are the local densities $\rho_{0}$ and $\rho_{1}$. In
RMF these can be associated with the zero-component of the four-vector
current, where $\rho_{0}$ is often called the vector density and $\rho_{1}$
the isovector-vector density. RMF uses one more ingredient, the isoscalar-
scalar density denoted here as $\rho_{\mathrm{S}}$. SHF instead employs the
kinetic-energy densities $\tau_{0/1}$ and the spin-orbit densities
$\mathbf{J}_{0/1}$. One can show that $\tau_{0}$ and $\mathbf{J}_{0}$ emerge
in the non-relativistic limit of $\rho_{\mathrm{S}}$ Rei89aR . The principal
difference between SHF and RMF is that the quantities $\tau_{0}$ and
$\mathbf{J}_{0}$ are independent in SHF, whereas they are tightly related
through $\rho_{\mathrm{S}}$ in RMF. Moreover, the RMF does not invoke an
isovector counterpart of $\rho_{\mathrm{S}}$ thus being more restricted in the
isovector channel.
The lower part of Table 1 displays the main components of the potential-energy
density. The underlying is to take all bi-linear isoscalar combinations of the
local densities and to associate a coupling constant with each term (Roh10) .
The SHF confines the combinations to have at most second order of derivatives
(the term $\mathbf{J}^{2}$ is also dropped). In the RMF approach one keeps
only terms that form a Lorentz scalar. Moreover, two bi-linear realizations of
RMF will be considered. First there is the straightforward point-coupling
(RMF-PC) realization that corresponds to contact interactions between nucleons
and, second, the meson-exchange folding (RMF-ME). The folding is motivated by
the traditional route to RMF as a model of nucleons coupled to classical meson
fields. Of course, at energies characteristic for nuclear binding meson
exchange represents just a convenient representation of the effective nuclear
interaction. In practice RMF-PC and RMF-ME present equivalent realizations of
the relativistic SCMF, differing in the range of effective interactions (zero-
range vs. finite-range) and the choice of density dependence for the
couplings. In practical applications one restricts the density dependence of
coupling (vertex) functions to keep the number of free parameters to a
minimum. In SHF, only the leading terms $\propto\rho_{0}^{2}$ and
$\rho_{1}^{2}$ are given a (simple) density dependence as shown in Table 1. In
RMF-PC and RMF-ME, each term has some density dependence, but not all of these
parameters are actually used. In RMF-PC, in particular, $c_{1},a_{\mathrm{S}}$
and $c_{\mathrm{S}}$ are set to zero (Nik08) . In RMF-ME, the parameters are
correlated by additional boundary conditions on $G_{i}$ (Nik02) ; (Lal05) . In
total, there are 11 adjustable parameters for SHF, 10 for RMF-PC, and 8 for
RMF-ME. From a formal perspective, SHF and RMF-PC are rather similar,
differing mainly in the relativistic kinematics, while RMF-ME includes a
significantly different density dependence of the couplings, in addition to
the finite range. These three models thus allow to display separately effects
of kinematics, density dependence, and range of the effective nuclear
interaction.
As far as particle-particle interaction, in the SHF we use the pairing
functional derived from a density-dependent zero-range force:
$\displaystyle E_{\rm pair}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{q\in\\{p,n\\}}\int
d^{3}r\tilde{\rho}^{2}_{q}\left[1-\frac{\rho(\mathbf{r})}{\rho_{\mathrm{pair}}}\right],$
(6a) $\displaystyle\tilde{\rho}_{q}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\alpha\in q}u_{\alpha}v_{\alpha}\big{|}\psi_{\alpha}({\bf
r})\big{|}^{2},$ (6b)
where $q$ runs over over protons and neutrons. It involves the pair-density
$\tilde{\rho}_{q}$ and is usually augmented by some density dependence. We
consider here $v_{0,p}$, $v_{0,n}$, and $\rho_{\mathrm{pair}}$ as free
parameters of the pairing functional in SHF. Note that we do not recouple to
isoscalar and isovector terms because pairing is considered independently for
protons and neutrons. Actually, the zero-range pairing force works only
together with a limited phase space for pairing. We use here a soft cut-off in
the space of single-nucleon energies Bon85a according to Ref. Ben00a .
In RMF calculations we use the recently developed separable pairing force
(Tia09a) ; (Tia09b) . It is separable in momentum space, and is completely
determined by two parameters that are adjusted to reproduce in symmetric
nuclear matter the pairing gap of the Gogny force. We have verified that both
pairing prescriptions yield comparable results for the pairing gaps.
## 3 Observables
In this section, we discuss observables pertaining to nuclear matter (NM) and
finite nuclei that are essential for discussion of NSE. Those observables can
be roughly divided Rei10 into good isovector indicators that correlate very
well with NSE (such as weak-charge form factor, neutron skins, dipole
polarizability, slope of the symmetry energy, and neutron pressure) and poor
isovector indicators (such as nuclear and neutron matter binding energy, giant
resonance energies, isoscalar and isovector effective mass, incompressibility,
and saturation density).
### 3.1 Nuclear matter properties
Bulk properties of symmetric nuclear matter, called nuclear matter properties
(NMP), are often used to characterize the properties of a model, or functional
respectively. Starting point for the definition of NMP is the binding energy
per nucleon in the symmetric nuclear matter
$E/A=E/A(\rho_{0},\rho_{1},\tau_{0},\tau_{1})$.
incompressibility: | $K_{\infty}$ | = | $\displaystyle 9\,\rho_{0}^{2}\,\frac{d^{2}}{d\rho_{0}^{2}}\,\frac{{E}}{A}\Big{|}_{\mathrm{eq}}$
---|---|---|---
symmetry energy: | $a_{\mathrm{sym}}$ | = | $\displaystyle\frac{1}{2}\frac{d^{2}}{d\rho_{1}^{2}}\frac{{E}}{A}\bigg{|}_{\mathrm{eq}}$
slope of $a_{\mathrm{sym}}$: | $L$ | = | $\displaystyle 3\rho_{0}\frac{da_{\mathrm{sym}}}{d\rho_{0}}\bigg{|}_{\mathrm{eq}}$
effective mass: | $\displaystyle\frac{\hbar^{2}}{2m^{*}}$ | = | $\displaystyle\frac{\hbar^{2}}{2m}+\frac{\partial}{\partial\tau_{0}}\frac{{E}}{A}\bigg{|}_{\mathrm{eq}}$
TRK sum-rule enhanc.: | $\kappa_{\mathrm{TRK}}$ | = | $\displaystyle\frac{2m}{\hbar^{2}}\frac{\partial}{\partial\tau_{1}}\frac{{E}}{A}\bigg{|}_{\mathrm{eq}}$
Table 2: Definitions of NMP used in this work. All derivatives are to be
taken at the equilibrium point corresponding to the saturation density
$\rho_{\mathrm{eq}}$.
Table 2 lists the NMP discussed in this work. It is important to note the
difference between total derivatives used for $K_{\infty}$,
$a_{\mathrm{sym}}$, $L$, and partial derivatives used for $m^{*}/m$ and
$\kappa_{\mathrm{TRK}}$. The latter take $E/A$ with $\tau_{T}$ as independent
variables while the total derivatives employ the dependence
$\tau_{T}=\tau_{T}(\rho_{0},\rho_{1})$. The slope of the symmetry energy $L$
parametrizes the density dependence of $a_{\mathrm{sym}}$. This quantity is
essential for the characterization of the EOS of neutron matter and the mass-
radius relation in neutron stars (Lat12) ; (Tsa12) ; (Fat12) ; (Fat12a) ;
(Erl13) ; (Ste13) . The enhancement factor $\kappa_{\mathrm{TRK}}$ for the
Thomas-Reiche-Kuhn (TRK) sum rule (Rin00) characterizes the isovector
effective mass.
Next to NMP come the corresponding bulk surface parameter, the (isoscalar)
surface energy $a_{\mathrm{surf}}$ and the (isovector) surface-symmetry energy
$a_{\mathrm{ssym}}$. These surface parameters can be determined from the
leptodermous expansion of the liquid drop model (LDM) energy per nucleon,
${\cal E}_{\rm LDM}=E_{\rm LDM}/A$, in terms of inverse radius ($\propto
A^{-1/3}$) and neutron excess $I$ (Rei06) :
$\begin{array}[]{rclclcl}{\cal E}_{\rm LDM}(A,I)&=&\displaystyle
a_{\mathrm{vol}}&+&\displaystyle a_{\mathrm{surf}}A^{-1/3}&+&\displaystyle
a_{\mathrm{curv}}A^{-2/3}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&&&+&a_{\mathrm{sym}}{I^{2}}&+&\displaystyle
a_{\mathrm{ssym}}A^{-1/3}{I^{2}}\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&&&&&+&\displaystyle a_{\mathrm{sym}}^{(2)}I^{4}.\end{array}$ (7)
The LDM energy ${\cal E}(A,I)$ is obtained from the DFT calculation by
subtracting the fluctuating shell correction energy. The general strategy
behind this correction and leptodermous expansion is detailed in Refs. (Rei06)
; (Kor12) . In essence, we combine NM calculations ($A=\infty$) with (shell
corrected) DFT calculations for a huge set of spherical nuclei and extract the
surface parameters by a fit to the expansion (7). Alternatively and simpler,
one can compute the surface energy and surface-symmetry energy thourgh a semi-
classical approximation (extended Thomas-Fermi) for the semi-infinite nuclear
matter Eif94a . In this survey, we shall apply both strategies, the semi-
classical approach whenever RMF is involved.
An important parameter characterizing the pure neutron matter is the neutron
pressure
$P(\rho_{n})=\rho_{n}^{2}\frac{d}{d\rho_{n}}\left({E\over A}\right)_{n},$ (8)
a quantity that is proportional to the slope of the binding energy of neutron
matter at a given neutron density (derivative of neutron EOS). As discussed
below, $P$ is excellent isovector indicator.
### 3.2 Observables from finite nuclei
The total energy of a nucleus $E(Z,N)$ is the most basic observable described
by SCMF. It is also the most important ingredient for calibrating the
functional, see Sec. 4.1. We ofter consider binding energy differences. Of
great importance for stability analysis are separation energies and
$Q_{\alpha}$ values. Another energy observable, potentially useful in the
context of NSE, is the indicator
$\displaystyle\delta V_{pn}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left[E(N,Z)-E(N-2,Z)\right.$ (9) $\displaystyle-$
$\displaystyle\left.E(N,Z-2)+E(N-2,Z-2)\right]$
involving the double difference of binding energies (Zha89) . Since $\delta
V_{pn}$ approximates the mixed partial derivative of binding energy with
respect to $N$ and $Z$, for nuclei with an appreciable neutron excess, the
average value of $\delta V_{pn}$ probes the symmetry energy term of LDM
(Sto07pn) : $\delta V_{pn}^{\rm LDM}\approx 2\left(a_{\rm sym}+a_{\rm
ssym}A^{-1/3}\right)/A.$ That is, the shell-averaged trend of $\delta V_{pn}$
is determined by the symmetry and surface symmetry energy coefficients.
It has been shown in Rei92c that effective SCMF provide a pertinent
description of the form factors in the momentum regime $q<2q_{\mathrm{F}}$
where $q_{\mathrm{F}}$ is the Fermi momentum. The key features of the nuclear
density are related to this low-$q$ range. The basic parameters characterizing
nuclear density distributions are: r.m.s. charge radius $r_{\mathrm{C}}$,
diffraction radius $R_{\mathrm{C}}$, and surface thickness
$\sigma_{\mathrm{C}}$ Fri82a . The diffraction radius $R_{\rm C}$, also called
the box-equivalent radius, parametrizes the gross diffraction pattern which
resemble those of a hard sphere of radius $R_{\mathrm{C}}$ Fri82a . The actual
charge form factor $F_{\mathrm{C}}(q)$ falls off faster than the box-
equivalent form factor $F_{\mathrm{box}}$. This is due to the finite surface
thickness $\sigma$ which, in turn, can be determined by comparing the height
of the first maximum of $F_{\mathrm{box}}$ with $F_{\mathrm{C}}$ from the
realistic charge distribution. The charge halo parameter $h_{\mathrm{C}}$ is
composed from the three basic charge form parameters and serves as a nuclear
halo parameter found to be a relevant measure of the outer surface diffuseness
Miz00a .
The charge distribution is basically a measure of the the proton distribution.
It is only recently that the parity-violating electron scattering experiment
PREX has provided some information on the weak-charge formfactor $F_{W}(q)$ of
208Pb (Abr12) ; (Hor12) . These unique data gives access to neutron
properties, such as the neutron r.m.s. radius $r_{\mathrm{n}}$. Closely
related and particularly sensitive to the asymmetry energy is the neutron skin
$r_{\mathrm{skin}}=r_{\mathrm{n}}-r_{\mathrm{p}}$, which is the difference of
neutron and proton r.m.s. radii. (As discussed in Ref. Miz00a , it is better
to define the neutron skin through neutron and proton diffraction radii and
surface thickness. However, for well-bound nuclei, which do not exhibit halo
features, the above definition of $r_{\mathrm{skin}}$ is practically
equivalent.) Neutron radii and skins are excellent isovector indicators
(Ton84) ; (Rei99) ; (Fur02) ; (Yos04) ; (Che05) ; (War09) ; Rei10 ; (Pie12) ;
(Fat12) that help to check and improve isovector properties of the nuclear
EDF Rei10 .
Nuclear excitations are characterized by the strength distributions
$S_{JT}(E)$ where $J$ is the angular momentum of the excitation, $T$ its
isospin, and $E$ the excitation energy. For example, the cross section for
photo-absorption is proportional to $S_{11}(E)$. The strengths functions can
be obtained from the excitation spectrum:
$\displaystyle S_{JT}(E)$ $\displaystyle=$
$\displaystyle\sum_{n}E_{n}B_{n}(EJT)\delta_{\Delta}(E-E_{n}),$ (10)
where $E_{n}$ is the excitation energy of state $n$, $B_{n}(EJT)$ the
corresponding transition matrix element of multipolarity $J$ and isospin $T$,
and $\delta_{\Delta}$ as finite width folding function – if $S_{JT}(E)$ is
calculated theoretically using, e.g., the random phase approximation (RPA). In
our RPA estimates, we use an energy dependent width
$\Delta=\mbox{max}(\Delta_{\mathrm{min}},(E_{n}-E_{\mathrm{thr}})/E_{\mathrm{slope}})$
which simulates the broadening mechanisms beyond RPA. The parameters for 208Pb
are $\Delta_{\mathrm{min}}=0.2$ MeV, $E_{\mathrm{thr}}=10$ MeV, and
$E_{\mathrm{slope}}=5$ MeV. The resulting spectral distributions for heavy
nuclei, as 208Pb, show one clear giant resonance peak at $E_{\rm GR}(JT)$ for
$(J,T)=(0,0),(1,1),(2,0)$. We will consider these resonance energies as
characteristic observables of dynamical response in heavy nuclei. The strength
functions $S_{JT}(E)$ in light nuclei are much more fragmented and cannot be
reduced to one single characteristic number.
There are other key observables that can be extracted from the strength
distributions, in particular for the dipole case $S_{11}(E)$, namely the
electric dipole polarizability
$\alpha_{\mathrm{D}}=\sum_{n}E_{n}^{-1}B_{n}(E11)\quad,$ (11)
and the TRK sum rule
$\sum_{n}E_{n}B_{n}(E11)=\frac{\hbar^{2}}{2m}\frac{NZ}{A}\left(1+\kappa_{\mathrm{TRK}}(Z,N)\right),$
(12)
which defines the sum-rule enhancement $\kappa_{\mathrm{TRK}}(Z,N)$. Note that
the latter is an observable in a specific finite nucleus and differs somewhat
from $\kappa_{\mathrm{TRK}}$ in nuclear matter. In the following, we will
consider $\alpha_{\mathrm{D}}$ and $\kappa_{\mathrm{TRK}}$ for 208Pb. In
particular, it has been demonstrated Rei10 ; (Pie12) that
$\alpha_{\mathrm{D}}$ strongly correlates with NSE; hence, it can serve as
excellent isovector indicator thant can be precisely extracted from measured
E1 strength (Tam11) . On the other hand, the low-energy E1 strength, sometimes
referred to as the pygmy dipole strength, exhibits weak collectivity. The
correlation between the accumulated low-energy strength and the symmetry
energy is weak, and depends on the energy cutoff assumed Rei10 ; (Dao12) ;
Rei13 .
Giant resonances are small amplitude excitations and belong to the regime of
linear response. The low energy branch of isoscalar quadrupole excitations is
often associated with large amplitude collective motion along nuclear shapes
with substantial quadrupole deformation. Of particular importance is nuclear
fission, which determines existence of heavy and superheavy nuclei. As a
simple and robust measure of fission, we shall consider the axial fission
barrier height in 266Hs. Unlike actinides, most superheavy nuclei have one
single fission barrier Erl12a ; (Sta13) ; (War12) , which simplifies the
analysis for our puroposes. It has to be kept in mind that the inner barrier
is often lowered by triaxial shapes, but this is not important for the study
of large-amplitude nuclear deformability.
## 4 Symmetry energy: constraints and correlations
### 4.1 Brief review of $\chi^{2}$ technique and correlation analysis
As discussed in Sec. 2, the nuclear EDF is characterized by about a dozen of
coupling constants $\mathbf{p}=(p_{1},...,p_{F})$ that are determined by
confronting DFT predictions with experiment. The standard procedure is to
adjust the parameters $\mathbf{p}$ to a large set of nuclear observables in
carefully selected nuclei (Ben03) ; (Klu09) ; (Kor10) ; (Fat11) ; (Gao13) .
This is usually done by the standard least-squares optimization technique.
Starting point is the $\chi^{2}$ objective function
$\chi^{2}(\mathbf{p})=\sum_{\mathcal{O}}\left(\frac{\mathcal{O}^{\mathrm{(th)}}(\mathbf{p})-\mathcal{O}^{\mathrm{(exp)}}}{\Delta\mathcal{O}}\right)^{2},$
(13)
where “th” stands for the calculated values, “exp” for experimental data, and
$\Delta\mathcal{O}$ for adopted errors. The optimum parametrization
$\mathbf{p}_{0}$ is the one which minimizes $\chi^{2}$ with the minimum value
$\chi^{2}_{0}=\chi^{2}(\mathbf{p}_{0})$. Around the minimum $\mathbf{p}_{0}$,
there is a range of “reasonable” parametrizations $\mathbf{p}$ that can be
considered as delivering a good fit, i.e.,
$\chi^{2}(\mathbf{p})\leq\chi^{2}_{0}+1$. As this range is usually rather
small, we can expand $\chi^{2}$ as
$\displaystyle\chi^{2}(\mathbf{p})\\!-\\!\chi^{2}_{\mathrm{0}}$
$\displaystyle\approx$
$\displaystyle\sum_{i,j=1}^{F}(p_{i}\\!-\\!p_{i,0})\mathcal{M}_{ij}(p_{j}\\!-\\!p_{j,0}),$
(14) $\displaystyle\mathcal{M}_{ij}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2}}\partial_{p_{i}}\partial_{p_{j}}\chi^{2}|_{\mathbf{p}_{0}}.$
(15)
The reasonable parametrizations thus fill the confidence ellipsoid given by
$(\mathbf{p}-\mathbf{p}_{0})\hat{\mathcal{M}}(\mathbf{p}-\mathbf{p}_{0})\leq
1,$ (16)
see Sec. 9.8 of [Bra97a] . Given a set of parameters $\mathbf{p}$, any
observable $A=\langle\hat{A}\rangle$ can be uniquely computed. In this way,
$A=A(\mathbf{p})$. The value $A$ thus varies within the confidence ellipsoid,
and this results in some uncertainty $\Delta A$. Let us assume for simplicity
that the observable varies weakly with $\mathbf{p}$ such that one can
linearize in the relevant range
$A(\mathbf{p})=A_{0}+(\mathbf{p}-\mathbf{p}_{0})\cdot\bm{\partial}_{\mathbf{p}}A$.
Let us, furthermore, associate a weight
$\propto\exp{\left(-\chi^{2}(\mathbf{p})\right)}$ with each parameter set. A
weighted average over the parameter space yields the covariance between two
observables $\hat{A}$ and $\hat{B}$, which represents their combined
uncertainty:
$\overline{\Delta A\,\Delta
B}=\sum_{ij}\partial_{p_{i}}A(\hat{\mathcal{M}}^{-1})_{ij}\partial_{p_{j}}B\quad.$
(17)
For $A$=$B$, Eq. (17) gives the variance $\overline{\Delta^{2}A}$ that defines
a statistical uncertainty of an observable. Variance and covariance are useful
quantities that allow to estimate the impact of an observable on the model and
its parametrization. We shall explore the covariance analysis in three
different ways:
1. 1.
We perform a constrained fit during which the observable of interest is kept
fixed at a desired value. In the present survey, we consider the symmetry
energy $a_{\mathrm{sym}}$ as constraining observable. Comparing uncertainties
from a constrained fit with those from an unconstrained fit provides a first
indicator on the impact of the constrained observable on other observables.
2. 2.
The next step is a trend analysis, in which one performs a series of
constrained fits with systematically varied values of the constraining
observable. One then studies other observables as a function of the
constrained quantity. This provides valuable information on possible inter-
dependences.
3. 3.
Finally, we compute correlation (17) between $a_{\mathrm{sym}}$ and other
observables. Here, a useful dimensionless measure is given by the Pearson
product-moment correlation coefficient: [Bra97a] :
${c}_{AB}=\frac{|\overline{\Delta A\,\Delta B}|}{\sqrt{\overline{\Delta
A^{2}}\;\overline{\Delta B^{2}}}}.$ (18)
A value ${c}_{AB}=1$ means fully correlated and ${c}_{AB}=0$ – uncorrelated.
In the following, we will apply these three ways of studying correlations with
$a_{\mathrm{sym}}$ to different groups of observables. To this end, we have
produced a series of parametrizations with systematically varied
$a_{\mathrm{sym}}$ for the SV Skyrme family and for the RMF-ME and RMF-PC
models.
The optimization and covariance analysis carried out in this survey is based
for all three EDFs (SHF-SV, RMF-PC, and RMF-ME) on the same standard set of
data on spherical nuclei (masses, diffraction radii, surface thickness, charge
radii, separation energies, isotope shifts, and odd-even mass differences)
that has originally been proposed in Ref. (Klu09) and recently employed in
Refs. (Erl10) ; (Erl13) . We wish to emphasize that this is the first time
that one consistent phenomenological input has been used to constrain SHF and
RMF EDFs. A slightly modified variant of the fitting protocol has been used
for RMF-ME. This EDF did not lead to stable results in the fits which were
unconstrained by NMP. Consequently, we included the nuclear matter information
on $(E/A)_{\mathrm{eq}}$ into the dataset. This is still much less than in the
previously published optimization protocols of RMF-ME, in which all NMP were
constrained (Typ99) ; (Nik02) ; (Lal05) .
### 4.2 Correlations with nuclear matter properties
The NMP corresponding to unconstrained optimization of SHF-SV, RMF-PC, and
RMF-ME EDFs – using the same standard dataset – are shown in Table 3. They are
compared with NMP of SHF-RD (Erl10) (employing a modified density dependence
and the standard dataset) and SHF-TOV (Erl13) (using neutron star data in
addition the standard dataset in the optimization process). As expected,
isoscalar effective mass is significantly lowered in RMF as compared to SHF,
and the opposite holds for $\kappa_{\rm TRK}$. The slope parameter $L$ is
predicted to be very different in all five models. In particular, RMF-ME has
very low value of $L$, and – at the same time – the uncertainty on
$a_{\mathrm{sym}}$ in this model is very small.
model | $\rho_{\rm eq}$ | $E/A$ | $K_{\infty}$ | $m^{*}/m$ | $a_{\mathrm{sym}}$ | $L$ | $\kappa_{\rm TRK}$
---|---|---|---|---|---|---|---
| (fm-3) | (MeV) | (MeV) | | (MeV) | (MeV) |
SHF-SV | 0.161(1) | -15.91(4) | 222(9) | 0.95(7) | 31(2) | 45(26) | 0.08(29)
RMF-PC | 0.159(1) | -16.14(3) | 185(18) | 0.57(1) | 35(2) | 82(17) | 0.75(2)
RMF-ME | 0.159(3) | -16.2(2) | 250(19) | 0.56(1) | 32.4(1) | 6(7) | 0.79(2)
SHF-RD | 0.161 | -15.93 | 231 | 0.90 | 32(2) | 60(32) | 0.04(32)
SHF-TOV | 0.161 | -15.93 | 222 | 0.94 | 32(1) | 76(15) | 0.21(26)
Table 3: Nuclear matter parameters of SHF-SV, RMF-PC, and RMF-ME EDFs used in
this survey (with error bars) obtained by means of unconstrained optimization.
Also shown are the values of NMP of SHF-RD (Erl10) and SHF-TOV (Erl13) .
Figure 1 shows the trends for selected properties of symmetric nuclear matter
with $a_{\mathrm{sym}}$.
Figure 1: Behavior of selected nuclear matter properties with symmetry energy
$a_{\mathrm{sym}}$ for the SV Skyrme family and for the ME and PC RMF model
families. The statistical uncertainties are indicated by error bars. The
result of the unconstrained fits are shown by large open symbols with
corresponding error bars.
The purpose of this analysis is to relate systematic variations with
$a_{\mathrm{sym}}$ to statistical uncertainties. The isoscalar properties
$K_{\infty}$, $m^{*}/m$ as well as the isovector dynamical response
$\kappa_{\rm TRK}$ are fairly insensitive to $a_{\mathrm{sym}}$. Their
variation with $a_{\mathrm{sym}}$ are much smaller than the typical
statistical uncertainties. This independency is also indicated by the fact
that the uncertainty obtained in the unconstrained fit is not visibly larger
than those from the constrained optimizations. The trend is markedly different
for the density dependence of the symmetry energy $L$: variations with
$a_{\mathrm{sym}}$ well exceed the statistical error bars and the
uncertainties from unconstrained fits are larger than those from constrained
calculations. It is to be noted that the dedicated variations of
$a_{\mathrm{sym}}$ stay within the uncertainty of $a_{\mathrm{sym}}$ in the
unconstrained optimization. The uncertainty of $L$ in the free fit thus covers
nicely the uncertainty of the constrained calculations plus the variation of
$L$ with $a_{\mathrm{sym}}$. Anyway, the results shows that $L$ cannot be used
as independent NMP although the formal structure of the EDF would allow that.
There seems to be a strong link established by the data which yet has to be
worked out.
Figure 2: Similar as in Fig. 1 but for selected properties of 208Pb: neutron
skin (top), dipole polarizability (middle), and weak-charge form factor
(bottom). The current experimental ranges are shaded grey: $r_{\rm
skin}$=0.33${}^{+0.16}_{-0.18}$ fm (Abr12) , $\alpha_{\rm D}=14.0\pm 0.4$
fm2/MeV (Tam11) , and $F_{W}(0.475)/F_{W}(0)=0.204\pm 0.028$ (Hor12) .
### 4.3 Correlations with properties of finite nuclei
Figure 2 illustrates the trends with $a_{\mathrm{sym}}$ and extrapolation
uncertainties for three observables in 208Pb: weak-charge form factor at
$q=0.475$ fm-1 ($q$-value of PREX), neutron skin, and dipole polarizability.
These observables are all known to be sensitive to isovector properties of EDF
Rei10 ; (Pie12) . This is confirmed by the trends in the present result. The
comparison of uncertainties shows a large growth when going from constrained
to unconstrained optimizations. This corroborates the close relation between
the symmetry energy and the three isovectors indicators shown in Fig. 2. It
is, furthermore, interesting to note that SHF and RMF-PC stay safely within
the bands given by experimental data and RMF-ME is not far away. A better
discrimination between models requires more precise data, a task on which
presently many experimental groups are heavily engaged.
Figure 3: Behavior of $\delta V_{pn}$ in 168Er with symmetry energy
$a_{\mathrm{sym}}$ for SHF-SV (solid line) as compared to experiment (dashed
line) and the LDM value (filled square). The result of the unconstrained fit
is marked by a large open square with corresponding error bars.
To explore the usefulness of $\delta V_{pn}$ as an isovector indicator, we
choose the heavy deformed nucleus 168Er, as its even-even neighbors have
similar structure and the calculated values of $\delta V_{pn}$ for even-even
Er isotopes show little variations around $N=100$. The results displayed in
Fig. 3 show a gradual decrease of this quantity with $a_{\mathrm{sym}}$, but
the magnitude of the variation is very small and cannot account for the
deviation from experiment (around 50 keV). It is apparent that this quantity
is too strongly influenced by shell effects (given by the deviation from the
LDM estimate; also around 50 keV) to probe NSE, see Refs. (Sto07pn) ; (Ben11)
and Sec. 4.4 below.
Figure 4 shows the trends of the three major giant resonances in 208Pb:
isoscalar monopole resonance (GMR), isovector dipole resonance (GDR), and
isoscalar quadrupole resonance (GQR). For technical reasons, we only show
results obtained with the SV Skyrme family.
Figure 4: Behavior of giant resonance energies in 208Pb with symmetry energy
$a_{\mathrm{sym}}$ for the SV Skyrme family (Klu09) . In order not to make the
graph too busy the uncertainties from the unconstrained fit are not shown;
they have the same size as those from the constrained fits.
The isoscalar resonances show no dependence on $a_{\mathrm{sym}}$ at all; this
is understandable for the symmetry energy belongs to the isovector sector.
Somewhat surprisingly, the GDR exhibits very little dependence on
$a_{\mathrm{sym}}$ as well, with the magnitude of variations well below the
statistical uncertainties. As demonstrated earlier (Klu09) ; (Pie12) , it is
the sum-rule enhancement factor $\kappa_{\mathrm{TRK}}$ that has the dominant
impact on the GDR peak frequency rather than $a_{\mathrm{sym}}$. The
covariance analysis of Fig. 4 confirms that the energies of GMR, GDR, and GQR
do not obviously relate to $a_{\mathrm{sym}}$.
Figure 5: Similar as in Fig. 1 but for surface energy (top) and fission
barrier (bottom) in 266Hs. The surface energy from RMF-ME is not shown.
Figure 5 shows behavior of surface energy $a_{\mathrm{surf}}$ and the inner
fission barrier $B_{f}$ in 266Hs with $a_{\mathrm{sym}}$. The surface energy
was computed by means of the extended Thomas-Fermi method. The trends of
$a_{\mathrm{surf}}$ predicted by SHF and RMF are similar. An offset of about 2
MeV is most likely due to very different effective masses in both models. Much
larger differences are seen for the fission barriers. The basic difference
between SHF and RMF can again be explained predominantly in terms of effective
masses. Barriers are produced by shell effects and shell effects are larger
for lower effective masses. There is also a difference between the two RMF
models. This could be due to a different handling of gradient terms (only RMF-
PC contains such) and a much different parametrization of density dependence.
All three models show not only different values as such, but also different
trends.
The statistical errors differ substantially between the models. RMF-ME shows a
small uncertainty in $B_{f}$. This may be due to the missing gradient term in
this model which would also restrict the uncertainty in the surface energy. We
note, however, that the gradient term in RMF-PC is to a certain extent
equivalent the mass term of the sigma meson in RMF-ME, which is considered a
free parameter. The plot of the $B_{f}$ demonstrates nicely the relative role
of statistical and systematic errors, with the statistical errors being much
smaller than inter-model differences. As discussed in Refs. Nikolov11 ;
(Kor12) , fission barriers are strongly affected by $a_{\mathrm{surf}}$ and
$a_{\mathrm{ssym}}$ of EDF. In particular, the recently developed EDF UNEDF1,
suitable for studies of strongly elongated nuclei, has relatively low values
of $a_{\mathrm{surf}}$ and $a_{\mathrm{ssym}}$ (see Fig. 7 below) that reflect
the constraints on the fission isomer data. The reduced surface energy
coefficients result in a reduced effective surface coefficient $a_{\rm
surf}^{\rm(eff)}=a_{\rm surf}+a_{\rm ssym}I^{2},$ which has profound
consequences for the description of fission barriers, especially in the
neutron-rich nuclei that are expected to play a role at the final stages of
the r-process through the recycling mechanism (Pan08) .
### 4.4 Correlations summary
The summary of our correlation analysis for $a_{\mathrm{sym}}$ is given in
Fig. 6.
Figure 6: The correlation (18) between symmetry energy and selected
observables ($Y$) for three models: SHF-SV, RMF-PC and RMF-ME. Results
correspond to unconstrained optimization employing the same strategy in all
three cases. For RMF-ME no reliable numbers could be obtained for
$a_{\mathrm{surf}}$; this is indicated by an open circle.
The first four entries concern the same nuclear matter properties as in Fig.
1. It is only for $L$, the density dependence of symmetry energy, that a
strong correlation with $a_{\mathrm{sym}}$ is seen. This complies nicely with
the findings of the trend analysis in Fig. 1. The next entry concerns the
neutron pressure (8) at $\rho_{n}=0.08$ neutrons/fm3. It is also strongly
correlated with $a_{\mathrm{sym}}$, which is no surprise because it is an
excellent isovector indicator (Bro00) ; (Typ01) ; (Fur02) ; (Yos04) ; Rei10 ;
(Fat12) . The diagram shows, furthermore, the (isoscalar) surface energy
$a_{\mathrm{surf}}$ computed in semi-classical approximation. This quantity is
well correlated with $a_{\mathrm{sym}}$ for SHF and practically uncorrelated
for RMF.
The next three entries are observables in 208Pb: weak-charge form factor,
neutron skin, and dipole polarizability. All three are known to be strong
isovector indicators Rei10 ; (Pie12) ; (Fat12) . This is confirmed here for
all three models.
The remaining four entries deal with exotic nuclei. These are: binding energy
and $\alpha$-decay energy in yet-to-be-measured superheavy nucleus
$Z=120,N=182$, binding energy in an extremely neutron rich 148Sn, and the
fission barrier in 266Hs (for which trends had been shown already in Fig. 5).
The data on $Z=120,N=182$ consistently do not correlate with
$a_{\mathrm{sym}}$. The binding energy of 148Sn shows some correlation with
$a_{\mathrm{sym}}$, about equally strong in the three models. This is expected
as a large neutron excess surely explores the static isovector sector.
Finally, the correlation with fission barrier in 266Hs exhibits an appreciable
model dependence with some correlation in SHF and practically none in RMF.
We also studied correlations between $\delta V_{pn}$ in 168Er and other
observables for finite nuclei and NM. We did not find a single observable that
would correlate well with this binding-energy indicator. In particular, the
correlation coefficient (18) with $a_{\mathrm{sym}}$ is 0.41, with
$\alpha_{\rm D}$ in 208Pb is 0.6, and with $r_{\rm skin}$ in 208Pb is 0.54.
This results demonstrates that $\delta V_{pn}$ in one single nucleus is too
strongly influenced by shell effects to be used as an isovector indicator.
## 5 Symmetry energy parameters of EDFs
The actual values of symmetry energy parameters depend on (i) the form of EDF
and (ii) the optimization strategy used. The first point is nicely illustrated
in Table 3, which compares NMP for different functional forms (SHF-SV, SHF-RD,
RMF-PC, and RMF-ME) using the same dataset and the same optimization
technique. As far as the second point, it is instructive to compare SHF-SV and
SHF-TOV NMP; namely, the inclusion of additional data on neutron stars in SHF-
TOV has significantly impacted $L$ and $\kappa_{\rm TRK}$. Many other examples
can be found in Refs. Sto07aR ; Dutra that demonstrate divergent predictions
of Skyrme EDFs for neutron and nuclear matter.
The range of $a_{\text{sym}}$ is fairly narrowly constrained by various data
and ab-initio theory (Lat12) ; it is $28\,{\rm MeV}<a_{\text{sym}}<34$ MeV.
The recent Finite-Range Droplet Model (FRDM) result (Mol12) is
$a_{\text{sym}}=(32.5\pm 0.5)$ MeV. All EDFs listed in Table 3 are consistent
with these expectations.
The values of $L$ are less precisely determined (Li08) ; (Tri08) ; (Ste12) ;
(Lat12) ; (Tsa12) ; (Fat12) ; (Fat12a) ; (Erl13) ; (Ste13) ; (Fat13) ; there
is more dependence on specific observables or methodology used. Recent surveys
(Lat12) ; (Ste13) suggest that a reasonable range of $L$ is $40\,{\rm
MeV}<L<80$ MeV, and FRDM gives $L=70\pm 15$ MeV (Mol12) . Except for RMF-ME,
all models shown in Table 3 are consistent with these estimates. The low value
of $L$ in RMF-ME is troublesome; here we note that while SHF-SV and RMF-PC
EDFs fall within the error bars of the current experimental data in Fig. 2,
RMF-ME (as defined by the present optimization protocol) does not.
As discussed in Ref. (Rei06) , the leading surface and symmetry terms appear
relatively similar within each family of EDFs, with a clear difference for
$a_{\mathrm{sym}}$ between SHF and RMF. By averaging over Skyrme-EDF results
of Refs. (Rei06) ; (Sat06) , one obtains: $a_{\mathrm{sym}}\approx 30.9\pm
1.7$ MeV, $a_{\mathrm{ssym}}\approx-48\pm 10$ MeV. Older relativistic models
provide systematically larger values (Rei06) : $a_{\mathrm{sym}}\approx
40.4\pm 2.7$ MeV and $a_{\mathrm{ssym}}\approx-103\pm 18$ MeV. (Codes for a
leptodermous expansion of the recent RMF-PC and RMF-ME models have yet to be
developed.)
The coefficient $a_{\mathrm{ssym}}$ is poorly constrained in the current EDF
parameterizations and there are large differences between models, see Fig. 7.
Figure 7: Correlation between the symmetry and surface symmetry coefficients
taken from Ref. Nikolov11 (Skyrme EDFs, dots; LDM values, stars) and Ref.
(Dan09) (Skyrme EDFs, circles). The UNEDF1 values (Kor12) are marked by a
square. (Adopted from Nikolov11 .)
In addition, the values of $a_{\mathrm{sym}}$ and $a_{\mathrm{ssym}}$ have
been shown to be systematically (anti)correlated (Far78) ; (Ton84) ; (Dan09) ;
Nikolov11 . Figure 7, displays the pairs $(a_{\text{sym}},a_{\text{ssym}})$
for various Skyrme EDFs and LDM parametrizations. While a correlation between
$a_{\text{sym}}$ and $a_{\text{ssym}}$ is apparent, a very large spread of
values is seen that demonstrates that the is indicative of the data on g.s.
nuclear properties are not able to constrain $a_{\mathrm{ssym}}$. It is
interesting to note that the LDM values and phenomenological estimates cluster
around $a_{\text{sym}}=30$ MeV and $a_{\text{ssym}}=-45$ MeV. The values for
UNEDF1 functional, additionally constrained by the data on very deformed
fission isomers (thus probing the surface-isospin sector of EDF) are
$a_{\text{sym}}=29$ MeV and $a_{\text{ssym}}=-29$ MeV.
## 6 Isospin physics and symmetry energy
The emergence of NSE is rooted in the isobaric symmetry and its breaking as a
function of neutron excess and mass. Single-reference DFT is essentially the
only framework allowing for understanding global behavior of isospin effects
throughout the entire nuclear landscape. While the nuclear interaction part of
the nuclear EDF is constructed to be an isoscalar (Per04) ; (Roh10) , the
Coulomb interaction breaks isospin manifestly. There are, therefore, two
different sources of isospin symmetry breaking in the nuclear DFT: spontaneous
isospin breaking associated with the self-consistent response to the neutron
excess, and the explicit breaking due to the electric charge of the protons
(Sat09) .
Effects related to isospin breaking and restoration are difficult to treat
theoretically within the nuclear DFT. Below, we discuss two ways of dealing
with this problem: isocranking and isospin projection.
### 6.1 1D- and 3D-isocranking
The isocranking model (Sat03) ; (Sat06) attributes the kinetic coefficient
$a_{\rm sym,kin}$ contribution to the mean level spacing at the Fermi energy
$\varepsilon(A)$ rather than to the total kinetic energy itself. The SHF
calculations also revealed that the isovector mean potential of the Skyrme EDF
can be quite well characterized by an effective $V_{TT}$ interaction (1)
characterized by a strength parameter $\kappa(A)$. The actual isovector part
of the Skyrme mean-field potential is composed of several terms (Per04) ;
(Roh10) . As seen from Table 1, in the uniform NM limit, two terms contribute
in SHF, $C_{1}^{\rho}\rho_{1}^{2}$ and $C_{1}^{\tau}\rho_{1}\tau_{1}$, and the
NSE strength reads:
$a_{\rm
sym}=\frac{1}{8}\frac{m}{m^{*}}\varepsilon_{FG}+\left[\left(\frac{3\pi^{2}}{2}\right)^{2/3}C_{1}^{\tau}\rho_{0}^{5/3}+C_{1}^{\rho}\rho_{0}\right],$
(19)
where $\varepsilon_{\rm{FG}}$ is the average level splitting in FGM.
Therefore, within this scenario, $a_{\rm sym}$ is non-trivially modified by
momentum-dependent effects introducing, in the leading order, the dependence
of $a_{\rm sym,kin}$ and $a_{\rm sym,int}$ on the isoscalar and isovector
effective mass, respectively.
Within the nuclear shell model, NSE appears through a contribution to the
binding energy proportional to $T(T+1)$ (Tal62) . However, the local
enhancement of binding around $N=Z$ (the Wigner energy) suggest an enhancement
of the linear term to $T(T+\lambda)$ with $\lambda\approx 1.26$ (Jan65) ;
(Jan03) ; (Glo04) . Since the Wigner energy is neither fully understood nor
included properly within the SCMF models (Sat97) , the microscopic origin of
$\lambda$ is still a matter of debate. Within the isocranking model, the Fock
exchange (isovector) potential gives rise to $\lambda\approx 0.5$, at variance
with enhancement seen in experimental data. The Wigner energy can be explained
by shell-model calculations (Sat97) in terms of configuration mixing. The
Wigner term is usually associated with the isoscalar neutron-proton (np)
pairing (Sat97a) ; (Roh10) , but its understanding is poor as realistic
calculations involving simultaneous np mixing in both the particle-hole (p-h)
and particle-particle (p-p) channels have not been carried out. It is only
very recently that 3D isocranking calculations including np mixing in the p-h
channel have been reported (Sat13) . This is the first step towards developing
the nuclear superfluid DFT including np mixing in both p-h and p-p channels.
An improved treatment of isospin within the 3D isocranking will open new
opportunities for quantitative studies of isobaric analogue states and, in
turn, the NSE.
### 6.2 Isospin projected DFT
The isospin and isospin-plus-angular-momentum projected DFT models have been
developed recently to describe isospin mixing effects. These new tools open
new avenues to probe NSE. To gain insight on this line of models, it is
instructive to to consider the spontaneous isospin symmetry breaking effect in
the so-called anti-aligned p-h configurations in $N=Z$ nuclei, which are
mixtures of $T=0$ and $T=1$ states (Sat11a) . Restoration of the isospin
symmetry results in the energy splitting, $\Delta E_{T}$, between the actual
$T=0$ and $T=1$ configurations. Since these states are projected from a single
mean-field determinant, the splitting is believed to be insensitive to
kinematics, and the method can be used to probe dynamical effects giving rise
to the interaction term $a_{\rm sym,int}$. The results of SHF calculations
(Sat11a) performed in finite nuclei confirm that $a_{\rm sym,int}$ is indeed
correlated with the isoscalar effective mass in agreement with the NM relation
(19).
The isospin and isospin plus angular momentum projected DFT were designed and
applied to study the isospin impurities (Sat09) and isospin symmetry breaking
corrections to the superallowed $0^{+}\rightarrow 0^{+}$ $\beta$-decay rates
(Sat11) . Unfortunately, the calculations show that these two observables are
not directly correlated with the symmetry energy. Ambiguities associated with
these calculations stimulated further development of the formalism in the
direction of the Resonating-group method. The scheme proceeds in three steps:
(i) First, a set of low-lying (multi)p-(multi)h SHF states $\\{\Phi_{i}\\}$ is
calculated. These states form a basis for a subsequent projection; (ii) Next,
the projection techniques are applied to calculate a family
$\\{\Psi_{I}^{(\alpha)}\\}$ of good angular momentum states with properly
treated $K$-mixing and isospin mixing; (iii) Finally, a configuration mixing
of $\\{\Psi_{I}^{(\alpha)}\\}$ states is performed using techniques suitable
for non-orthogonal ensambles.
Although at present the calculations can be realized only for the SkV EDF, the
preliminary results (Sat13a) are encouraging, as shown in Fig. 8. Since the
projected approach treats rigorously the angular momentum conservation and the
long-range polarization due to the Coulomb force, it opens up a possibility of
detailed studies of the isovector terms of the nuclear EDF that are sources of
the NSE.
Figure 8: Energies of $I^{\pi}=1^{+}$ states in 32S normalized to the
isobaric analogue state $I^{\pi}=1^{+},T=1$. The results of projected SHF-SkV
calculations involving configuration mixing (Sat13a) (left) are compared to
experiment (right). The calculations are based on 24 $I=1^{+}$ states
projected from 6 HF determinants representing low-lying 1p-1h configurations.
## 7 Conclusions
This work surveys various aspects of NSE within the nuclear DFT represented by
non-relativistic and relativistic self-consistent mean-field frameworks. After
defining the models and statistical tools, we reviewed key observables
pertaining to bulk nucleonic matter and finite nuclei. Using the statistical
covariance technique, constraints on the symmetry energy were studied,
together with correlations between observables and symmetry-energy parameters.
Through the systematic correlation analysis, we scrutinized various
observables from finite nuclei that are accessible by current and future
experiments. We confirm that by far the most sensitive isovector indicators
are observables related to the neutron skin (neutron radius, diffraction
radius, weak charge form factor) and the dipole polarizability Rei10 ; (Pie12)
. In this context, PREX-II measurement of the neutron skin in 208Pb Prex2 (a
follow-up measurement to PREX (Abr12) designed to improve the experimental
precision), CREX measurement of the neutron skin in 48Ca Crex , and on-going
measurements of $\alpha_{\rm D}$ in neutron-rich nuclei (Tam13) are
indispensable.
The masses of heavy neutron-rich nuclei also seem to correlate well with NSE
parameters. Other observables, such as $Q_{\alpha}$-values, $\delta V_{pn}$,
barrier heights, and low-energy dipole strength Rei10 ; (Dao12) ; Rei13 are
too strongly impacted by shell effects to be useful as global isovector
indicators.
A major challenge is to develop the universal nuclear EDF with improved
isovector properties. Various improvements are anticipated in the near future.
Those include constraining the EDF at sub-saturation densities using ab initio
models (Bog11) ; (Mar13) and using the density matrix expansion to develop an
EDF based on microscopic nuclear interactions (Sto10) . This work will be
carried out under the Nuclear Low Energy Computational Initiative (NUCLEI)
NUCLEI . Other exciting avenues are related to multi-reference isospin
projected DFT, which will enable us to make reliable predictions for isobaric
analogues, isospin mixing, and mirror energy differences.
###### Acknowledgements.
This work was supported by the U.S. Department of Energy under Contract No.
DE-FG02-96ER40963 (University of Tennessee), No. DE-SC0008499 (NUCLEI SciDAC
Collaboration); by BMBF under Contract No. 06 ER 142D; and by NCN under
Contract No. 2012/07/B/ST2/03907
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|
arxiv-papers
| 2013-07-22T17:11:01 |
2024-09-04T02:49:48.288122
|
{
"license": "Public Domain",
"authors": "W. Nazarewicz, P.-G. Reinhard, W. Satula, D. Vretenar",
"submitter": "Witold Nazarewicz",
"url": "https://arxiv.org/abs/1307.5782"
}
|
1307.5860
|
# Direct Evaluation of the Helium Abundances in Omega Centauri
A. K. Dupree and E. H. Avrett Harvard-Smithsonian Center for Astrophysics,
Cambridge, MA 02138, USA [email protected]; [email protected]
###### Abstract
A direct measure of the helium abundances from the near-infrared transition of
He I at 1.08$\mu$m is obtained for two nearly identical red giant stars in the
globular cluster Omega Centauri (catalog ). One star exhibits the He I line;
the line is weak or absernt in the other star. Detailed non-LTE semi-empirical
models including expansion in spherical geometry are developed to match the
chromospheric H$\alpha$, H$\beta$, and Ca II K lines, in order to predict the
helium profile and derive a helium abundance. The red giant spectra suggest a
helium abundance of $Y\leq 0.22$ (LEID 54064) and $Y=0.39-0.44$ (LEID 54084)
corresponding to a difference in the abundance $\Delta Y\geq 0.17$. Helium is
enhanced in the giant star (LEID 54084) that also contains enhanced aluminum
and magnesium. This direct evaluation of the helium abundances gives
observational support to the theoretical conjecture that multiple populations
harbor enhanced helium in addition to light elements that are products of
high-temperature hydrogen burning. We demonstrate that the 1.08$\mu$m He I
line can yield a helium abundance in cool stars when constraints on the semi-
empirical chromospheric model are provided by other spectroscopic features.
stars: individual (LEID 54064 (catalog Cl*NGC5139 LEID54064), LEID 54084
(catalog Cl*NGC5139 LEID54084) ) - stars: abundances - stars: atmospheres -
globular clusters: individual (Omega Centauri)
††slugcomment: ApJ Letters, in press
## 1 Introduction
Our current understanding of stellar populations in globular clusters has
dramatically changed with the discoveries of multiple stellar generations in a
single globular cluster. While variations in color and a spread in the [Fe/H]
values of red giants in massive clusters have been long recognized (Woolley
1966, Geyer 1967) along with variations of light elements (Martell 2011), the
firm identification of multiple populations on the main sequence in Omega
Centauri (Anderson 1997; Bedin et al. 2004; Bellini et al. 2010), and
subsequently several other clusters (cf. Gratton et al. 2012), was surprising
and continues to present theoretical challenges. Norris (2004) suggested,
based on isochrone calculations, that dwarf stars on the ‘blue’ main sequence
in Omega Cen would be enhanced in helium by $\Delta$Y$\sim$0.10$-$0.15. The
lowered hydrogen opacity causes stars of the same mass to appear hotter and
more luminous (Valcarce et al. 2012). Subsequently, the assessment of metals
in dwarfs on the bifurcated main sequence in Omega Cen, showed that the hotter
objects (the ‘blue’ dwarfs) were less metal-poor than the ‘red’ dwarf stars
(Piotto et al. 2005). Stellar models suggest that increased metals also signal
the presence of enhanced helium in the ‘blue’ main sequence. The source (or
sources) of such an enhancement remains elusive. One attractive explanation
appears to be a second stellar generation formed from the material lost by the
first generation of intermediate mass stars during their asymptotic giant
phases (D’Ercole et al. 2010; Johnson & Pilachowski 2010; Renzini 2013),
although other possibilities such as fast-rotating massive stars (Charbonnel
et al. 2013) or massive binary-star mass overflow (de Mink et al. 2009) may
well contribute (cf. Gratton et al. 2012). The formation of cluster
populations with several generations of star formation also impacts an
understanding of the halo of the Milky Way, satellites of our Galaxy, and the
star formation and assembly history of other galaxies (cf. Gratton et al.
2012; Brodie & Strader 2006).
It is obviously of great interest if a helium enhancement could be verified in
globular cluster stars in our Galaxy. A direct measure of the helium abundance
from a spectrum would provide confirmation of Norris’ conjecture. Such a
measurement is challenging because useful lines of helium are generally absent
in the optical spectra of cool stars. Moreover, in hotter stars, such as blue
horizontal branch objects, sedimentation caused by diffusion and element
stratification occur. Helium abundances from the spectroscopy of hot
horizontal branch stars in Omega Cen demonstrate the effects of surface
diffusion, or mixing during late helium core flashes (DaCosta et al. 1986;
Moehler et al. 2011; Moni Bidin et al. 2012) and derived abundance values vary
widely from Y $\leq$0.02 to Y=0.9.
In cool stars, a transition in He I occurs in the near-infrared at 1.08$\mu$m
and has been identified in many metal-poor field stars, where, in addition to
abundances, it can indicate atmospheric dynamics because the lower level of
the transition is metastable (Dupree et al. 1992, 2009; Smith et al. 2012). In
Omega Centauri, a closely matched group of first-ascent red giant stars
displays strong and weak helium absorption that correlates (Dupree et al.
2011) with increased [Al/Fe] and [Na/Fe] abundance, more than with [Fe/H].
This result gave direct observational support to the idea that products of
high-temperature hydrogen burning in a previous stellar generation had, in
fact, occurred. A quantitative measure of the helium abundance in these
objects is the goal of this Letter.
Pasquini et al. (2011) calculated profiles of the He I 1.08$\mu$m line in an
approximate way based on a stationary plane-parallel model applied to two very
cool luminous stars in NGC 2808. They showed that a change in the
chromospheric structure itself can strengthen or weaken helium absorption. In
fact, chromospheric line profiles are highly sensitive to the structure and
dynamics of the atmospheric model. In this paper, we have selected similar
stars and first constrained the atmospheric structure and dynamics using other
chromospheric lines. A model for the radiative transfer must be used that is
appropriate to the stars. Following that, the abundance of helium can be
inferred from line synthesis using the semi-empirical atmospheric model that
is anchored by other chromospheric lines.
Here we focus on two ‘identical’ red giants in Omega Centauri, LEID 54064 and
LEID 54084 (van Leeuwen et al. 2000). They are located $\sim$5.7 arc min to
the SW from the cluster center, and are separated by 1.6 arcminutes on the
sky. These giants have very similar temperatures, luminosities, and values of
[Fe/H] (Table 1). However they differ remarkably in [Na/Fe] and [Al/Fe]
abundances and the strength of the helium line (Dupree et al. 2011). The star
LEID 54084 exhibits enhanced light elements as compared to LEID 54064.
## 2 Modeling Chromospheric Lines
The PANDORA code (Avrett & Loeser 2003, 2008) is used to develop the semi-
empirical, spherical model of the chromosphere where the temperature
distribution, the turbulent velocities, and the expansion velocities are
adjusted to obtain optimum agreement between calculated profiles and
observations of chromospheric lines (H$\alpha$, H$\beta$, and Ca II-K). The
initial model consists of a static LTE photosphere corresponding to a
effective temperature of 4740K (Kurucz 2011), gravity $log~{}g$= 1.75, a
stellar radius of 20$R_{\odot}$, and [Fe/H]=$-$1.72 with the
$\alpha$-abundances enhanced by $+$0.44 dex. Chromospheric line emission is
essentially unaffected by the photospheric model. A chromospheric structure
similar to other metal-poor models (Mészáros et al. 2009) was added to begin
the iterations, and expansion started in the low chromosphere. Our
calculations assume multi-level atoms (H I:15 levels, Ca II: 5 levels, He I:
13 levels), and the iterations explicitly consider the velocity field in the
evaluation of the line source functions and as a contribution to the pressure
in the hydrostatic equilibrium equations. The total model is iterated with
full and complete non-LTE calculations in order to match the chromospheric
line profiles. The Ca II-K line profile is computed with partial frequency
redistribution; complete frequency redistribution is used for the hydrogen and
helium lines. These flux profiles are calculated with an integration over the
apparent spherical stellar disk including the extended chromosphere.
The profiles of the optical lines, H$\alpha$, H$\beta$, and Ca II-K were taken
from spectra obtained with the MIKE double echelle spectrograph (Bernstein et
al. 2003) mounted on the Magellan/CLAY telescope at Las Campanas Observatory.
These spectra were used previously to derive elemental abundances (Dupree et
al. 2011). The spectra and the calculated stellar profiles for H$\alpha$,
H$\beta$, and Ca II-K are shown in Fig. 1. The observed profiles are
effectively identical between the two giants, signaling that the activity
levels of the stars are similar. The spectra are well matched by the
calculated profiles. Note the asymmetry in the H$\alpha$ line core; the core
is formed higher in the atmosphere than the rest of the profile and is
sensitive to the outflow. However the line itself is narrow, and demands a
relatively low turbulent velocity, which increases with height in the
chromosphere. The final model (Fig. 2) has a temperature that extends to 105K
(although such high temperatures do not affect the profiles evaluated here),
and an outflow velocity that reaches 100 km s-1, which yields a mass outflow
rate of $\sim 3\times 10^{-9}$ M⊙ yr-1. This rate follows straightforwardly
from the atmospheric model (Fig. 2) and is proportional to $Nvr^{2}$ in the
chromosphere, where $r$ is the radial distance at which the wind has a
velocity $v$, and $N$ is the hydrogen density. This value exceeds by a factor
of 1.3–1.5 the rate estimated from an extension of the Mészáros et al. (2009)
fit to H$\alpha$ profiles of cooler stars shown in their Fig. 10. For more
luminous stars in the more metal-rich NGC 2808, Mauas et al. (2006) find
values of $0.7-3.8\times 10^{-9}$ M⊙ yr-1 from the H$\alpha$ line. Field
metal-poor giants, comparable in $M_{bol}$ to our targets possess a mass loss
rate spanning $1.3\times 10^{-9}-10^{-8}$ M⊙ yr-1(Dupree et al. 2009). While
mass loss rates have been measured (Mészaros et al. 2009) to vary with time by
factors of 1.5 to 6 in metal-poor red giants with luminosity
$logL/L_{\odot}\sim 3.0$, the values inferred from semi-empirical model fits
are less by an order of magnitude than the Reimers (1977), Origlia et al.
(2007), or the Schröder & Cuntz (2005) approximations.
This temperature and velocity model (Fig. 2) is used to evaluate the profile
of the He I 1.08$\mu$m line. The near-ir He I lines measured with PHOENIX on
Gemini-S were reported earlier (Dupree et al. 2011). The populations,
ionization fraction, and continuum emission, are evaluated in separate models
for each value of the helium abundance, and the profile is calculated assuming
spherical geometry in an expanding atmosphere. The contribution of the
extended chromosphere can be noted in the weak emission present on the long
wavelength side of the line. The helium absorption extends substantially
towards shorter wavelengths due to scattering in the expanding atmosphere and
is enhanced by the metastable nature of the lower level of the transition. The
helium lines are essentially P Cygni profiles since the red giants have
extended atmospheres. The population of the lower level of the 1.08$\mu$m
transition peaks at T=18,000K, but lies within a factor of two of its maximum
value between 14,000 and 25,000K; the outflow velocity doubles over this
temperature span.
Various values of the helium abundance, from Y=0.15 to Y=0.50 [log(nHe/nH)
ranging from 10.65 to 11.4], were assumed and 9 models calculated. The
abundance selected minimizes the residuals between the observed and calculated
profiles. The star LEID 54084 clearly exhibits a broad helium line which could
extend to shorter wavelengths beyond the Si I absorption at 1.027$\mu$m but is
compromised by the presence of the water vapor blend with Si I. A value of
Y=0.39 to Y=0.44 well represents the depth of the observed profile
representing the minimum range in the residuals. Helium is not clearly
detected in LEID 54064. The calculated profiles for Y$\leq$0.22 give a minimum
in the residuals, and we adopt this value as an upper limit to Y. Inspection
of the helium profiles shows that a value of Y=0.25 overpredicts the strength
of the line in LEID 54064, and the residuals of the fit are larger than for
Y=0.22. These simulations suggest that the helium abundance difference is
$\Delta Y\geq 0.17$ between the two stars.
## 3 Discussion and Conclusions
This spectroscopic value of helium from LEID 54084, namely Y=0.39–0.44 can be
compared to values obtained from models of stellar structure and evolution. In
Omega Cen, Norris (2004) estimated the presence of helium from isochrones
matching the lower main sequence with values of Y ranging from 0.23 to 0.38.
Piotto et al. (2005) noted the blue main sequence could only be matched with
stellar models with helium abundance ranging from 0.35 $<$ Y $<$ 0.45 and
concluded that Y=0.38 best fit the ridgelines in the color-magnitude diagram
of Omega Cen. HST photometry of an outer field in the cluster (King et al.
2012), reveals a helium abundance for the blue main sequence of
Y=0.39$\pm$0.02. Recent Yonsei-Yale isochrones for several subpopulations in
Omega Cen (Joo & Lee 2013) suggest a range in Y from 0.38 to 0.41. Thus the
spectroscopic value of helium for LEID 54084, a star with enhanced light
element abundances is in harmony with the abundance inferred from stellar
structure models. In a more metal rich cluster, NGC 2808, the approximate
model of Pasquini et al (2011) suggested one star may have a similar value of
Y=0.39 to 0.5.
The Y value for LEID 54064 where the helium line is weak (or not detected) has
an upper limit (Y $\leq$ 0.22) that is slightly less than the cosmic value
(Y=0.24). These abundances suggest the helium enhancement, $\Delta$Y, is
$\geq$0.17. King et al. (2012) concluded from plausible fits to the color
magnitude diagram of Omega Cen that $\Delta$Y$\sim$0.15 where a value for the
primeval abundance of helium (Y=0.24) was chosen for the red main sequence.
Piotto et al. (2005) required $\Delta$Y=0.14 to explain the differences in
metal abundances found for the blue and red main sequences. It is interesting
to note that the Sun requires a helium abundance of Y=0.27$-$0.28 to match the
solar luminosity, but due to diffusion and settling, the helium abundance in
the envelope is less, Y=0.24$-$0.25 (Christensen-Dalsgaard 2002; Guzik & Cox
1993), and Y=0.16 (corresponding to nHe/nH=0.05) in the steady-state solar
wind (Kasper et al. 2007). It may be that spectroscopy will yield different
values for the helium abundance from those inferred from stellar isochrone
models, although currently we do not know if the characteristics of the solar
abundance pattern occur in these metal-poor giant stars.
The optical and near-infrared spectra used here were acquired about 3 months
apart, and a variation in the line profiles might occur. However, these giants
have log $L/L_{\odot}~{}\sim$ 2.2, and $M_{V}\sim-$0.45 and lie on the red
giant branch below the stars that exhibit H$\alpha$ wing emission. It is this
emission which can vary in strength in first-ascent red giants (Mészáros et
al. 2008; Cacciari et al. 2004). The remarkable similarity of H$\alpha$,
H$\beta$, and Ca II-K profiles between the two giants suggests that activity
does not cause significant changes.
Another consideration might be the presence of X-rays or EUV emission from a
high temperature plasma. Because neutral helium can be photoionized and then
recombine preferentially into the lower level of the 1.08$\mu$m line, this
process would enhance the strength of the observed helium line. Red giants
need substantial magnetic confinement of material to produce hot plasma;
magnetic signatures in the spectra of similar stars have not been detected,
and the coronae appear absent (Rosner et al. 1995). The slightly metal-poor K
giant, $\alpha$ Boo, has a ‘tentative detection’ (Ayres et al. 2003) of X-rays
but, if indeed present, they are a factor of 104 weaker in $L_{X}/L_{bol}$
than the average solar value and would seem to have little effect on the
profile.111CHANDRA images of Omega Cen (Cool et al. 2013) do not reach faint
sources ($L_{X}\lesssim 10^{29}\ erg\ s^{-1}$). The identified optical
counterparts of the X-ray sources are binaries, and not the single red giants
that are targeted here. In $\alpha$ Boo, the equivalent width of the
1.083$\mu$m line varies in absorption strength which could be caused by wind
variation as well as chromospheric excitation conditions (O’Brien & Lambert
1986). Single metal-poor red giants in the field also display a very weak
1.08$\mu$m absorption line, and though these stars are optically brighter,
they have not been detected in X-rays. Population I giants, which generally
exhibit X-rays, have stronger helium absorption as compared to their metal-
poor field counterparts (cf. Dupree et al. 2009). This suggests the line is
not influenced by X-rays in the metal-poor stars. The Ca II-K lines are very
similar in the two giants (Fig. 1) indicating that these stars have similar
chromospheres such that X-rays would not be present in only one star causing
the strengthening of the helium absorption. Thus it does not appear likely
that X-rays contribute to the line formation for the targets considered here.
Several epochs of measurement would clearly be useful to determine if
variation occurs in the helium lines.
Pasquini et al. (2011) carried out a similar calculation for two stars in the
globular cluster NGC 2808. The 2 luminous stars ($logL/L_{\odot}\ \sim 3.2$)
selected by Pasquini et al. (2011) have different levels of activity as
indicated by the Ca K line which underscores the ubiquitous variability of
such luminous giants. These differences demand different semi-empirical models
for the two stars yet only one model was used; in addition the observations of
the optical and infrared spectra were separated by some weeks which brings
uncertainty when modeling such luminous active objects. Computation of the
line profiles in Pasquini et al. invokes models that do not adequately
represent the stars nor the conditions in their atmospheres. The use of a
plane-parallel approximation is questionable when modeling a star of radius
$\sim$84 R⊙. The computation assumed a static atmosphere, and the authors
simply shifted the calculated line in wavelength to match observations.
However, the spectra show that the chromosphere, as measured by H$\alpha$, Ca
K, and the helium line, exhibits signatures of outflow. We have taken our
model and calculated the helium profile under the same assumptions adopted by
Pasquini et al. (2011) (plane parallel and static) for comparison to a model
with the appropriate assumptions for these stars, namely spherical geometry
and expanding. The results of this calculation show substantial differences.
Not only does the spherical model exhibit emission, but the absorption is
larger than the static model due to the expanding atmosphere. A larger star,
with extended chromosphere and/or wind, might be expected to exhibit more
substantial changes. For the same value of the helium abundance, the
equivalent width of the absorption in the expanding spherical model is larger
by 5 to 19% than the static plane-parallel model depending on the value of Y.
(Here we assumed Y=0.28 and Y=0.44.) Thus, interpreting the observed profile
formed in an expanding large giant star, by ’matching it’ to a static, plane
parallel profile, as did Pasquini et al. (2011), will lead to an overestimate
of the abundance of helium. Pasquini et al. (2011) do not compare the computed
profiles of Ca K and H$\alpha$, to the stellar spectra so the adequacy of the
models is unknown. Consideration of all of these facts indicates that the
determination of the helium abundance in Pasquini et al. (2011) must be
approached with caution.
The targets selected in this paper are of much lower luminosity where
variability is absent or greatly minimized. Moreover, the 2 stars are
effectively identical in temperature, luminosity, iron abundance, activity,
and in chromospheric features - with the exception of helium and enhanced Al
and Mg. The treatment of the radiative transfer is state of the art with a
spherical atmosphere, assuming an outflow, where the outflow is incorporated
into the source function for the lines.
The abundance of helium and its variation between these two giant stars in
Omega Cen gives quantitative observational confirmation of a helium
enhancement to accompany the enhanced light metals. The near-ir line of He I
can provide a probe of the helium abundance in cool stars when additional
chromospheric profiles are available to constrain the atmospheric structure
and dynamics and appropriate radiative transfer calculations are employed.
We are grateful to Bob Kurucz who calculated specific photospheric models to
initiate the calculations. Facilities: Gemini:South (PHOENIX), Magellan:Clay
(MIKE)
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Figure 1: H$\alpha$ , H$\beta$, and Ca II K-line region shown for the two
giant stars with the model fit overlaid (indicated by a solid line which is
colored red in the electronic edition). The spectra of the two stars are
virtually identical. The Ca II spectrum displays an interstellar absorption
feature blended with the Fe I line at $-$3Å from the line core in the LEID
54064 spectrum, but distinct from the Fe I feature in the LEID 54084 spectrum.
Weak emission may be present on the long-wavelength wing of the H$\alpha$ line
and on both wings of the H$\beta$ line. The success of the model can be seen
from the agreement between observed and calculated chromospheric profiles. A
color version of this figure is in the electronic edition. Figure 2: Final
model. The top panel displays the total hydrogen density (left axis) and the
temperature (right axis). The lower panel shows the turbulent and outflow
velocities needed to match the observed profiles.
Figure 3: Observed Helium lines binned to a resolution element in the two
matched red giants with the 10830Å profile as calculated for several values of
the helium abundance (shown by a solid smoothly varying line that is colored
red in the electronic edition). Values of Y are given for the calculated
curves arranged top to bottom and the corresponding log (nHe/nH) is shown in
parentheses where log $n_{H}$=12.00. Table 1: Characteristics of Target Stars
Quantity | LEID 54064 | LEID 54084 | Refs.
---|---|---|---
V | 13.27 | 13.21 | 1
B$-$V | 1.048 | 1.044 | 1
Ks | 10.62 | 10.56 | 2
Teff [K] | 4741 | 4745 | 3
log g [cm s-2] | 1.76 | 1.74 | 3
MV | $-$0.43 | $-$0.49 | 4
log $L/L_{\odot}$ | 2.21 | 2.23 | 5
$[Fe/H]$ | $-$1.86 | $-$1.79 | 3
$[Na/Fe]$ | $-$0.14 | 0.37 | 3
$[Al/Fe]$ | $\leq$0.36 | 1.12 | 3
EW (He I) [mÅ] | $\leq$9.2 | 89.5 | 3
References. — (1) van Leeuwen et al. 2000 ;(2) 2MASS All Sky Survey; Skrutskie
et al. 2006; (3) Dupree et al. 2011; (4) Distance modulus from Johnson &
Pilchowski 2010; (5) Bolometric correction from Alonso et al. 1999.
|
arxiv-papers
| 2013-07-22T20:01:06 |
2024-09-04T02:49:48.309195
|
{
"license": "Public Domain",
"authors": "A. K. Dupree and E. H. Avrett (Harvard-Smithsonian Center for\n Astrophysics)",
"submitter": "Andrea Dupree",
"url": "https://arxiv.org/abs/1307.5860"
}
|
1307.5891
|
# Quantum Synchronization of Two Ensembles of Atoms
Minghui Xu, D. A. Tieri, E. C. Fine, James K. Thompson, and M. J. Holland
JILA, National Institute of Standards and Technology and Department of
Physics, University of Colorado, Boulder, Colorado 80309-0440, USA
###### Abstract
We propose a system for observing the correlated phase dynamics of two
mesoscopic ensembles of atoms through their collective coupling to an optical
cavity. We find a dynamical quantum phase transition induced by pump noise and
cavity output-coupling. The spectral properties of the superradiant light
emitted from the cavity show that at a critical pump rate the system undergoes
a transition from the independent behavior of two disparate oscillators to the
phase-locking that is the signature of quantum synchronization.
###### pacs:
05.45.Xt, 42.50.Lc, 37.30.+i, 64.60.Ht
Synchronization is an emergent phenomenon that describes coupled objects
spontaneously phase-locking to a common frequency in spite of differences in
their natural frequencies book1 . It was famously observed by Huygens, the
seventeenth century clock maker, in the antiphase synchronization of two
maritime pendulum clocks Huygens . Dynamical synchronization is now recognized
as ubiquitous behavior occurring in a broad range of physical, chemical,
biological, and mechanical engineering systems book1 ; book2 ; book3 .
Theoretical treatments of this phenomenon are often based on the study of
phase models kuramoto0 ; kuramoto , and as such have been applied to an
abundant variety of classical systems, including the collective blinking of
fireflies, the beating of heart cells, and audience clapping. The concept can
be readily extended to systems with an intrinsic quantum mechanical origin
such as nanomechanical resonators Cross04 ; Milburn12 , optomechanical arrays
Marquardt11 , and Josephson junctions Jain84 ; Wiesenfeld96 . When the number
of coupled oscillators is large, it has been demonstrated that the onset of
classical synchronization is analogous to a thermodynamic phase transition
Winfree67 and exhibits similar scaling behavior Oleg09 .
Recently, there has been increasing interest in exploring manifestations in
the quantum realm. Small systems have been considered, e.g., one qubit
Zhirov06 and two qubits Zhirov09 coupled to a quantum dissipative driven
oscillator, two dissipative spins Orth10 , two coupled cavities Tony12 , and
two micromechanical oscillators mian12 ; Mari12 . Connections between quantum
entanglement and synchronization have been revealed in continuous variable
systems Mari12 . It has been shown that quantum synchronization may be
achieved between two canonically conjugate variables Hriscu13 . Since the
phenomenon is inherently non-equilibrium, all of these systems share the
common property of competition between coherent and incoherent driving and
dissipative forces.
In this paper, we propose a modern-day realization of the original Huygens
experiment Huygens . We consider the synchronization of two active atomic
clocks coupled to a common single-mode optical cavity. It has been predicted
that in the regime of steady-state superradiance Meiser09 ; Meiser101 ;
Thompson12 ; Thompson121 a neutral atom lattice clock could produce an
ultracoherent optical field with a quality factor (ratio of frequency to
linewidth) that approaches $10^{18}$. We show that two such clocks may exhibit
a dynamical phase transition Zoller10 ; Zoller11 ; Cirac12 ; Cirac13 from two
disparate oscillators to quantum phase-locked dynamics. The onset of
synchronization at a critical pump strength is signified by an abruptly
increased relative phase diffusion that diverges in the thermodynamic limit.
Besides being of fundamental importance in nonequilibrium quantum many-body
physics, this work could have broad implications for many practical
applications of ultrastable lasers and precision measurements Meiser09 .
Figure 1: (color online) Two ensembles of driven two-level atoms coupled to a
single-mode cavity field. The atoms in ensemble $A$ are detuned above the
cavity resonance (dashed line). Ensemble $B$ contains atoms detuned below the
cavity resonance by an equivalent amount.
The general setup is shown schematically in Fig. 1. Two ensembles, each
containing $N$ two-level atoms with excited state $|e\rangle$ and ground state
$|g\rangle$, are collectively coupled to a high-quality optical cavity. The
transition frequencies of the atoms in ensembles $A$ and $B$ are detuned from
the cavity resonance by $\delta/2$ and $-\delta/2$ respectively. This could be
achieved by spatially separating the ensembles and applying an inhomogeneous
magnetic field to induce a differential Zeeman shift. The atoms in both
ensembles are pumped incoherently to the excited state, as could be realized
by driving a transition to a third state that rapidly decays to $|e\rangle$
Thompson12 ; Thompson121 .
This system is described by the Hamiltonian in the rotating frame of the
cavity field:
$\hat{H}=\frac{\hbar\delta}{2}(\hat{J}_{A}^{z}-\hat{J}_{B}^{z})+\frac{\hbar\Omega}{2}(\hat{a}^{\dagger}\hat{J}_{A}^{-}+\hat{J}_{A}^{+}\hat{a}+\hat{a}^{\dagger}\hat{J}_{B}^{-}+\hat{J}_{B}^{+}\hat{a})\,,$
(1)
where $\Omega$ is the atom-cavity coupling, and $\hat{a}$ and
$\hat{a}^{\dagger}$ are annihilation and creation operators for cavity
photons. Here
$\hat{J}_{A,B}^{z}=\frac{1}{2}\sum_{j=1}^{N}\hat{\sigma}_{(A,B)j}^{z}$ and
$\hat{J}_{A,B}^{-}=\sum_{j=1}^{N}\hat{\sigma}_{(A,B)j}^{-}$ are the collective
atomic spin operators, written in terms of the Pauli operators for the two-
level system $\hat{\sigma}_{(A,B)j}^{z}$ and
$\hat{\sigma}_{(A,B)j}^{-}=(\hat{\sigma}_{(A,B)j}^{+})^{\dagger}$.
In addition to the coherent atom-cavity coupling, incoherent processes are
critical and include: the cavity intensity decay at rate $\kappa$, the pump
rate $w$, the free-space spontaneous emission rate $\gamma$, and a background
dephasing of the $|e\rangle$–$|g\rangle$ transition at rate $T_{2}^{-1}$. The
total system is then described using a master equation for the reduced density
operator $\rho$:
$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{i\hbar}[\hat{H},\rho]+\kappa\mathcal{L}[\hat{a}]\,\rho+\sum_{\mathcal{T}=A,B}\sum_{j=1}^{N}\Bigl{(}\gamma_{s}\mathcal{L}[\hat{\sigma}_{\mathcal{T}j}^{-}]$
(2)
$\displaystyle{}+w\mathcal{L}[\hat{\sigma}_{\mathcal{T}j}^{+}]+\frac{1}{2T_{2}}\mathcal{L}[\hat{\sigma}_{\mathcal{T}j}^{z}]\Bigr{)}\,\rho,$
where
$\mathcal{L}[\hat{O}]\,\rho=(2\hat{O}\rho\hat{O}^{\dagger}-\hat{O}^{\dagger}\hat{O}\rho-\rho\hat{O}^{\dagger}\hat{O})/2$
denotes the Lindblad superoperator.
The regime of steady-state superradiance is defined by the cavity decay being
much faster than all other incoherent processes Meiser09 ; Meiser101 ;
Thompson12 ; Thompson121 . In this regime, the cavity can be adiabatically
eliminated Meiser101 , resulting in a field that is slaved to the collective
atomic dipole of the two ensembles of atoms:
$\hat{a}\simeq-\frac{i\Omega}{\kappa+i\delta}\hat{J}_{A}^{-}-\frac{i\Omega}{\kappa-i\delta}\hat{J}_{B}^{-}.$
(3)
For small detuning on the scale of the cavity linewidth, $\delta\ll\kappa$,
Eq. (3) reduces to $\hat{a}\simeq-i\Omega\hat{J}^{-}/\kappa$, where
$\hat{J}^{-}=\hat{J}_{A}^{-}+\hat{J}_{B}^{-}$ is the total collective spin-
lowering operator. In this limit, the net effect of the cavity is to provide a
collective decay channel for the atoms, with rate
$\gamma_{c}=\Omega^{2}/\kappa$. This collective decay should be dominant over
other atomic decay processes Meiser101 , i.e.,
$N\gamma_{c}\gg\gamma_{s},T_{2}^{-1}$, so that the time evolution is
effectively given by a superradiance master equation containing only atoms:
$\frac{d\rho}{dt}=\frac{\delta}{2i\hbar}[J_{A}^{z}-J_{B}^{z},\rho]+\gamma_{c}\mathcal{L}[\hat{J}^{-}]\,\rho+w\sum_{j=1}^{N}(\mathcal{L}[\hat{\sigma}_{Aj}^{+}]+\mathcal{L}[\hat{\sigma}_{Bj}^{+}])\,\rho.$
(4)
With this system we naturally provide the three necessary ingredients for
quantum synchronization; a controllable difference between the oscillation
frequencies of two mesoscopic ensembles, a dissipative coupling generated by
the emission of photons into the same cavity mode, and a driving force
produced by optical pumping.
The photons emitted by the cavity provide directly measurable observables.
Synchronization is evident in the properties of the photon spectra. In the
case of two independent ensembles in the unsynchronized phase, each ensemble
radiates photons at its own distinct transition frequency. This leads to two
Lorentzian peaks that are typically well-separated. In the synchronized phase,
all of the atoms radiate at a common central frequency resulting in a single
peak.
To solve this problem and find the steady state, we use a semiclassical
approximation that is applicable to large atom numbers. Cumulants for the
expectation values of system operators
$\\{\hat{\sigma}_{(A,B)j}^{z},\hat{\sigma}_{(A,B)j}^{\pm}\\}$ are expanded to
second order Meiser09 ; Meiser101 . All expectation values are symmetric with
respect to exchange of atoms within each ensemble, i.e.
$\langle\hat{\sigma}_{Bi}^{+}\hat{\sigma}_{Bj}^{-}\rangle=\langle\hat{\sigma}_{B1}^{+}\hat{\sigma}_{B2}^{-}\rangle$,
for all $i\neq j$. Due to the U(1) symmetry,
$\langle\hat{\sigma}_{(A,B)j}^{\pm}\rangle=0$. Therefore, all nonzero
observables can be expressed in terms of
$\langle\hat{\sigma}_{(A,B)j}^{z}\rangle$,
$\langle\hat{\sigma}_{(A,B)i}^{+}\hat{\sigma}_{(A,B)j}^{-}\rangle$, and
$\langle\hat{\sigma}_{(A,B)i}^{z}\hat{\sigma}_{(A,B)j}^{z}\rangle$.
Expectation values involving only one ensemble are the same for both ensembles
and for these cases we omit the superfluous $A$,$B$ subscripts. The equations
of motion can then be found from Eq. (4):
$\displaystyle\frac{d}{dt}\langle\hat{\sigma}_{1}^{z}\rangle$
$\displaystyle=-\gamma_{c}\left(\langle\hat{\sigma}_{1}^{z}\rangle+1\right)-w\left(\langle\hat{\sigma}_{1}^{z}\rangle-1\right)$
(5)
$\displaystyle{}-2\gamma_{c}(N-1)\langle\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}\rangle-\gamma_{c}N\left(\langle\hat{\sigma}_{A1}^{+}\hat{\sigma}_{B1}^{-}\rangle+{\rm
c.c}\right),$
$\displaystyle\frac{d}{dt}\langle\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}\rangle$
$\displaystyle=-(w+\gamma_{c})\langle\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}\rangle+\frac{\gamma_{c}}{2}\left(\langle\hat{\sigma}_{1}^{z}\hat{\sigma}_{2}^{z}\rangle+\langle\hat{\sigma}_{1}^{z}\rangle\right)$
(6)
$\displaystyle{}+\gamma_{c}(N-2)\langle\hat{\sigma}_{1}^{z}\rangle\langle\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}\rangle$
$\displaystyle{}+\frac{\gamma_{c}}{2}N\langle\hat{\sigma}_{1}^{z}\rangle\left(\langle\hat{\sigma}_{A1}^{+}\hat{\sigma}_{B1}^{-}\rangle+{\rm
c.c}\right),$
$\displaystyle\frac{d}{dt}\langle\hat{\sigma}_{A1}^{+}\hat{\sigma}_{B1}^{-}\rangle$
$\displaystyle=-(w+\gamma_{c}-i\delta)\langle\hat{\sigma}_{A1}^{+}\hat{\sigma}_{B1}^{-}\rangle+\frac{\gamma_{c}}{2}\left(\langle\hat{\sigma}_{A1}^{z}\hat{\sigma}_{B1}^{z}\rangle+\langle\hat{\sigma}_{1}^{z}\rangle\right)$
(7)
$\displaystyle{}+\gamma_{c}(N-1)\langle\hat{\sigma}_{1}^{z}\rangle\left(\langle\hat{\sigma}_{A1}^{+}\hat{\sigma}_{B1}^{-}\rangle+\langle\hat{\sigma}_{1}^{+}\hat{\sigma}_{2}^{-}\rangle\right),$
describing population inversion, spin-spin coherence within each ensemble, and
correlation between ensembles, respectively. In deriving Eq. (6) and (7), we
have dropped third order cumulants semi . We also factorize
$\langle\hat{\sigma}_{(A,B)i}^{z}\hat{\sigma}_{(A,B)j}^{z}\rangle\approx\langle\hat{\sigma}_{1}^{z}\rangle^{2}$,
which we find to be valid outside the regime of very weak pumping where a non-
factorizable subradiant dark state plays an important role Meiser101 . After
making these approximations, Eq. (5) to (7) form a closed set of equations.
The steady state is found by setting the time derivatives to zero and the
resulting algebraic equations can be solved exactly. These solutions are the
basis for the figures shown below.
Figure 2: (color online) Steady-state relative phase precession for two
ensembles as a function of detuning at $w=N\gamma_{c}/2$ for $N=100$ (blue
dashed line), $N=500$ (purple dot dashed line) and $N=10^{6}$ (red solid
line). The straight dotted line is $\delta=\Delta$.
In order to calculate the photon spectrum, we employ the quantum regression
theorem qr to obtain the two-time correlation function of the light field,
$\langle\hat{a}^{\dagger}(\tau)\hat{a}(0)\rangle$, where time 0 denotes an
arbitrary time-origin in steady-state. In the limit $\delta\ll\kappa$,
according to Eq. (3), the phase diffusion of the atoms and light are the same,
i.e.
$\langle\hat{a}^{\dagger}(\tau)\hat{a}(0)\rangle\sim\langle\hat{J}^{+}(\tau)\hat{J}^{-}(0)\rangle$.
We begin by deriving equations of motion for
$\langle\hat{\sigma}_{A1}^{+}(\tau)\hat{\sigma}_{B1}^{-}(0)\rangle$ and
$\langle\hat{\sigma}_{B1}^{+}(\tau)\hat{\sigma}_{B2}^{-}(0)\rangle$:
$\frac{d}{d\tau}\begin{pmatrix}\langle\hat{\sigma}_{A1}^{+}(\tau)\hat{\sigma}_{B1}^{-}(0)\rangle\\\
\langle\hat{\sigma}_{B1}^{+}(\tau)\hat{\sigma}_{B2}^{-}(0)\rangle\end{pmatrix}=\frac{1}{2}\begin{pmatrix}X&Y\\\
Y&X^{*}\end{pmatrix}\begin{pmatrix}\langle\hat{\sigma}_{A1}^{+}(\tau)\hat{\sigma}_{B1}^{-}(0)\rangle\\\
\langle\hat{\sigma}_{B1}^{+}(\tau)\hat{\sigma}_{B2}^{-}(0)\rangle\end{pmatrix},$
(8)
where
$X=\gamma_{c}(N-1)\langle\hat{\sigma}_{1}^{z}(0)\rangle-\gamma_{c}-w+i\delta\,,Y=\gamma_{c}N\langle\hat{\sigma}_{1}^{z}(0)\rangle\,.$
We have systematically factorized:
$\displaystyle\langle\hat{\sigma}_{1}^{z}(\tau)\hat{\sigma}_{A1}^{+}(\tau)\hat{\sigma}_{B1}^{-}(0)\rangle$
$\displaystyle\approx$
$\displaystyle\langle\hat{\sigma}_{1}^{z}(0)\rangle\langle\hat{\sigma}_{A1}^{+}(\tau)\hat{\sigma}_{B1}^{-}(0)\rangle\,,$
$\displaystyle\langle\hat{\sigma}_{1}^{z}(\tau)\hat{\sigma}_{B1}^{+}(\tau)\hat{\sigma}_{B2}^{-}(0)\rangle$
$\displaystyle\approx$
$\displaystyle\langle\hat{\sigma}_{1}^{z}(0)\rangle\langle\hat{\sigma}_{B1}^{+}(\tau)\hat{\sigma}_{B2}^{-}(0)\rangle\,.$
(9)
Similarly, one finds that
$\langle\hat{\sigma}_{A1}^{+}(\tau)\hat{\sigma}_{A2}^{-}(0)\rangle$ and
$\langle\hat{\sigma}_{B1}^{+}(\tau)\hat{\sigma}_{A1}^{-}(0)\rangle$ satisfy
the same equation of motion as Eq. (8). The solution of this coupled set is
straightforward and shows that both
$\langle\hat{\sigma}_{A1}^{+}(\tau)\hat{\sigma}_{B1}^{-}(0)\rangle$ and thus
also $\langle\hat{a}^{\dagger}(\tau)\hat{a}(0)\rangle$ evolve in proportion to
the exponential:
$\exp\left[-\frac{1}{2}\left(w+\gamma_{c}-(N-1)\gamma_{c}\langle\hat{\sigma}_{1}^{z}\rangle-\sqrt{(N\gamma_{c}\langle\hat{\sigma}_{1}^{z}\rangle)^{2}-\delta^{2}}\right)\tau\right],$
(10)
which we parametrize by $\exp\left[-(\Gamma+i\Delta)\tau/2\right]$, where
$\Gamma$ represents the decay of the first-order correlation and $\Delta$ the
modulation frequency. Laplace transformation yields the photon spectrum which
consists of Lorentzians of halfwidth $\Gamma/2$ centered at frequencies
$\pm\Delta/2$.
Figure 3: (color online) (a) Nonequilibrium phase diagram of the quantum
synchronization represented by $\Gamma$ (in units of $\gamma_{c}$) on the
$w$-$\delta$ parameter plane, where the dissipative coupling $N\gamma_{c}$
($N=10^{4}$) is fixed. An abrupt peak is observed at the boundary between the
synchronized and unsynchronized phases. (b) As for (a) but on the
$w$-$N\gamma_{c}$ parameter plane.
The importance of the two-time correlation function is that it provides direct
access to the correlated phase dynamics of the two ensembles. The parameter
$\Delta$ physically represents the precession frequency of the phase of the
collective mesoscopic dipoles with respect to one another. In Fig. 2, we show
$\Delta$ as a function of $\delta$ at $w=N\gamma_{c}/2$ for several values of
$N$. For large detuning, $\Delta$ approaches $\delta$, indicating that the
dipoles precess independently at their uncoupled frequency. Below a critical
$\delta$, we find $\Delta$ to be zero, indicating synchronization and phase
locking.
The fact that this system undergoes a synchronization transition that is
fundamentally quantum mechanical and thus quite distinct from the classical
synchronization previously discussed for coupled oscillators is evident in the
observed properties of the linewidth $\Gamma$ of the Lorenzian peak(s),
representing the relative quantum phase diffusion of the collective dipoles.
This system has three independent control variables; the detuning $\delta$,
the dissipative coupling $N\gamma_{c}$ and the pumping $w$, so we show
$\Gamma$ on the $w$-$\delta$ parameter plane in Fig. 3(a) and on the
$w$-$N\gamma_{c}$ parameter plane in Fig. 3(b).
In the region of no quantum correlation, the quantum noise due to pumping
destroys the coherences between spins faster than the collective coupling
induced by the cavity field can reestablish them. Therefore the mesoscopic
dipole is destroyed and the observed spectra are broad. In both the
synchronized and unsynchronized regions, spins within each ensemble are well-
correlated so that the corresponding Lorenzian peaks have ultranarrow
linewidth. As is apparent in Fig. 3(a), the two ensembles cannot be
synchronized when $N\gamma_{c}<\delta$ since then the coherent coupling is not
sufficient to overcome the relative precession that arises from the detuning.
For strong coupling, $N\gamma_{c}>\delta$, the synchronization transition
occurs as the pump rate passes a critical value. The two phases on either side
of the critical region are abruptly separated. As one approaches the
synchronized phase from the unsynchronized one by variation of either $\delta$
or $w$, the linewidth increases rapidly, showing amplification of the effect
of quantum noise in vicinity of the critical point. After passage of the
critical region, the linewidth drops rapidly, leading to rigid phase locking
between the two collective dipoles.
We emphasize that the synchronization dynamics shown in Fig. 2 and 3 is a
dynamical phase transition Zoller10 ; Zoller11 ; Cirac12 ; Cirac13 that is
reminiscent of a second-order quantum phase transition.
Figure 4: (color online) Finite size scaling behavior of the quantum
criticality for $\delta=N\gamma_{c}/2$. For $N\rightarrow\infty$, the critical
pump rate is $w_{c}=\delta$. The red dots show the offset between the critical
pump rate $w_{N}$ for finite $N$ and $w_{c}$. The blue squares show $\Gamma$
(in units of $\gamma_{c}$) at $w_{N}$. Both exhibit linear scalings on the
log-log plot.
To capture features of the quantum criticality, we numerically study the
finite size scaling behavior. Fig. 4 shows both the critical pump rate $w_{N}$
for finite $N$ and the corresponding $\Gamma$ at $w_{N}$. The scaling laws of
$(w_{N}-w_{c})/w_{c}\simeq N^{-0.34}$ and $\Gamma/\gamma_{c}\simeq N^{0.66}$
can be identified.
In Hamiltonian systems, a quantum phase transition results from the
competition between two noncommuting Hamiltonian components with different
symmetries on changing their relative weight. The transition between the two
distinct quantum phases can be identified from the nonanalytical behavior of
an order parameter, and the scaling behavior of certain correlation functions
that diverge at the critical point. By analogy, the synchronization phase
transition is caused by the competition between unitary dynamics that is
parametrized by $\delta$ and enters asymmetrically for the two ensembles, and
driven-dissipative dynamics parametrized by $\gamma_{c}$ that is symmetric.
The order parameter $\Delta$ is zero in the synchronized phase and non-zero in
the unsynchronized phase. The critical behavior is encapsulated by the
divergence of the relative quantum phase diffusion. It should be emphasized
that the treatment given here is quite different to the typical analysis since
the transition is embodied by the characteristic features of the two-time
correlation functions, rather than the behavior of an energy gap or
correlation length.
In the thermodynamic limit, simple expressions for
$\langle\hat{\sigma}_{1}^{z}\rangle$ to leading order in $1/N$ can be
obtained:
$\langle\hat{\sigma}_{1}^{z}\rangle=\left\\{\begin{array}[]{rl}\frac{w}{2N\gamma_{c}},&\mbox{
if $\delta=0$}\\\ \frac{w^{2}+\delta^{2}}{2wN\gamma_{c}},&\mbox{ if
$0<\delta<w$}\\\ \frac{w}{N\gamma_{c}},&\mbox{ if $\delta\geq
w$}\end{array},\right.$ (11)
where $w$ should be such that $\langle\hat{\sigma}_{1}^{z}\rangle<1$. A
critical point at $w_{c}=\delta$ can be found by substituting Eq. (11) into
Eq. (10). In particular, $\Delta=(\delta^{2}-w^{2})^{1/2}$ in the
unsynchronized phase, which shows an analogous critical exponent to that of a
second-order quantum phase transition, i.e., $\beta=1/2$.
In conclusion, we have presented a system that exhibits quantum
synchronization as a modern analogue of the Huygens experiment but is
implemented using state-of-the-art neutral atom lattice clocks of the highest
precision. It will be intriguing in future work to study the many possible
extensions that are inspired by these results, such as the effect of an atom
number imbalance on the synchronization dynamics, and the sensitivity of the
phase-locking to external perturbation.
We acknowledge stimulating discussions with J. Cooper, J. G. Restrepo, D.
Meiser, K. Hazzard, and A. M. Rey. This work has been supported by the DARPA
QuASAR program, the NSF, and NIST.
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* (25) We have validated the closed set of Eq. (5)–Eq. (7) by comparison with exact solutions of the quantum master equation based on applying the SU(4) group theory (see Minghui Xu, D. A. Tieri, M. J. Holland, Phys. Rev. A 87, 062101 (2013)). Due to the presence of multiple ensembles it is difficult to implement exact calculations for more than about ten atoms.
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|
arxiv-papers
| 2013-07-22T21:14:41 |
2024-09-04T02:49:48.319276
|
{
"license": "Public Domain",
"authors": "Minghui Xu, D. A. Tieri, E. C. Fine, James K. Thompson, and M. J.\n Holland",
"submitter": "Murray Holland",
"url": "https://arxiv.org/abs/1307.5891"
}
|
1307.5902
|
# Algebraic cycles and local quantum cohomology
Charles F. Doran and Matt Kerr
Department of Mathematical and Statistical Sciences
University of Alberta, Canada
_e-mail_ : [email protected]
Department of Mathematics, Campus Box 1146
Washington University in St. Louis
St. Louis, MO63130, USA __
_e-mail_ : [email protected]
###### Abstract.
We review the Hodge theory of some classic examples from mirror symmetry, with
an emphasis on what is intrinsic to the A-model. In particular, we illustrate
the construction of a quantum $\mathbb{Z}$-local system on the cohomology of
$K_{\mathbb{P}^{2}}$ and suggest how this should be related to the higher
algebraic cycles studied in [DK].
###### 2000 Mathematics Subject Classification:
14D05, 14D07, 14N35, 32G20, 53D37
This note concerns three types of polarized variations of mixed Hodge
structure (PVMHS) which arise in mirror symmetry:
In each case, at the large complex structure boundary point one obtains a
limiting mixed Hodge structure (LMHS) of Hodge-Tate type. It follows that
replacing $W_{\bullet}$ by the relative weight filtration $M_{\bullet}$
produces a new PVMHS of the form
occurring simultaneously in the A and B models. In particular, the $F^{p}\cap
M_{2p}$ subspaces identify with $H^{3-p,3-p}$ in quantum cohomology.
Let $\Delta^{*}$ denote the punctured unit disk and write
$\mathcal{O}_{\Delta^{*}}=:\mathcal{O}$,
$\Omega_{\Delta^{*}}^{1}=:\Omega^{1}$. A PVMHS
$(\mathbb{V},\mathcal{V},\mathcal{F}^{\bullet},W_{\bullet},\nabla,Q)$ over
$\Delta^{*}$ comprises
* •
a $\mathbb{Z}$-local system $\mathbb{V}$ on $\Delta^{*}$,
* •
the holomorphic vector bundle $\mathcal{V}$ with sheaf of sections
$\mathbb{V}\otimes\mathcal{O}$,
* •
a decreasing filtration by holomorphic subbundles
$\mathcal{F}^{j}\subset\mathcal{V}$,
* •
an increasing filtration by sub local systems
$W_{i}\subset\mathbb{V}_{\mathbb{Q}}:=\mathbb{V}\otimes\mathbb{Q}$,
* •
a flat connection $\nabla:\mathcal{V}\to\mathcal{V}\otimes\Omega^{1}$ with
$\nabla(\mathcal{F}^{\bullet})\subset\mathcal{F}^{\bullet-1}$ and
$\nabla(\mathbb{V})=0$, and
* •
bilinear forms $Q_{i}:\left(Gr_{i}^{W}\mathbb{V}\right)^{\otimes
2}\to\mathbb{Z}$,
such that each
$(Gr_{i}^{W}\mathbb{V}_{s},Gr_{i}^{W}\mathcal{F}_{s}^{\bullet},Q_{i})$
($s\in\Delta^{*}$) yields a polarized Hodge structure. The PVMHS considered
here, as well as all PVMHS arising from geometry, are _admissible_ – i.e. have
well-defined LMHS at $0$.
In the above pictures, the number of bullets in position $(p,q)$ signifies the
dimension of the summand in the Deligne bigrading on $\mathcal{V}$ defined
pointwise by
$I^{p,q}(\mathcal{V}_{s}):=F^{p}\cap
W_{p+q}\cap\left(\overline{F^{q}}+\sum_{j\geq
0}\left\\{\overline{F^{q-j-1}}\cap W_{p+q-j-2}\right\\}\right).$
This bigrading is uniquely determined by the properties
1. (1)
$\oplus_{p\geq j}\oplus_{q}I^{p,q}(\mathcal{V}_{s})=\mathcal{F}_{s}^{j}$
2. (2)
$\oplus_{p+q\leq i}I^{p,q}(\mathcal{V}_{s})=(W_{i})_{s}\otimes\mathbb{C}$
3. (3)
$\overline{I^{b,a}(\mathcal{V}_{s})}\equiv I^{a,b}(\mathcal{V}_{s})$ modulo
$\oplus_{p<a}\oplus_{q<b}I^{p,q}(\mathcal{V}_{s})$.
In passing to the limit, heuristically one may visualize the bullets in each
line $p+q=i$ moving up and down in such a way that the end result remains
symmetric about this line.
#### Notation:
Set $\ell(s):=\frac{\log(s)}{2\pi i}$. We shall often write $\mathcal{V}$
(instead of the 6-tuple) for a PVMHS.
#### Acknowledgments:
We thank E. Zaslow for a helpful conversation. The first author wishes to
recognize support from the NSERC Discovery Grants Program and the second
author from the NSF under Standard Grant DMS-1068974.
## 1\. Closed string
Beginning on the B-model side, recall how the LMHS construction works for a
pure ($\mathbb{Z}$-)VHS $\mathcal{V}$ of weight 3 over $\Delta^{*}$ with
unimodular polarization $Q$. The weight filtration is the trivial one
$W_{3}=\mathcal{V}\supset W_{2}=\\{0\\}$. Denote the (unipotent part of the)
monodromy operator by $T$, with nilpotent logarithm
$N:=\log(T):\,\mathbb{V_{Q}}\to\mathbb{V_{Q}}.$
There exists an unique filtration
$M_{-1}=\\{0\\}\subset M_{0}\subset M_{1}\subset\cdots\subset
M_{6}=\mathbb{V_{Q}}$
satisfying $N(M_{\alpha})\subset M_{\alpha-2}$ and
$N^{\ell}:Gr_{3+\ell}^{M}\overset{\cong}{\to}Gr_{3-\ell}^{M}$. Untwisting the
local system by
$\tilde{\mathbb{V}}:=e^{-\ell(s)N}\mathbb{V},$
we obtain the canonical extension
$\mathcal{V}_{e}:=\tilde{\mathbb{V}}\otimes\mathcal{O}_{\Delta}.$
Let $\\{\gamma_{i}\\}$ be a multivalued basis of $\mathbb{V}$ generating the
steps of the integral filtration $M_{m}^{\mathbb{Z}}:=\mathbb{V}\cap M_{m}$,
and set
$\tilde{\gamma}_{i}:=e^{-\ell(s)N}\gamma_{i}\in\Gamma(\Delta,\tilde{\mathbb{V}})$.
###### Definition 1.1.
The LMHS of $\mathcal{V}$, denoted informally $V_{lim}$, is given by the data
$V_{lim}^{\mathbb{Z}}:=\mathbb{Z}\langle\\{\tilde{\gamma}_{i}(0)\\}\rangle$,
$\mathcal{F}_{lim}^{\bullet}:=\mathcal{F}_{e}^{p}(0)$, and (monodromy weight)
filtration $M_{\bullet}$ on $V_{lim}:=\mathcal{V}_{e}(0)$.
Assume that $V_{lim}$ is _Hodge-Tate_ , i.e.
$Gr_{2j}^{M}\cong\mathbb{Z}(-j)^{\oplus d_{j}}$ for $j=0,1,2,3$ and $\\{0\\}$
otherwise. (For example, the LMHS for $H^{3}$ of the quintic mirror is of this
type, while that for the Fermat quintic family is _not_.) In the rank $4$
setting, where we must have all $d_{j}=1$, we may pick (for each $j$) a
holomorphic section $e_{j}\in\Gamma(\Delta,\mathcal{F}_{e}^{j}\cap
M_{2j}^{\mathbb{C}})$ mapping to the image of
$\gamma_{j}\in\Gamma(\mathfrak{H},M_{2j}^{\mathbb{Z}})$ in
$\Gamma(\Delta^{*},Gr_{2j}^{M}\mathcal{V})$ hence generating the latter. Write
$e=\\{e_{3},e_{2},e_{1},e_{0}\\}$ and
$\gamma=\\{\gamma_{3},\gamma_{2},\gamma_{1},\gamma_{0}\\}$ for the two bases.
To make things explicit, we have (for some $a,b\in\mathbb{Z}$ and
$e,f\in\mathbb{Q}$)
(1.1) $[Q]_{\gamma}=\left(\begin{array}[]{cccc}0&0&0&1\\\ 0&0&1&0\\\
0&-1&0&0\\\ -1&0&0&0\end{array}\right)=[Q]_{e}\;\text{ and
}\;[N]_{\gamma}=\left(\begin{array}[]{cccc}0&0&0&0\\\ a&0&0&0\\\ e&b&0&0\\\
f&e&-a&0\end{array}\right)$
(cf. [GGK1]), in which we shall demand that $|a|=1$. Replacing the local
coordinate $s$ by $q:=e^{2\pi\sqrt{-1}\tau}$, where
$\tau:=Q(\gamma_{1},e_{3})$, and making full use of the bilinear relations
(e.g.
$Q(\mathcal{F}^{1},\mathcal{F}^{3})=0=Q(\mathcal{F}^{2},\mathcal{F}^{2})$),
the limiting period matrix becomes (cf. [op. cit.])
(1.2)
$_{\tilde{\gamma}(0)}[\mathbf{1}]_{e(0)}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ \frac{f}{2}&e&1&0\\\ \alpha_{0}&\frac{f}{2}&0&1\end{array}\right).$
###### Example 1.2.
For the mirror quintic family, we have (cf. [op. cit.], where the computation
is based on [CdOGP]) $a=-1$, $b=5$, $e=\frac{11}{2}$, $f=-\frac{25}{6}$, and
$\alpha_{0}=\frac{25i}{\pi^{3}}\zeta(3)=:C$.
Following Deligne [De], the $e_{j}(q)|_{\Delta^{*}}$ provide the Hodge(-Tate)
basis of a PVMHS
$(\mathbb{V},\mathcal{V},\mathcal{F}^{\bullet},M_{\bullet},\nabla)$ on
$\Delta^{*}$, denoted $\mathcal{V}_{rel}$ for short. For the connection, we
have
$[\nabla]_{e}=d+\left(\begin{array}[]{cccc}0&0&0&0\\\ 1&0&0&0\\\
0&-Y(q)&0&0\\\ 0&0&-1&0\end{array}\right)\otimes\frac{dq}{(2\pi\sqrt{-1})q}$
where $Y(q)$ defines the Yukawa coupling. In the event that $\mathcal{V}$
comes from $H^{3}(X)$, and $\Phi$ denotes the Gromov-Witten prepotential of
the mirror $X^{\circ}$ (composed with the inverse mirror map), according to
mirror symmetry we have
$Y=\Phi^{\prime\prime\prime}:=\frac{d^{3}\Phi}{d\tau^{3}}.$
###### Example 1.3.
The mirror quintic VHS arises from $H^{3}$ of $X_{\xi}$, which is a smooth
compactification of
$\left\\{1-\xi\left(\sum_{i=1}^{4}x_{i}+\frac{1}{\prod_{i=1}^{4}x_{i}}\right)=0\right\\}\subset\left(\mathbb{C}^{*}\right)^{\times
4}.$
Taking $s:=\xi^{5}$, we obtain $\tau$ and $q$ as above, and
$\Phi(q)=\frac{5}{6}\tau^{3}+\Phi_{h}(q),$
where the holomorphic part
$\Phi_{h}(q)=\frac{1}{(2\pi i)^{3}}\sum_{d\geq 1}N_{d}q^{d}.$
From [CdOGP, GGK1, Pe], we have the mixed Hodge basis
$\displaystyle e_{0}=\gamma_{0}$ $\displaystyle
e_{1}=\gamma_{1}-\tau\gamma_{0}$ $\displaystyle
e_{2}=\gamma_{2}-\left(5\tau+\frac{11}{2}+\Phi_{h}^{\prime\prime}\right)\gamma_{1}+\left(\frac{5}{2}\tau^{2}+\frac{25}{12}+\tau\Phi_{h}^{\prime\prime}-\Phi_{h}^{\prime}\right)\gamma_{0}$
$\displaystyle
e_{3}=\gamma_{3}+\tau\gamma_{2}-\left(\frac{5}{2}\tau^{2}+\frac{11}{2}\tau-\frac{25}{12}+\Phi_{h}^{\prime}\right)\gamma_{1}$
$\displaystyle\mspace{200.0mu}+\left(\frac{5}{6}\tau^{3}+\frac{25}{12}\tau-C+\tau\Phi_{h}^{\prime}-2\Phi_{h}\right)\gamma_{0}.$
Here $e_{3}$ can also be viewed as the class of a holomorphic $3$-form in the
original VHS, whose LMHS is reflected by the presence of $C$. The mirror
$X^{\circ}$ is the Fermat quintic.
Turning to the A-model, we need to define an integral structure, Hodge and
weight filtrations on
$H^{\text{even}}(X^{\circ})=H^{3,3}\oplus H^{2,2}\oplus H^{1,1}\oplus H^{0,0}$
which will lead to VHS, LMHS, and VMHS isomorphic to those on $H^{3}(X)$.
These variations will be defined over a small disk $0<|q|<\epsilon$. For
constructing them, the general idea is to use the family of algebraic
structures on $H^{even}$ parametrized by $\tau[H]\in H^{1,1}(X^{\circ})$,
known as the _(small) quantum cohomology_. (Here $[H]$ the the class of a
hyperplane section and $\tau=\ell(q)$, and we are working in the rank $4$
setting.)
For the filtrations, we set
$F^{a}H^{\text{even}}=\oplus_{i\leq
3-a}H^{i,i}\;,\;\;\;M_{b}H^{\text{even}}=\oplus_{j\geq 3-\frac{b}{2}}H^{j,j}$
so that $\mathcal{F}^{3-k}\cap M_{6-2k}=H^{i,i}(X^{\circ},\mathbb{C})$ as a
subspace of $H^{\text{even}}$. This is where the “naive” fundamental classes
of coherent sheaves or algebraic cycles of codimension $i$ lie. In contrast,
the integral local system will be generated by _quantum-deformed_ fundamental
classes of algebraic cycles on $X^{\circ}$. Alternately, we can regard the
flat structure as given by the solution to a quantum differential equation
$\nabla=d+E\otimes\frac{dq}{(2\pi\sqrt{-1})q},$
which gives the integral structure up to a constant. (Note that $d$
differentiates with respect to $\oplus_{i}H^{i,i}(X^{\circ},\mathbb{C})$.)
Since $E$ kills $M$-graded pieces, we get a natural identification between
$Gr_{2i}^{M}$ of this “integral structure” and
$H^{i,i}(X^{\circ},\mathbb{Z})$.
###### Example 1.4.
For $X^{\circ}$ the Fermat quintic, we have Hodge basis
$[X^{\circ}]=e_{3},\;[H]=e_{2},\;-[L]=e_{1},\;[p]=e_{0}$
where $H$ is a hyperplane section, $L$ a line and $p$ a point. The minus sign
on $[L]$ ensures that the form
$Q(\alpha,\beta):=(-1)^{\frac{\deg(\alpha)}{2}}\int_{X^{\circ}}\alpha\cup\beta$
has matrix $[Q]_{e}$ as above, which is necessary for equality of _polarized_
VHS.
For the quantum deformed classes, we invert the relations of Example 1.3 to
obtain
$\displaystyle[X^{\circ}]_{\mathcal{Q}}=\gamma_{3}=[X^{\circ}]-\tau[H]+\left(\frac{5}{2}\tau^{2}+\frac{25}{12}+\tau\Phi_{h}^{\prime\prime}-\Phi_{h}^{\prime}\right)[L]$
$\displaystyle\mspace{200.0mu}+\left(-\frac{5}{6}\tau^{3}-\frac{25}{12}\tau+C-\tau\Phi_{h}^{\prime}+2\Phi_{h}\right)[p],$
$\displaystyle[H]_{\mathcal{Q}}=\gamma_{2}=[H]-\left(5\tau+\frac{11}{2}+\Phi_{h}^{\prime\prime}\right)[L]+\left(\frac{5}{2}\tau^{2}+\frac{11}{2}\tau-\frac{25}{12}+\Phi_{h}^{\prime}\right)[p],$
$\displaystyle[L]_{\mathcal{Q}}=-\gamma_{1}=[L]-\tau[p],$
$\displaystyle[p]_{\mathcal{Q}}=\gamma_{0}=[p].$
These are solutions to the above differential equation with $E$ given by the
(small) quantum product $[H]*$ defined by
$[H]*[X^{\circ}]=[H],\;\;[H]*[H]=\Phi^{\prime\prime\prime}[L],\;\;[H]*[L]=[p],\text{
and }[H]*[p]=0.$
(Note that this is consistent with cup product, in the sense that
$[H]\cup[H]=5[L]=\Phi^{\prime\prime\prime}(0)[L]$.) The resulting variations
of HS on $H^{\text{even}}(X^{\circ})$ and $H^{3}(X)$ match by construction.
The natural question at this point is: _how much of this “common
$\mathbb{Z}$-VHS” is intrinsic to the A-model, and not just the B-model?_
Clearly the issue lies not in the Hodge and monodromy weight filtrations
(given by the grading of $H^{even}$ by degree), or the polarizing form $Q$, or
the $\nabla$-flat complex local system (given by the quantum product), but in
the integral structure on the latter. Another way to think of this (cf. [De])
is that we must determine the “constant of integration” of the VHS, or
equivalently the LMHS (1.2).
Naively, one could try to find a basis $\delta$ of the local system with
integral $[Q]_{\delta}$ and integral monodromy matrices (which are computable
in principle by analytic continuation). Unfortunately the result may not be
unique, even after identifying bases related by a rational symplectic matrix.
In the above example, one could have
$\delta_{3}=\frac{\gamma_{3}}{\sqrt{5}}+\frac{\gamma_{2}}{\sqrt{5}}\,,\;\delta_{2}=\frac{\gamma_{2}}{\sqrt{5}}-\frac{3\gamma_{1}}{\sqrt{5}}-\frac{3\gamma_{0}}{\sqrt{5}}\,,\;\delta_{1}=\sqrt{5}\xi_{1}\,,\;\delta_{0}=\sqrt{5}\gamma_{0},$
which produces the (distinct) quintic _twin_ mirror $\mathbb{Z}$-VHS. Indeed,
in [DM] this phenomenon is responsible for the bifurcation of each
$\mathbb{R}$-VHS into finitely many distinct $\mathbb{Z}$-VHS.
Instead, what is needed is a direct construction of an integral structure on
quantum cohomology, which has only recently been realized by Iritani [Ir1,
Ir2] and Katzarkov-Kontsevich-Pantev [KKP]. We illustrate how this works in
the setting where $X^{\circ}$ is a smooth CY 3-fold, and $\dim
H^{even}(X^{\circ})=4$. A map $\sigma$ from $H^{even}$ to multivalued
$\nabla$-flat sections (in a neighborhood of $q=0$), defined in terms of
Gromov-Witten theory, has been known for some time (cf. [CK, secs. 8.5.3,
10.2.2]). If $\alpha_{i}\in H^{2(3-i)}(X^{\circ})$ ($i=0,1,2,3$) denote a
$Q$-symplectic basis with $\alpha_{2}=[H]$, this boils down to first setting
$\tilde{\sigma}(\alpha_{0}):=\alpha_{0},\;\;\tilde{\sigma}(\alpha_{1}):=\alpha_{1},\;\;\tilde{\sigma}(\alpha_{2}):=\alpha_{2}+\Phi_{h}^{\prime\prime}\alpha_{1}+\Phi_{h}^{\prime}\alpha_{0},$
$\tilde{\sigma}(\alpha_{3}):=\alpha_{3}+\Phi_{h}^{\prime}\alpha_{1}+2\Phi_{h}\alpha_{0}$
and then
$\sigma(\alpha):=\tilde{\sigma}\left(e^{-\tau[H]}\cup\alpha\right):=\sum_{k\geq
0}\frac{(-1)^{k}}{k!}\tilde{\sigma}\left([H]^{k}\cup\alpha\right).$
(In our running example, we obviously have in mind $\alpha_{3}=[X^{\circ}]$,
$\alpha_{2}=[H]$, $\alpha_{1}=-[L]$, and $\alpha_{0}=[p]$.) These are
$\nabla$-flat sections with monodromy
(1.3) $T(\sigma(\alpha))=\sigma\left(e^{-[H]}\cup\alpha\right).$
We also set $\sigma_{\infty}(\alpha):=\tilde{\sigma}(\alpha)|_{q=0}.$
The key new ingredient introduced by [Ir1, KKP] is a characteristic class
defined using the $\Gamma$-function, and which in our setting specializes to
(1.4) $\hat{\Gamma}(X^{\circ}):=\exp\left(\sum_{k\geq
2}\frac{(-1)^{k}(k-1)!}{(2\pi i)^{k}}\zeta(k)ch_{k}(TX^{\circ})\right)\in
H^{even}(X^{\circ}).$
Using it, we may assign a flat section
(1.5) $\gamma(\xi):=\sigma\left(\hat{\Gamma}(X^{\circ})\cup ch(\xi)\right)$
to each $\xi\in K_{0}^{num}(X^{\circ})$, which defines a $\mathbb{Z}$-local
system. (Similarly, we can define $\tilde{\gamma}(\xi)$,
$\gamma_{\infty}(\xi)$ by applying $\tilde{\sigma}$, $\sigma_{\infty}$.) A
strong indication that $\hat{\Gamma}$ gives the right “correction” is
Iritani’s result (cf. [Ir1, Prop. 2.10]) that the Mukai pairing
$\left\langle\xi,\xi^{\prime}\right\rangle:=\int_{X^{\circ}}ch(\xi^{\vee}\otimes\xi^{\prime})\cup
Td(X^{\circ})\;=\;Q(\gamma(\xi),\gamma(\xi^{\prime})).$
Moreover, since $ch(\mathcal{O}(-1))=e^{-[H]}$, (1.3) implies that
$T(\gamma(\xi))=\gamma(\mathcal{O}(-1)\otimes\xi)$
— an elementary example of how a categorical autoequivalence of
$D^{b}(X^{\circ})$ corresponds to monodromy. The autoequivalences
corresponding to monodromies arising away from $q=0$ have been explicitly
identified in [CIR].
###### Example 1.5.
Once more we take $X^{\circ}$ to be the Fermat quintic, which has total Chern
class $c(X^{\circ})=1+50[L]-200[p]$ and Todd class
$Td(X^{\circ})=1+\frac{25}{6}[L]$. A Mukai-symplectic basis of
$K_{0}^{num}(X^{\circ})$ is
$\xi_{3}:=\mathcal{O}_{X^{\circ}},\;\xi_{2}:=\mathcal{O}_{H}-3\mathcal{O}_{L}-8\mathcal{O}_{p},\;\xi_{1}:=-\mathcal{O}_{L}-\mathcal{O}_{p}\equiv-\mathcal{O}_{L}(1),\;\xi_{0}:=\mathcal{O}_{p};$
this in fact (referring to Example 1.2 and (1.1)) satisfies
$[\mathcal{O}(-1)\otimes]_{\xi}=\exp\left([N]_{\gamma}\right)$. (Note that
taking $\xi_{2}=\mathcal{O}_{H}$ and $\xi_{1}=\mathcal{O}_{L}$ does _not_
yield a symplectic basis.) From
$ch(\xi_{3})=[X^{\circ}],\;ch(\xi_{2})=[H]-\frac{11}{2}[L]-\frac{25}{6}[p],\;ch(\xi_{1})=-[L],\;ch(\xi_{0})=[p]$
and $\hat{\Gamma}(X^{\circ})=[X^{\circ}]+\frac{25}{12}[L]+C[p]$, a
straightforward computation gives that
$\gamma(\xi_{i})=\gamma_{i}\;\;\;(i=0,1,2,3),$
with the $\\{\gamma_{i}\\}$ exactly as in Example 1.4. Moreover, the
$\\{\gamma_{\infty}(\xi_{i})\\}$ recover the LMHS matrix (1.2) (with
$e,f,\alpha_{0}$ as in Example 1.2), including the crucial constant $C$ which
visibly comes from $\hat{\Gamma}$.
###### Remark 1.6.
The toric-hypersurface CY 3-fold families from which B-model VHS’s are often
produced are intrinsically defined over $\mathbb{Q}$. Moreover, by virtue of
its toric nature, the large complex structure limit may be regarded as a
$\mathbb{Q}$-semistable degeneration. The general conjectural framework
surrounding the limiting motive (cf. [GGK1, (III.B.5)]) therefore predicts
that the class $\alpha_{0}\in
Ext_{\text{MHS}}^{1}(\mathbb{Q}(-3),\mathbb{Q}(0))\cong\mathbb{C}/\mathbb{Q}$
arising in the corresponding LMHS is always a rational multiple of the
constant $\frac{\zeta(3)}{(2\pi i)^{3}}$, motivating its appearance in
(1.4).111Note that we are interested in the arithmetic of locally complete CY
families; taking irrational “slices” of such to force an extension both misses
the point and will not affect $\alpha_{0}$.
The “non-toric” degenerations at the conifold and Gepner points, on the other
hand, produce singular fibers whose desingularization may introduce an
algebraic extension of $\mathbb{Q}$, leading to an arithmetically richer LMHS.
One should try to use mirror symmetry to get at this, perhaps beginning with
###### Problem 1.7.
Adapt the (A-model) $\hat{\Gamma}$-integral structure on FJRW theory
introduced in [CIR] to the explicit computation of the periods of (B-model)
LMHS at the Gepner point ($s=\infty$).
See $\S 4$ for another source of algebraic extensions.
## 2\. Local string
This section is based on a simple example studied by [CKYZ], [MOY], [Ho], and
[DK]. Once and for all we set
(2.1)
$Y_{\xi}:=\left\\{(x,y;u,v)\in(\mathbb{C}^{*})^{2}\times\mathbb{C}^{2}\right.\left|1-\xi\left(x+y+\frac{1}{xy}\right)+u^{2}+v^{2}=0\right\\},$
the so-called _Hori-Vafa mirror_ of $Y^{\circ}=K_{\mathbb{P}^{2}}$. The
canonical holomorphic $(3,0)$ form on $Y_{\xi}$ is given by
$\eta_{\xi}=2\sqrt{-1}\text{Res}_{Y_{\xi}}\left(\frac{\frac{dx}{x}\wedge\frac{dy}{y}\wedge
du\wedge dv}{1-\xi(x+y+\frac{1}{xy})+u^{2}+v^{2}}\right).$
The $3$-cycles are spanned in homology by (a) a real $3$-torus
$\mathbb{T}^{3}$ and (b) circle-bundles over membranes in
$(\mathbb{C}^{*})^{\times 2}$ bounding $1$-cycles on the thrice-punctured
elliptic curve
$W_{\xi}^{*}:=\left\\{(x,y)\in(\mathbb{C}^{*})^{2}\right.\left|1-\xi(x+y+\frac{1}{xy})=0\right\\}.$
The circle is pinched to a point over the $1$-cycles.
We write $W_{\xi}$ for the complete elliptic curve,
$\tilde{\omega}_{\xi}:=\frac{1}{2\pi
i}\text{Res}_{W_{\xi}}\left(\frac{\frac{dx}{x}\wedge\frac{dy}{y}}{1-\xi(x+y+\frac{1}{xy})}\right)$
for the canonical holomorphic $1$-form, and $\varphi_{0}$,$\varphi_{1}$ for
$1$-cycles spanning $H_{1}(W_{\xi},\mathbb{Z})$ with periods
$\pi_{i}:=\int_{\varphi_{i}}\tilde{\omega}_{\xi}$. In particular, we let
$\varphi_{0}$ be the vanishing cycle and
$\omega_{\xi}:=\tilde{\omega}_{\xi}/\pi_{0}$ the normalization of the $1$-form
so that $\int_{\varphi_{0}}\omega_{\xi}\equiv 1$.
Denoting the membrane construction (b) by $\mathcal{M}$, we have the short
exact sequence
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\langle\mathbb{T}^{3}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{3}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker\left\\{H_{1}(W^{*})\to
H_{1}((\mathbb{C}^{*})^{2})\right\\}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\mathcal{M}}$$\textstyle{0}$$\textstyle{H_{1}(W)(1)}$
(cf. [DK, sec. 5]).222The isomorphism is valid only rationally, but can be
made integral by replacing $H_{1}(W,\mathbb{Z})$ by $\mathbb{Z}\langle
3\varphi_{0},\varphi_{1}\rangle$, which is done tacitly below. Its dual
$\textstyle{0}$$\textstyle{\mathbb{Z}(-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(W)(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
yields an extension class
$\varepsilon\in\text{Ext}_{\text{MHS}}^{1}\left(\mathbb{Z}(-2),H^{1}(W)\right)\cong\text{Hom}\left(H_{1}(W),\mathbb{C}/\mathbb{Z}(2)\right).$
Miraculously, this is the image of a higher cycle $\Xi\in
K_{2}^{\text{alg}}(W)$ by a generalized Abel-Jacobi map [DK], and the periods
of $\eta$ may be described by
$\frac{1}{2\pi\sqrt{-1}}\int_{\mathcal{M}(\gamma)}\eta\underset{\mathbb{Z}(2)}{\equiv}\langle
AJ(\Xi),\gamma\rangle_{W}\;,\;\;\;\;\frac{1}{(2\pi\sqrt{-1})^{3}}\int_{\mathbb{T}^{3}}\eta=1.$
Normalizing the local coordinate $s:=\xi^{3}$ to $q$ where
$\ell(q):=\tau:=\frac{\pi_{1}}{\pi_{0}}=\int_{\varphi_{1}}\omega_{\xi},$
we remark that $s\mapsto q$ gives the mirror map for the family $W$ of
elliptic curves. Similarly, if we set
$\ell(Q):=\mathcal{T}:=\frac{1}{(2\pi\sqrt{-1})^{3}}\int_{\mathcal{M}(3\varphi_{0})}\eta,$
then $s\mapsto Q$ is the local mirror map for $Y$. The initial VMHS
$\mathcal{V}$ is that on $H^{3}(Y)$, with integral basis333We will ignore for
now the fact that $\gamma_{1}$ is really $\frac{1}{3}$ of an integral class;
it is a more convenient choice for our purposes than
$\mathcal{M}(\varphi_{1})^{\vee}$.
$\gamma=\\{\gamma_{3},\gamma_{2},\gamma_{1}\\}$ where
$\gamma_{3}:=\mathbb{T}^{\vee},\;\gamma_{2}=\mathcal{M}(3\varphi_{0})^{\vee},\;\gamma_{1}=\mathcal{M}(3\varphi_{1})^{\vee}.$
From the exact sequence we can read off the weight filtration
$W_{6}=\mathcal{V}\supset
W_{5}=W_{4}=W_{3}=\langle\gamma_{2},\gamma_{1}\rangle=\text{im}\\{\mu\\}\supset
W_{2}=\\{0\\},$
and Hodge filtration (except for $\mathcal{F}^{3}=\langle\eta\rangle$). The
extension data are recorded by $\mathcal{T}=\langle
AJ(\Xi),3\varphi_{0}\rangle$ and $\Phi:=\langle AJ(\Xi),3\varphi_{1}\rangle$.
The monodromy logarithm
$[N]_{\gamma}=\left(\begin{array}[]{ccc}0&0&0\\\ -1&0&0\\\
\frac{1}{2}&-1&0\end{array}\right)$
leads to a relative weight filtration $M_{\bullet}$. The resulting
$\mathcal{V}_{rel}$ has Hodge-Tate basis
$\displaystyle
e_{3}:=\frac{\eta}{(2\pi\sqrt{-1})^{3}}=\gamma_{3}+\mathcal{T}\gamma_{2}+\Phi\gamma_{1}\in\mathcal{F}^{3}\cap
M_{6},$ $\displaystyle
e_{2}:=\mu(\omega)=\gamma_{2}+\tau\gamma_{1}\in\mathcal{F}^{2}\cap M_{4},$
$\displaystyle e_{1}=\gamma_{1}\in\mathcal{F}^{1}\cap M_{2}.$
From transversality
$\gamma_{2}+\frac{d\Phi}{d\mathcal{T}}\gamma_{1}=\nabla_{\partial_{\mathcal{T}}}e_{3}\in\mathcal{F}^{2}$
we deduce that $\frac{d\Phi}{d\mathcal{T}}=\tau$, which may also be derived
from the fact that logarithmic derivatives of the extension classes give
periods444That is, we have $\delta_{s}\mathcal{T}=\frac{1}{2\pi i}\pi_{0}$,
$\delta_{s}\Phi=\frac{1}{2\pi i}\pi_{1}$. of $\tilde{\omega}_{\xi}$ [op.
cit.]:
$\frac{d\Phi}{d\mathcal{T}}=\frac{s\cdot d\Phi/ds}{s\cdot
d\mathcal{T}/ds}=\frac{\pi_{1}}{\pi_{0}}=\tau.$
This equality has the important consequence
$\Phi^{\prime\prime}:=\frac{d^{2}\Phi}{d\mathcal{T}^{2}}=\frac{d\tau}{d\mathcal{T}}=\frac{\delta_{s}(\pi_{1}/\pi_{0})}{\delta_{s}\mathcal{T}}=\frac{2\pi\sqrt{-1}(\pi_{0}\delta_{s}\pi_{1}-\pi_{1}\delta_{s}\pi_{0})}{\pi_{0}^{3}}=\frac{\mathcal{Y}}{\pi_{0}^{3}},$
where $\mathcal{Y}$ is the (suitably normalized) Yukawa coupling for the
family $\\{W_{\xi}\\}$ of elliptic curves. Noting as well that
$\nabla_{\partial_{\mathcal{T}}}e_{2}=\frac{d\tau}{d\mathcal{T}}e_{1}$, we
conclude that
$[\nabla]_{e}=d+\left(\begin{array}[]{ccc}0&0&0\\\ 1&0&0\\\
0&\Phi^{\prime\prime}&0\end{array}\right)\otimes\frac{dQ}{(2\pi\sqrt{-1})Q}$
where $e=\\{e_{3},e_{2},e_{1}\\}$.
Turning to the A-model, we shall seek a quantum interpretation of $\nabla$.
Before doing so, we remark that by [Ho] and [DK], under the local mirror map
$\Phi$ may be identified as the local Gromov-Witten prepotential
(2.2)
$\Phi\equiv\frac{1}{2}\mathcal{T}^{2}-\frac{1}{(2\pi\sqrt{-1})^{2}}\sum_{d}3dN_{d}Q^{d}.$
modulo lower order terms in $\mathcal{T}$.555A different form of this result
is already present in $\S$6.2 of [CKYZ], about which we shall say more in the
next section. Differentiating (2.2) twice, we have
$1-\sum_{d}3d^{3}N_{d}Q^{d}=\frac{\mathcal{Y}}{\pi_{0}^{3}},$
in which the right-hand side has a pole where the family $W$ degenerates.
Directly computing $\langle AJ(\Xi),\varphi_{0}\rangle$ at this singular
elliptic curve gives
$\Im(\mathcal{T}_{0})=\frac{27\sqrt{3}}{8\pi^{2}}L(\chi_{-3},2)$ [DK], and
hence $Q_{0}=|e^{2\pi\sqrt{-1}\mathcal{T}_{0}}|=e^{-2\pi\Im(\mathcal{T}_{0})}$
for the radius of convergence. This ties the asymptotic growth rate
$\limsup_{d\to\infty}|N_{d}|^{\frac{1}{d}}=e^{2\pi\Im(\mathcal{T}_{0})}$
of the local Gromov-Witten numbers directly to the Beilinson regulator of an
algebraic cycle.
For the quantum interpretation, we consider the dual VMHS $\mathcal{V}^{\vee}$
on $H_{3}(Y)$ under the pairing $H^{3}(Y)\times H_{3}(Y)\to
H_{0}(Y)=\mathbb{Z}.$ The dual integral (flat) basis is of course
$\gamma_{1}^{\vee}=\mathbb{T}^{3},\;\;\gamma_{2}^{\vee}=\mathcal{M}(3\varphi_{0}),\;\;\gamma_{1}^{\vee}=\mathcal{M}(3\varphi_{1}),$
and in the dual Hodge basis
$e^{\vee}=\\{e_{3}^{\vee},e_{2}^{\vee},e_{1}^{\vee}\\}$ we have
(2.3) $[\nabla]_{e^{\vee}}=d-\left(\begin{array}[]{ccc}0&1&0\\\
0&0&\Phi^{\prime\prime}\\\ 0&0&0\\\
\end{array}\right)\otimes\frac{dQ}{2\pi\sqrt{-1}Q}.$
Now recalling that $Y^{\circ}=K_{\mathbb{P}^{2}}$, Hosono [Ho] proposed a
homological mirror map
$\text{mir}:\,K_{0}^{c}(Y^{\circ})\to H_{3}(Y,\mathbb{Z})$
from coherent sheaves with compact support to homology classes of Lagrangian
3-cycles, given explicitly by
(2.4)
$\mathcal{O}_{p}\mapsto\gamma_{3}^{\vee},\;\;\mathcal{O}_{\mathbb{P}^{1}}(-1)\mapsto\gamma_{2}^{\vee},\;\;\mathcal{O}_{\mathbb{P}^{2}}(-2)\mapsto\gamma_{1}^{\vee}.$
(The sheaves are all supported on the zero-section $\mathbb{P}^{2}\subset
Y^{\circ}$.) Making the identifications $e_{3}^{\vee}=[p]$,
$e_{2}^{\vee}=[\mathbb{P}^{1}]$, $e_{1}^{\vee}=[\mathbb{P}^{2}]$ under
$\overline{\text{mir}}:\,H_{\text{even}}(Y^{\circ})\overset{\cong}{\to}H_{3}(Y)$,
we impose as before an integral structure on the A-model side by means of the
quantum deformed classes
$([p]=)\,[p]_{\mathcal{Q}}:=\gamma_{3}^{\vee},\;\;[\mathbb{P}^{1}]_{\mathcal{Q}}:=\gamma_{2}^{\vee},\;\;[\mathbb{P}^{2}]_{\mathcal{Q}}:=\gamma_{1}^{\vee}.$
Together with the filtrations $W_{-6}=W_{-5}=W_{-4}=\langle[p]\rangle\subset
W_{-3}=H_{\text{even}}$, and $\langle[p]\rangle=\mathcal{F}^{-3}\cap M_{-6}$,
$\langle[\mathbb{P}^{1}]\rangle=\mathcal{F}^{-2}\cap M_{-4}$,
$\langle[\mathbb{P}^{2}]\rangle=\mathcal{F}^{-1}\cap M_{-2}$, this determines
the A-model (relative) variation matching that on the B-model.
Finally, consider the formal quantum product
(2.5)
$\begin{matrix}e_{1}^{\vee}*e_{3}^{\vee}=0,\;e_{1}^{\vee}*e_{2}^{\vee}=-3e_{3}^{\vee},\;e_{1}^{\vee}*e_{1}^{\vee}=-3\Phi^{\prime\prime}e_{2}^{\vee},\\\
e_{2}^{\vee}*e_{3}^{\vee}=0,\;e_{3}^{\vee}*e_{3}^{\vee}=0,\;e_{2}^{\vee}*e_{2}^{\vee}=0,\end{matrix}$
where we continue to identify classes under $\overline{\text{mir}}$. This is
compatible with the ordinary cup product in the sense that
$\displaystyle e_{1}^{\vee}\cup e_{3}^{\vee}=[\mathbb{P}^{2}]\cup[p]=0,$
$\displaystyle e_{1}^{\vee}\cup
e_{2}^{\vee}=[\mathbb{P}^{2}]\cup[\mathbb{P}^{1}]=(\mathbb{P}^{2}\cdot\mathbb{P}^{1})_{Y^{\circ}}[p]=-3[p]=-3e_{3}^{\vee},\;\text{and}$
$\displaystyle e_{1}^{\vee}\cup
e_{1}^{\vee}=[\mathbb{P}^{2}]\cup[\mathbb{P}^{2}]=-3[\mathbb{P}^{1}]=-3e_{2}^{\vee},$
the last of which contains the leading term of
$-3\Phi^{\prime\prime}=-3+\cdots$.
###### Proposition 2.1.
With the product (2.5), (2.3) may be rewritten
$\nabla=d+\left(\frac{1}{3}e_{1}^{\vee}*\right)\otimes\frac{dQ}{2\pi\sqrt{-1}Q}$
in terms of the quantum product with the zero-section $\mathbb{P}^{2}\subset
K_{\mathbb{P}^{2}}$.
This motivates the following
###### Problem 2.2.
Develop a general theory of quantum cohomology for the local setting that
produces $\nabla$ on $H_{\text{even}}(Y^{\circ})$ as Prop. 2.1.
We will obtain a solution for our running example in the next section.
The Abel-Jacobi maps from [DK] touched on above may be viewed as maps from
$K_{2}^{\text{alg}}(W)=K_{2}(Coh(W))$ to ($\mathbb{C}/\mathbb{Z}(2)$-valued)
functionals on (classes of) Lagrangian 1-cycles on $W$. Noting that
$W^{\circ}$ is also an elliptic curve, we propose
###### Problem 2.3.
Derive (in general) a homological mirror to $AJ$. This would produce a
“symplectic regulator” map from $K_{2}(Fuk(W^{\circ}))$ to functionals on
coherent sheaves on $W^{\circ}$. The functional mirroring the $AJ$ class in
our example would send
$\mathcal{O}_{p}\mapsto\frac{(2\pi\sqrt{-1})^{2}}{3}\mathcal{T}$ and
$\mathcal{O}_{W^{\circ}}\mapsto\frac{(2\pi\sqrt{-1})^{2}}{3}\Phi$.
The motivation for such a quantum $AJ$ map is clear: it would bring
Beilinson’s conjectures directly to bear upon the arithmetic of GW invariants,
in the context of the A-model VHS on quantum cohomology. A first step might be
to construct, in our example, a mirror in $K_{2}(Fuk(W^{\circ}))$ to the toric
symbol $\\{x,y\\}\in K_{2}^{\text{alg}}(W)$ (i.e. the higher cycle), by
representing $K_{2}^{\text{alg}}(W)$ using the Quillen category of $Coh(W)$
and applying homological mirror symmetry for elliptic curves.
## 3\. Closed to Local
We begin by summarizing a computation from [CKYZ]. The setting is a
2-parameter family $X_{\xi_{1},\xi_{2}}$ of $h^{2,1}=2$ CY 3-folds over a
product of punctured disks, with $\hat{\eta}\in\Omega^{3}(X)$. The mirror
($h^{1,1}=2$) CY has an elliptic fibration
$X^{\circ}\overset{\bar{\rho}}{\to}\mathbb{P}^{2}$
with
* •
zero-section $D_{2}\cong\mathbb{P}^{2}$,
* •
a line $C_{2}\cong\mathbb{P}^{1}\subset D_{2}$ with preimage
$D_{1}=\bar{\rho}^{-1}(C_{2})$, and
* •
a fiber $C_{1}=\bar{\rho}^{-1}(p)$.
We will use the bases
$\left\\{\begin{array}[]{ccc}J_{1}=[D_{2}]+3[D_{1}],\;J_{2}=[D_{1}]&\text{for}&H^{1,1}(X^{\circ})\\\
C_{1},\;C_{2}&\text{for}&H^{2,2}(X^{\circ})\end{array}\right.$
which are dual under cup product. The period vector for $\hat{\eta}$ takes the
form
$\left(\Pi_{0},\tau_{1}\Pi_{0},\tau_{2}\Pi_{0},\partial_{\tau_{1}}\tilde{\Phi},\partial_{\tau_{2}}\tilde{\Phi},2\tilde{\Phi}-\delta_{\tau_{1}}\tilde{\Phi}-\delta_{\tau_{2}}\tilde{\Phi}\right)$
where $\Pi_{0}$ is the “holomorphic period” and
$\textstyle{\tilde{\Phi}:=\frac{3}{2}\tau_{1}^{3}+\frac{3}{2}\tau_{1}^{2}\tau_{2}+\frac{1}{2}\tau_{1}\tau_{2}^{2}+\left\\{\frac{17}{4}\tau_{1}+\frac{3}{2}\tau_{2}+C\right\\}+\frac{1}{(2\pi\sqrt{-1})^{3}}\sum_{d_{1},d_{2}}\tilde{N}_{d_{1},d_{2}}q_{1}^{d_{1}}q_{2}^{d_{2}}}$
is the prepotential.666This is the usual G-W prepotential _plus_ the bracketed
lower-order correction terms. Here, $q_{j}=e^{2\pi\sqrt{-1}\tau_{j}}$ are the
disk-coordinates and $\tilde{N}_{d_{1},d_{2}}$ is the G-W invariant of the
class $d_{1}[C_{1}]+d_{2}[C_{2}]$ on $X^{\circ}$; the Kähler class is simply
$\tau_{1}J_{1}+\tau_{2}J_{2}$.
Now we take $\tau_{1}\to i\infty$ ($q_{1}\to 0$) considered as the “large
volume limit” for the fibers of $\bar{\rho}$. For the purposes of G-W theory
on the A-model, in this limit $X^{\circ}$ is equivalent to the total space of
$\mathcal{N}_{D_{2}/X^{\circ}}\cong K_{\mathbb{P}^{2}}$, i.e. $Y^{\circ}$ in
the last section (with the map
$\rho:Y^{\circ}\twoheadrightarrow\mathbb{P}^{2}$). On the B-model, which we
shall henceforth ignore, the periods remaining finite are $\Pi_{0}$,
$\tau_{2}\Pi_{0}$, and
(3.1)
$(\partial_{\tau_{1}}-3\partial_{\tau_{2}})\tilde{\Phi}\;=\;\frac{1}{2}\tau_{2}^{2}-\frac{1}{4}+\frac{1}{(2\pi\sqrt{-1})^{2}}\sum_{d_{1},d_{2}}\tilde{N}_{d_{1},d_{2}}(d_{1}-3d_{2})q_{1}^{d_{1}}q_{2}^{d_{2}}.$
Indeed, actually taking the limit of (3.1) (and writing
$\mathcal{T}:=\tau_{2}$, $Q:=e^{2\pi\sqrt{-1}\mathcal{T}}$,
$N_{d}:=\tilde{N}_{0,d}$) defines the local prepotential
$\Phi_{\text{loc}}:=\frac{1}{2}\mathcal{T}^{2}-\frac{1}{4}-\frac{1}{(2\pi\sqrt{-1})^{2}}\sum_{d}3dN_{d}Q^{d}$
in agreement with (2.2).777In fact, by a computation in [Ho],
$\Phi=\Phi_{\text{loc}}-\frac{1}{2}\mathcal{T}+\frac{1}{2}$.
The next step is to consider the limit of the quantum products of classes in
$H^{\text{even}}(X^{\circ})$ which come from
$H_{c}^{\text{even}}(Y^{\circ})$($\cong H_{\text{even}}(Y^{\circ})$), namely
$[p]$, $[C_{2}]$, and
$[D_{2}]=J_{1}-3J_{2}.$
In general, the only interesting products (not given by the cup product) are
$J_{j}*J_{k}=\sum_{\ell}\left(\partial_{\tau_{j}}\partial_{\tau_{k}}\partial_{\tau_{\ell}}\tilde{\Phi}\right)[C_{\ell}].$
So (using (3.1)) we have
$[D_{2}]*[D_{2}]=\left(\partial_{\tau_{1}}-3\partial_{\tau_{2}}\right)^{2}\left(\partial_{\tau_{1}}\tilde{\Phi}[C_{1}]+\partial_{\tau_{2}}\tilde{\Phi}[C_{2}]\right)$
$=-3[C_{2}]+\sum_{d_{1},d_{2}}\tilde{N}_{d_{1},d_{2}}(d_{1}-3d_{2})^{2}(d_{1}[C_{1}]+d_{2}[C_{2}])q_{1}^{d_{1}}q_{2}^{d_{2}},$
whereupon taking the limit $\lim_{q_{1}\to 0}[D_{2}]*[D_{2}]=$
$\left\\{-3+\sum_{d}N_{d}(-3d)^{2}dQ^{d}\right\\}[C_{2}]=$
$-3\left\\{1-\sum_{d}3d^{3}N_{d}Q^{d}\right\\}[C_{2}]$
gives
$[\mathbb{P}^{2}]*[\mathbb{P}^{2}]=-3\Phi^{\prime\prime}[\mathbb{P}^{1}]$,
which is exactly what we wanted.
This makes a case for the general principle that the “local restriction” of
the quantum product in a closed CY should remain finite under an appropriate
large volume limit. Beyond establishing this, a solution to Problem 2.2 would
have to show the result is consistent with a formula of the shape888For
$Y^{\circ}\cong K_{\mathbb{P}}\to\mathbb{P}$ with $\mathbb{P}$ a toic Fano
surface, negativity of $K_{\mathbb{P}}$ allows us to express the local
invariants as closed invariants
$\langle\alpha,\beta,\phi^{k}\rangle_{0,3,\iota_{*}(d)}$for
$\overline{Y^{\circ}}:=\mathbb{P}(\mathcal{O}\oplus
K_{\mathbb{P}})\overset{\iota}{\supset}Y^{\circ}$, cf. [CI, sec. 9].
(3.2)
$\alpha*_{\text{loc}}\beta:=\sum_{k}\sum_{d}\langle\alpha,\beta,\phi^{k}\rangle_{0,3,d}^{\text{loc}}\phi_{k}e^{\langle
d,\mathcal{T}\rangle}$
for $\alpha,\beta\in H_{even}\cong H_{c}^{even}$, $\mathcal{T}\in H^{2}$,
$d\in H_{2}$, and $\phi^{k}$ resp. $\phi_{k}$ dual bases of $H^{even}$ resp.
$H_{even}.$
The resulting local quantum cohomology would then provide a direct A-model
approach to “most” of the variation of mixed Hodge structure (the
$\\{I^{p,q}\\}$ and $\nabla$-flat structure), leaving only the
###### Problem 3.1.
Extend Iritani’s construction of an integral structure on $\nabla$-flat
sections to the local CY setting.
This is easily accomplished in our running example by “taking LMHS along
$q_{1}=0$” of the $\mathbb{Z}$-VHS over $(\Delta^{*})^{2}$ (common to both the
A- and B-models). More precisely, if $T_{1}$ denotes the monodromy about
$q_{1}=0$, with logarithm $N_{1}$, then the limiting variation of MHS takes
the form
where the circled bullets denote $\ker(N_{1})=\ker(T_{1}-\text{id})$. For our
purposes, then it will suffice to compute the limit of the $T_{1}$-invariant
“cycles” in the $\hat{\Gamma}$-integral structure on the closed A-model VHS
$H^{even}(X^{\circ})$.
Indeed, together with the Clemens-Schmid sequence, the assumption that
“$Y^{\circ}$ is the A-model limit of $X^{\circ}$” implies that
$\textstyle{0\to
H_{3}(Y)(-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{q_{1}\to
0}{\lim}H^{3}(X)(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{1}}$$\textstyle{\underset{q_{1}\to
0}{\lim}H^{3}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\to
0}$$\textstyle{0\to
H_{even}(Y^{\circ})(-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{q_{1}\to
0}{\lim}H^{even}(X^{\circ})(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{1}}$$\textstyle{\underset{q_{1}\to
0}{\lim}H^{even}(X^{\circ})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{even}(Y^{\circ})\to
0}$
is an exact sequence of VMHS (in $q_{2}$). Iritani’s procedure necessarily
gives integral $\nabla$-flat sections $\\{\hat{\gamma_{i}}\\}_{i=1}^{6}$ in
$H^{even}(X^{\circ})$, with
$\\{\hat{\gamma}_{4},\hat{\gamma}_{5},\hat{\gamma}_{6}\\}\subset\text{im}(N_{1})$
and
$\\{\hat{\gamma}_{1}^{\vee},\hat{\gamma}_{2}^{\vee},\hat{\gamma}_{3}^{\vee}\\}\subset\ker(N_{1})$,
such that
$\frac{\hat{\eta}}{\Pi_{0}}=\hat{\gamma}_{1}+\tau_{2}\hat{\gamma}_{2}+\left\\{(\partial_{\tau_{1}}-3\partial_{\tau_{2}})\tilde{\Phi}\right\\}\hat{\gamma}_{3}+\sum_{j=4}^{6}\hat{\pi}_{j}(\underline{\tau})\hat{\gamma_{j}}.$
Taking the limit whilst killing $\text{im}(N_{1})$, then making the change of
basis
$\\{\hat{\gamma}_{1},\hat{\gamma}_{2},\hat{\gamma}_{3}\\}=:\\{\gamma_{1}+\frac{1}{2}\gamma_{3},\gamma_{2}-\frac{1}{2}\gamma_{3},\gamma_{3}\\}$,
recovers
$e_{3}=\gamma_{1}+\mathcal{T}\gamma_{2}+\Phi\gamma_{3}$
in $H^{even}(Y^{\circ})$.
Of course, in analogy to (3.2), it would be better to solve Problem 3.1 in a
manner intrinsic to the local A-model. That is, there should be a direct
construction as in (1.5) assigning flat sections of $H_{even}(Y^{\circ})$ to
classes in $K_{0}^{c}(Y^{\circ})$, and “compatible with monodromy”. In our
example, (2.4) has this compatibility, since
$\otimes\mathcal{O}_{Y^{\circ}}(-J_{2})$ on the coherent sheaves and monodromy
about $q=0$ on the cycles have the same matrix
$\left(\begin{array}[]{ccc}1&1&0\\\ 0&1&1\\\ 0&0&1\end{array}\right).$
Apparently, either solution still leaves us a long way from the “holy grail”
of Problem 2.3.
## 4\. Open string
Problem 2.3 is probably intractable without major theoretical developments.
However, its rough analogue in the _relative_ situation studied by Morrison
and Walcher [MW] appears to be more accessible. In particular, there is
nothing mysterious about the mirror of the (usual, not higher) algebraic cycle
– it is just a Lagrangian.
The B-model in the example we consider (following [op. cit.]) comprises:
* •
$X=$ a double-cover of the mirror quintic family, with holomorphic form
$\omega\in\Omega^{3}(X)$;
* •
$Z\in CH^{2}(X)_{\text{hom}}$ a family of algebraic 1-cycles (for analogy to
$\S 2$, think “$K_{0}(Coh(X))$”); and
* •
$\langle AJ_{X}^{2}(Z),\omega\rangle=$ the resulting “truncated normal
function”, solving
* •
the inhomogeneous Picard-Fuchs equation $D_{\text{PF}}^{\omega}\langle
AJ_{X}^{2}(Z),\omega\rangle=:g$.
On the A-model side these data mirror to:
* •
$X^{\circ}$= the Fermat quintic;
* •
$Z^{\circ}\cong\mathbb{RP}^{3}$ the real quintic, viewed as a Lagrangian
3-cycle (think “$K_{0}(Fuk(X))$”); and
* •
the Gromov-Witten generating function whose coefficients count holomorphic
disks bounding on $Z^{\circ}$,
which (under the mirror map) solves the same PF equation.
As in the closed and local stories, GW numbers are therefore obtained as
power-series coefficients of a Hodge-theoretic function, with (in this latter
role) the Yukawa coupling replaced by the truncated normal function.
###### Problem 4.1.
Work out (in analogy with $\S\S$1-2) the $[\nabla]_{e}$ story. This will
require the _full_ normal function (not considered in [op. cit.]), which means
computing also $\langle
AJ_{X}^{2}(Z),\nabla_{\partial_{\tau}}\omega^{3,0}\rangle$.
Since the B-model VMHS is an extension of the constant variation
$\mathbb{Z}(-2)$ by the pure VHS $H^{3}(X)$, the extension class is defined
over $\mathbb{R}$ hence given completely by $\langle
AJ_{X}^{2}(Z),\omega^{3,0}\rangle$ and $\langle
AJ_{X}^{2}(Z),\nabla_{\partial_{\tau}}\omega^{3,0}\rangle$. The extension
arises geometrically from the residue exact sequence
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(X\setminus|Z|)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ker\left(H^{4}_{|Z|}(X)\to
H^{4}(X)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Q}(-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
Completely missing, however, is an approach to the following.
###### Problem 4.2.
Can one produce the extension class from the pair $X^{\circ}$,$Z^{\circ}$ from
the standpoint of quantum cohomology and the A-model VMHS?
To illustrate its difficulty, a naive attempt to mirror the exact sequence
approach, viz.
$0\to\frac{H^{\text{even}}(X^{\circ})}{H_{Z^{\circ}}^{6}(X^{\circ})}\to
H^{\text{even}}(X^{\circ}\backslash
Z^{\circ})\to\ker\left(H_{Z^{\circ}}^{3}(X^{\circ})\to
H^{3}(X^{\circ})\right)\to 0,$
fails due to the vanishing of the third term. The result of [op. cit.],
however, that the “truncated” extension class is given by the open GW
generating function, gives one reason to believe the problem has interesting
content.
###### Remark 4.3.
We briefly note another interesting phenomenon that arises in the open
setting, related to Remark 1.6. Even on a family of CY 3-folds defined oveer
$\mathbb{Q}$, algebraic cycles often force an algebraic extension
$L/\mathbb{Q}$ upon us, as in the case of the van Geemen lines on the mirror
quintic family studied by Laporte and Walcher [LW]. The resulting limits of
truncated normal functions can then often be expressed in terms of the Borel
regulator on $K_{3}^{ind}(L)$ (see [GGK2] for the theoretical reason). This
makes the open setting ideal terrain for exploring generalizations of the
A-model $\hat{\Gamma}$-construction where the B-model LMHS does not correspond
to a $\mathbb{Q}$-rational limiting motive.
### 4.1. Local to open
Recent work of Chan, Lau, Leung, Tseng and Wu [CLL, CLT, LLW] has brought to
light an interesting relation between the (local) mirror map and certain open
Gromov-Witten invariants for a toric Calabi-Yau manifold $Y^{\circ}$. The
first three authors conjecture in [CLL] that the SYZ mirror construction
(applied to $Y^{\circ}$) inverts the mirror map given by a normalized integral
basis of single-log-divergent periods of the Hori-Vafa mirror $Y$. With the
integrality hypothesis dropped, the conjecture is established in [CLT] for
$Y^{\circ}=K_{Z}$ with $Z$ a compact toric Fano variety; it is known
integrally for toric surfaces [LLW] and a handful of other examples [CLL],
including $K_{\mathbb{P}^{2}}$.
We briefly describe the case $Y^{\circ}=K_{\mathbb{P}^{2}}$ in the notation of
$\S 2$. Take $\beta_{0}$ to denote the class of a holomorphic disk bounding on
the zero section $D\,(\cong\mathbb{P}^{2})\subset Y^{\circ}$, $\ell$ the class
of a line $L\,(\cong\mathbb{P}^{1})\subset D$; and let
$\mathcal{T}[\rho^{-1}(L)]\in H^{2}(Y,\mathbb{C})$ be th Kähler class with
corresponding Kähler parameter $Q=e^{2\pi i\mathcal{T}}$. Then the SYZ
construction in [op. cit] produces produces the noncompact Calabi-Yau in
$(\mathbb{C}^{*})^{2}\times\mathbb{C}^{2}$ given by
(4.1) $UV=c(Q)+X+Y+\frac{Q}{XY},$
where $c(Q)=1+\sum_{k\geq 1}n_{\beta_{0}+k\ell}Q^{k}$ is a local Gromov-Witten
generating series. An easy change of coordinates exhibits (4.1) as the Hori-
Vafa manifold $Y_{\xi}$ of (2.1), with $\xi=-\frac{Q}{c(Q)^{3}}$; taking the
cube gives
(4.2) $s(Q)=-\frac{Q}{c(Q)^{3}}.$
The observation of [op. cit.] is that (4.2) inverts the local mirror map
$Q(s)=e^{2\pi\sqrt{-1}\mathcal{T}(s)}=\exp\left(\frac{1}{(2\pi\sqrt{-1})^{2}}\int_{\mathcal{M}(3\varphi_{0})}\eta\right)$
in $\S 2$. So just as for $\Phi$, we have an enumerative interpretation for
$\mathcal{T}$, and one can use the computation999up to the sign and term
$\frac{1}{2}$ which are required for consistency with $\S 2$ and [Ho]
$\mathcal{T}(s)=\ell(s)+\frac{1}{2}+\frac{1}{2\pi\sqrt{-1}}\sum_{k\geq
1}\frac{\binom{3k}{k,k,k}}{k}s^{k}$
in [CLL] or [DK] to compute $c(Q)=1-2Q+5Q^{2}-32Q^{3}+\cdots.$
We conclude with one final
###### Problem 4.4.
Can one use the formulae in $\S 5$ of [DK] for the integral periods of Hori-
Vafa mirrors, to establish integrality in [CLT]?
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* [CLL] K. Chan, S.-C. Lau, and N. C. Leung, _SYZ mirror symmetry for toric Calabi-Yau manifolds_ , J. Differential Geom. 90 (2012), No. 2, 177-250.
* [CLT] K. Chan, S.-C. Lau, and H.-H. Tseng, _Enumerative meaning of mirror maps for toric Calabi-Yau manifolds_ , math.AG/:1110.4439v3
* [CIR] A. Chiodo, H. Iritani and Y. Ruan, _Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence_ , math.AG/1201.0813v2.
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* [De] P. Deligne, _Local Behavior of Hodge Structures at Infinity_ , in “Mirror Symmetry II (B. Green, S.–T. Yau, eds.)”, 683-699, AMS/IP Stud. Adv. Math., American Mathematical Society, Providence, RI, 1997.
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|
arxiv-papers
| 2013-07-22T22:11:29 |
2024-09-04T02:49:48.327727
|
{
"license": "Public Domain",
"authors": "Charles F. Doran and Matt Kerr",
"submitter": "Matt Kerr",
"url": "https://arxiv.org/abs/1307.5902"
}
|
1307.6024
|
# Cyclotrons with Fast Variable and/or Multiple Energy Extraction
C. Baumgarten Paul Scherrer Institute, Switzerland
[email protected]
###### Abstract
We discuss the principle possibility of stripping extraction in combination
with reverse bends in isochronous separate sector cyclotrons (and/or FFAGs).
If one uses reverse bends between the sectors (instead of drifts) and places
stripper foils at the sector exit edges, the stripped beam has a reduced
bending radius and it should be able to leave the cyclotron within the range
of the reverse bend - even if the beam is stripped at less than full energy.
We are especially interested in $H_{2}^{+}$-cyclotrons, which allow to double
the charge to mass ratio by stripping. However the principle could be applied
to other ions or ionized molecules as well. For the production of proton beams
by stripping extraction of an $H_{2}^{+}$-beam, we discuss possible designs
for three types of machines: First a low-energy cyclotron for the simultaneous
production of several beams at multiple energies - for instance 15 MeV, 30 MeV
and 70 MeV - thus allowing to have beam on several isotope production targets.
In this case it is desired to have a strong energy dependence of the direction
of the extracted beam thus allowing to run multiple target stations
simultaneously. Second we consider a fast variable energy proton machine for
cancer therapy that should allow extraction (of the complete beam) at all
energies in the range of about 70 MeV to about 250 MeV into the same beam
line. And third, we consider a high intensity high energy machine, where the
main design goals are extraction with low losses, low activation of components
and high reliability. Especially if such a machine is considered for an
accelerator driven system (ADS), this extraction mechanism has severe
advantages: Beam trips by the failure of electrostatic elements could be
avoided and the turn separation could be reduced, thus allowing to operate at
lower main cavity voltages. This would in turn reduce the number of RF-trips.
The price that has to be paid for these advantages is an increase in size
and/or in field strength compared to proton machines with standard extraction
at the final energy.
Cyclotrons, Particle Accelerators, Accelerators in Radiation therapy,
Accelerator Driven Transmutation
###### pacs:
29.20.dg,45.50.Dd,87.56.bd,28.65.+a
## I Introduction
A major fraction of the practical problems in the operation of cyclotrons are
related to beam extraction: the activation of extraction elements increases
the personal dose during maintenance work and sometimes requires to shutoff
the beam long before the scheduled work. The electrostatic elements are
frequently the cause of beam interruptions due to high voltage trips, they
require regular maintenance like cleaning and conditioning. In order to
increase the extraction efficiency, the energy gain per turn must be
maximized, which requires to run cavity and resonators at the limit of what
can be achieved. This in turn increases the frequency of cavity trips and
amplifier failures.
In this work, we propose to utilize the mechanism of stripping extraction,
which is fairly well-established in many machines worldwide in nearly the
complete energy and intensity range that can be achieved by cyclotrons strip0
; strip1 ; strip2 ; strip3 ; strip4 ; strip5 . The extraction mechanism that
we present here might help to avoid most extraction problems completely. In
the case of variable energy extraction as we propose for proton therapy
machines, energy degraders and energy selection systems can be omitted, thus
reducing the costs and the facility footprint significantly. Since the
required beam intensities are typically in the order of $1\,\mathrm{nA}$ at
the patient only, it should be possible to keep a cyclotron with variable
energy extraction almost free from activation of components. Higher beam
currents – of up to $1\,\mathrm{\mu A}$ – are mainly required to compensate
the losses of energy degradation and collimation G2 .
Variable energy extraction by stripping has been proposed and used at the
Manitoba cyclotron by stripping of $H^{-}$-ions hminusvarenergystripper and
in the RACCAM cyclotron Hminus . Unfortunately, the $H^{-}$-ion is not stable
in strong magnetic fields at high energy, so that $H^{-}$-cyclotron are either
limited in energy or restricted magnetic field values. This requires large
radius machines like the TRIUMF cyclotron CraddockSymon . Furthermore, the use
of $H^{-}$ ions in accelerators is more demanding with respect to the machine
vacuum and the production of $H^{-}$ in ion sources.
Cyclotrons (and/or FFAGs) with reverse bends have been proposed in the past
ffag ; revbend0 ; revbend1 – mainly in order to achieve the focusing
conditions that are required for energies of $1\,\mathrm{GeV}$ and above.
However there is no publication known to the authors that proposes the use of
reverse bends in combination with stripping extraction.
In most (if not all) cases where stripping extraction of $H_{2}^{+}$-ions is
used, the proposed extraction schemes lead to complicated orbits that circle
one or even multiple times within the cyclotron before the beam exits strip0 ;
strip1 ; strip2 ; strip3 ; strip4 ; strip5 . The use of this method for
multiple or even for continously variable energy extraction is difficult - if
at all possible.
Another method to achieve beam extraction at variable energy is the variation
of the main field of the cyclotron and to use a sequence of trim coils to
achieve isochronism for the desired extraction energy. This method is the most
“natural” way and it is known to work. However the minimal time to switch
between energies is given by the ramping of the main field and the magnetic
relaxation time of the yoke. In the optimal case it might be possible to
realize energy switching within minutes. We are aiming for a millisecond
range, i.e. energy switching times that are compatable to the time that is
required to adjust a beamline with laminated magnets to the new energy.
The goal of this work is to present first basic geometrical and beam dynamical
studies in order to investigate the feasibility of variable energy cyclotrons
and to explore the energy ranges that could be achieved. Concerning the beam
dynamics we restrict ourselves to the minimum, which we consider to be the
verification of the stability of motion of a coasting beam and of the
extraction mechanism. In order to survey the parameter space for such machines
we restrict ourselves to the so-called hard edge approximation of the magnets.
And we further simplify this approach by assuming homogeneous magnetic fields
within sectors and valleys. Isochronism is achieved exclusively by a variation
of the azimuthal sector width along the orbit schatz .
In Sec. II, we give a description of the geometry and the calculation of the
transfer matrices. The equations given there have been used in Mathematica® to
analyze the orbits and the traces of the transfer matrices in hard edge
approximation in order to find stable solutions with the desired extraction
orbits. Base on the results, a “C”-program was used to generate smoothed
magnetic field maps, which have then been analyzed with an equilibrium orbit
code Gordon and a cartesic tracking code to verify the analytical results of
the hard edge approximation numerically.
In Sections III-V we present the results of the calculations.
## II Geometry of a Separate Sector Cyclotron with Reverse Bends
We consider $H_{2}^{+}$-cyclotrons that are composed of $N$ identical sections
which are each composed of a sequence of homogeneous sector magnets, reverse
bends with homogeneous fields and (optionally) drifts. We do not consider beam
injection nor other details of central regions in detail. We are not concerned
about the question, if these machines might need pre-accelerators or can be
made “compact”. We first consider machines that are composed of exclusively
positive and negative bends as shown in Fig. 1, where we call the positive
bends “sector” and the negative bends “valley”.
Figure 1: Geometry of a cyclotron “section” with reverse bends. The spiralled
“sector” is indicated by a gray polygon. Since it has a constant field, the
equilibrium orbit for a certain energy is composed of two arcs: Within the
sector magnet it has bending radius $R_{s}$ and within the reverse bend
(“valley”), it has a larger radius $R_{v}$. Due to the increasing complexity
of a graphical analysis of the geometry we present an Ansatz for an
algorithmic method in the App. A.
The absolute values of the sector (valley) field is $B_{s}$ ($B_{v}$), the
corresponding bending angle is $2\,\phi_{s}$ ($2\,\phi_{v}$), then we have for
an ion of mass $m$, charge $q$ and momentum $p=m\,c\,\gamma\,\beta$:
$\begin{array}[]{rcl}{\pi\over N}&=&\phi_{s}-\phi_{v}\\\ R_{s}&=&{p\over
q\,B_{s}}={m\,c\,\gamma\,\beta\over q\,B_{s}}\\\ R_{v}&=&{p\over
q\,B_{v}}={m\,c\,\gamma\,\beta\over q\,B_{v}}\\\
L_{tot}&=&2\,N\,(R_{s}\,\phi_{s}+R_{v}\,\phi_{v})=2\,N\,{m\,c\over
q}\,(\frac{\phi_{s}}{B_{s}}+\frac{\phi_{v}}{B_{v}})\,\beta\,\gamma\,.\end{array}$
(1)
Isochronism requires that the velocity $v$, the orbital angular frequency
$\omega_{o}={2\,\pi\over T}$ and the total length of the orbit $L_{tot}$ are
related by
$\begin{array}[]{rcl}v&=&{L_{tot}\over T}={\omega_{o}\,L_{tot}\over 2\,\pi}\\\
\beta&=&{v\over c}={\omega_{o}\,L_{tot}\over 2\,\pi\,c}={L_{tot}\over
2\,\pi\,a}\,,\end{array}$ (2)
where $a={c\over\omega_{o}}$ is the cyclotron length unit. In combination with
Eqn. (1) this yields:
$\begin{array}[]{rcl}2\,\pi\,a\,\beta&=&2\,N\,{m\,c\over
q}\,(\frac{\phi_{s}}{B_{s}}+\frac{\phi_{v}}{B_{v}})\,\beta\,\gamma\\\
a&=&{N\over\pi}\,{m\,c\over
q}\,(\frac{\phi_{s}}{B_{s}}+\frac{\phi_{s}-\pi/N}{B_{v}})\,\gamma\\\
\phi_{s}&=&{\pi\over N\,(1+\lambda)}\,\left({B_{v}\over
B_{0}\,\gamma}+1\right)\,,\end{array}$ (3)
where we used $\lambda={B_{v}\over B_{s}}={R_{s}\over R_{v}}$ and defined the
“nominal field” $B_{0}$ by
$B_{0}={m\,c\over a\,q}={m\over q}\,\omega_{o}\,.$ (4)
From Fig. (1) we pick the following equations
$\begin{array}[]{rcl}R_{v}\,\sin{(\phi_{v})}&=&R\,\sin{(\frac{\pi}{N}-\frac{\alpha}{2})}\\\
&=&R\,\left(\sin{(\frac{\pi}{N})}\,\cos{(\frac{\alpha}{2})}-\cos{(\frac{\pi}{N})}\,\sin{(\frac{\alpha}{2})}\right)\\\
R_{s}\,\sin{(\phi_{s})}&=&R\,\sin{(\frac{\alpha}{2})}\,,\end{array}$ (5)
from which we obtain in a few steps
$\tan{(\frac{\alpha}{2})}={\lambda\,\tan{(\frac{\pi}{N})}\,\tan{(\phi_{s})}\over(1+\lambda)\,\tan{(\phi_{s})}-\tan{(\frac{\pi}{N})}}\,.$
(6)
If $\theta_{c}$ is the azimuthal angle of the sector center and $\theta_{i}$
($\theta_{f}$) are the angles of entrance (exit) of the orbit into the sector,
then
$\begin{array}[]{rcl}\theta_{i}&=&\theta_{c}-\frac{\alpha}{2}\\\
\theta_{f}&=&\theta_{c}+\frac{\alpha}{2}\\\ \end{array}$ (7)
The angles $\varepsilon_{i}$ ($\varepsilon_{f}$) between the sector edges and
the radial direction can be obtained by
$\begin{array}[]{rcl}\tan{(\varepsilon_{i})}&=&R\,{d\theta_{i}\over
dR}=R\,{d\theta_{i}\over d\gamma}/{dR\over d\gamma}\\\
\tan{(\varepsilon_{f})}&=&R\,{d\theta_{f}\over dR}=R\,{d\theta_{i}\over
d\gamma}/{dR\over d\gamma}\\\ \end{array}$ (8)
where $R$ is the radius of the orbit entering (exiting) the sector. The angles
between the orbit and the sector edges (which are required for the transfer
matrices) can be computed by
$\begin{array}[]{rcl}\beta_{i}&=&\phi_{s}+\varepsilon_{i}-\frac{\alpha}{2}\\\
\beta_{f}&=&\phi_{s}-\varepsilon_{f}-\frac{\alpha}{2}\,.\end{array}$ (9)
The radii of the arc centers in the sector $\rho_{s}$ and the valley
$\rho_{v}$ are:
$\begin{array}[]{rcl}\rho_{s}&=&R_{s}\,({\sin{\phi_{s}}\over\tan{(\frac{\alpha}{2})}}-\cos{\phi_{s}})\\\
\rho_{v}&=&R_{v}\,({\sin{\phi_{v}}\over\tan{(\frac{\pi}{N}-\frac{\alpha}{2})}}+\cos{\phi_{v}})\\\
\end{array}$ (10)
The cartesic coordinates $x_{s}(\phi),y_{s}(\phi)$ of the orbit inside the
sector in dependence of the angle $\phi$ can be written as
$\begin{array}[]{rcl}x_{s}(\phi)&=&(\rho_{s}+R_{s}\,\cos{(\phi)})\,\cos{(\theta_{c})}-R_{s}\,\sin{(\phi)}\,\sin{(\theta_{c})}\\\
y_{s}(\phi)&=&(\rho_{s}+R_{s}\,\cos{(\phi)})\,\sin{(\theta_{c})}+R_{s}\,\sin{(\phi)}\,\cos{(\theta_{c})}\,,\end{array}$
(11)
where $\phi$ ranges from $-\phi_{s}$ to $\phi_{s}$. Correspondingly one finds
for the valley:
$\begin{array}[]{rcl}x_{v}(\phi)&=&(\rho_{v}-R_{v}\,\cos{(\phi)})\,\cos{(\theta_{c})}-R_{v}\,\sin{(\phi)}\,\sin{(\theta_{c})}\\\
y_{v}(\phi)&=&(\rho_{v}-R_{v}\,\cos{(\phi)})\,\sin{(\theta_{c})}+R_{v}\,\sin{(\phi)}\,\cos{(\theta_{c})}\,,\end{array}$
(12)
where $\phi$ ranges from $-\phi_{v}$ to $\phi_{v}$.
If we assume that a stripper foil is placed exactly at the sector exit (i.e.
at radius $R$ and azimuthal angle $\theta_{f}$), then the center coordinates
$x_{c}(\phi),y_{c}(\phi)$ of the arc described by the extracted orbit are
$\begin{array}[]{rcl}x_{c}&=&(\rho_{s}+(R_{s}+R_{v}/2)\,\cos{(\phi_{s})})\,\cos{(\theta_{c})}\\\
&-&(R_{s}+R_{v}/2)\,\sin{(\phi_{s})}\,\sin{(\theta_{c})}\\\
y_{c}&=&(\rho_{s}+(R_{s}+R_{v}/2)\,\cos{(\phi_{s})})\,\sin{(\theta_{c})}\\\
&+&(R_{s}+R_{v}/2)\,\sin{(\phi_{s})}\,\cos{(\theta_{c})}\\\ \end{array}$ (13)
coordinates $x_{x}(\phi),y_{x}(\phi)$ of the extracted orbit are:
$\begin{array}[]{rcl}x_{x}(\phi)&=&x_{c}+R_{v}/2\,\cos{(\theta_{c}+\phi_{s}+\pi-\phi)}\\\
y_{x}(\phi)&=&y_{c}+R_{v}/2\,\sin{(\theta_{c}+\phi_{s}+\pi-\phi)}\,,\end{array}$
(14)
where $\phi$ starts at zero. With the above equations, we analyzed the
geometry of the orbit and the extraction for different choices of $B_{s}$,
$B_{v}$, $B_{0}$ and $\theta_{c}(\gamma)$ as a function of $\gamma=1+E/E_{0}$.
### II.1 The Transfer Matrices
The horizontal transfer matrices ${\bf M}_{s,v}$ for the sector (valley) are
given by
${\bf
M}_{s,v}=\left(\begin{array}[]{cc}\cos{(2\,\phi_{s,v})}&R_{s,v}\,\sin{(2\,\phi_{s,v})}\\\
-{\sin{(2\,\phi_{s,v})}\over
R_{s,v}}&\cos{(2\,\phi_{s,v})}\end{array}\right)\,.$ (15)
The horizontal transfer matrix that describes the edge focusing effect is
${\bf M}_{i,f}=\left(\begin{array}[]{cc}1&0\\\ {\tan{(\beta_{i,f})}\over
R_{\mathrm{eff}}}&1\end{array}\right)\,.$ (16)
where $R_{\mathrm{eff}}=(R_{s}^{-1}+R_{v}^{-1})^{-1}$. Starting with the
entrance into the sector magnet the horizontal transfer matrix ${\bf M}$ for a
single section is the product
${\bf M}_{sec}={\bf M}_{v}\,{\bf M}_{f}\,{\bf M}_{s}\,{\bf M}_{i}\,.$ (17)
The radial focusing frequency $\nu_{r}$ can be obtained from the
parametrization by the twiss-parameters $\alpha_{t}$, $\beta_{t}$ and
$\gamma_{t}$:
${\bf M}_{sec}={\bf
1}\,\cos{(2\,\pi\,\nu_{r})}+\left(\begin{array}[]{cc}\alpha_{t}&\beta_{t}\\\
-\gamma_{t}&-\alpha_{t}\\\ \end{array}\right)\,\sin{(2\,\pi\,\nu_{r})}\,,$
(18)
from which one obtains with $\beta_{t}\,\gamma_{t}-\alpha_{t}^{2}=1$
$\begin{array}[]{rcl}\cos{(2\,\pi\,\nu_{r})}&=&Tr({\bf M}_{sec})/2\\\
\sin{(2\,\pi\,\nu_{r})}^{2}&=&Det({\bf M}_{sec}-{\bf 1}\,Tr({\bf
M}_{sec})/2)\\\ \end{array}$ (19)
The matrices for the vertical motion are
$\begin{array}[]{rcl}{\bf
T}_{s,v}&=&\left(\begin{array}[]{cc}1&2\,R_{s,v}\,\phi_{s,v}\\\ 0&1\\\
\end{array}\right)\\\ {\bf T}_{i,f}&=&\left(\begin{array}[]{cc}1&0\\\
-{\tan{(\beta_{i,f})}\over
R_{\mathrm{eff}}}&1\end{array}\right)\,.\end{array}$ (20)
so that correspondingly
${\bf T}_{sec}={\bf T}_{v}\,{\bf T}_{f}\,{\bf T}_{s}\,{\bf T}_{i}\,.$ (21)
The motion is stable, if
$\begin{array}[]{rcl}|Tr(M_{sec})/2|&\leq&1\\\
|Tr(T_{sec})/2|&\leq&1\,.\end{array}$ (22)
In case of cyclotrons with reverse bends, one has a huge flutter $F={\langle
B^{2}\rangle-\langle B\rangle^{2}\over\langle B\rangle^{2}}$ due to the
negative field regions. If one uses the dimensionless ratios
$\lambda={B_{v}\over B_{s}}={R_{s}\over R_{v}}$ and $\mu={B_{0}\over B_{s}}$,
then the flutter yields
$F={(1-\mu\,\gamma)\,(\mu\,\gamma+\lambda)\over\mu^{2}\,\gamma^{2}}\,,$ (23)
and can approch easily values of above $4$. Therefore care must be taken to
not have too strong focusing, i.e. to avoid the $N/2$-stopband. Due to the
huge flutter, there is no need for large spiral angles. In contrary, the
spiral angle must be kept sufficiently small to avoid the stopband.
## III A multibeam isotope production cyclotron
The described simplified cyclotron description allows a first analysis, if
extraction at various energies can be combined with stability of axial and
horizontal motion. Fig. 2 shows a topview layout for a isotope production
machine with maximal $H_{2}^{+}$-energy of $140\,\mathrm{MeV}$ that allows
stripping extraction of proton beams with energies between $15$ and
$70\,\mathrm{MeV}$.
Figure 2: Geometry of a $H_{2}^{+}$-cyclotron for isotope production with
reverse bends. The $H_{2}^{+}$-beam is stripped at the sector edge (indicated
by symbols). The orbits of stripped proton beams are shown from
$15\,\mathrm{MeV}$ to $70\,\mathrm{MeV}$ in steps of $2.75\,\mathrm{MeV}$. The
arrows indicate the extracted beams for $15$, $26$, $37$, $48$, $59$ and
$70\,\mathrm{MeV}$. The nominal field $B_{0}$ is $0.75\,\mathrm{T}$, the
sector field $B_{s}$ $2\,\mathrm{T}$ and the field strength $B_{v}$ in the
reverse bends is $0.55\,\mathrm{T}$. Without further provision, the directions
of the extracted beams differ enough to allow for energy specific targets. If
the beam is partially stripped at lower energy, simultaneous irradiation at
several targets should be possible. If superconducting coils are used to
increase the field strength, the size of the cyclotron reduces accordingly.
The sector entrance edge has been chosen straight ${d\theta_{i}\over
d\gamma}=0$ so that $\theta_{c}=\theta_{i}+\alpha/2$ and
$\theta_{f}=\theta_{i}+\alpha$.
Fig. 3 shows the corresponding tune diagram. Due to the strong flutter, the
axial tune $\nu_{z}$ is very large. As a consequence the minimum number of
sectors is likely $4$, so that the $N/2$-stopband starts at $\nu=2$. Higher
sector numbers are in principle possible, but more expensive and not required
for this energy range.
Figure 3: Tune diagram of the isotope production cyclotron. The radial tune
is (except for the last turns) approximately constant at about $\nu_{r}\approx
1.5$. The vertical tune $\nu_{z}$ starts at low energy at about $1.9$ and
decreases smoothly to about $1.75$ at maximal energy. Due to the large
flutter, a stable solution for cyclotron with only 3 sectors has not been
found and we assume that stable solutions can not be found for a comparable
energy range of the extracted beam.
Partial stripping of the beam could allow simultaneous extraction at multiple
energies. For this purpose one would move a stripper foil vertically towards
the median plane until it strips off the desired beam current for the
corresponding energy. The remaining beam (with reduced emittance) could be
accelerated to higher energies (see Fig. 4).
Figure 4: Partial beam stripping by vertical positioning of a stripper foil
keeping a certain distance to the median plane (MP).
## IV A variable energy cyclotron for proton therapy
Commercially available cyclotrons for proton therapy deliver beams with an
energy of $235\dots 250\,\mathrm{MeV}$ Klein ; Jongen . Since the presently
available cyclotron technology delivers the beam at fixed energy, the beam
energy must be reduced to the value that is required for the treatment. This
is typically done by energy degradation at the cost of significant emittance
increase and energy straggling in the degradation process Deg0 ; Deg1 ; Deg2 .
In order to deliver a beam of the required quality most of the degraded beam
has to be cut off by collimators and an energy selection system (ESS). The
intensity is (depending on energy) reduced by up to three orders of magnitude.
Even though there are strong arguments for the use of cyclotrons in proton
therapy, there are also disadvantages of the combination of a fixed-energy-
cyclotron, degrader and ESS:
1. 1.
the strong energy dependence of the beam intensity which makes fast and save
energy variations (without intensity variations) of the beam difficult to
achieve.
2. 2.
the activation of the accelerator, the degrader material, the collimators and
other components, which could be reduced by orders of magnitude, if one could
extract high quality beam at various energies.
3. 3.
the cost for the degrader and the ESS which typically consists of two dipoles,
eight quadrupoles, moveable slits, beam diagnostics and vacuum components for
about $10\,\mathrm{m}$ beamline.
4. 4.
the need to use large aperture quadrupoles and dipoles in order to achieve a
suitable transmission efficiency of beam line and gantry.
The list is certainly incomplete, but it suffices to argue that one has to
take the over-all costs of an accelerator concept into account. A separate
sector cyclotron with reverse bends is certainly more expensive than a compact
cyclotron. It will also have a larger footprint and a higher power
consumption. However the footprint of the accelerator itsself is only a small
fraction of a complete proton therapy facility.
Figure 5: Geometry of a $H_{2}^{+}$-cyclotron for proton therapy with reverse
bends and drifts. The equilibrium orbits and stripped proton trajectories for
energies from $70\,\mathrm{MeV}$ to $250\,\mathrm{MeV}$ in steps of $\approx
5.5\,\mathrm{MeV}$ are also shown. They been computed by an equilibrium orbit
code Gordon and by Runge-Kutta tracking, respectively. The nominal field
$B_{0}$ is $0.7\,\mathrm{T}$, the sector field $B_{s}$ $2\,\mathrm{T}$ and the
field strength $B_{v}$ in the reverse bends is $0.55\,\mathrm{T}$. With a bit
more fine-shaping of the magnetic field of the reverse bend, it should be
possible to make all extracted beams pass a region small enough to install a
fast “catcher” magnet, which allows to bend the extracted beams of all
energies into the same beamline. The resulting tunes are shown in the right
graph. The small spiral angle has been introduced to avoid the
$\nu_{r}=\nu_{z}$-resonance shown as a solid straight line. Using
superconducting coils and correspondingly higher field values, the size could
be reduced accordingly. If partial stripping would be applied, it should be
possible to extract up to four beams simultaneously.
We found that variable energy extraction by a moveable stripper foil before a
reverse bend allows in principle to extract beams with energies between
$70\,\mathrm{MeV}$ and $250\,\mathrm{MeV}$. Fig. 5 shows the layout of an
$H_{2}^{+}$-cyclotron with $2\,\mathrm{T}$ sector magnets and
$0.55\,\mathrm{T}$ reverse bend magnets, the equilibrium and extraction orbits
for energies from $70\,\mathrm{MeV}$ to $250\,\mathrm{MeV}$. The use of
superconducting coils would allow to increase the field and the radius would
reduce by the same factor.
Figure 6: Tune diagram of the medical proton cyclotron with variable energy
extraction as shown in Fig. 5.
The time required for a change of the beam energy is then determined by the
ramping time of the beamline magnets and the time for the positioning of the
stripper foil. If a series of foils at different radii would be inserted
vertically into the beam, then the actuator would need just a few millimeters
of motion for the insertion of the foil as shown in Fig. 4. Other mechanisms
using radial motion with the advantage of continous energy adjustment are also
possible. Even though the design of fast moveable parts in vacuum is not
trivial, we believe that mechanisms should be feasible with a response time in
the order of $100\,\mathrm{ms}$ or below. Since the extracted beam current
that is required for radiation therapy is of the order of $1\,\mathrm{nA}$,
cooling of the stripper foil is (for this application) not necessary.
More challenging (in terms of costs and engineering time) is the design of a
central region that allows either to use an internal ion source or a spiral
inflector. An internal ion source causes a higher rest gas pressure compared
to an external source. However the beam current in such a PT machine is very
low so that even a high relative beam loss by rest gas stripping could be
accepted.
Certainly the presented extraction mechanism could also be used in combination
with a pre-accelerator, but the stripping process itsself can be used only
once. The preaccelerator would necessarily have a different extraction
mechanism.
The design scetched in Fig. 5 has four sectors so that with an appropriate
design of rf-resonators one might use at maximum four exit ports in four
directions. They might (but don’t have to) be used simultaneously in order to
deliver beam for four treatment rooms located around the cyclotron bunker.
Since a direct beam from the cyclotron has a small emittance and energy
spread, the beam transport system does not require magnets with large
aperture. Hence beamline and gantry might be smaller and cheaper than those of
conventional systems. If the beam size and energy spread are too small for
fast painting of the tumor, one could insert scatterers into the beam path -
or one might directly use “thick” stripper foils, which increase the beam size
by scattering and make the beam shape more Gaussian.
We used the flat field design since it allows to calculate the desired
properties analytically in very good approximation. However a cyclotron with a
flat field has also practical advantages. It allows for instance to make
precise online field measurements by NMR-probes. The results could be used to
stabilize the magnetic field without beam extraction as it is required for a
phase probe phase . This would not only reduce start up time and simplify beam
quality management, but it might also reduce activation of an external beam
dump. Furthermore the mechanism that places the stripper foil - if fast enough
- could be made “fail-save”: if a spring retracts the foil off the median
plane in case of emergency, extraction immediately stops. Without stripper
foil but with an appropriate shaping of the edge field with enough phase shift
per turn, the cyclotron could operate in a stand-by mode without activation
and extraction but with contineous beam in the median plane. The beam would be
accelerated to maximal energy, phase shifted in the fringe field, decelerated
back to the cyclotron center and dumped there without activation of
components. In this way, the equipment could stay “warm” in stand-by mode. If
beam is requested, the only action to be taken is to insert the stripper foil
at the desired location for the requested energy.
## V A high energy high intensity proton cyclotron
Recently there has been renewed interest in high intensity cyclotrons not only
for the potential use in accelerator driven systems (ADS) for transmutation of
nuclear waste or as “energy amplifier” EA1 , but also for physical experiments
like $DAE\delta ALUS$ Daedalus1 ; Daedalus2 ; Daedalus3 . Typically the
cyclotron should be able to deliver $10\,\mathrm{mA}$ or more proton beam
current at $800$ and $1000\,\mathrm{MeV}$. Such cyclotrons have never been
build, but the PSI ring machine which delivers $2.2\,\mathrm{mA}$ at
$590\,\mathrm{MeV}$ often serves as a proof-of-principle machine ring.
However, there is still a factor of $8$ between the beam power of the PSI
machine ($1.3\,\mathrm{MW}$) and the desired $10\,\mathrm{MW}$ (or more) for
an ADS driver. We are not going to discuss this in detail here, but we give an
example of an $H_{2}^{+}$-cyclotron with stripping extraction between $500$
and $950\,\mathrm{MeV}$. The major advantage of the proposed extraction method
is the increased reliability of extraction without electrostatic elements.
Furthermore the concept allows to reduce the turn separation, i.e. the
required voltage of the accelerating cavities is not a major design issue. A
flattop system could also be obsolete.
Fig. 7 shows a machine layout for a homogenous sector (valley) field of
$4\,\mathrm{T}$ and $1.05\,\mathrm{T}$. As shown in Fig. 8, major resonances
could be avoided by an adequat choice of the spiral angle.
Figure 7: Geometry of an 8-sector $H_{2}^{+}$-cyclotron for extraction
energies in the range between $500$ and $950\,\mathrm{MeV}$ for ADS. The
orbits of the protons after stripping are shown from $470\,\mathrm{MeV}$ to
$950\,\mathrm{MeV}$. The nominal field $B_{0}$ is $0.88\,\mathrm{T}$, the
sector field $B_{s}$ $4\,\mathrm{T}$ and the field strength $B_{v}$ in the
reverse bends is $1.05\,\mathrm{T}$. The tune diagram is shown in the right
graph and covers the $H_{2}^{+}$-energy range from $220\,\mathrm{MeV}$ to
$1.9\,\mathrm{GeV}$. A more advanced field shaping with reduced flutter at
lower energies would keep the vertical $\nu_{z}$ below $2\,\nu_{r}$ also at
lower injection energy and would allow to stay above $\nu_{r}$ at higher
energies. The spiral angle has been chosen to be $\theta_{c}=(\gamma-1)/4.2$,
$\theta_{i}=\theta_{c}-\alpha/2$ and $\theta_{f}=\theta_{c}+\alpha/2$.
There are two major differences between the cyclotron design here and the one
proposed in Ref. Daedalus3 , the first being the difference in the vertical
tune, which is in our design considerably increased by the reverse bends. The
second is the trajectory of the stripped beam. The design proposed in Ref.
Daedalus3 uses the conventional scheme in which the stripped beam is bend
inwards and passes the cyclotron median plane at nearly all radii before
exiting the field. There is no principle problem with this scheme, but it has
disadvantages. First the exact position and direction of the extracted beam
depends on the cyclotron field all along the extraction orbit which is more or
less the complete median plane area. It is therefore influenced by trim coils,
cavities and main field changes. Second, this beam path has to be free from
obstacles. Fig. 7 shows a machine layout for a homogeneous sector (valley)
field of $4\,\mathrm{T}$ and $1.05\,\mathrm{T}$, Fig. 8 the tune-diagram. The
spiral angle has been optimized to avoid major resonances.
Figure 8: Tune diagram of the high intensity cyclotron with variable energy
extraction as shown in Fig. 7.
The extraction path with reverse bends is as short as possible and passes only
the area between two sectors. It is therefore much less sensitive to changes
of main field and/or trim coil settings.
## VI Some final remarks
The discussed machine layouts allow some further optimization with respect to
the direction of the extracted beam by an appropriate shaping of the fields of
the sectors magnets and the reverse bends. This would go beyond the scope of
this paper, since our intention was to survey the principle possiblities of
the extraction mechanism. Certainly the flat field approach used above is
neither necessary for this extraction scheme to work nor do we consider it to
be the optimal choice. It has been chosen as it allows for a fairly simple
analytical description of cyclotron beam optics.
The “inner region” of such cyclotrons, i.e. the energy range in which beam
extraction is not possible, might be designed very different from what is
scetched above. The negative field in the reverse bends is not required at
small radii. Therefore it is possible (and unavoidable) to reduce the effect
of the reverse bend towards the cyclotron center (compensating this with
reduced sector field or sector width).
The beam in a cyclotron like the ones described above should be centered so
that the energy and radius are related in a predictable and reproducable way.
This is especially important for PT applications, where the position of the
stripper foil selects the beam energy.
Since resonant beam extraction is not required, the phase curve may be chosen
flat up to the cyclotron fringe field. This allows to accelerate beams with
relatively low cavity voltages.
Since neither a low energy spread nor a high turn separation is essential in
order to minimize extraction losses (depending on the acceptance of the beam
line transporting the beam to target), even the high intensity machine might
be operational without flattop cavity. The space saved this way could be used
to improve the vacuum conditions by the installation of cryogenic pumps. The
beam loss by rest gas stripping has to be minimized when such cyclotrons are
operated with high currents.
We discussed a long list of advantages of the new extraction mechanism, but
the discussed method has it’s price: $H_{2}^{+}$ has half the charge to mass
ratio of protons and therefore on has to use double size and/or field strength
to reach the same final proton energy. The use of reverse bends has a
comparable effect. A discussion, if and when the increase in size or field
strength pays off by the mentioned advantages, is beyond the scope of this
paper, but depends certainly on the purpose of the machine.
## VII Summary
The geometry of cyclotrons with reverse bends has been analyzed and the
resulting transfer matrices have been given. We investigated some of the
design options involving the use of reverse bends in combination with
stripping extraction of $H_{2}^{+}$. We proved the principle feasability of
variable energy/multiple beam extraction from cyclotrons with reverse bends
and verified the analytical beam stability by a numerical calculation of the
tunes.
We presented three potential applications for the described extraction
mechanism, an isotope production cyclotron with simultaneous extraction at
several energies between $15$ and $70\,\mathrm{MeV}$, a medical cyclotron with
a variable energy extraction in the range between $70$ and $250\,\mathrm{MeV}$
and a high intensity ring cyclotron with beam extraction at energies between
$500$ and $950\,\mathrm{MeV}$.
## VIII Acknowledgements
We thank Nada Fakhoury for her help in writing the Mathematica® notebooks used
for this work.
Software has been written in “C” and been compiled with the GNU©-C++ compiler
on Scientific Linux. The figures have been generated with the cern library
(PAW) and XFig.
## Appendix A An algebraic method for the analysis of accelerator floor
layouts
The floor layout of cyclotrons is just a special case of the general problem
of the calculation of floor layouts, which itself is a special case of the
geometry of curves in the plane. In the general case, a planar smooth curve
can be described by a “state vector” $\psi$ that contains the coordinates and
the direction derivatives ${\bf\psi}=(x,y,x^{\prime},y^{\prime})$, where $x$
and $y$ are the Cartesic coordinates of the orbit (planar curve) and
$x^{\prime}={dx\over ds}$ and $y^{\prime}={dy\over ds}$ are the derivatives
with respect to the pathlength $s$. By definition one has
$x^{\prime 2}+y^{\prime 2}=1\,,$ (24)
so that one may also write $(x^{\prime},y^{\prime})=(\cos{\phi},\sin{\phi})$
with the direction angle $\phi$ of the orbit.
The state vector is a function of the pathlength $s$ of the orbit and the
general evolution of this vector can be described by a differential equation
of the form:
$\begin{array}[]{rcl}{\bf\psi^{\prime}}&=&{d{\bf\psi}\over ds}={\bf
F}({1\over\rho})\,{\bf\psi}\\\ \left(\begin{array}[]{c}x\\\ y\\\ x^{\prime}\\\
y^{\prime}\\\ \end{array}\right)&=&\left(\begin{array}[]{cccc}0&0&1&0\\\
0&0&0&1\\\ 0&0&0&-\frac{1}{\rho}\\\ 0&0&\frac{1}{\rho}&0\\\
\end{array}\right)\,\left(\begin{array}[]{c}x\\\ y\\\ x^{\prime}\\\
y^{\prime}\\\ \end{array}\right)\,,\end{array}$ (25)
where $\rho=\rho(s)$ is the local bending radius of the curve. In the hard
edge approximation, we assume that $\frac{1}{\rho}={q\,B\over p}$ is piecewise
constant. In this case, a transfer matrix method can be used and the solution
is given by a transfer (or transport) matrix ${\bf M}(s)$:
${\bf\psi}(s)={\bf M}(s)\,{\bf\psi}(0)\,,$ (26)
where the matrix ${\bf M}$ is the product of the transfer matrices for the
individual segments:
${\bf M}(s)=\prod\limits_{k=0}^{n-1}\,{\bf M}_{k}$ (27)
In hard edge approximation, there are basically two transfer matrices, the
matrix ${\bf M}_{d}(L)$ for a drift of length $L$ and the matrix ${\bf
M}_{b}(\rho,\alpha)$ for a bending magnet for a bending radius $\rho$ and
angle $\alpha$:
$\begin{array}[]{rcl}{\bf M}_{d}(L)&=&\exp{({\bf F}(0)\,L)}\\\ {\bf
M}_{b}(\rho,\alpha)&=&\exp{({\bf F}({1\over\rho})\,\alpha\,\rho)}\\\
\end{array}$ (28)
The matrix powers of ${\bf F}$ are readily computed:
$\begin{array}[]{rcl}{\bf
F}^{2}&=&\left(\begin{array}[]{cccc}0&0&0&-{1\over\rho}\\\
0&0&{1\over\rho}&0\\\ 0&0&-{1\over\rho^{2}}&0\\\ 0&0&0&-{1\over\rho^{2}}\\\
\end{array}\right)\\\ {\bf F}^{3}&=&-{1\over\rho^{2}}\,{\bf F}\\\ {\bf
F}^{4}&=&-{1\over\rho^{2}}\,{\bf F}^{2}\\\ \end{array}$ (29)
from which one finds within a few steps
$\begin{array}[]{rcl}{\bf
M}_{b}(\rho,\alpha)&=&\left(\begin{array}[]{cccc}1&0&\rho\,s&-\rho\,(1-c)\\\
0&1&\rho\,(1-c)&\rho\,s\\\ 0&0&c&-s\\\ 0&0&s&c\\\
\end{array}\right)\,,\end{array}$ (30)
where $s=\sin{\alpha}$, $c=\cos{\alpha}$ and $\alpha={L\over\rho}$. A reverse
bend (i.e. a bend into the opposite direction), is described by a negative
radius and a negative angle, yielding a positve length $L=\alpha\,\rho$. If
${1\over\rho}=0$, then the transfer matrix simplifies to the transfer matrix
of a drift:
$\begin{array}[]{rcl}{\bf M}_{d}(L)&=&\exp{({\bf
F}\,L)}=\left(\begin{array}[]{cccc}1&0&L&0\\\ 0&1&0&L\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{array}\right)\,.\end{array}$ (31)
These two matrices are sufficient to compute the floor layout of most
accelerator beamlines. But they are also usefull for the geometrical analysis
of separate sector cyclotrons with homogeneous field magnets in hard edge
approximation as described above.
In addition to the above transfer matrices, we will use the familiar
coordinate rotation matrix ${\bf M}_{rot}(\theta)$:
${\bf
M}_{rot}(\theta)=\left(\begin{array}[]{cccc}\cos{\theta}&-\sin{\theta}&0&0\\\
\sin{\theta}&\cos{\theta}&0&0\\\ 0&0&\cos{\theta}&-\sin{\theta}\\\
0&0&\sin{\theta}&\cos{\theta}\\\ \end{array}\right)\,.$ (32)
If we consider an accelerator with $N$ equal sectors (or sections), then -
given an arbitrary starting position ${\bf\psi}(0)$ \- the position and
direction change after one sector relative to some center point
${\bf\psi}_{c}$ can be described by a rotation with an angle of
$\theta={2\,\pi\over N}$. Hence we may write
$({\bf\psi}(L)-{\bf\psi}_{c})={\bf
M}_{rot}(\theta)\,({\bf\psi}(0)-{\bf\psi}_{c})\,,$ (33)
so that by the use of Eqn. 26 one finds
$\begin{array}[]{rcl}({\bf M}_{sec}{\bf\psi}(0)-{\bf\psi}_{c})&=&{\bf
M}_{rot}(\theta)\,({\bf\psi}(0)-{\bf\psi}_{c})\\\ ({\bf M}_{sec}-{\bf
M}_{rot}(\theta))\,{\bf\psi}(0)&=&({\bf 1}-{\bf
M}_{rot}(\theta))\,{\bf\psi}_{c}\,.\end{array}$ (34)
The coordinates of the accelerator center can be obtained by:
${\bf\psi}_{c}=({\bf 1}-{\bf M}_{rot}(\theta))^{-1}\,({\bf M}_{sec}-{\bf
M}_{rot}(\theta))\,{\bf\psi}(0)\,.$ (35)
The matrix ${\bf M}_{x}(\theta)\equiv({\bf 1}-{\bf M}_{rot}(\theta))^{-1}$ can
be directly computed and is explicitely given by
$\begin{array}[]{rcl}{\bf M}_{x}(\theta)&=&{1\over
2\,\sin{({\theta/2})}}\,{\bf M}_{rot}(\pi/2-\theta/2)\\\
&=&\frac{1}{2}\,\left(\begin{array}[]{cccc}1&-\cot{\theta/2}&0&0\\\
\cot{\theta/2}&1&0&0\\\ 0&0&1&-\cot{\theta/2}\\\ 0&0&\cot{\theta/2}&1\\\
\end{array}\right)\end{array}$ (36)
If one computes the center of motion of a bending magnet for an angle
$\theta$, the result is given by
$\begin{array}[]{rcl}{\bf\psi}_{c}&=&{\bf M}_{x}(\theta)\,({\bf
M}_{b}(\rho,\theta)-{\bf M}_{rot}(\theta))\,{\bf\psi}(0)\\\
&=&\left(\begin{array}[]{cccc}1&0&0&-\rho\\\ 0&1&\rho&0\\\ 0&0&0&0\\\
0&0&0&0\\\ \end{array}\right)\,{\bf\psi}(0)\\\
&=&(x(0)-\rho\,y^{\prime}(0),y(0)+\rho\,x^{\prime}(0),0,0)^{T}\,,\end{array}$
(37)
which is easy to verify.
The computation of the center coordinates is therefore straightforward - and
yields a result even, if the matrix ${\bf M}_{sec}$ does not describe a
“valid” sector. Such a non-valid situation is given, if the the “velocity”
components of ${\bf\psi}_{0}$ do not vanish, which happens, if the sum of the
bending angles entering ${\bf M}_{sec}$ does not equal $\theta$.
In the following we use Eqn. 35 to compute the starting conditions for a
cyclotron centered at $x_{c}=y_{c}=0$, i.e. the radius and direction of an
equilibrium orbit. If we let the orbit start at $(0,0)$ in arbitrary
direction, i.e. we choose for instance ${\bf\psi}(0)=(0,0,0,1)$, then the
orbit with starting position ${\bf\psi}(0)-{\bf\psi}_{c}$ is centered. Hence
the starting position
${\bf\psi}(0)=\left({\bf 1}-{\bf M}_{x}(\theta)\,({\bf M}_{sec}-{\bf
M}_{rot}(\theta))\right)\,(0,0,0,1)^{T}\,.$ (38)
is centered. The orbit still starts at an “arbitrary” angle $\theta_{0}$, i.e.
${\bf\psi}(0)$ as given by Eqn. 38 can be written as
${\bf\psi}(0)=(R\,\cos{\theta_{0}},R\,\sin{\theta_{0}},0,1)^{T}=(x_{0},y_{0},0,1)^{T}\,.$
(39)
If one aims for a specific orientation of the orbit with respect to the floor
coordinates - for instance on the x-axis - then one may use the rotation
matrix with $\theta_{0}=\arctan{({y_{0}\over x_{0}})}$:
${\bf\psi}(0)\to{\bf
M}_{rot}(-\theta_{0})\,{\bf\psi}(0)=(R,0,\frac{y_{0}}{R},\frac{x_{0}}{R})^{T}\,.$
(40)
The angular width of the magnet can then be calculated by computing the
position angle of ${\bf M}_{b}\,{\bf\psi}(0)$.
Figure 9: Geometry of the cyclotron sector in case of a sector magnet (gray
area) with a constant field along the closed orbit (shown in as a thick dashed
line). $\varepsilon_{1}$ and $\varepsilon_{2}$ are the spiral angles of the
entrance and exit of the magnet. $\gamma_{1}$ and $\gamma_{2}$ are the angles
between the sector entrance and exit and the orbit normal vector. It is
obvious from the drawing that $R\,\sin{(\alpha/2)}=r\,\sin{(\pi/N)}$.
If this method is applied to a cyclotron sector composed of a dipole with
bending radius $r$ and bend angle ${2\,\pi\over N}$ and a drift of length $L$
as shown in Fig. 9, then one obtains:
$\begin{array}[]{rcl}\tan{\phi}&=&{2\,r\over L}+\cot{\pi\over N}\\\
R&=&\sqrt{\frac{L^{2}}{4}+(r+\frac{L}{2}\,\cot{\pi\over
N})^{2}}\,.\end{array}$ (41)
Both conditions could also be derived from Fig. 9. The advantage of the
algebraic method is, that it gives an algorithm at hand that allows to
determine the essential geometric conditions directly from the parameters $r$,
$N$ and $L$, without the need to analyze a “hand-made” drawing. Furthermore
the algebraic algorithm enables to produce the drawing.
In case of the simple situation scetched in Fig. 9, the drawing might do as
well. But in the case of the medical cyclotron as shown in Fig. 5, the
symmetrie of the equilibrium orbit for a given energy is broken by the drift
between the reverse bend (valley) and the next sector. In this case and in
case of more complex configurations, the analysis of the layout by a handmade
scetch becomes cumbersome due to the increasing number of angles and
geometrical relations. In fact, the geometry of the medical cyclotron
presented above has been analyzed by a “C”-program and a Mathematica® notebook
based on the above algebraic ansatz. The main reason was the desire to create
a map of the magnetic field in cylindrical coordinates for the numerical (and
hence more “realistic”) computation of the tunes. In case of a cyclotron that
is composed of $N$ sections each containing a sector magnet, a reverse bend
and a drift, it turns out, that the entrance and exit radius for a given
energy are not equal.
For a given radius of the grid, we had to determine the energy (i.e. the
$\gamma$-value) of the equilibrium orbit entering the sector, a second
$\gamma$-value for the orbit existing the sector and a third one at the exit
of the reverse bend. This was done by an iterative numerical interval search.
## References
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|
arxiv-papers
| 2013-07-23T11:33:14 |
2024-09-04T02:49:48.341578
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Baumgarten",
"submitter": "Christian Baumgarten",
"url": "https://arxiv.org/abs/1307.6024"
}
|
1307.6084
|
# An algorithm to compute the Hilbert depth
Adrian Popescu Adrian Popescu, Department of Mathematics, University of
Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany
[email protected]
###### Abstract.
We give an algorithm which computes the Hilbert depth of a graded module based
on a theorem of Uliczka. Partially answering a question of Herzog, we see that
the Hilbert depth of a direct sum of modules can be strictly greater than the
Hilbert depth of all the summands.
Key words : depth, Hilbert depth, Stanley depth.
2010 Mathematics Subject Classification : Primary 13C15, Secondary 13F20,
13F55, 13P10.
The support from the Department of Mathematics of the University of
Kaiserslautern is gratefully acknowledged.
## Introduction
Let $K$ be a field and $R=K[x_{1}\ldots,x_{n}]$ be the polynomial algebra over
$K$ in $n$ variables. On $R$ consider the following two grading structures:
the $\operatorname{\mathbb{Z}}-$grading in which each $x_{i}$ has degree $1$
and the multigraded structure, i.e. the
$\operatorname{\mathbb{Z}}^{n}-$grading in which each $x_{i}$ has degree the
$i-$th vector $e_{i}$ of the canonical basis.
After Bruns-Krattenthaler-Uliczka [4] (see also [11]), a Hilbert decomposition
of a $\operatorname{\mathbb{Z}}-$graded $R-$module $M$ is a finite family
${\mathcal{H}}=(R_{i},s_{i})_{i\in I}$
in which $s_{i}\in{\operatorname{\mathbb{Z}}}$ and $R_{i}$ is a
$\operatorname{\mathbb{Z}}-$graded $K-$algebra retract of $R$ for each $i\in
I$ such that
$M\cong\displaystyle\bigoplus_{i\in I}R_{i}(-s_{i})$
as a graded $K-$vector space.
The Hilbert depth of $\mathcal{H}$ denoted by
$\operatorname{hdepth}_{1}\mathcal{H}$ is the depth of the $R-$module
$\displaystyle\bigoplus_{i\in I}R_{i}(-s_{i})$. The Hilbert depth of $M$ is
defined as
$\operatorname{hdepth}_{1}(M)=\operatorname{max}\\{\operatorname{hdepth}_{1}\mathcal{H}\
|\ \textnormal{$\mathcal{H}$ is a Hilbert decomposition of }M\\}.$
We set $\operatorname{hdepth}_{1}(0)=\infty$.
###### Theorem 0.1.
(Uliczka [13]) $\operatorname{hdepth}_{1}(M)=\operatorname{max}\\{e\ |\
{(1-t)}^{e}HP_{M}(t)\textnormal{ is positive}\\}$, where
$\operatorname{HP}_{M}(t)$ is the Hilbert$-$Poincaré series of $M$ and a
Laurent series in $\operatorname{\mathbb{Z}}[[t,t^{-1}]]$ is called positive
if it has only nonnegative coefficients.
If $M$ is a multigraded $\operatorname{\mathbb{Z}}^{n}-$module, then one can
define $\operatorname{hdepth}_{n}(M)$ as above by considering the
$\operatorname{\mathbb{Z}}^{n}-$grading instead of the standard one. There
exists an algorithm for computing the $\operatorname{hdepth}_{n}$ of a
finitely generated multigraded module $M$ over the standard multigraded
polynomial ring $K[x_{1},\ldots,x_{n}]$ in Ichim and Moyano-Fernández’s paper
[8] (see also [9]).
The main purpose of this paper is to provide an algorithm to compute
$\operatorname{hdepth}_{1}(M)$, where $M$ is a graded $R-$module (see
Algorithm 1.3). This is part of the author’s Master Thesis [10].
A Stanley decomposition (see [12]) of a $\operatorname{\mathbb{Z}}-$graded
(resp. ${\operatorname{\mathbb{Z}}}^{n}-$graded) $R-$module $M$ is a finite
family
${\mathcal{D}}=(R_{i},u_{i})_{i\in I}$
in which $u_{i}$ are homogeneous elements of $M$ and $R_{i}$ is a graded
(resp. ${\operatorname{\mathbb{Z}}}^{n}-$graded) $K-$algebra retract of $R$
for each $i\in I$ such that $R_{i}\cap\operatorname{Ann}(u_{i})=0$ and
$M=\displaystyle\bigoplus_{i\in I}R_{i}u_{i}$
as a graded $K-$vector space.
The Stanley depth of $\mathcal{D}$ denoted by
$\operatorname{sdepth}\mathcal{D}$ is the depth of the $R-$module
$\displaystyle\bigoplus_{i\in I}R_{i}u_{i}$. The Stanley depth of $M$ is
defined as
$\operatorname{sdepth}(M)=\operatorname{max}\\{\operatorname{sdepth}\mathcal{D}\
|\ \textnormal{$\mathcal{D}$ is a Stanley decomposition of }M\\}.$
We set $\operatorname{sdepth}(0)=\infty$.
We talk about $\operatorname{sdepth}_{1}(M)$ and
$\operatorname{sdepth}_{n}(M)$ if we consider the
$\operatorname{\mathbb{Z}}-$grading respectively the
${\operatorname{\mathbb{Z}}}^{n}-$grading of $M$. The Hilbert depth of $M$ is
greater than the Stanley depth of $M$ and can be strictly greater (an example
can be found in [4]).
Herzog posed the following question (see also [1, Problem 1.67]): is
$\operatorname{sdepth}_{n}(R\oplus m)=\operatorname{sdepth}_{n}(m)$, where $m$
is the maximal ideal in $R$? Since we implemented an algorithm to compute
$\operatorname{hdepth}_{1}$, we have tested whether
$\operatorname{hdepth}_{1}(R\oplus m)=\operatorname{hdepth}_{1}(m)$ and as a
consequence when $\operatorname{sdepth}_{n}(R\oplus
m)=\operatorname{sdepth}_{n}(m)$. Proposition 2.6 says that Herzog’s question
holds for $n\in\\{1,\ldots,5,7,9,11\\}$, but Remark 2.4 says that for $n=6$ it
holds $\operatorname{hdepth}_{1}(R\oplus m)>\operatorname{hdepth}_{1}m$, which
is a sign that in this case $\operatorname{sdepth}_{n}(R\oplus
m)>\operatorname{sdepth}_{n}m$ and so Herzog’s question could have a negative
answer for $n=6$. This is indeed the case as it was shown later by Ichim and
Zarojanu in [9]. Meanwhile Bruns et. al. [5] found another algorithm computing
$\operatorname{hdepth}_{1}$ and Chen [6] gave another one in the frame of
ideals.
We owe thanks to Ichim who suggested us this problem and to Uliczka who found
a mistake in a previous version of our algorithm.
## 1\. hdepth Computation
In this section we introduce an algorithm which computes
$\operatorname{hdepth}_{1}$ (Algorithm 1.3) and prove its correctness (Theorem
1.4). In the next section we provide some examples and some results related to
[1, Problem 1.67].
###### Remark 1.1.
The algorithm presented in this section is based on Theorem 0.1 and at a first
glance it might look trivial. The difficulty lies in the fact that it is not
clear how many coefficients of the infinite Laurent series have to be checked
for positivity. This paper provides a bound up to which it suffices to check.
Recall first [3, Corollary 4.1.8] the definition of the Hilbert$-$Poincaré
series of a module $M$
$\operatorname{HP}_{M}(t)=\displaystyle\frac{Q(t)}{{(1-t)}^{n}}=\displaystyle\frac{G(t)}{{(1-t)}^{d}}\
,$ (1)
where $d=\dim M$ and $Q(t),\ G(t)\in\operatorname{\mathbb{Z}}[t],\ G(1)\neq
0$. In fact, note that $G(1)$ is equal to the multiplicity of the module which
is known to be positive.
The algorithm which we construct requires the module $M$ as the input.
Actually we only need the $G(t)$ from (1) and the dimension of $M$.
###### Definition 1.2.
Let $p(t)=\displaystyle\sum_{i=0}^{\infty}a_{i}\cdot
t^{i}\in\operatorname{\mathbb{Z}}[[t]]$ be a formal power series. By jetj($p$)
we understand the polynomial
jet${}_{j}(p)=\displaystyle\sum_{i=0}^{j}a_{i}\cdot t^{i}$.
###### Algorithm 1.3.
We now present the algorithm that computes the $\operatorname{hdepth}_{1}$ of
a $\operatorname{\mathbb{Z}}-$graded module $\verb"M"$. The algorithm uses the
following procedures which can easily be constructed in any computer algebra
system:
* $\circ$
`inverse(poly p, int bound)`: computes the inverse of a power series `p` till
the degree `bound`,
* $\circ$
`hilbconstruct(module M)`: computes the second Hilbert series of the module
`M` \- a way to do this in $\operatorname{\textsc{Singular}}$ is to use the
already built-in function `hilb(module M, 2)` which returns the list of
coefficients of the second Hilbert series and construct the series,
* $\circ$
`positive(poly f)`: returns `1` if $f$ has all the coefficients nonnegative
and `0` else,
* $\circ$
`sumcoef(poly f)`: returns the sum of the coefficients of `f`,
* $\circ$
`jet(poly p, int j)`: returns the jetj `p`. This procedure is already
implemented in $\operatorname{\textsc{Singular}}$,
* $\circ$
`dim(module M)`: returns the dimension of `M`. This procedure is already
implemented in $\operatorname{\textsc{Singular}}$.
Below we give the algorithm `hdepth(poly g, int dim__M)`. Hence in order to
compute $\operatorname{hdepth}_{1}\verb"M"$, one considers $\verb"g(t) =
hilbconstruct( M )"$ and $\verb"dim__M = dim(M)"$.
Algorithm $\operatorname{hdepth}_{1}$ (poly g, int dim__M)
0:
0: a polynomial $g(t)\in\operatorname{\mathbb{Z}}[t]$ (equal to
$\operatorname{HP}_{M}(t)$)
0: an integer $dim\\_\\_M=\dim M$
0:
0: $\operatorname{hdepth}_{1}M$
1: if positive($g$) = 1 then
2: return $dim\\_\\_M$;
3: end if
4: poly $f=g$;
5: int $c$, $d$, $\beta$;
6: $\beta$ = $\operatorname{deg}(g)$;
7: for $d=dim\\_\\_M$ to $d=0$ do
8: $d=d-1$;
9: $f=\textnormal{jet}(\ g\cdot\textnormal{inverse}({\ (1-t)}^{dim\\_\\_M-d},\
\beta\ )\ );$
10: if positive($f$) = 1 then
11: return $d$;
12: end if
13: $c$ = sumcoef($f$);
14: if $c<0$ then
15: while $c<0$ do
16: $\beta=\beta+1$;
17: $f=\textnormal{jet}(\ g\cdot\textnormal{inverse}(\
{(1-t)}^{dim\\_\\_M-d},\ \beta\ )\ );$
18: $c$ = sumcoef($f$);
19: end while
20: end if
21: end for
###### Theorem 1.4.
Given a $\operatorname{\mathbb{Z}}-$graded module $M$, Algorithm 1.3 correctly
computes
$\operatorname{max}\left\\{n\ \middle|\
{(1-t)}^{n}\cdot\operatorname{HP}_{M}(t)\textnormal{ is positive }\right\\}$
(2)
where $\operatorname{HP}_{M}(t)=\displaystyle\frac{G(t)}{{(1-t)}^{\dim M}}$ is
the Hilbert-Poincaré series of $M$. Hence, by Theorem 0.1, the algorithm
computes the Hilbert depth of a module $M$ for $g=G(t)$ and $dim\\_\\_M=\dim
M$.
###### Proof.
Note that $G(1)$ is the multiplicity of the module $M$ and hence $G(1)>0$.
Assume that $M\neq 0$. Denote the bound $\beta$ at the end of the loop where
$d=i$ by $\beta_{i}$. In order to prove this theorem one has to show the
following two claims:
* $\circ$
the maximum from (2) does not exceed $\dim M$,
* $\circ$
after the bound $\beta_{i}$ degree, the coefficients are nonnegative.
For the first part consider $G(t)=\displaystyle\sum_{\mu=0}^{g}a_{\mu}\cdot
t^{\mu}$. Note that
$(1-t)^{\dim M+1}\cdot\operatorname{HP}_{M}(t)=(1-t)\cdot
G(t)=a_{0}+(a_{1}-a_{0})\cdot t+\ldots+(a_{g}-a_{g-1})\cdot t^{g}-a_{g}\cdot
t^{g+1}.$
If all coefficients would be nonnegative, we would obtain
$0\geq a_{g}\geq a_{g-1}\geq a_{g-2}\geq\ldots\geq a_{2}\geq a_{1}\geq
a_{0}\geq 0$
which implies that $G(t)=0$. This will lead to a contradiction with $M\neq 0$.
The same holds for $(1-t)^{\dim M+\alpha}\cdot\operatorname{HP}_{M}(t)$ by
considering $(1-t)^{\dim M+\alpha-1}\cdot\operatorname{HP}_{M}(t)$ instead of
$G(t)$, where $\alpha\geq 0$. Thus the maximum from (2) is smaller or equal
than $\dim M$.
Note that if $G(t)$ already has all the coefficients nonnegative, then the
algorithm stops by returning $\dim M$, and the result is correct since in this
case $\operatorname{hdepth}_{1}M=\dim M$.
For the second part we need to show that at each step $i$ the coefficient of
the term of order $\beta_{i}$ in $\displaystyle\frac{G(t)}{{(1-t)}^{\dim
M-i}}$ is nonnegative and the coefficients of the terms of higher order are
increasing (and hence nonnegative). Apply induction on $i$. For the first
step, $d=\dim M-1$, $f=\displaystyle\frac{G(t)}{(1-t)}$ and all the
coefficients of the terms of order $\geq\beta_{\dim
M-1}=\operatorname{deg}G(t)$ are equal to the sum of the coefficients
$G(1)>0$. For the general step $i$, assume that at the beginning of loop
$d=i$, we started with $\displaystyle\frac{G(t)}{{(1-t)}^{\dim
M-i}}=\displaystyle\sum_{\mu=0}^{\infty}b_{\mu}\cdot t^{\mu}$ which satisfied
all the desired properties by induction: the bound $\beta_{i}$ was increased
(if required), such that the coefficient sum
$c_{i}:=\displaystyle\sum_{\mu=0}^{\beta_{i}}b_{\mu}>0$ and all coefficients
of higher order terms are nonnegative, i.e. $b_{\mu}\geq 0$ for
$\mu\geq\beta_{i-1}$. We now consider the next step, $d=i-1$, and compute the
new $f$ as in line 9 of the algorithm. In order to check that the coefficients
of the terms of order higher than the bound $\beta_{i}$ are nonnegative. We
have:
$\displaystyle\frac{G(t)}{{(1-t)}^{\dim
M-(i-1)}}=\overbrace{b_{0}+(b_{0}+b_{1})\cdot
t+\ldots+\underbrace{\left(\displaystyle\sum_{\mu=0}^{\beta_{i}}b_{\mu}\right)}_{c_{i}>0}\cdot
t^{\beta_{i}}}^{=\textnormal{ jet}_{\beta_{i}}}+(c_{i}+b_{\beta_{i}+1})\cdot
t^{\beta_{i}+1}+\ldots$
By induction, $0<b_{\beta_{i}}\leq b_{\beta_{i}+1}\leq
b_{\beta_{i}+2}\leq\ldots$ and since $c_{i}>0$ we obtain
$c_{i}+b_{\beta_{i}+\nu}>0$ for $\nu\geq 0$.
The termination of the algorithm is trivial since we know that in the last
loop we would consider $\displaystyle\frac{G(t)}{{(1-t)}^{\dim
M}}=\operatorname{HP}_{M}(t)$ which is positive by the definition, and hence
it will return $\operatorname{hdepth}_{1}M=0$.
###### Remark 1.5.
The maximum from the statement of [13, Theorem 3.2] (see here Theorem 0.1) is
always smaller than $\dim M$. This was not shown in Uliczka’s proof and it has
to be proved in Theorem 1.4.
## 2\. Computational Experiments
The following examples illustrate the usage of the implementation of the
algorithm in $\operatorname{\textsc{Singular}}$, which can be found in the
Appendix. Note that in the outputs we print exactly the jet we considered in
our computations followed by “`+...`”.
###### Example 2.1.
Consider the ring $\operatorname{\mathbb{Q}}[x,y_{1},\ldots,y_{5}]$ and
consider the ideal $I=(x)\cap(y_{1},\ldots,y_{5})$.
ring R=0,(x,y(1..5)),ds;
ideal i=intersect(x,ideal(y(1..5)));
module m=i;
"dim M = ",dim(m);
// dim M = 5
hdepth( hilbconstruct( m ), dim(m) );
// G(t)= 1+t-4t2+6t3-4t4+t5
// G(t)/(1-t)^ 1 = 1+2t-2t2+4t3+t5 +...
// G(t)/(1-t)^ 2 = 1+3t+t2+5t3+5t4+6t5 +...
// hdepth= 3
###### Example 2.2.
Consider a module $M$ for which
$\operatorname{HP}_{M}(t)=\displaystyle\frac{2-3t-2t^{2}+2t^{3}+4t^{4}}{{(1-t)}^{\dim
M}}$. Denote by $\verb"dim__M"$ the dimension of $M$.
ring R = 0, t, ds;
poly g = 2-3*t-2*t^2+2*t^3+4*t^4;
hdepth( g, dim__M);
// G(t)= 2-3t-2t2+2t3+4t4
// G(t)/(1-t)^ 1 = 2-t-3t2-t3+3t4+3t5 +...
// G(t)/(1-t)^ 2 = 2+t-2t2-3t3+3t5 +...
// G(t)/(1-t)^ 3 = 2+3t+t2-2t3-2t4+t5 +...
// G(t)/(1-t)^ 4 = 2+5t+6t2+4t3+2t4+3t5 +...
Hence, it results $\operatorname{hdepth}_{1}M=\dim M-4$.
As seen in the proof, we had to increase our bound if the coefficient sum was
$\leq 0$. Note that in this example, the coefficient sum of
jet4$\left(\displaystyle\frac{G(t)}{(1-t)}\right)$ is zero and thus we
increase the bound to $5$ (the coefficient sum of the jet5 will be equal to
$3>0$).
###### Example 2.3.
Consider $R=K[x_{1},\ldots,x_{n}]$ for $n\in\\{4,5,\ldots,19\\}$ and $m$ the
maximal ideal. We computed $\operatorname{hdepth}_{1}m$,
$\operatorname{hdepth}_{1}(R\oplus m)$, $\ldots$,
$\operatorname{hdepth}_{1}(R^{6}\oplus m)$ and
$\operatorname{hdepth}_{1}(R^{100}\oplus m)$. We obtain the following results:
n | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\operatorname{hdepth}_{1}(m)$ | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 10
$\operatorname{hdepth}_{1}(R\oplus m)$ | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 9 | 10 | 11 | 11
$\operatorname{hdepth}_{1}(R^{2}\oplus m)$ | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 10 | 10 | 11 | 11 | 12
$\operatorname{hdepth}_{1}(R^{3}\oplus m)$ | 3 | 3 | 4 | 5 | 5 | 6 | 7 | 7 | 8 | 9 | 9 | 10 | 10 | 11 | 12 | 12
$\operatorname{hdepth}_{1}(R^{4}\oplus m)$ | 3 | 3 | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 9 | 10 | 11 | 11 | 12 | 12
$\operatorname{hdepth}_{1}(R^{5}\oplus m)$ | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 10 | 10 | 11 | 11 | 12 | 13
$\operatorname{hdepth}_{1}(R^{6}\oplus m)$ | 3 | 4 | 4 | 5 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 | 11 | 11 | 12 | 13
$\operatorname{hdepth}_{1}(R^{100}\oplus m)$ | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 9 | 10 | 11 | 11 | 12 | 13 | 13 | 14 | 15
Figure 1.
###### Remark 2.4.
Note that for $n=6$ we have $\operatorname{hdepth}_{1}(R\oplus
m)=4>3=\operatorname{hdepth}_{1}m$. This is a sign that in this case
$\operatorname{sdepth}_{n}(R\oplus m)>\operatorname{sdepth}_{n}(m)$ and so
Herzogs’s question could have a negative answer for $n=6$. The difference
$\operatorname{hdepth}_{1}(R\oplus m)-\operatorname{hdepth}_{1}m$ can be $>1$
as one can see for $n=18$.
Note that $\operatorname{hdepth}_{1}(R^{s}\oplus
m)-\operatorname{hdepth}_{1}m$ increases when $s$ and $n$ increase. For
example $\operatorname{hdepth}_{1}(R^{100}\oplus
m)-\operatorname{hdepth}_{1}m=5$ for $s=100$ and $n=19$.
###### Lemma 2.5.
Let $n\in\mathbb{N}$ be such that
$\operatorname{hdepth}_{1}m=\operatorname{hdepth}_{1}(R\oplus m)$. Then
$\operatorname{sdepth}_{n}m=\operatorname{sdepth}_{n}(R\oplus m)$.
###### Proof.
By [4] and [2] we have
$\operatorname{hdepth}_{1}m=\left\lceil\displaystyle\frac{n}{2}\right\rceil=\operatorname{sdepth}_{n}m$.
It is enough to see that the following inequalities hold:
$\operatorname{hdepth}_{1}m=\operatorname{sdepth}_{n}m\leq\operatorname{sdepth}_{n}(R\oplus
m)\leq\operatorname{hdepth}_{n}(R\oplus
m)\leq\operatorname{hdepth}_{1}(R\oplus m).$
###### Proposition 2.6.
If $n\in\\{1,\ldots,5,7,9,11\\}$ then
$\operatorname{sdepth}_{n}m=\operatorname{sdepth}_{n}(R\oplus m)$, that is
Herzog’s question has a positive answer.
###### Proof.
Note that $\operatorname{hdepth}_{1}m=\operatorname{hdepth}_{1}(R\oplus m)$
for $n$ as above and apply Lemma 2.5.
## Appendix
As stated before, Algorithm 1.3 was implemented as a procedure for the
computer algebra system $\operatorname{\textsc{Singular}}$ [7]. This procedure
was used in order to obtain the results from Figure 1. The additional
procedures which have been used were defined in Algorithm 1.3. In addition, we
printed some information which we find useful for understanding the algorithm.
⬇
proc hdepth(poly g, int dim__M)
{
int d;
ring T = 0,t,ds;
”G(t)=”,g;
if(positiv(g)==1)
{return(”hdepth=”,dim__M);}
poly f=g;
number ag;
int c1;
int bound;
bound = deg(g);
for(d = dim__M; d>=0; d–)
{
f = jet( g*inverse( (1-t)^(dim__M-d),bound ) , bound );
if(positiv(f) == 1)
{
”G(t)/(1-t)^”,dim__M-d,”=”,f,”+…”;
”hdepth=”,d;
return();
}
c1=sumcoef(f);
if(c1<=0)
{
while( c1<0 )
{
bound = bound + 1;
f = jet( g*inverse( (1-t)^(dim__M-d),bound ) , bound );
c1 = sumcoef(f);
}
”G(t)/(1-t)^”,dim__M-d,”=”,g,”+…”;
}
}
}
## References
* [1] A.M. Bigatti, P. Gimenez, E. Sáenz-de-Cabezón: _Monomial Ideals, Computations and Applications_ , Springer, 2013
* [2] C. Biro, D.M. Howard, M.T. Keller, W.T. Trotter, S.J. Young, _Interval partitions and Stanley depth_ , J. Combin. Theory Ser. A 117 (2010), 475-482.
* [3] W. Bruns, J. Herzog: _Cohen-Macaulay rings_ , Revised edition, Cambridge University Press (1998).
* [4] W. Bruns, C. Krattenthaler, J. Uliczka: _Stanley decompositions and Hilbert depth in the Koszul complex_ , J. Commut. Algebra 2 (2010), 327-357
* [5] W. Bruns, J. Moyano-Fernández, J. Uliczka: Hilbert regularity of ZZ-graded modules over polynomial rings, (2013), arXiv:AC/1308.2917
* [6] R.-X. Chen: How to compute the Hilbert depth of a graded ideal, (2013), arXiv:AC/1308.3205
* [7] W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann: Singular 3-1-6 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2013).
* [8] B. Ichim, J. J. Moyano-Fernández, How to compute the multigraded Hilbert depth of a module, to appear in Mathematische Nachrichten, arXiv:AC/1209.0084.
* [9] B. Ichim, A. Zarojanu: An algorithm for computing the multigraded Hilbert depth of a module, (2013), to appear in Experimental Mathematics, arXiv:AC/1304.7215
* [10] A. Popescu: Standard Bases over Principal Ideal Rings, Master Thesis at Technische Universität Kaiserslautern (2013).
* [11] Y.H. Shen: Lexsegment ideals of Hilbert depth 1, (2012), arXiv:AC/1208.1822v1.
* [12] R.P. Stanley: Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
* [13] J. Uliczka: _Remarks on Hilbert series of graded modules over polynomial rings_ , Manuscripta Math. 132 (2010), 159-168.
|
arxiv-papers
| 2013-07-23T14:00:58 |
2024-09-04T02:49:48.352622
|
{
"license": "Public Domain",
"authors": "Adrian Popescu",
"submitter": "Adrian Popescu",
"url": "https://arxiv.org/abs/1307.6084"
}
|
1307.6165
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-125 LHCb-PAPER-2013-031 July, 23 2013
Studies of the decays $B^{+}\rightarrow p\overline{}ph^{+}$ and observation of
$B^{+}\rightarrow\kern 1.20007pt\overline{\kern-1.20007pt\mathchar
28931\relax}(1520)p$
The LHCb collaboration
Dynamics and direct $C\\!P$ violation in three-body charmless decays of
charged $B$ mesons to a proton, an antiproton and a light meson (pion or kaon)
are studied using data, corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$, collected by the LHCb experiment in $pp$ collisions at
a center-of-mass energy of $7$ TeV. Production spectra are determined as a
function of Dalitz-plot and helicity variables. The forward-backward asymmetry
of the light meson in the $p\overline{}p$ rest frame is measured. No
significant $C\\!P$ asymmetry in $B^{+}\rightarrow p\overline{}pK^{+}$ decay
is found in any region of the Dalitz plane. We present the first observation
of the decay $B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)(\rightarrow
K^{+}\overline{}p)p$ near the $K^{+}\overline{}p$ threshold and measure
$\mathcal{B}(B^{+}\rightarrow\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p)=(3.9^{+1.0}_{-0.9}~{}(\mathrm{stat})\pm
0.1~{}(\mathrm{syst})\pm 0.3~{}(\mathrm{BF}))\times 10^{-7}$, where BF denotes
the uncertainty on secondary branching fractions.
Submitted to Phys. Rev. D
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S. Farry51, D.
Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S.
Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R.
Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph.
Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30,
A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A. Hicheur1, E.
Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4, W. Hulsbergen40,
P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50, V.
Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40,
P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, C.
Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M. Karacson37, T.M.
Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7,
I. Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40,
L. Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F.
Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N.
La Thi38, D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W.
Lambert41, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30,
F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J.
Maratas5, U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G.
Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez
Santos41, D. Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z.
Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A.
McNab53, R. McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M.
Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S.
Monteil5, D. Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R.
Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
S. Neubert37, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o,
M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54,
A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S.
Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora
Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
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Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B.
Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8,
G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N.
Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E.
Santovetti23,k, M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M.
Savrie16,e, D. Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M.
Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A.
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Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires9, R.
Silva Coutinho47, M. Sirendi46, N. Skidmore45, T. Skwarnicki58, N.A. Smith51,
E. Smith54,48, J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
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Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stevenson54, S.
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Szczypka38,37, T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F.
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Tobin38, S. Tolk41, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E.
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P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, D. Urner53, A.
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Dijk45, R. Vazquez Gomez18, P. Vazquez Regueiro36, C. Vázquez Sierra36, S.
Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B.
Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10,
D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C.
Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
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Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z.
Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
Evidence of inclusive direct $C\\!P$ violation in three-body charmless decays
of $B^{+}$ mesons111Throughout the paper, the inclusion of charge conjugate
processes is implied, except in the definition of $C\\!P$ asymmetries. has
recently been found in the modes $B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}$,
$B^{+}\rightarrow K^{+}K^{+}K^{-}$, $B^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$,
and $B^{+}\rightarrow K^{+}K^{-}\pi^{+}$ [1, 2]. In addition, very large
$C\\!P$ asymmetries were observed in the low $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$
mass regions, without clear connection to a resonance. The localization of the
asymmetries and the correlation of the $C\\!P$ violation between the decays
suggest that $\pi^{+}\pi^{-}\leftrightarrow K^{+}K^{-}$ rescattering may play
an important role in the generation of the strong phase difference needed for
such a violation to occur [3, 4]. Conservation of $C\\!PT$ symmetry imposes a
constraint on the sum of the rates of final states with the same flavour
quantum numbers, providing the possibility of entangled long-range effects
contributing to the $C\\!P$ violating mechanism [5]. In contrast,
$h^{+}h^{-}\leftrightarrow p\overline{}p$ ($h=\pi$ or $K$ throughout the
paper) rescattering is expected to be suppressed compared to
$\pi^{+}\pi^{-}\leftrightarrow K^{+}K^{-}$, and thus is not expected to play
an important role.
The leading quark-level diagrams for the modes $B^{+}\rightarrow
p\overline{}ph^{+}$ are shown in Fig. 1. The $B^{+}\rightarrow
p\overline{}pK^{+}$ mode is expected to be dominated by the $b\rightarrow s$
loop (penguin) transition while the mode $B^{+}\rightarrow
p\overline{}p\pi^{+}$ is likely to be dominated by the $b\rightarrow u$ tree
decay, which is CKM suppressed compared to the former. Since the short
distance dynamics are similar to that of the $B^{+}\rightarrow
h^{+}h^{+}h^{-}$ modes, a $C\\!P$ analysis of $B^{+}\rightarrow
p\overline{}ph^{+}$ decays could help to clarify the role of long-range
scatterings in the $C\\!P$ asymmetries of $B^{+}\rightarrow h^{+}h^{+}h^{-}$
decays.
Figure 1: Leading tree and penguin diagrams for $B^{+}\rightarrow
p\overline{}ph^{+}$ decays.
First studies were performed at the $B$ factories on the production and
dynamics of $B^{+}\rightarrow p\overline{}ph^{+}$ decays [6, 7, 8]. The
results have shown a puzzling opposite behaviour of $B^{+}\rightarrow
p\overline{}pK^{+}$ and $B^{+}\rightarrow p\overline{}p\pi^{+}$ decays in the
asymmetric occupation of the Dalitz plane. Charmonium contributions to the
$B^{+}\rightarrow p\overline{}pK^{+}$ decay have been studied by LHCb [9].
This paper reports a detailed study of the dynamics of the $B^{+}\rightarrow
p\overline{}ph^{+}$ decays and a systematic search for $C\\!P$ violation, both
inclusively and in regions of the Dalitz plane. The charmless region, defined
for the invariant mass
$m_{p\overline{}p}<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, is of
particular interest. The relevant observables are the differential production
spectra of Dalitz-plot variables and the global charge asymmetry $A_{C\\!P}$,
defined as
$A_{C\\!P}=\frac{N(B^{-}\rightarrow f^{-})-N(B^{+}\rightarrow
f^{+})}{N(B^{-}\rightarrow f^{-})+N(B^{+}\rightarrow f^{+})},$ (1)
where $f^{\pm}=p\overline{}ph^{\pm}$. The mode
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)K^{+}$ serves as a control channel. The first observation of the
decay $B^{+}\rightarrow\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p$ is presented. Its branching fraction is derived through
the ratio of its yield to the measured yield of the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)K^{+}$ decay.
## 2 Detector and software
The LHCb detector [10] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing o̱r q̧uarks. The detector includes a high precision tracking system
consisting of a silicon-strip vertex detector surrounding the $pp$ interaction
region, a large-area silicon-strip detector located upstream of a dipole
magnet with a bending power of about $4{\rm\,Tm}$, and three stations of
silicon-strip detectors and straw drift tubes placed downstream. The combined
tracking system has momentum resolution $\Delta p/p$ that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP)
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors
(RICH) [11]. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers.
The trigger [12] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage which applies a
full event reconstruction. Events triggered both on objects independent of the
signal, and associated with the signal, are used. In the latter case, the
transverse energy of the hadronic cluster is required to be at least
3.5$\mathrm{\,Ge\kern-1.00006ptV}$. The software trigger requires a two-,
three- or four-track secondary vertex with a large sum of the transverse
momentum, $p_{\rm T}$, of the tracks and a significant displacement from all
primary $pp$ interaction vertices. At least one track must have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, track fit $\chi^{2}$ per degree
of freedom less than 2, and an impact parameter $\chi^{2}$
($\chi^{2}_{\mathrm{IP}}$) with respect to any primary interaction greater
than 16. The $\chi^{2}_{\mathrm{IP}}$ is defined as the difference between the
$\chi^{2}$ of the primary vertex reconstructed with and without the considered
track. A multivariate algorithm is used to identify secondary vertices [13].
The simulated $pp$ collisions are generated using Pythia 6.4 [14] with a
specific LHCb configuration [15]. Decays of hadronic particles are described
by EvtGen [16] in which final state radiation is generated using Photos [17].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [18, *Agostinelli:2002hh] as
described in Ref. [20]. Non-resonant $B^{+}\rightarrow p\overline{}ph^{+}$
events are simulated, uniformly distributed in phase space, to study the
variation of efficiencies across the Dalitz plane, as well as resonant samples
such as $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow p\overline{}p)K^{+}$, $B^{+}\rightarrow\eta_{c}(\rightarrow
p\overline{}p)K^{+}$, $B^{+}\rightarrow\psi{(2S)}(\rightarrow
p\overline{}p)K^{+}$, $B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)(\rightarrow
K^{+}\overline{}p)p$, and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)\pi^{+}$.
## 3 Signal reconstruction and determination
Candidate $B^{+}\rightarrow p\overline{}ph^{+}$ decays are formed by combining
three charged tracks, with appropriate mass assignments. The tracks are
required to satisfy track fit quality criteria and a set of loose selection
requirements on their momenta, transverse momenta, $\chi^{2}_{\mathrm{IP}}$,
and distance of closest approach between any pair of tracks. The requirement
on the momentum of the proton candidates,
$p>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, is larger than for the kaon and
pion candidates, $p>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $B^{+}$
candidates formed by the combinations are required to have $p_{\mathrm{T}}>$
1.7 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\mathrm{IP}}<10$.
The distance between the decay vertex and the primary vertex is required to be
greater than 3 mm, and the vector formed by the primary and decay vertices
must align with the $B^{+}$ candidate momentum. Particle identification (PID)
is applied to the proton, kaon and pion candidates, using combined subdetector
information, the main separation power being provided by the RICH system. The
PID efficiencies are derived from data calibration samples of kinematically
identified pions, kaons and protons originating from the decays
$D^{*+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ and $\mathchar
28931\relax\rightarrow p\pi^{-}$.
Signal and background are extracted using unbinned extended maximum likelihood
fits to the mass of the $p\overline{}ph^{+}$ combinations. The
$B^{+}\rightarrow p\overline{}pK^{+}$ signal is modelled by a double Gaussian
function. The combinatorial background is represented by a second-order
polynomial function. A Gaussian function accounting for a partially
reconstructed component from $B\rightarrow p\overline{}pK^{*}$ decays is used.
A possible $p\overline{}p\pi^{+}$ cross-feed contribution is included in the
fit and is found to be small. An asymmetric Gaussian function with power law
tails is used to estimate the uncertainties related to the variation of the
signal yield.
In the case of the $B^{+}\rightarrow p\overline{}p\pi^{+}$ decay, the signal
yield is smaller and the background is larger. The ranges of the signal and
cross-feed parameters are constrained to the values obtained in the simulation
within their uncertainties. The signal and the $p\overline{}pK^{+}$ cross-feed
contribution are modelled with Gaussian functions. The combinatorial
background is represented by a third-order polynomial function.
The $B^{+}\rightarrow p\overline{}ph^{+}$ invariant mass spectra are shown in
Fig. 2.
Figure 2: Invariant mass distributions of (left) $p\overline{}pK^{+}$ and
(right) $p\overline{}p\pi^{+}$ candidates. The points with error bars
represent data. The solid black line represents the total fit function. Blue
dashed, purple dotted, red long-dashed and green dashed-dotted curves
represent the signal, cross-feed, combinatorial background and partially
reconstructed background, respectively.
The signal yields obtained from the fits are $N(p\overline{}pK^{\pm})=7029\pm
139$ and $N(p\overline{}p\pi^{\pm})=656\pm 70$, where the uncertainties are
statistical only.
## 4 Dynamics of $B^{+}\rightarrow p\overline{}ph^{+}$ decays
To probe the dynamics of the $B^{+}\rightarrow p\overline{}ph^{+}$ decays,
differential production spectra are derived as a function of
$m_{p\overline{}p}$ and $\cos\theta_{p}$, where $\theta_{p}$ is the angle
between the charged meson $h$ and the opposite-sign baryon in the rest frame
of the $p\overline{}p$ system. The $p\overline{}ph^{+}$ invariant mass is
fitted in bins of the aforementioned variables and the signal yields are
corrected for trigger, reconstruction and selection efficiencies. They are
estimated with simulated samples and corrected to account for discrepancies
between data and simulation. The signal yields are determined with the fit
models described in the previous section, but allowing the combinatorial
background parameters to vary. The systematic uncertainties are determined for
each bin and include uncertainties related to the PID correction, fit model,
trigger efficiency, and the size of the simulated samples. The latter is
evaluated from the differences between data and simulation as a function of
the Dalitz-plot variables. No trigger-induced distortions are found.
### 4.1 Invariant mass of the $p\overline{}p$ system
Table 1: Fitted $B^{+}\rightarrow p\overline{}pK^{+}$ signal yield, including the charmonium modes, efficiency and relative systematic uncertainty, in bins of $p\overline{}p$ invariant mass. The error on the efficiency includes all the sources of uncertainty. $m_{p\overline{}p}$ $[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}]$ | $B^{+}\rightarrow p\overline{}pK^{+}$ yield | Efficiency (%) | Syst. (%)
---|---|---|---
$<$ 2.85 | 3315$\pm$ | 83 | 1.74$\pm$0.04 | 2.9
$<$ 2 | 446$\pm$ | 32 | 1.80$\pm$0.08 | 8.1
$[2,2.2]$ | 1001$\pm$ | 42 | 1.77$\pm$0.05 | 4.4
$[2.2,2.4]$ | 732$\pm$ | 39 | 1.77$\pm$0.03 | 4.0
$[2.4,2.6]$ | 550$\pm$ | 35 | 1.67$\pm$0.03 | 3.4
$[2.6,2.85]$ | 580$\pm$ | 34 | 1.67$\pm$0.02 | 2.9
$[2.85,3.15]$ | 2768$\pm$ | 58 | 1.61$\pm$0.02 | 2.6
$[3.15,3.3]$ | 125$\pm$ | 18 | 1.57$\pm$0.03 | 3.8
$[3.3,4]$ | 585$\pm$ | 37 | 1.47$\pm$0.01 | 2.2
$>$ 4 | 233$\pm$ | 32 | 1.22$\pm$0.01 | 2.3
Table 2: Fitted $B^{+}\rightarrow p\overline{}p\pi^{+}$ signal yield, including the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mode, $B^{+}\rightarrow p\overline{}pK^{+}$ cross-feed yield, signal efficiency, and relative systematic uncertainty in bins of $p\overline{}p$ invariant mass. $m_{p\overline{}p}$ $[{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}]$ | $B^{+}\rightarrow p\overline{}p\pi^{+}$ yield | $B^{+}\rightarrow p\overline{}pK^{+}$ cross-feed | Efficiency (%) | Syst. (%)
---|---|---|---|---
$<$ 2.85 | 564$\pm$ | 61 | 114$\pm$ | 62 | 1.31$\pm$0.10 | 7.6
$<$ 2 | 140$\pm$ | 26 | 64$\pm$ | 26 | 1.34$\pm$0.15 | 11
$[2,2.2]$ | 261$\pm$ | 31 | 10$\pm$ | 29 | 1.30$\pm$0.10 | 7.9
$[2.2,2.4]$ | 95$\pm$ | 30 | 0$\pm$ | 39 | 1.33$\pm$0.09 | 7.1
$[2.4,2.6]$ | 48$\pm$ | 28 | 14$\pm$ | 30 | 1.35$\pm$0.09 | 6.4
$[2.6,2.85]$ | 21$\pm$ | 20 | 35$\pm$ | 23 | 1.26$\pm$0.07 | 5.9
$[2.85,3.15]$ | 72$\pm$ | 19 | 12$\pm$ | 18 | 1.28$\pm$0.07 | 5.5
$[3.15,3.3]$ | 19$\pm$ | 11 | 0$\pm$ | 3 | 1.24$\pm$0.08 | 6.7
$[3.3,4]$ | 0$\pm$ | 7 | 0$\pm$ | 23 | 1.24$\pm$0.06 | 4.7
$>$ 4 | 23$\pm$ | 21 | 57$\pm$ | 23 | 0.94$\pm$0.05 | 4.9
The yields and total efficiency for $B^{+}\rightarrow p\overline{}ph^{+}$ in
$m_{p\overline{}p}$ bins are shown in Tables 1 and 2. The charmonium
contributions originate from the decays
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)K^{+}$, $B^{+}\rightarrow\eta_{c}(\rightarrow
p\overline{}p)K^{+}$ and $B^{+}\rightarrow\psi{(2S)}(\rightarrow
p\overline{}p)K^{+}$ for the $B^{+}\rightarrow p\overline{}pK^{+}$ mode, and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)\pi^{+}$ for the $B^{+}\rightarrow p\overline{}p\pi^{+}$ mode.
Before deriving the distributions, the charmonium contributions are unfolded
by performing two dimensional extended unbinned maximum likelihood fits to the
$p\overline{}ph^{+}$ and $p\overline{}p$ invariant masses. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ resonances are
modelled by Gaussian functions and the $\eta_{c}$ resonance is modelled by a
convolution of Breit-Wigner and Gaussian functions. The non-resonant
$p\overline{}p$ component and the combinatorial background are modelled by
polynomial shapes. Table 3 shows the yields of contributing charmonium modes.
The results are consistent with those reported in Ref. [9].
Table 3: Yields, efficiencies and relative systematic uncertainties of the charmonium modes from the combined $(m_{p\overline{}ph^{+}},m_{p\overline{}p})$ fits for the regions $m_{p\overline{}p}\in[2.85,3.15]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (for both $B^{+}\rightarrow p\overline{}pK^{+}$ and $B^{+}\rightarrow p\overline{}p\pi^{+}$) and $[3.60,3.75]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (for $B^{+}\rightarrow p\overline{}pK^{+}$). Mode | Yield | Efficiency (%) | Syst. (%)
---|---|---|---
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow p\overline{}p)K^{+}$ | 1413$\pm$ | 40 | 1.624$\pm$0.005 | 1.8
$B^{+}\rightarrow\eta_{c}(\rightarrow p\overline{}p)K^{+}$ | 722$\pm$ | 36 | 1.660$\pm$0.005 | 2.0
$B^{+}\rightarrow\psi{(2S)}(\rightarrow p\overline{}p)K^{+}$ | 132$\pm$ | 16 | 1.475$\pm$0.011 | 1.5
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow p\overline{}p)\pi^{+}$ | 59$\pm$ | 11 | 1.328$\pm$0.011 | 4.2
After unfolding, the efficiency-corrected differential distributions are shown
in Fig. 3. An enhancement is observed at low $p\overline{}p$ mass both for
$B^{+}\rightarrow p\overline{}pK^{+}$ and $B^{+}\rightarrow
p\overline{}p\pi^{+}$, with a more sharply peaked distribution for
$B^{+}\rightarrow p\overline{}p\pi^{+}$. This accumulation of events at low
$m_{p\overline{}p}$ is a well known feature that has also been observed in
different contexts such as $\Upsilon(1S)\rightarrow\gamma p\overline{}p$ [21],
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\gamma p\overline{}p$
[22] and $B^{0}\rightarrow D^{(*)0}p\overline{}p$ [23] decays. It appears to
be caused by proton-antiproton rescattering and is modulated by the particular
kinematics of the decay from which the $p\overline{}p$ pair originates [24].
Figure 3: Efficiency-corrected differential yield as a function of
$m_{p\overline{}p}$ for (left) $B^{+}\rightarrow p\overline{}pK^{+}$ and
(right) $B^{+}\rightarrow p\overline{}p\pi^{+}$. The data points are shown
with their statistical and total uncertainties. For comparison, the solid
lines represent the expectations for a uniform phase space production,
normalized to the efficiency-corrected area.
### 4.2 Invariant mass squared of the $Kp$ system
The $B^{+}\rightarrow p\overline{}pK^{+}$ signal yield as a function of the
Dalitz-plot variable $m_{Kp}^{2}$ is considered, where $Kp$ denotes the
neutral combinations $K^{-}p$ or $K^{+}\overline{}p$. Table 4 shows the yields
and efficiencies, after the charmonium bands have been vetoed in the ranges
$m_{p\overline{}p}\in[2.85,3.15]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
and $[3.60,3.75]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The differential
spectrum derived after efficiency correction is shown in Fig. 4. Contrary to
the situation for $m_{p\overline{}p}$, the data distribution is in reasonable
agreement with the uniform phase space distribution, with some discrepancies
in the region
$m_{Kp}^{2}\in[4,12]~{}({\mathrm{Ge\kern-1.00006ptV\\!/}c^{2}})^{2}$.
Table 4: Fitted $B^{+}\rightarrow p\overline{}pK^{+}$ yields after subtracting the charmonium bands, efficiencies and relative systematic uncertainties in bins of $Kp$ invariant mass squared. $m_{Kp}^{2}$ $[({\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})^{2}]$ | $B^{+}\rightarrow p\overline{}pK^{+}$ yield | Efficiency (%) | Syst. (%)
---|---|---|---
$<$ 4 | 454$\pm$37 | 1.40$\pm$0.02 | 3.3
$[4,6]$ | 522$\pm$36 | 1.43$\pm$0.02 | 2.5
$[6,8]$ | 797$\pm$37 | 1.45$\pm$0.01 | 2.6
$[8,10]$ | 702$\pm$42 | 1.51$\pm$0.01 | 2.6
$[10,12]$ | 445$\pm$32 | 1.53$\pm$0.01 | 2.8
$[12,14]$ | 526$\pm$34 | 1.66$\pm$0.01 | 2.8
$[14,16]$ | 338$\pm$29 | 1.67$\pm$0.02 | 3.4
$>$ 16 | 305$\pm$28 | 1.66$\pm$0.02 | 3.5
Figure 4: Efficiency-corrected differential yield as a function of
$m_{Kp}^{2}$ for $B^{+}\rightarrow p\overline{}pK^{+}$. The data points are
shown with their statistical and total uncertainties. The solid line
represents the expectation for a uniform phase space production, normalized to
the efficiency-corrected area, for comparison.
### 4.3 Helicity angle of the $p\overline{}p$ system
The $B^{+}\rightarrow p\overline{}ph^{+}$ signal yields are considered as a
function of $\cos\theta_{p}$. Tables 5 and 6 show the corresponding yields and
efficiencies. The differential distributions are shown in Fig. 5.
Table 5: Fitted $B^{+}\rightarrow p\overline{}pK^{+}$ yields, efficiencies and relative systematic uncertainties in bins of $\cos\theta_{p}$. $\cos\theta_{p}$ range | $B^{+}\rightarrow p\overline{}pK^{+}$ yield | Efficiency (%) | Syst. (%)
---|---|---|---
$[-1,-0.75]$ | 508$\pm$ | 34 | 1.54$\pm$0.01 | 2.7
$[-0.75,-0.5]$ | 497$\pm$ | 31 | 1.51$\pm$0.02 | 3.0
$[-0.5,-0.25]$ | 309$\pm$ | 27 | 1.48$\pm$0.01 | 2.9
$[-0.25,0]$ | 381$\pm$ | 28 | 1.49$\pm$0.01 | 2.6
$[0,0.25]$ | 640$\pm$ | 46 | 1.51$\pm$0.01 | 2.9
$[0.25,0.5]$ | 799$\pm$ | 42 | 1.52$\pm$0.01 | 2.2
$[0.5,0.75]$ | 976$\pm$ | 41 | 1.56$\pm$0.01 | 2.8
$[0.75,1]$ | 1346$\pm$ | 51 | 1.55$\pm$0.01 | 2.7
Table 6: Fitted $B^{+}\rightarrow p\overline{}p\pi^{+}$ signal yields, efficiencies and relative systematic uncertainties in bins of $\cos\theta_{p}$. $\cos\theta_{p}$ range | $B^{+}\rightarrow p\overline{}p\pi^{+}$ yield | Efficiency(%) | Syst. (%)
---|---|---|---
$[-1,-0.75]$ | 150$\pm$ | 31 | 1.23$\pm$0.02 | 5.5
$[-0.75,-0.5]$ | 85$\pm$ | 27 | 1.15$\pm$0.02 | 5.5
$[-0.5,-0.25]$ | 104$\pm$ | 24 | 1.19$\pm$0.02 | 5.5
$[-0.25,0]$ | 77$\pm$ | 23 | 1.19$\pm$0.02 | 5.5
$[0,0.25]$ | 43$\pm$ | 21 | 1.14$\pm$0.02 | 5.5
$[0.25,0.5]$ | 24$\pm$ | 20 | 1.16$\pm$0.02 | 5.5
$[0.5,0.75]$ | 10$\pm$ | 12 | 1.19$\pm$0.02 | 5.5
$[0.75,1]$ | 93$\pm$ | 26 | 1.19$\pm$0.02 | 5.2
Figure 5: Efficiency-corrected differential yields as functions of
$\cos\theta_{p}$ for (left) $B^{+}\rightarrow p\overline{}pK^{+}$ and (right)
$B^{+}\rightarrow p\overline{}p\pi^{+}$ modes, after subtraction of the
charmonium contributions. The data points are shown with their statistical and
total uncertainties.
The forward-backward asymmetries are derived by comparing the yields for
$\cos\theta_{p}>0$ and $\cos\theta_{p}<0$, accounting for the averaged
efficiencies in each region
$A_{\mathrm{FB}}=\frac{\frac{N_{\mathrm{pos}}}{\epsilon_{\mathrm{pos}}}-\frac{N_{\mathrm{neg}}}{\epsilon_{\mathrm{neg}}}}{\frac{N_{\mathrm{pos}}}{\epsilon_{\mathrm{pos}}}+\frac{N_{\mathrm{neg}}}{\epsilon_{\mathrm{neg}}}}=\frac{N_{\mathrm{pos}}-fN_{\mathrm{neg}}}{N_{\mathrm{pos}}+fN_{\mathrm{neg}}},$
(2)
where $\epsilon_{\mathrm{pos}}=\epsilon(\cos\theta_{p}>0)$ and
$\epsilon_{\mathrm{neg}}=\epsilon(\cos\theta_{p}<0)$ are the averaged
efficiencies, $f=\epsilon_{\mathrm{pos}}/\epsilon_{\mathrm{neg}}$ and
$N_{\mathrm{pos}}=N(\cos\theta_{p}>0)$,
$N_{\mathrm{neg}}=N(\cos\theta_{p}<0)$. The values obtained are:
$A_{\mathrm{FB}}(p\overline{}pK^{+})=0.370\pm 0.018~{}(\mathrm{stat})\pm
0.016~{}(\mathrm{syst})$ and $A_{\mathrm{FB}}(p\overline{}p\pi^{+})=-0.392\pm
0.117~{}(\mathrm{stat})\pm 0.015~{}(\mathrm{syst})$.
A clear opposite angular correlation between $B^{+}\rightarrow
p\overline{}pK^{+}$ and $B^{+}\rightarrow p\overline{}p\pi^{+}$ decays is
observed; the light meson $h$ tends to align with the opposite-sign baryon for
$B^{\pm}\rightarrow p\overline{}pK^{\pm}$ while it aligns with the same-sign
baryon for the $B^{\pm}\rightarrow p\overline{}p\pi^{\pm}$ mode. A quark level
analysis suggests that the meson should align with the same-sign baryon, since
the opposite-sign baryon has larger momentum, being formed by products from
the decaying q̱uark [25]. This is in agreement with the angular spectrum of
$B^{+}\rightarrow p\overline{}p\pi^{+}$ but not for $B^{+}\rightarrow
p\overline{}pK^{+}$ decays.
### 4.4 Dalitz plot
From the fits to the $B$-candidate invariant mass, shown in Fig. 2, signal
weights are calculated with the sPlot technique [26] and are used to produce
the signal Dalitz-plot distributions shown in Fig. 6. To ease the comparison,
the $\cos\theta_{p}$ curves corresponding to the boundaries of the eight bins
used to make the angular distributions in Fig. 5 are superimposed.
Figure 6: Signal weighted Dalitz-plot distributions for (left)
$B^{+}\rightarrow p\overline{}pK^{+}$ and (right) $B^{+}\rightarrow
p\overline{}p\pi^{+}$. Also shown are the iso-$\cos\theta_{p}$ lines
corresponding to the $\cos\theta_{p}$ bin boundaries; $\cos\theta_{p}=-1$ (+1)
is the uppermost (lowermost) line. The distributions are not corrected for
efficiency.
With the exception of the charmonium bands ($\eta_{c}$,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\psi{(2S)}$ for
$B^{+}\rightarrow p\overline{}pK^{+}$, and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ for $B^{+}\rightarrow
p\overline{}p\pi^{+}$), the structure of the low $p\overline{}p$ mass
enhancement is very different between $B^{+}\rightarrow p\overline{}pK^{+}$
and $B^{+}\rightarrow p\overline{}p\pi^{+}$. The $B^{+}\rightarrow
p\overline{}pK^{+}$ events are distributed in the middle and lower
$m_{Kp}^{2}$ half, exhibiting a possible $p\overline{}p$ band structure near
$4~{}\mathrm{GeV}^{2}/c^{4}$. An enhancement at low $m_{Kp}$ is also observed
and is caused to a large extent by a $\mathchar 28931\relax(1520)$ signal, as
will be shown in the next section. The $B^{+}\rightarrow p\overline{}p\pi^{+}$
events are mainly clustered in the upper $m_{\pi p}^{2}$ half, with also a few
events on the doubly-charged top diagonal $(p\pi)^{++}$ (near the
$\cos\theta_{p}=-1$ boundary). These distributions of events are consistent
with the angular distributions and asymmetries reported earlier.
## 5 Measurement of $A_{C\\!P}$ for $B^{+}\rightarrow p\overline{}pK^{+}$
decays
The raw charge asymmetry is obtained by performing a simultaneous extended
unbinned maximum likelihood fit to the $B^{-}$ and $B^{+}$ samples. The
$B^{\pm}$ yields are defined as a function of the total yield $N$ and the raw
asymmetry, $A_{\rm raw}$, by $N^{\mp}=N(1\pm A_{\rm raw})/2$.
The $C\\!P$ asymmetry is then derived after correcting for the $B^{\pm}$
production asymmetry $A_{\rm P}(B^{\pm})$ and the kaon detection asymmetry
$A_{\rm D}(K^{\pm})$
$A_{C\\!P}=A_{\rm raw}-A_{\rm P}(B^{\pm})-A_{\rm D}(K^{\pm}).$ (3)
The correction $A_{\Delta}=A_{\rm P}(B^{\pm})+A_{\rm D}(K^{\pm})$ is measured
from data with the decay
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)K^{\pm}$ which is part of the data sample
$A_{\Delta}=A_{\rm raw}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow
p\overline{}p)K^{\pm})-A_{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}),$ (4)
where $A_{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm})=(1\pm
7)\times 10^{-3}$ [27].
Another correction has been applied to account for the proton antiproton
asymmetry, which exactly cancels for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow p\overline{}p)K^{\pm}$ but not necessarily in the full
phase space of $p\overline{}pK^{\pm}$ events. This effect has been estimated
in simulation studying the difference in the interactions of protons and
antiprotons with the detector material between
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)K^{\pm}$ and $p\overline{}pK^{\pm}$ events generated uniformly
over phase space. We obtained a $m_{Kp}^{2}$-dependent bias, up to 3% for the
highest bin, for $A_{\rm raw}$.
To measure $A_{\rm raw}$ for charmonium modes, and in particular
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow
p\overline{}p)K^{\pm}$, a two dimensional $(m_{B},m_{p\overline{}p})$
simultaneous fit to the $B^{+}$ and $B^{-}$ samples is performed. The
systematic uncertainties are estimated by varying the fit functions and
splitting the data sample according to trigger requirements or magnet
polarities, and recombining the results from the sub-samples. The procedure is
applied to obtain a global value of $A_{C\\!P}$ as well as the variation of
the asymmetry as a function of the Dalitz-plot variables. The results are:
$A_{C\\!P}=-0.022\pm 0.031~{}(\mathrm{stat})\pm 0.007~{}(\mathrm{syst})$ for
the full $p\overline{}pK^{\pm}$ spectrum, and $A_{C\\!P}=-0.047\pm
0.036~{}(\mathrm{stat})\pm 0.007~{}(\mathrm{syst})$ for the region
$m_{p\overline{}p}<2.85~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Figure 7
shows the variation of $A_{C\\!P}$ as a function of the Dalitz-plot variables.
Figure 7: Distribution of $A_{C\\!P}$ for the Dalitz-plot projections on
$m_{p\overline{}p}$ and $m_{Kp}^{2}$ for $B^{\pm}\rightarrow
p\overline{}pK^{\pm}$ events. In the $m_{p\overline{}p}$ projection (left),
the bin $[2.85,3.15]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ contains only
the value of the charmless $p\overline{}pK^{\pm}$ after subtraction of the
$\eta_{c}$-${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ contribution. The
$m_{Kp}^{2}$ projection (right) has been obtained after removing the charmonia
bands.
For the charmonium resonances, the values are:
$A_{C\\!P}(\eta_{c}K^{\pm})=0.046\pm 0.057~{}(\mathrm{stat})\pm
0.007~{}(\mathrm{syst})$ and $A_{C\\!P}(\psi{(2S)}K^{\pm})=-0.002\pm
0.123~{}(\mathrm{stat})\pm 0.012~{}(\mathrm{syst})$. All results indicate no
significant $C\\!P$ asymmetries.
## 6 Observation of the $B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)p$ decay
In the $p\overline{}pK^{+}$ spectrum, near the threshold of the neutral $Kp$
combination, a peak in invariant mass at
1.52${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is observed, as shown in Fig.
8, corresponding to the $\overline{}u\overline{}d\overline{}s$ resonance
$\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)$. The
possible presence of higher $\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}$ and $\overline{}\mathchar 28934\relax$ resonances may explain
the enhancement in the range of
$[1.6,1.7]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Figure 8: Invariant mass $m_{Kp}$ for the $B^{+}\rightarrow
p\overline{}pK^{+}$ candidates near threshold.
To identify the $\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)$ signal, the $B^{+}$ signal is analyzed in the region
$m_{Kp}\in[1.44,1.585]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Figure 9
shows the $B$ signal weighted $Kp$ invariant mass, and the expected $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)$ shape obtained
from a model based on an asymmetric Breit-Wigner function derived from an
EvtGen [16] simulation of the decay $B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)p$, convolved
with a Gaussian resolution function, and a second-order polynomial function
representing the tail of the non-$\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)$
$B^{+}\rightarrow p\overline{}pK^{+}$ decays.
Figure 9: Fit to the $B$ signal weighted $m_{Kp}$ distribution.
These shapes are then used in a two dimensional
$(m_{p\overline{}pK^{+}},m_{Kp})$ extended unbinned maximum likelihood fit to
obtain the $B^{+}\rightarrow\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p$ yield. The fit results in $N(B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p)=47^{+12}_{-11}$ with a statistical significance of $5.3$
standard deviations, obtained by comparing the likelihood at its maximum for
the nominal fit and for the background-only hypothesis. Figure 10 shows the
projections of the fit for the $Kp$ and $p\overline{}pK^{+}$ invariant masses.
Figure 10: Projections of (left) $m_{Kp}$ and (right) $m_{p\overline{}pK^{+}}$
of the two dimensional fit used to obtain the $B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)p$ signal yield.
To test the robustness of the observation, different representations of the
$Kp$ background have been used, combining first or second order polynomials
and a contribution modelled by a Breit-Wigner function, for which the mean
($\mu$) and width ($\Gamma$) are allowed to vary within the known values of
the $\mathchar 28931\relax(1600)$ baryon
($\mu\in[1.56,1.7]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
$\Gamma\in[0.05,0.25]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$). Fits in a
wider $m_{Kp}$ range were also considered. In all cases the yield was stable
with a statistical significance similar to the nominal fit case.
The branching fraction for the decay $B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)p$ is derived
from the ratio
$\frac{\mathcal{B}(B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)(\rightarrow
K^{+}\overline{}p)p)}{\mathcal{B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow p\overline{}p)K^{+})}=\frac{N_{\mathchar
28931\relax(1520)\rightarrow Kp}}{N_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow
p\overline{}p}}\times\frac{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow p\overline{}p}^{\mathrm{gen}}}{\epsilon_{\mathchar
28931\relax(1520)\rightarrow
Kp}^{\mathrm{gen}}}\times\frac{\epsilon_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow p\overline{}p}^{\mathrm{sel}}}{\epsilon_{\mathchar
28931\relax(1520)\rightarrow Kp}^{\mathrm{sel}}},$ (5)
where $N_{i}$ is the yield of the decay chain $i$, $\epsilon^{\mathrm{gen}}$
denotes the efficiency after geometrical acceptance and simulation
requirements. The global selection efficiency $\epsilon^{\mathrm{sel}}$
includes the reconstruction, the trigger, the offline selection, and the
particle identification requirements. The ratio of branching fractions
obtained is
$\frac{\mathcal{B}(B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)(\rightarrow
K^{+}\overline{}p)p)}{\mathcal{B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow
p\overline{}p)K^{+})}=0.041^{+0.011}_{-0.010}~{}(\mathrm{stat})\pm
0.001~{}(\mathrm{syst}).$
The systematic uncertainties include effects of the $Kp$ background model, the
particle identification, the limited simulation sample size, the uncertainties
on the relative trigger efficiencies, and are summarized in Table 7.
Convolving the systematic uncertainty with the statistical likelihood profile,
the global significance is 5.1 standard deviations.
Table 7: Systematic uncertainties for the $\mathcal{B}(B^{+}\rightarrow\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)(\rightarrow K^{+}\overline{}p)p)/\mathcal{B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}(\rightarrow p\overline{}p)K^{+})$ branching fraction ratio. The total uncertainty is the sum in quadrature of the individual sources. Source | Uncertainty (%)
---|---
$Kp$ background | 2.1
PID | 1.7
Simulation sample size | 0.5
Trigger | 1.0
Total | 2.9
Using $\mathcal{B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})=(1.016\pm 0.033)\times 10^{-3}$,
$\mathcal{B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow
p\overline{}p)=(2.17\pm 0.07)\times 10^{-3}$ [27], and
$\mathcal{B}(\Lambda(1520)\rightarrow K^{-}p)=0.234\pm 0.016$ [28], the
branching fraction is
$\mathcal{B}(B^{+}\rightarrow\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p)=(3.9^{+1.0}_{-0.9}~{}(\mathrm{stat})\pm
0.1~{}(\mathrm{syst})\pm 0.3~{}(\mathrm{BF}))\times 10^{-7}$.
The last error corresponds to the uncertainty on the secondary branching
fractions. This result is in agreement with the upper limit set in Ref. [6],
$\mathcal{B}(B^{+}\rightarrow\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p)<1.5\times 10^{-6}$.
Considering the separate $B^{\pm}$ signals in the range
$m_{Kp}\in[1.44,1.585]~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the yields
are $N(B^{-})=50\pm 12$ and $N(B^{+})=27\pm 11$.
## 7 Summary
Based on a data sample, corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$, collected in 2011 by the LHCb experiment, an analysis
of the three body $B^{+}\rightarrow p\overline{}ph^{+}$ decays ($h=K$ or
$\pi$) has been performed. The dynamics of the decays has been probed using
differential spectra of Dalitz-plot variables and signal-weighted Dalitz
plots. The charmless $B^{+}\rightarrow p\overline{}pK^{+}$ decay populates
mainly the low $m_{p\overline{}p}^{2}$ and lower
$m_{K^{+}\overline{}p}^{2}$-half regions whereas the $B^{+}\rightarrow
p\overline{}p\pi^{+}$ decay has a similar enhancement at low
$m_{p\overline{}p}^{2}$ but with an upper $m_{\pi^{+}\overline{}p}^{2}$-half
occupancy. From the occupation pattern of the Dalitz plots, it is likely that
the $B^{+}\rightarrow p\overline{}pK^{+}$ decay is primarily driven by
$p\overline{}p$ rescattering with a secondary contribution from neutral $Kp$
rescattering while the $B^{+}\rightarrow p\overline{}p\pi^{+}$ decay is also
dominated by $p\overline{}p$ rescattering but with a secondary contribution
from doubly-charged $(p\pi)^{++}$ rescattering, along the lines of the
rescattering amplitude analysis performed in Ref. [29]. This difference of
behaviour is reflected in the values of the forward-backward asymmetry of the
light meson in the $p\overline{}p$ rest frame
$A_{\mathrm{FB}}(p\overline{}pK^{+})=\phantom{-}0.370\pm
0.018~{}(\mathrm{stat})\pm 0.016~{}(\mathrm{syst})$,
$A_{\mathrm{FB}}(p\overline{}p\pi^{+})=-0.392\pm 0.117~{}(\mathrm{stat})\pm
0.015~{}(\mathrm{syst})$.
$C\\!P$ asymmetries for the $B^{+}\rightarrow p\overline{}pK^{+}$ decay have
been measured and no significant deviation from zero observed:
$A_{C\\!P}=-0.047\pm 0.036~{}(\mathrm{stat})\pm 0.007~{}(\mathrm{syst})$ for
the charmless region
$m_{p\overline{}p}<2.85~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
$A_{C\\!P}(\eta_{c}K^{\pm})=0.046\pm 0.057~{}(\mathrm{stat})\pm
0.007~{}(\mathrm{syst})$ and $A_{C\\!P}(\psi{(2S)}K^{\pm})=-0.002\pm
0.123~{}(\mathrm{stat})\pm 0.012~{}(\mathrm{syst})$. These measurements are
consistent with the current known values, $A_{C\\!P}(B^{\pm}\rightarrow
p\overline{}pK^{\pm},m_{p\overline{}p}<2.85~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}})=-0.16\pm
0.07$ [27], $A_{C\\!P}(\eta_{c}K^{\pm})=-0.16\pm 0.08~{}(\mathrm{stat})\pm
0.02~{}(\mathrm{syst})$ [8], and $A_{C\\!P}(\psi{(2S)}K^{\pm})=-0.025\pm
0.024$ [27]. The absence of any significant charge asymmetry, contrary to the
situation for $B^{+}\rightarrow h^{+}h^{+}h^{-}$ decays [1, 2], may be due to
different long range behaviour. Final state interactions in the
$B^{+}\rightarrow p\overline{}ph^{+}$ case do not change the nature of the
particles, such as $p\overline{}p\rightarrow p\overline{}p$ or $ph\rightarrow
ph$, while $B^{+}\rightarrow h^{+}h^{+}h^{-}$ modes can be affected by
$\pi^{+}\pi^{-}\leftrightarrow K^{+}K^{-}$ scattering.
Finally, the observation of the decay $B^{+}\rightarrow\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}(1520)p$ is reported,
with the branching fraction
$\mathcal{B}(B^{+}\rightarrow\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)p)=(3.9^{+1.0}_{-0.9}~{}(\mathrm{stat})\pm
0.1~{}(\mathrm{syst})\pm 0.3~{}(\mathrm{BF}))\times 10^{-7}$,
in agreement with the current existing upper limit [6].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-07-23T17:21:02 |
2024-09-04T02:49:48.362730
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, C. Gotti, M. Grabalosa G\\'andara, R. Graciani Diaz,\n L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, A. Hicheur, E. Hicks, D. Hill, M. Hoballah, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I.\n Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H.\n Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J.\n Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D.\n Martinez Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z.\n Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R.\n McNulty, B. McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I. Mous, F. Muheim, K.\n M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar,\n I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C.\n Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch,\n A. Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P.\n Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin,\n U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez,\n P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, M. Witek, S.A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Adlene Hicheur",
"url": "https://arxiv.org/abs/1307.6165"
}
|
1307.6379
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-130 LHCb-PAPER-2013-008 24 July 2013
Measurement of $J/\psi$ polarization
in $pp$ collisions at $\sqrt{s}=7$ TeV
The LHCb collaboration111Authors are listed on the following pages.
An angular analysis of the decay ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ is performed to measure the polarization
of prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons produced in
$pp$ collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The dataset
corresponds to an integrated luminosity of 0.37 $\mbox{\,fb}^{-1}$ collected
with the LHCb detector. The measurement is presented as a function of
transverse momentum, $p_{\rm T}$, and rapidity, $y$, of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, in the kinematic region
$2<\mbox{$p_{\rm T}$}<15~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$.
Published in Eur. Phys. J. C
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso57,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso58, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,p, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz25, K.
Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V.
Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di
Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil
Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, A. Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S.
Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L.
Eklund50,37, I. El Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C.
Färber11, G. Fardell49, C. Farinelli40, S. Farry12, V. Fave38, D. Ferguson49,
V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32,
M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O.
Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini57,
Y. Gao3, J. Garofoli57, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph.
Ghez4, V. Gibson46, V.V. Gligorov37, C. Göbel58, D. Golubkov30, A.
Golutvin52,30,37, A. Gomes2, H. Gordon54, C. Gotti20, M. Grabalosa Gándara5,
R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A.
Grecu28, E. Greening54, S. Gregson46, O. Grünberg59, B. Gui57, E. Gushchin32,
Yu. Guz34,37, T. Gys37, C. Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C.
Haines46, S. Hall52, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T.
Harnew45, J. Harrison53, T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A. Hicheur1, E.
Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4, W. Hulsbergen40,
P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50, V.
Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40,
P. Jaton38, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, C. Joram37, B.
Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37, T.M. Karbach37, I.R.
Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I.
Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L.
Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9,
M. Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38,
D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18,37, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38,
H. Lu3, D. Lucchesi21,p, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53, P.
Morawski25, M.J. Morello22,r, R. Mountain57, I. Mous40, F. Muheim49, K.
Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R.
Nandakumar48, I. Nasteva1, M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D.
Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T.
Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal57, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L.
Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M.
Perrin-Terrin6, K. Petridis52, A. Petrolini19,i, A. Phan57, E. Picatoste
Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo
Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D.
Popov10, B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A.
Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,q, W.
Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42,
N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B.
Saitta15,d, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42,
P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O.
Shevchenko42, V. Shevchenko30, A. Shires52, R. Silva Coutinho47, T.
Skwarnicki57, N.A. Smith51, E. Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P.
Soler50, F. Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49,
P. Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone57, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G.
Valenti14, R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J.
Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12,
S. Wandernoth11, J. Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, Z. Xing57, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pUniversità di Padova, Padova, Italy
qUniversità di Pisa, Pisa, Italy
rScuola Normale Superiore, Pisa, Italy
## 1 Introduction
Studies of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in
hadronic collisions provide powerful tests of QCD. In $pp$ collisions,
quarkonium resonances can be produced directly, through feed-down from higher
quarkonium states (such as the $\psi{(2S)}$ or $\chi_{c}$ resonances [1]), or
via the decay of $b$ hadrons. The first two production mechanisms are
generically referred to as prompt production. The mechanism for prompt
production is not yet fully understood and none of the available models
adequately predicts the observed dependence of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-section and
polarization on its transverse momentum $p_{\rm T}$ [1]. This paper describes
the measurement of the polarization of the prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ component in $pp$ collisions at
$\sqrt{s}=7$ $\mathrm{\,Te\kern-1.00006ptV}$, using the dimuon decay mode. The
measured polarization is subsequently used to update the LHCb measurement of
the cross-section given in Ref. [2]. This improves the precision of the cross-
section measurement significantly as the polarization and overall
reconstruction efficiency are highly correlated.
The three polarization states of a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ vector meson are specified in terms of a chosen coordinate system in
the rest frame of the meson. This coordinate system is called the polarization
frame and is defined with respect to a particular polarization axis. Defining
the polarization axis to be the $Z$-axis, the $Y$-axis is chosen to be
orthogonal to the production plane (the plane containing the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum and the beam axis) and
the $X$-axis is oriented to create a right-handed coordinate system.
Several polarization frame definitions can be found in the literature. In the
helicity frame [3] the polarization axis coincides with the flight direction
of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the centre-of-mass
frame of the colliding hadrons. In the Collins-Soper frame [4] the
polarization axis is the direction of the relative velocity of the colliding
beams in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame.
The angular decay distribution, apart from a normalization factor, is
described by
$\frac{d^{2}N}{d\cos\theta\;d\phi}\propto
1+\lambda_{\theta}\cos^{2}\\!\theta+\lambda_{\theta\phi}\sin
2\theta\cos\phi+\lambda_{\phi}\sin^{2}\\!\theta\cos 2\phi,$ (1)
where $\theta$ is the polar angle between the direction of the positive lepton
and the chosen polarization axis, and $\phi$ is the azimuthal angle, measured
with respect to the production plane. In this formalism, the polarization is
completely longitudinal if the set of polarization parameters
($\lambda_{\theta}$, $\lambda_{\theta\phi}$, $\lambda_{\phi}$) takes the
values $(-1,0,0)$ and it is completely transverse if it takes the values
$(1,0,0)$. In the zero polarization scenario the parameters are $(0,0,0)$. In
the general case, the values of ($\lambda_{\theta}$, $\lambda_{\theta\phi}$,
$\lambda_{\phi}$) depend on the choice of the spin quantization frame and
different values can be consistent with the same underlying polarization
states. However, the combination of parameters
$\lambda_{\mathrm{inv}}=\frac{\lambda_{\theta}+3\lambda_{\phi}}{1-\lambda_{\phi}}$
(2)
is invariant under the choice of polarization frame [5, 6]. The natural
polarization axis for the measurement is that where the lepton azimuthal angle
distribution is symmetric ($\lambda_{\phi}=\lambda_{\theta\phi}=0$) and
$\lambda_{\theta}$ is maximal [7].
Several theoretical models are used to describe quarkonium production,
predicting the values and the kinematic dependence of the cross-section and
polarization. The colour-singlet model (CSM) at leading order [8, 9]
underestimates the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production
cross-section by two orders of magnitude [2, 10] and predicts significant
transverse polarization. Subsequent calculations at next-to-leading order and
at next-to-next-to-leading order change these predictions dramatically. The
cross-section prediction comes close to the observed values and the
polarization is expected to be large and longitudinal [11, 12, 13, 14].
Calculations performed in the framework of non-relativistic quantum
chromodynamics (NRQCD), where the $c\overline{}c$ pair can be produced in
colour-octet states (color-octet model, COM [15, 16, 17]), can explain the
shape and magnitude of the measured cross-section as a function of $p_{\rm
T}$. COM predicts a dependence of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ polarization on the $p_{\rm T}$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. In the low $p_{\rm T}$
region ($p_{\rm T}$ $<M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})/c$ with
$M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ the mass of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson), where the gluon fusion
process dominates, a small longitudinal polarization is expected [18]. For
$p_{\rm T}$ $\gg M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$, where gluon
fragmentation dominates, the leading order predictions [19, 20] and next-to-
leading order calculations [21] suggest a large transverse component of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization.
The polarization for inclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
production (including the feed-down from higher charmonium states) in hadronic
interactions has been measured by several experiments at Fermilab [22],
Brookhaven [23] and DESY [24]. The CDF experiment, in $p\overline{p}$
collisions at $\sqrt{s}=1.96$ TeV, measured a small longitudinal
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization, going to zero at
small $p_{\rm T}$. This measurement is in disagreement with the COM
calculations and does not support the conclusion that the colour-octet terms
dominate the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in the
high $p_{\rm T}$ region. The PHENIX experiment measured the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization in $pp$ collisions
at $\sqrt{s}=200\mathrm{\,Ge\kern-1.00006ptV}$, for $p_{\rm T}$ $<3$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The HERA-B experiment studied
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization in 920
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ fixed target proton-nucleus ($p$-$C$
and $p$-$W$) collisions. The explored kinematic region is defined for $p_{\rm
T}$ $<5.4$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and Feynman variable
$x_{\mathrm{F}}$ between $-0.34$ and $0.14$. Also in these cases a small
longitudinal polarization is observed. Recently, at the LHC, ALICE [25] and
CMS [26] have measured the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
polarization in $pp$ collisions at $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$, in the kinematic ranges of $2<\mbox{$p_{\rm
T}$}<8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $2.5<y<4.0$, and
$14<\mbox{$p_{\rm T}$}<70~{}{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$\left|y\right|<1.2$, respectively. The ALICE collaboration finds a small
longitudinal polarization vanishing at high values of $p_{\rm T}$ 111In the
ALICE measurement the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $b$
decays are also included., while the CMS results do not show evidence of large
transverse or longitudinal polarizations.
The analysis presented here is performed by fitting the efficiency-corrected
angular distribution of the data. Given the forward geometry of the LHCb
experiment, the polarization results are presented in the helicity frame and,
as a cross-check, in the Collins-Soper frame. The polarization is measured by
performing a two-dimensional angular analysis considering the distribution
given in Eq. (1) and using an unbinned maximum likelihood fit. To evaluate the
detector acceptance, reconstruction and trigger efficiency, fully simulated
events are used. The measurement is performed in six bins of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum and five
rapidity bins. The edges of the bins in ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ $p_{\rm T}$ and $y$ are defined respectively as [2, 3, 4, 5, 7, 10,
15] ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $p_{\rm T}$ and [2.0, 2.5, 3.0,
3.5, 4.0, 4.5] in ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ $y$.
The remainder of the paper is organized as following. In Sec. 2 a brief
description of the LHCb detector and the data sample used for the analysis is
given. In Sec. 3 the signal selection is defined. In Sec. 4 and Sec. 5
respectively, the fit procedure to the angular distribution and the
contributions to the systematic uncertainties on the measurement are
described. The results are presented in Sec. 6 and in Sec. 7 the update of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section, including the
polarization effect, is described. Finally in Sec. 8 conclusions are drawn.
## 2 LHCb detector and data sample
The LHCb detector [27] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of hadrons containing
$b$ or $c$ quarks. A right-handed Cartesian coordinate system is used, centred
on the nominal $pp$ collision point with $z$ pointing downstream along the
nominal beam axis and $y$ pointing upwards. The detector includes a high
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about 4 Tm, and
three stations of silicon-strip detectors and straw drift tubes placed
downstream. The combined tracking system provides momentum measurement with
relative uncertainty that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high $p_{\rm T}$. Charged hadrons are
identified using two ring-imaging Cherenkov detectors. Photon, electron and
hadron candidates are identified by a calorimeter system consisting of
scintillating-pad and pre-shower detectors, an electromagnetic calorimeter and
a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers [28].
The trigger [29] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. Candidate events are selected by the hardware
trigger requiring the $p_{\rm T}$ of one muon to be larger than 1.48
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, or the products of the $p_{\rm T}$ of
the two muons to be larger than 1.68 $(\mathrm{Ge\kern-1.00006ptV\\!/}c)^{2}$.
In the subsequent software trigger [29], two tracks with $\mbox{$p_{\rm
T}$}>0.5$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and momentum $p>6$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are required to be identified as muons
and the invariant mass of the two muon tracks is required to be within $\pm
120{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal mass of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson [30]. The data used for
this analysis correspond to an integrated luminosity of 0.37
$\mbox{\,fb}^{-1}$ of $pp$ collisions at a center-of-mass energy of
$\sqrt{s}=7$ TeV, collected by the LHCb experiment in the first half of 2011.
The period of data taking has been chosen to have uniform trigger conditions.
In the simulation, $pp$ collisions are generated using Pythia 6.4 [31] with a
specific LHCb configuration [32]. Decays of hadronic particles are described
by EvtGen [33], in which final state radiation is generated using Photos [34].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [35, *Agostinelli:2002hh] as
described in Ref. [37]. The prompt charmonium production is simulated in
Pythia according to the leading order colour-singlet and colour-octet
mechanisms.
## 3 Signal selection
The selection requires that at least one primary vertex is reconstructed in
the event. Candidate ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons are
formed from pairs of opposite-sign tracks reconstructed in the tracking
system. Each track is required to have $p_{\mathrm{T}}>0.75$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and to be identified as a muon. The two
muons must originate from a common vertex and the $\chi^{2}$ probability of
the vertex fit must be greater than 0.5%.
Figure 1: (${\it Left}$) Invariant mass distribution of muon pairs passing
the selection criteria. In the plot, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ candidates are required to have 5 $<$ $p_{\rm T}$ $<$ 7
${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and $3.0<y<3.5$. The solid (dashed)
vertical lines indicate the signal (sideband) regions. (${\it Right}$) Pseudo
decay-time significance ($S_{\tau}$) distribution for background subtracted
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates in the same
kinematic bin. The solid vertical lines indicate the $S_{\tau}$ selection
region. The right tail of the distribution is due to
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production through the decay of
$b$ hadrons.
In Fig. 1 (left), the invariant mass distribution of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates for 5 $<$ $p_{\rm
T}$ $<$ 7 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $3.0<y<3.5$ is shown as
an example. A fit to the mass distribution has been performed using a Crystal
Ball function [38] for the signal and a linear function for the background,
whose origin is combinatorial. The Crystal Ball parameter describing the
threshold of the radiative tail is fixed to the value obtained in the
simulation. The Crystal Ball peak position and resolution determined in the
fit shown in Fig. 1 (left) are respectively $\mu=3090.5$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $\sigma=14.6$
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The signal region is defined as
$\left[\mu-3\sigma,\;\mu+3\sigma\right]$ and the two sideband regions as
$\left[\mu-7\sigma,\mu-4\sigma\right]$ and
$\left[\mu+4\sigma,\mu+7\sigma\right]$ in the mass distribution.
Prompt $J/\psi$ mesons and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mesons from $b$-hadron decays can be discriminated by the pseudo-decay-time
$\tau$, which is defined as:
$\tau=\frac{(z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}-z_{\mathrm{\,PV}})M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu})}{p_{z}}\;,$ (3)
where $z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $z_{\mathrm{PV}}$
are the positions of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay
vertex and the associated primary vertex along the $z$-axis,
$M({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})$ is the nominal
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass and $p_{z}$ is the
measured $z$ component of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
momentum in the center-of-mass frame of the $pp$ collision. For events with
several primary vertices, the one which is closest to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ vertex is used. The uncertainty
$\sigma_{\tau}$ is calculated for each candidate using the measured covariance
matrix of $z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $p_{z}$ and
the uncertainty of $z_{\mathrm{PV}}$. The bias induced by not refitting the
primary vertex removing the two tracks from the reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson is found to be negligible
[2]. The pseudo decay-time significance $S_{\tau}$ is defined as
$S_{\tau}=\tau/\sigma_{\tau}$. In order to suppress the component of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons from $b$-hadron decays,
it is required that $|S_{\tau}|<4$. With this requirement, the fraction of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from $b$-hadron decays reduces
from about 15% to about 3%. The distribution of the pseudo-decay-time
significance in one kinematic bin is shown in Fig. 1 (right).
## 4 Polarization fit
The polarization parameters are determined from a fit to the angular
distribution ($\cos\theta,\phi$) of the $J/\psi\rightarrow\mu^{+}\mu^{-}$
decay. The knowledge of the efficiency as a function of the angular variables
($\cos\theta,\phi$) is crucial for the analysis. The detection efficiency
$\epsilon$ includes geometrical, detection and trigger efficiencies and is
obtained from a sample of simulated unpolarized
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons decaying in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
channel, where the events are divided in bins of $p_{\rm T}$ and $y$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. The efficiency is
studied as a function of four kinematic variables: $p_{\rm T}$ and $y$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, and $\cos\theta$ and
$\phi$ of the positive muon. As an example, Fig. 2 shows the efficiency as a
function of $\cos\theta$ (integrated over $\phi$) and $\phi$ (integrated over
$\cos\theta$) respectively, for two different bins of $p_{\rm T}$ and all five
bins of $y$. The efficiency is lower for $\cos\theta\approx\pm 1$, as one of
the two muons in this case has a small momentum in the center-of-mass frame of
the $pp$ collision and is often bent out of the detector acceptance by the
dipole field of the magnet. The efficiency is also lower for $|\phi|\approx 0$
or $\pi$, because one of the two muons often escapes the LHCb detector
acceptance.
Figure 2: Global efficiency (area normalized to unity) as a function of (top)
$\cos\theta$ and (bottom) $\phi$ for (left) 3 $<$ $p_{\rm T}$ $<$ 4
${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ and for (right) 7 $<$ $p_{\rm T}$ $<$
10 ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons in the helicity frame.
The efficiency is determined from simulation.
To fit the angular distribution in Eq. (1), a maximum likelihood (ML) approach
is used. The logarithm of the likelihood function, for data in each $p_{\rm
T}$ and $y$ bin, is defined as
$\displaystyle\log L$ $\displaystyle=$
$\displaystyle\sum^{N_{\mathrm{tot}}}_{i=1}w_{i}\times\log\left[\frac{P(\cos\theta_{i},\phi_{i}|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})\;\epsilon(\cos\theta_{i},\phi_{i})}{N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})}\right]$
(4) $\displaystyle=$
$\displaystyle\sum^{N_{\mathrm{tot}}}_{i=1}w_{i}\times\log\left[\frac{P(\cos\theta_{i},\phi_{i}|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})}{N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})}\right]+\sum^{N_{\mathrm{tot}}}_{i=1}w_{i}\times\log\left[\epsilon(\cos\theta_{i},\phi_{i})\right]\;,$
(5)
where
$P(\cos\theta_{i},\phi_{i}|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})=1+\lambda_{\theta}\cos^{2}\theta_{i}+\lambda_{\theta\phi}\sin
2\theta_{i}\cos\phi_{i}+\lambda_{\phi}\sin^{2}\theta_{i}\cos 2\phi_{i}$,
$w_{i}$ are weighting factors and the index $i$ runs over the number of the
candidates, $N_{\mathrm{tot}}$. The second sum in Eq. (5) can be ignored in
the fit as it has no dependence on the polarization parameters.
$N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})$ is a normalization
integral, defined as
$N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})=\int d\Omega
P(\cos\theta,\phi|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})\times\epsilon(\cos\theta,\phi)\;.$
(6)
In the simulation where ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons
are generated unpolarized, the $(\cos\theta$, $\phi)$ two-dimensional
distribution of selected candidates is the same as the efficiency
$\epsilon(\cos\theta,\phi)$, so Eq. (6) can be evaluated by summing
$P(\cos\theta_{i},\phi_{i}|\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})$
over the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates in the
simulated sample. The normalization
$N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})$ depends on all three
polarization parameters. The weighting factor $w_{i}$ is chosen to be $+1$
($-1$) if a candidate falls in the signal region (sideband regions) shown in
Fig. 1. In this way the background component in the signal window is
subtracted on a statistical basis.222The signal window and the sum of the
sideband regions have the same width. For this procedure it is assumed that
the angular distribution $(\cos\theta,\phi)$ of background events in the
signal region is similar to that of the events in sideband regions, and that
the mass distribution of the background is approximately linear.
The method used for the measurement of the polarization is tested by measuring
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization in two
simulated samples with a fully transverse and fully longitudinal polarization,
respectively. In both cases the results reproduce the simulation input within
the statistical sensitivity.
To evaluate the normalization function
$N(\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi})$ on the simulated
sample of unpolarized ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons,
we rely on the correct simulation of the efficiency. In order to cross check
the reliability of the efficiency obtained from the simulation, the control-
channel $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,K^{+}$
is studied. The choice of this channel is motivated by the fact that, due to
angular momentum conservation, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ must be longitudinally polarized and any difference between the
angular distributions measured in data and in the simulation must be due to
inaccuracies in the simulation.
To compare the kinematic variables of the muons in data and simulation, a
first weighting procedure is applied to the simulated sample to reproduce the
$B^{+}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ kinematics in the
data. In Fig. 3, $\cos\theta$ distributions for
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,K^{+}$
candidates for data and simulation are shown, as well as their ratio. A small
difference between the distributions for data and simulation is observed,
which is attributed to an overestimation of the efficiency in the simulation
for candidates with values of $\left|\cos\theta\right|\approx 1$. To correct
for the acceptance difference, an additional event weighting is applied where
the weighting factors are obtained by comparing the two-dimensional muon
$p_{\rm T}$ and $y$ distribution in the center-of-mass frame of $pp$
collisions in data and simulation. This weighting corrects for the observed
disagreement in the $\cos\theta$ distribution. The weights as a function of
muon $p_{\rm T}$ and $y$ obtained from the
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,K^{+}$ sample
are subsequently applied in the same way to the simulated prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sample, which is used to
determine the efficiency for the polarization measurement.
Figure 3: (Left) Distributions of $\cos\theta$ in the helicity frame for
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons from
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays in
data (open circles) and simulated sample (open squares) after the weighting
based on the $B^{+}$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
kinematics and (right) their ratio.
## 5 Systematic uncertainties
The largest systematic uncertainty is related to the determination of the
efficiency and to the weighting procedure used to correct the simulation,
using the $B^{+}\rightarrow J/\psi\,K^{+}$ control channel. The weighting
procedure is performed in bins of $p_{\rm T}$ and $y$ of the two muons and,
due to the limited number of candidates in the control channel, the
statistical uncertainties of the correction factors are sizeable (from 1.3% up
to 25%, depending on the bin). To propagate these uncertainties to the
polarization results, the following procedure is used. For each muon
($\mbox{$p_{\rm T}$},y$) bin, the weight is changed by one standard deviation,
leaving all other weights at their nominal values. This new set of weights is
used to redetermine the detector efficiency and then perform a new fit of the
polarization parameters. The difference of the obtained parameters with
respect to the nominal polarization result is considered as the contribution
of this muon $(\mbox{$p_{\rm T}$},y)$ bin to the uncertainty. The total
systematic uncertainty is obtained by summing all these independent
contributions in quadrature. In the helicity frame, the average absolute
uncertainty over all the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
$(\mbox{$p_{\rm T}$},y)$ bins due to this effect is 0.067 on
$\lambda_{\theta}$.
Concerning the background subtraction, the choice of the sidebands and the
background model are checked. A systematic uncertainty is evaluated by
comparing the nominal results for the polarization parameters, and those
obtained using only the left or the right sideband, or changing the background
fit function (as alternatives to the linear function, exponential and
polynomial functions are used). In both cases the maximum variation with
respect to the nominal result is assigned as systematic uncertainty.
Typically, the absolute size of this effect is 0.012 on $\lambda_{\theta}$ for
$\mbox{$p_{\rm T}$}>5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
The effect of the ($\mbox{$p_{\rm T}$},y$) binning for the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson could also introduce an
uncertainty, due to the difference of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ kinematic distributions between
data and simulation within the bins. To investigate this effect, each bin is
divided in four sub-bins ($2\times 2$) and the polarization parameters are
calculated in each sub-bin. The weighted average of the results in the four
sub-bins is compared with the nominal result and the difference is quoted as
the systematic uncertainty. As expected, this effect is particularly important
in the rapidity range near the LHCb acceptance boundaries, where the
efficiency has a strong dependence on the kinematic properties of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. It however depends on
$p_{\rm T}$ only weakly and the average effect on $\lambda_{\theta}$ is 0.018
(absolute).
Two systematic uncertainties related to the cut on the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay time significance are
evaluated. The first is due to the residual
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates from $b$-hadron
decays, 3% on average and up to 5% in the highest $p_{\rm T}$ bins, that
potentially have different polarization. The second is due to the efficiency
difference in the $S_{\tau}$ requirement in data and simulation. The average
size of these effects, over the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
$(\mbox{$p_{\rm T}$},y)$, is 0.012.
The limited number of events in the simulation sample, used to evaluate the
normalization integrals of Eq. (6), is a source of uncertainty. This effect is
evaluated by simulating a large number of pseudo-experiments and the average
absolute size is 0.015.
Finally, the procedure used to statistically subtract the background
introduces a statistical uncertainty, not included in the standard likelihood
maximization uncertainty. A detailed investigation shows that it represents a
tiny correction to the nominal statistical uncertainty, reported in Tables 2
and 3.
The main contributions to the systematic uncertainties on $\lambda_{\theta}$
are summarized in Table 1 for the helicity and the Collins-Soper frames. While
all uncertainties are evaluated for every $p_{\rm T}$ and $y$ bin separately,
we quote for the individual contributions only the average, minimum and
maximum values. The systematic uncertainties on $\lambda_{\theta\phi}$ and
$\lambda_{\phi}$ are similar to each other and a factor two lower than those
for $\lambda_{\theta}$. Apart from the binning and the simulation sample size
effects, the uncertainties of adjacent kinematic bins are strongly correlated.
To quote the global systematic uncertainty (Tables 2 and 3) in each kinematic
bin of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, the different
contributions for each bin are considered to be uncorrelated and are added in
quadrature.
Table 1: Main contributions to the absolute systematic uncertainty on the parameter $\lambda_{\theta}$ in the helicity and Collins-Soper frames. While the systematic uncertainties are evaluated separately for all $p_{\rm T}$ and $y$ bins, we give here only the average, the minimum and the maximum values of all bins. Source | helicity frame | Collins-Soper frame
---|---|---
average (min. – max.) | average (min. – max.)
Acceptance | 0.067 (0.045 – 0.173) | 0.044 (0.025 – 0.185)
Binning effect | 0.018 (0.001 – 0.165) | 0.016 (0.001 – 0.129)
Simulation sample size | 0.015 (0.005 – 0.127) | 0.015 (0.007 – 0.170)
Sideband subtraction | 0.016 (0.001 – 0.099) | 0.029 (0.001 – 0.183)
$b$-hadron contamination | 0.012 (0.002 – 0.019) | 0.006 (0.002 – 0.029)
## 6 Results
The fit results for the three parameters $\lambda_{\theta}$,
$\lambda_{\theta\phi}$ and $\lambda_{\phi}$, with their uncertainties, are
reported in Tables 2 and 3 for the helicity frame and the Collins-Soper frame,
respectively. The parameter $\lambda_{\theta}$ is also shown in Fig. 4 as a
function of the $p_{\rm T}$ of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson, for different $y$ bins.
Figure 4: Measurements of $\lambda_{\theta}$ in bins of $p_{\rm T}$ for five
rapidity bins in (left) the helicity frame and (right) the Collins-Soper
frame. The error bars represent the statistical and systematic uncertainties
added in quadrature. The data points are shifted slightly horizontally for
different rapidities to improve visibility.
The polarization parameters $\lambda_{\phi}$ and $\lambda_{\theta\phi}$ in the
helicity frame are consistent with zero within the uncertainties. Following
the discussion in Sec.1, the helicity frame represents the natural frame for
the polarization measurement in our experiment and the measured
$\lambda_{\theta}$ parameter is an indicator of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization, since it is equal
to the invariant parameter defined in Eq. (2).
The measured value of $\lambda_{\theta}$ shows a small longitudinal
polarization. A weighted average is calculated over all the $(\mbox{$p_{\rm
T}$},y)$ bins, where the weights are chosen according to the number of events
in each bin in the data sample. The average is $\lambda_{\theta}=-0.145\pm
0.027$. The uncertainty is statistical and systematic uncertainties added in
quadrature. Since the correlations of the systematic uncertainties are
observed to be relevant only between adjacent kinematic bins, when quoting the
average uncertainty, we assume the different kinematic bins are uncorrelated,
apart from the adjacent ones, which we treat fully correlated.
A cross-check of the results is performed by repeating the measurement in the
Collins-Soper reference frame (see Sec. 1). As LHCb is a forward detector, the
Collins-Soper and helicity frames are kinematically quite similar, especially
in the low $p_{\rm T}$ and $y$ regions. Therefore, the polarization parameters
obtained in Collins-Soper frame are expected to be similar to those obtained
in the helicity frame, except at high $p_{\rm T}$ and low $y$ bins.
Calculating the frame-invariant variable, according to Eq. (2), the
measurements performed in the two frames are in agreement within the
uncertainty.
The results can be compared to those obtained by other experiments at
different valuses of $\sqrt{s}$. Measurements by CDF [22], PHENIX [23] and
HERA-B [24], also favour a negative value for $\lambda_{\theta}$. The HERA-B
experiment has also published results on $\lambda_{\phi}$ and
$\lambda_{\theta\phi}$, which are consistent with zero. At the LHC, the ALICE
[25] and the CMS [26] collaboration studied the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization in $pp$ collisions
at $\sqrt{s}=7$ $\mathrm{\,Te\kern-1.00006ptV}$. The CMS results, determined
in a different kinematic range, disfavour large transverse or longitudinal
polarizations. The analysis by ALICE is based on the $\cos\theta$ and $\phi$
projections and thus only determines $\lambda_{\theta}$ and $\lambda_{\phi}$.
Furthermore it also includes ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mesons from $b$-hadron decays. The measurement has been performed in bins of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum integrating
over the rapidity in a range very similar to that of LHCb, being
$2<\mbox{$p_{\rm T}$}<8$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.5<y<4.0$. To compare our results with the ALICE measurements, averages over
the $y$ region are used for the different $p_{\rm T}$ bins and good agreement
is found for $\lambda_{\theta}$ and $\lambda_{\phi}$. The comparison for
$\lambda_{\theta}$ is shown in Fig. 5 for the helicity and Collins-Soper
frames, respectively.
Figure 5: Comparison of LHCb and ALICE results for $\lambda_{\theta}$ in
different $p_{\rm T}$ bins integrating over the rapidity range $2.5<y<4.0$ in
(left) the helicity frame and (right) the Collins-Soper frame. Error bars
represent the statistical and systematic uncertainties added in quadrature.
In Fig. 6 our measurements of $\lambda_{\theta}$ are compared with the NLO CSM
[39] and NRQCD predictions of Refs. [39], [40] and [41, *Shao2012fs]. The
comparison is done in the helicity frame and as a function of the $p_{\rm T}$
of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson (integrating over
$2.5<y<4.0$). The theoretical calculations in Refs. [39], [40] and [41,
*Shao2012fs] use different selections of experimental data to evaluate the
non-perturbative matrix elements. Our results are not in agreement with the
CSM predictions and the best agreement is found between the measured values
and the NRQCD predictions of Ref. [41, *Shao2012fs]. It should be noted that
our analysis includes a contribution from feed-down, while the theoretical
computations from CSM and NRQCD [39] do not include feed-down from excited
states. It is known that, among all the feed-down contributions to prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production from higher
charmonium states, the contribution from $\chi_{c}$ mesons can be quite
important (up to 30%) and that $\psi{(2S)}$ mesons also can give a sizable
contribution [43, 40, 41, *Shao2012fs], depending on the yields and their
polarizations. The NLO NRQCD calculations [40, 41, *Shao2012fs] include the
feed-down from $\chi_{c}$ and $\psi{(2S)}$ mesons.
Figure 6: Comparison of LHCb prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ polarization measurements of $\lambda_{\theta}$ with direct NLO color
singlet (magenta diagonal lines [39]) and three different NLO NRQCD (blue
diagonal lines (1) [39], red vertical lines (2) [40] and green hatched (3)
[41, *Shao2012fs]) predictions as a function of the $p_{\rm T}$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson in the rapidity range
$2.5<y<4.0$ in the helicity frame.
## 7 Update of the $\mathbf{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
cross-section measurement
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-section in $pp$
collisions at $\sqrt{s}=7~{}\mathrm{\,Te\kern-1.00006ptV}$ was previously
measured by LHCb in 14 bins of $p_{\rm T}$ and five bins of $y$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson [2]. The uncertainty on
the prompt cross-section measurement is dominated by the unknown
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization, resulting in
uncertainties of up to 20%:
$\sigma_{\mathrm{prompt}}(2<y<4.5,\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c})=10.52\pm 0.04\pm
1.40\,^{+1.64}_{-2.20}~{}\rm\,\upmu b$
where the first uncertainty is statistical, the second is systematic and the
third one is due to the unknown polarization.
The previous measurement of the prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ cross-section can be updated in the range of the polarization
analysis, $2<\mbox{$p_{\rm T}$}<14$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$, by applying the measured polarization and its uncertainty to the
efficiency calculation in the cross-section measurement. To re-evaluate the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-section, the
same data sample, trigger and selection requirements as in Ref. [2] are used.
Technically the polarization correction is done by reweighting the muon
angular distribution of a simulated sample of unpolarized
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
events to reproduce the expected distribution, according to Eq. (1), for
polarized ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons. The
polarization parameters $\lambda_{\theta}$, $\lambda_{\theta\phi}$ and
$\lambda_{\phi}$ are set to the measured values, quoted in Table 2 for each
bin of $p_{\rm T}$ and $y$ of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson.
In addition to the polarization update, the uncertainties on the luminosity
determination and on the track reconstruction efficiency are updated to take
into account the improvements described in Refs. [44, 45]. For the tracking
efficiency it is possible to reduce the systematic uncertainty to 3%, compared
to an 8% uncertainty assigned in the original measurement[2]. Taking advantage
of the improvements described in [44] the uncertainty due to the luminosity
measurement has been reduced from the 10%, quoted in [2] to the 3.5%. The
results obtained for the double-differential cross-section are shown in Fig. 7
and reported in Table 4. The integrated cross-section in the kinematic range
of the polarization analysis, 2 $<\mbox{$p_{\rm T}$}<14$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$, is
$\sigma_{\mathrm{prompt}}(2<y<4.5,2<\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c})=4.88\pm 0.01\pm 0.27\pm
0.12\;\rm\,\upmu b$
and for the range $\mbox{$p_{\rm T}$}<14$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$, it is
$\sigma_{\mathrm{prompt}}(2<y<4.5,\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c})=9.46\pm 0.04\pm
0.53\,^{+0.86}_{-1.10}\;\rm\,\upmu b.$
For the two given cross-section measurements, the first uncertainty is
statistical, the second is systematic, while the third arises from the
remaining uncertainty due to the polarization measurement and is evaluated
using simulated event samples. For the $p_{\rm T}$ range $\mbox{$p_{\rm
T}$}<2$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, where no polarization
measurement exists, we assume zero polarization and assign as systematic
uncertainty the difference between the zero polarization hypothesis and fully
transverse (upper values) or fully longitudinal (lower values) polarization.
For $\mbox{$p_{\rm T}$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ the
uncertainties on the polarization measurement coming from the various sources
are propagated to the cross-section measurement fluctuating the values of the
polarization parameters in Eq. 1 with a Gaussian width equal to one standard
deviation. The relative uncertainty due to the polarization effect on the
integrated cross-section in $2<\mbox{$p_{\rm T}$}<14$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$ is $2.4\%$. The
relative uncertainty on the integrated cross-section in the range of Ref. [2],
$\mbox{$p_{\rm T}$}<14$ ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$, is reduced to $12\%$ (lower polarization uncertainty) and to
$9\%$ (upper polarization uncertainty) with respect to the value published in
Ref. [2].
Figure 7: Differential cross-section of prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production as a function of
$p_{\rm T}$ and in bins of $y$. The vertical error bars show the quadratic sum
of the statistical and systematic uncertainties.
## 8 Conclusion
A measurement of the prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
polarization obtained with $pp$ collisions at $\sqrt{s}=7$
$\mathrm{\,Te\kern-1.00006ptV}$, performed using a dataset corresponding to an
integrated luminosity of 0.37 $\mbox{\,fb}^{-1}$, is presented. The data have
been collected by the LHCb experiment in the early 2011. The polarization
parameters ($\lambda_{\theta},\lambda_{\theta\phi},\lambda_{\phi}$) are
determined by studying the angular distribution of the two muons from the
decay ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$ with respect to the polar and azimuthal
angle defined in the helicity frame. The measurement is performed in five bins
of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rapidity $y$ and six bins of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum $p_{\rm T}$
in the kinematic range
$2<p_{\mathrm{T}}<15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$.
The results for $\lambda_{\theta}$ indicate a small longitudinal polarization
while the results for $\lambda_{\theta\phi}$ and $\lambda_{\phi}$ are
consistent with zero. Although a direct comparison is not possible due to the
different collision energies and analysis ranges, the measurements performed
by CDF [22], PHENIX [23], HERA-B [24] and CMS [26] show no significant
transverse or longitudinal polarization. Good agreement has also been found
with ALICE measurements [25], performed in a $p_{\rm T}$ and rapidity range
very similar to that explored by LHCb.
Our results, that are obtained for prompt
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production, including the feed-
down from higher excited states, contradict the CSM predictions for direct
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production, both in the size of
the polarization parameters and the $p_{\rm T}$ dependence. Concerning the
NRQCD models, predictions from Ref. [41, *Shao2012fs] give the best agreement
with the LHCb measurement.
This evaluation of the polarization is used to update the measurement of the
integrated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production cross-
section [2] in the range $\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$, resulting in a
reduction of the corresponding systematic uncertainty to $9\%$ and $12\%$. The
result is
$\sigma_{\mathrm{prompt}}(2<y<4.5,\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c})=9.46\pm 0.04\pm
0.53\,^{+0.86}_{-1.10}\;\rm\,\upmu b.$
The integrated cross-section has also been measured in the polarization
analysis range $2<\mbox{$p_{\rm T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $2.0<y<4.5$:
$\sigma_{\mathrm{prompt}}(2<y<4.5,2<\mbox{$p_{\rm
T}$}<14{\mathrm{\,Ge\kern-1.00006ptV\\!/}c})=4.88\pm 0.01\pm 0.27\pm
0.12\;\rm\,\upmu b.$
with an uncertainty due to polarization of $2.4\%$.
## Acknowledgements
We wish to thank M. Butenschoen, B. Gong and Y.-Q. Ma for providing us the
theoretical calculations and helpful discussions. We are grateful for fruitful
discussions with S. P. Baranov. We express our gratitude to our colleagues in
the CERN accelerator departments for the excellent performance of the LHC. We
thank the technical and administrative staff at the LHCb institutes. We
acknowledge support from CERN and from the national agencies: CAPES, CNPq,
FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne
(France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM
and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom,
RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT
and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain),
GridPP (United Kingdom). We are thankful for the computing resources put at
our disposal by Yandex LLC (Russia), as well as to the communities behind the
multiple open source software packages that we depend on.
Appendices
Table 2: Measured ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization parameters in bins of $p_{\rm T}$ and $y$ in the helicity frame. The first uncertainty is statistical (from the fit and the background subtraction) while the second is the systematic uncertainty. $p_{\rm T}$ (${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | $\lambda$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---|---
| $\lambda_{\theta}$ | -0.306 $\pm$ | 0.095 $\pm$ | 0.288 | -0.207 $\pm$ | 0.010 $\pm$ | 0.101 | -0.169 $\pm$ | 0.006 $\pm$ | 0.066 | -0.161 $\pm$ | 0.005 $\pm$ | 0.059 | -0.081 $\pm$ | 0.008 $\pm$ | 0.092
2-3 | $\lambda_{\theta\phi}$ | 0.057 $\pm$ | 0.052 $\pm$ | 0.114 | -0.055 $\pm$ | 0.004 $\pm$ | 0.039 | -0.054 $\pm$ | 0.003 $\pm$ | 0.034 | 0.004 $\pm$ | 0.003 $\pm$ | 0.043 | 0.052 $\pm$ | 0.006 $\pm$ | 0.050
| $\lambda_{\phi}$ | 0.034 $\pm$ | 0.016 $\pm$ | 0.075 | 0.023 $\pm$ | 0.003 $\pm$ | 0.043 | 0.009 $\pm$ | 0.002 $\pm$ | 0.027 | 0.036 $\pm$ | 0.003 $\pm$ | 0.026 | 0.048 $\pm$ | 0.005 $\pm$ | 0.041
| $\lambda_{\theta}$ | -0.419 $\pm$ | 0.073 $\pm$ | 0.218 | -0.077 $\pm$ | 0.010 $\pm$ | 0.100 | -0.173 $\pm$ | 0.006 $\pm$ | 0.056 | -0.149 $\pm$ | 0.006 $\pm$ | 0.054 | -0.125 $\pm$ | 0.010 $\pm$ | 0.086
3-4 | $\lambda_{\theta\phi}$ | -0.055 $\pm$ | 0.044 $\pm$ | 0.094 | -0.024 $\pm$ | 0.004 $\pm$ | 0.030 | -0.029 $\pm$ | 0.003 $\pm$ | 0.023 | 0.022 $\pm$ | 0.003 $\pm$ | 0.026 | 0.045 $\pm$ | 0.005 $\pm$ | 0.046
| $\lambda_{\phi}$ | 0.021 $\pm$ | 0.016 $\pm$ | 0.045 | -0.014 $\pm$ | 0.003 $\pm$ | 0.018 | -0.002 $\pm$ | 0.003 $\pm$ | 0.019 | 0.029 $\pm$ | 0.003 $\pm$ | 0.025 | 0.013 $\pm$ | 0.006 $\pm$ | 0.034
| $\lambda_{\theta}$ | -0.390 $\pm$ | 0.056 $\pm$ | 0.174 | -0.022 $\pm$ | 0.010 $\pm$ | 0.077 | -0.149 $\pm$ | 0.007 $\pm$ | 0.050 | -0.129 $\pm$ | 0.007 $\pm$ | 0.055 | -0.158 $\pm$ | 0.012 $\pm$ | 0.099
4-5 | $\lambda_{\theta\phi}$ | -0.059 $\pm$ | 0.037 $\pm$ | 0.075 | -0.013 $\pm$ | 0.004 $\pm$ | 0.029 | -0.037 $\pm$ | 0.004 $\pm$ | 0.023 | 0.003 $\pm$ | 0.004 $\pm$ | 0.026 | 0.078 $\pm$ | 0.007 $\pm$ | 0.048
| $\lambda_{\phi}$ | 0.032 $\pm$ | 0.015 $\pm$ | 0.038 | -0.004 $\pm$ | 0.003 $\pm$ | 0.015 | -0.009 $\pm$ | 0.003 $\pm$ | 0.017 | 0.025 $\pm$ | 0.004 $\pm$ | 0.022 | -0.015 $\pm$ | 0.008 $\pm$ | 0.031
| $\lambda_{\theta}$ | -0.126 $\pm$ | 0.037 $\pm$ | 0.133 | -0.072 $\pm$ | 0.009 $\pm$ | 0.067 | -0.158 $\pm$ | 0.007 $\pm$ | 0.048 | -0.104 $\pm$ | 0.008 $\pm$ | 0.055 | -0.045 $\pm$ | 0.013 $\pm$ | 0.098
5-7 | $\lambda_{\theta\phi}$ | -0.051 $\pm$ | 0.024 $\pm$ | 0.064 | -0.010 $\pm$ | 0.004 $\pm$ | 0.026 | 0.007 $\pm$ | 0.004 $\pm$ | 0.022 | -0.022 $\pm$ | 0.005 $\pm$ | 0.026 | 0.005 $\pm$ | 0.008 $\pm$ | 0.053
| $\lambda_{\phi}$ | -0.016 $\pm$ | 0.010 $\pm$ | 0.031 | -0.014 $\pm$ | 0.003 $\pm$ | 0.012 | -0.035 $\pm$ | 0.003 $\pm$ | 0.014 | 0.027 $\pm$ | 0.003 $\pm$ | 0.018 | 0.030 $\pm$ | 0.007 $\pm$ | 0.026
| $\lambda_{\theta}$ | 0.009 $\pm$ | 0.037 $\pm$ | 0.120 | -0.217 $\pm$ | 0.012 $\pm$ | 0.064 | -0.162 $\pm$ | 0.011 $\pm$ | 0.055 | -0.042 $\pm$ | 0.013 $\pm$ | 0.066 | -0.057 $\pm$ | 0.020 $\pm$ | 0.100
7-10 | $\lambda_{\theta\phi}$ | 0.027 $\pm$ | 0.023 $\pm$ | 0.048 | -0.016 $\pm$ | 0.005 $\pm$ | 0.026 | 0.029 $\pm$ | 0.005 $\pm$ | 0.022 | 0.006 $\pm$ | 0.007 $\pm$ | 0.028 | -0.005 $\pm$ | 0.012 $\pm$ | 0.053
| $\lambda_{\phi}$ | 0.003 $\pm$ | 0.010 $\pm$ | 0.024 | -0.008 $\pm$ | 0.004 $\pm$ | 0.011 | -0.025 $\pm$ | 0.004 $\pm$ | 0.013 | 0.007 $\pm$ | 0.005 $\pm$ | 0.016 | 0.034 $\pm$ | 0.010 $\pm$ | 0.027
| $\lambda_{\theta}$ | -0.248 $\pm$ | 0.047 $\pm$ | 0.115 | -0.267 $\pm$ | 0.020 $\pm$ | 0.075 | -0.040 $\pm$ | 0.022 $\pm$ | 0.077 | -0.076 $\pm$ | 0.028 $\pm$ | 0.082 | -0.089 $\pm$ | 0.046 $\pm$ | 0.115
10-15 | $\lambda_{\theta\phi}$ | -0.088 $\pm$ | 0.027 $\pm$ | 0.054 | -0.012 $\pm$ | 0.009 $\pm$ | 0.028 | 0.018 $\pm$ | 0.010 $\pm$ | 0.023 | 0.010 $\pm$ | 0.014 $\pm$ | 0.035 | -0.043 $\pm$ | 0.025 $\pm$ | 0.042
| $\lambda_{\phi}$ | 0.009 $\pm$ | 0.014 $\pm$ | 0.029 | 0.008 $\pm$ | 0.007 $\pm$ | 0.013 | -0.018 $\pm$ | 0.009 $\pm$ | 0.017 | -0.014 $\pm$ | 0.012 $\pm$ | 0.019 | -0.027 $\pm$ | 0.021 $\pm$ | 0.040
Table 3: Measured ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarization parameters in bins of $p_{\rm T}$ and $y$ in Collins-Soper frame. The first uncertainty is statistical (from the fit and the background subtraction) while the second is the systematic uncertainty. $p_{\rm T}$ (${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | $\lambda$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---|---
| $\lambda_{\theta}$ | -0.305 $\pm$ | 0.118 $\pm$ | 0.338 | -0.176 $\pm$ | 0.009 $\pm$ | 0.108 | -0.130 $\pm$ | 0.004 $\pm$ | 0.058 | -0.051 $\pm$ | 0.005 $\pm$ | 0.067 | -0.043 $\pm$ | 0.011 $\pm$ | 0.085
2-3 | $\lambda_{\theta\phi}$ | 0.152 $\pm$ | 0.044 $\pm$ | 0.158 | 0.114 $\pm$ | 0.006 $\pm$ | 0.058 | 0.102 $\pm$ | 0.004 $\pm$ | 0.035 | 0.098 $\pm$ | 0.003 $\pm$ | 0.036 | 0.037 $\pm$ | 0.005 $\pm$ | 0.050
| $\lambda_{\phi}$ | -0.031 $\pm$ | 0.011 $\pm$ | 0.125 | 0.014 $\pm$ | 0.003 $\pm$ | 0.059 | 0.008 $\pm$ | 0.002 $\pm$ | 0.038 | -0.001 $\pm$ | 0.002 $\pm$ | 0.031 | -0.005 $\pm$ | 0.003 $\pm$ | 0.036
| $\lambda_{\theta}$ | -0.180 $\pm$ | 0.086 $\pm$ | 0.215 | -0.076 $\pm$ | 0.007 $\pm$ | 0.067 | -0.064 $\pm$ | 0.004 $\pm$ | 0.034 | 0.017 $\pm$ | 0.005 $\pm$ | 0.042 | -0.001 $\pm$ | 0.011 $\pm$ | 0.070
3-4 | $\lambda_{\theta\phi}$ | 0.223 $\pm$ | 0.042 $\pm$ | 0.095 | 0.090 $\pm$ | 0.006 $\pm$ | 0.047 | 0.109 $\pm$ | 0.004 $\pm$ | 0.031 | 0.081 $\pm$ | 0.004 $\pm$ | 0.032 | 0.015 $\pm$ | 0.006 $\pm$ | 0.049
| $\lambda_{\phi}$ | -0.070 $\pm$ | 0.014 $\pm$ | 0.065 | -0.027 $\pm$ | 0.004 $\pm$ | 0.036 | -0.033 $\pm$ | 0.003 $\pm$ | 0.028 | -0.017 $\pm$ | 0.004 $\pm$ | 0.026 | -0.049 $\pm$ | 0.005 $\pm$ | 0.040
| $\lambda_{\theta}$ | -0.084 $\pm$ | 0.068 $\pm$ | 0.171 | -0.000 $\pm$ | 0.007 $\pm$ | 0.040 | -0.035 $\pm$ | 0.005 $\pm$ | 0.030 | 0.031 $\pm$ | 0.006 $\pm$ | 0.037 | 0.051 $\pm$ | 0.012 $\pm$ | 0.071
4-5 | $\lambda_{\theta\phi}$ | 0.240 $\pm$ | 0.041 $\pm$ | 0.092 | 0.067 $\pm$ | 0.006 $\pm$ | 0.041 | 0.081 $\pm$ | 0.004 $\pm$ | 0.027 | 0.065 $\pm$ | 0.004 $\pm$ | 0.030 | -0.028 $\pm$ | 0.008 $\pm$ | 0.052
| $\lambda_{\phi}$ | -0.104 $\pm$ | 0.017 $\pm$ | 0.055 | -0.042 $\pm$ | 0.005 $\pm$ | 0.032 | -0.050 $\pm$ | 0.005 $\pm$ | 0.027 | -0.033 $\pm$ | 0.005 $\pm$ | 0.029 | -0.095 $\pm$ | 0.007 $\pm$ | 0.047
| $\lambda_{\theta}$ | -0.110 $\pm$ | 0.037 $\pm$ | 0.081 | 0.008 $\pm$ | 0.006 $\pm$ | 0.032 | 0.005 $\pm$ | 0.005 $\pm$ | 0.027 | 0.054 $\pm$ | 0.006 $\pm$ | 0.033 | 0.089 $\pm$ | 0.012 $\pm$ | 0.072
5-7 | $\lambda_{\theta\phi}$ | 0.160 $\pm$ | 0.029 $\pm$ | 0.070 | 0.056 $\pm$ | 0.005 $\pm$ | 0.032 | 0.041 $\pm$ | 0.004 $\pm$ | 0.023 | 0.063 $\pm$ | 0.004 $\pm$ | 0.028 | -0.000 $\pm$ | 0.008 $\pm$ | 0.053
| $\lambda_{\phi}$ | -0.068 $\pm$ | 0.014 $\pm$ | 0.051 | -0.056 $\pm$ | 0.005 $\pm$ | 0.031 | -0.085 $\pm$ | 0.005 $\pm$ | 0.026 | -0.051 $\pm$ | 0.005 $\pm$ | 0.031 | -0.056 $\pm$ | 0.008 $\pm$ | 0.052
| $\lambda_{\theta}$ | 0.079 $\pm$ | 0.032 $\pm$ | 0.061 | 0.035 $\pm$ | 0.009 $\pm$ | 0.035 | 0.032 $\pm$ | 0.009 $\pm$ | 0.030 | 0.031 $\pm$ | 0.011 $\pm$ | 0.036 | 0.072 $\pm$ | 0.020 $\pm$ | 0.071
7-10 | $\lambda_{\theta\phi}$ | 0.014 $\pm$ | 0.028 $\pm$ | 0.061 | 0.073 $\pm$ | 0.006 $\pm$ | 0.026 | 0.036 $\pm$ | 0.005 $\pm$ | 0.023 | 0.022 $\pm$ | 0.007 $\pm$ | 0.029 | 0.007 $\pm$ | 0.013 $\pm$ | 0.045
| $\lambda_{\phi}$ | -0.074 $\pm$ | 0.018 $\pm$ | 0.053 | -0.078 $\pm$ | 0.007 $\pm$ | 0.032 | -0.076 $\pm$ | 0.007 $\pm$ | 0.029 | -0.027 $\pm$ | 0.009 $\pm$ | 0.036 | -0.022 $\pm$ | 0.014 $\pm$ | 0.055
| $\lambda_{\theta}$ | 0.064 $\pm$ | 0.037 $\pm$ | 0.076 | 0.099 $\pm$ | 0.016 $\pm$ | 0.046 | -0.004 $\pm$ | 0.018 $\pm$ | 0.044 | -0.009 $\pm$ | 0.024 $\pm$ | 0.050 | 0.019 $\pm$ | 0.042 $\pm$ | 0.086
10-15 | $\lambda_{\theta\phi}$ | 0.105 $\pm$ | 0.033 $\pm$ | 0.057 | 0.070 $\pm$ | 0.010 $\pm$ | 0.024 | 0.004 $\pm$ | 0.010 $\pm$ | 0.024 | 0.021 $\pm$ | 0.014 $\pm$ | 0.028 | 0.033 $\pm$ | 0.026 $\pm$ | 0.041
| $\lambda_{\phi}$ | -0.093 $\pm$ | 0.026 $\pm$ | 0.059 | -0.108 $\pm$ | 0.013 $\pm$ | 0.040 | -0.024 $\pm$ | 0.013 $\pm$ | 0.040 | -0.024 $\pm$ | 0.017 $\pm$ | 0.048 | -0.084 $\pm$ | 0.030 $\pm$ | 0.064
Table 4: Double-differential cross-section $d^{2}\sigma/d\mbox{$p_{\rm T}$}\,dy$ in nb/(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) for prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production in bins of $p_{\rm T}$ and $y$, with statistical, systematic and polarization uncertainties. $p_{\rm T}$ (${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | $2.0<y<2.5$ | | $2.5<y<3.0$ | | $3.0<y<3.5$ |
---|---|---|---|---|---|---
2-3 | 1083 $\pm$ 18 $\pm$ 64 $\pm$ 210 | | 1055 $\pm$ 8 $\pm$ 61 $\pm$ 47 | | 918 $\pm$ 6 $\pm$ 53 $\pm$ 28 |
3-4 | 639 $\pm$ 9 $\pm$ 41 $\pm$ 93 | | 653 $\pm$ 5 $\pm$ 39 $\pm$ 28 | | 541 $\pm$ 4 $\pm$ 32 $\pm$ 17 |
4-5 | 370 $\pm$ 5 $\pm$ 24 $\pm$ 46 | | 359.1 $\pm$ 3.1 $\pm$ 22.3 $\pm$ 14.1 | | 285.1 $\pm$ 2.4 $\pm$ 17.7 $\pm$ 8.5 |
5-6 | 199.0 $\pm$ 3.0 $\pm$ 13.8 $\pm$ 17.4 | | 185.9 $\pm$ 2.0 $\pm$ 12.2 $\pm$ 6.2 | | 146.4 $\pm$ 1.7 $\pm$ 9.3 $\pm$ 4.2 |
6-7 | 101.2 $\pm$ 1.9 $\pm$ 7.3 $\pm$ 8.0 | | 94.1 $\pm$ 1.3 $\pm$ 6.4 $\pm$ 2.9 | | 71.7 $\pm$ 1.1 $\pm$ 4.8 $\pm$ 1.9 |
7-8 | 62.2 $\pm$ 1.4 $\pm$ 4.1 $\pm$ 4.6 | | 50.6 $\pm$ 0.9 $\pm$ 3.7 $\pm$ 1.7 | | 37.8 $\pm$ 0.7 $\pm$ 2.4 $\pm$ 1.2 |
8-9 | 32.5 $\pm$ 0.9 $\pm$ 2.1 $\pm$ 2.2 | | 28.1 $\pm$ 0.7 $\pm$ 1.8 $\pm$ 0.9 | | 20.3 $\pm$ 0.5 $\pm$ 1.3 $\pm$ 0.6 |
9-10 | 18.5 $\pm$ 0.7 $\pm$ 1.2 $\pm$ 1.3 | | 15.8 $\pm$ 0.5 $\pm$ 1.0 $\pm$ 0.5 | | 10.8 $\pm$ 0.4 $\pm$ 0.7 $\pm$ 0.3 |
10-11 | 10.8 $\pm$ 0.5 $\pm$ 0.7 $\pm$ 0.9 | | 8.7 $\pm$ 0.4 $\pm$ 0.6 $\pm$ 0.3 | | 7.70 $\pm$ 0.34 $\pm$ 0.50 $\pm$ 0.31 |
11-12 | 5.65 $\pm$ 0.32 $\pm$ 0.37 $\pm$ 0.41 | | 5.04 $\pm$ 0.26 $\pm$ 0.32 $\pm$ 0.18 | | 4.03 $\pm$ 0.23 $\pm$ 0.26 $\pm$ 0.13 |
12-13 | 4.16 $\pm$ 0.27 $\pm$ 0.27 $\pm$ 0.32 | | 3.42 $\pm$ 0.23 $\pm$ 0.22 $\pm$ 0.14 | | 2.64 $\pm$ 0.18 $\pm$ 0.17 $\pm$ 0.09 |
13-14 | 2.82 $\pm$ 0.26 $\pm$ 0.19 $\pm$ 0.21 | | 2.68 $\pm$ 0.20 $\pm$ 0.17 $\pm$ 0.11 | | 1.37 $\pm$ 0.15 $\pm$ 0.09 $\pm$ 0.06 |
$p_{\rm T}$ (${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | $3.5<y<4.0$ | | $4.0<y<4.5$ | | |
2-3 | 762 $\pm$ 5 $\pm$ 46 $\pm$ 23 | | 549 $\pm$ 5 $\pm$ 36 $\pm$ 27 | | |
3-4 | 422.9 $\pm$ 3.4 $\pm$ 26.2 $\pm$ 12.9 | | 284 $\pm$ 3 $\pm$ 19 $\pm$ 16 | | |
4-5 | 219.1 $\pm$ 2.3 $\pm$ 13.9 $\pm$ 6.7 | | 145.4 $\pm$ 2.4 $\pm$ 9.2 $\pm$ 8.7 | | |
5-6 | 107.2 $\pm$ 1.4 $\pm$ 7.5 $\pm$ 3.2 | | 69.2 $\pm$ 1.5 $\pm$ 4.4 $\pm$ 3.5 | | |
6-7 | 54.6 $\pm$ 1.0 $\pm$ 3.5 $\pm$ 1.6 | | 30.6 $\pm$ 1.0 $\pm$ 1.9 $\pm$ 1.4 | | |
7-8 | 26.2 $\pm$ 0.6 $\pm$ 1.7 $\pm$ 0.9 | | 16.71 $\pm$ 0.69 $\pm$ 1.06 $\pm$ 0.92 | | |
8-9 | 14.3 $\pm$ 0.5 $\pm$ 0.9 $\pm$ 0.5 | | 7.78 $\pm$ 0.43 $\pm$ 0.49 $\pm$ 0.39 | | |
9-10 | 7.18 $\pm$ 0.32 $\pm$ 0.46 $\pm$ 0.22 | | 3.96 $\pm$ 0.31 $\pm$ 0.25 $\pm$ 0.24 | | |
10-11 | 4.15 $\pm$ 0.24 $\pm$ 0.27 $\pm$ 0.18 | | 2.47 $\pm$ 0.25 $\pm$ 0.16 $\pm$ 0.18 | | |
11-12 | 2.24 $\pm$ 0.17 $\pm$ 0.14 $\pm$ 0.08 | | - | | |
12-13 | 0.97 $\pm$ 0.11 $\pm$ 0.06 $\pm$ 0.04 | | - | | |
13-14 | - | | - | | |
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|
arxiv-papers
| 2013-07-24T10:56:33 |
2024-09-04T02:49:48.379506
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D.\n Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H.\n Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo Garcia,\n M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini,\n M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M.\n Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras,\n P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau,\n G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C.\n D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De\n Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone,\n D. Decamp, M. Deckenhoff, L. Del Buono, D. Derkach, O. Deschamps, F. Dettori,\n A. Di Canto, H. Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M.\n Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao,\n J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, C. Gotti, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A.\n Hernando Morata, E. van Herwijnen, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R.\n Jacobsson, A. Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach,\n I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, I.\n Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, M.J. Morello, R.\n Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster,\n P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G.\n Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. 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Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu,\n K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M.\n Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A.\n Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Yanxi Zhang",
"url": "https://arxiv.org/abs/1307.6379"
}
|
1307.6500
|
# Strong optical self-focusing effect in coherent light scattering with
condensates
Chengjie Zhu National Institute of Standards & Technology, Gaithersburg,
Maryland USA 20899 East China Normal University, Shanghai, China 200062 L.
Deng National Institute of Standards & Technology, Gaithersburg, Maryland USA
20899 E.W. Hagley National Institute of Standards & Technology,
Gaithersburg, Maryland USA 20899 G.X. Huang East China Normal University,
Shanghai, China 200062
###### Abstract
We present a theoretical investigation of optical self-focusing effects in
light scattering with condensates. Using long ($>200\ \mu s$), red-detuned
pulses we show numerically that a non-negligible self-focusing effect is
present that causes rapid optical beam width reduction as the scattered field
propagates through a medium with an inhomogeneous density distribution. The
rapid growth of the scattered field intensity and significant local density
feedback positively to further enhance the wave generation process and
condensate compression, leading to highly efficient collective atomic recoil
motion.
###### pacs:
03.75.-b, 42.65.-k, 42.50.Gy
Introduction. $-$ Effects of a strongly-driven medium on the propagation of a
near resonant light field have been extensively studied in both linear and
nonlinear optics. In linear optics, a medium with a non uniform index of
refraction, such as an optical fiber agarwal , can lead to a lensing effect
that causes the light field traversing the medium to be focused or defocused,
depending on the detuning of the light with respect to some general
transitions of the medium. In nonlinear optics shen , however, a significant
local light field intensity can itself substantially alter the local optical
index of refraction. This process, known as the Kerr effect, can result in
laser beam self-focusing/defocusing, and even material break down and laser
beam filamentation. These effects have been widely observed both in gaseous
phase and solid-state media at room temperature. Theoretically, the general
practice is to begin with the material equations without considering the
center-of-mass motion (CM) of individual atoms or molecules participating in
the wave generation and propagation process. This makes sense because in a
room-temperature gaseous-phase medium the random thermal motion of the
scatterers completely dwarfs any possible collective CM. In a solid-state
medium, on the other hand, the scatterers are tightly bounded to their lattice
sites, so again the CM motion is not important.
Self-focusing of an optical field in a medium is a non-linear process that
arises from the local change of the refractive index of the material induced
by the intensity of an optical field. In typical solid state material this
often requires an intense electromagnetic field Cumberbatch1970 ; Mourou2006 .
In room-temperature dilute gaseous phase media this effect is generally
unimportant even with an intense parallel-beam light pulse of a relatively
short pulse length. This is, however, not the case with an ultra cold quantum
gas where the extremely narrow optical transition line width between momentum
states can lead to highly efficient generation of a light field within a very
small propagation distance. The spatial inhomogeneity of the density
distribution of a trapped condensate, the extremely small medium cross
section, and the confinement of a fast growing optical field result in an
extraordinary optical self-focusing phenomenon that has never been seen before
in a room temperature dilute gas. We further note that in an ultra-cold
quantum gas, such as a Bose condensate trapped in a magnetic trap, the
collective CM recoil motion of atoms is of paramount importance. This new
feature leads to modified material equations and therefore phenomena that have
not been examined previously.
In this work, we present a numerical study that investigates the optical self-
focusing effect by considering both dynamic medium density evolution and the
impact of local field growth due to an abnormally rapid local field cross
section change. We first derive a (2+1)-D nonlinear Schrödinger (NLS) equation
from the Gross-Pitaevskii equation and the Maxwell equation describing the
dynamic propagation effects due to an internally generated field in a Bose
condensate by stimulated Raman scattering. We show by extensive numerical
simulations that under long-pulse, red-detuned laser excitation significant
coherent growth of the scattered field by a wave mixing process leads to a
rapid reduction of the local field cross section and also results in a self-
focusing effect that significantly alters the spatial inhomogeneity of a
gaseous phase Bose condensate. Before describing our work, we first point out
that many early experimental Inouye1999A ; Schneble2003 ; Schneble2004 ;
Kuga2004 ; Inouye1999B ; Kozuma1999 and theoretical Moore1999 ; Li2000 ;
Piovella2001 ; Pu2003 ; Bonifacio2004 ; Fallani2005 ; Sarlo2005 ; Yu2004 ;
Uys2007 ; Benedek2004 ; Robb2005 ; Ketterle2001 ; Zobay2006 ; trifonov ;
sorenson ; Deng2010A ; Deng2010B ; buchmann2010 studies have been devoted to
light scattering in a Bose condensate. These works, which mostly considered
the linear regime of the scattering process, have contributed substantially to
the understanding of the light scattering in condensates.
Thoery. $-$ We start with a set of equations of motion describing the atomic
mean field amplitudes and the propagation of the generated electric field
inside the condensate. We consider a longitudinal pump scheme where a pump
beam (field amplitude $E_{L}$) polarized in the $x-$direction propagates along
the long axis of the condensate which is aligned with the $+z-$direction. In
addition, a new field $E_{G}$, (see Fig. 1) is generated inside the medium and
it counter-propagates relative to the pump laser. More specifically, we assume
that
Figure 1: (Color online) Energy levels with laser couplings (left) and
scattering geometry in a cylindrical coordinate system (lower-right). The red
wavy arrow depicts the coherently scattered field with the largest gain. An
atom absorbs a photon from the pump and then emits a photon via stimulated
emission in the direction opposite to the pump, acquiring a net $2\hbar k_{\rm
L}$ momentum in the direction of the pump laser.
$\displaystyle\mathbf{E}_{L,G}^{(+)}$ $\displaystyle=$ $\displaystyle
E_{L,G}^{(0)(+)}e^{i{\bf{k}}_{L,G}\cdot{\bf{r}}-i\omega_{L}t}{\mathbf{e}}_{x},$
$\displaystyle\psi(\rho,z,t)$ $\displaystyle=$
$\displaystyle\sum_{m}\psi_{m}(\rho,t)e^{imKz-i\omega_{m}t},$
where ${\bf{k}}_{L,G}\\!\cdot\\!{\bf{r}}\\!=\\!\pm k_{L,G}z$,
$K\\!=\\!k_{L}+k_{G}$ and ${\mathbf{e}}_{x}$ is the polarization direction of
the light fields. For what follows, we assume a uniform and constant pump
$E_{L}^{(0)(+)}$ and a generated field of $E_{G}^{(0)(+)}=E_{G}^{(0)(+)}({\bf
r},t)$. Without loss of generality, we also assume the condensate is
cylindrically shaped and has a uniform density distribution along the long
$z-$axis. However, the initial transverse density profile taken to be
$n(\rho)=n_{0}(1-\rho^{2}/\rho_{0}^{2})$ where $\rho^{2}=x^{2}+y^{2}$
($r^{2}=\rho^{2}+z^{2}$) and $n_{0}$ is the peak density. Here, $\rho$ is the
radial coordinate and $\rho_{0}$ is the initial transverse radius of the
condensate (i.e., the short axis, see Fig. 1). In the case of a true two-level
system this longitudinal pump scheme is isomorphic to the transverse pumping
scheme which yields two end-fire modes.
With respect to Fig. 1, the equation of motion for the $n$-th order mean field
atomic wave function is given by
$\displaystyle\frac{\partial\psi_{n}}{\partial t}$ $\displaystyle=$
$\displaystyle
i\frac{\hbar}{2M}\nabla_{\bot}^{2}\psi_{n}-iV_{T}\psi_{n}-ig_{0}\delta|\epsilon^{(+)}|^{2}\psi_{n}$
(1) $\displaystyle-$ $\displaystyle
ig\sum_{m_{1},m_{2}}\psi_{m_{1}}\psi_{m_{2}}^{\ast}\psi_{n-m_{1}+m_{2}}S(n,m_{1},m_{2},t)$
$\displaystyle-$ $\displaystyle
ig_{0}\delta\epsilon^{(-)}\psi_{n-1}e^{-i(\omega_{n-1}-\omega_{n})t-i\Delta_{L}t}$
$\displaystyle-$ $\displaystyle
ig_{0}\delta\epsilon^{(+)}\psi_{n+1}e^{-i(\omega_{n+1}-\omega_{n})t+i\Delta_{L}t},$
where
$S(n,m_{1},m_{2},t)=e^{i(\omega_{n}-\omega_{m_{1}}+\omega_{m_{2}}-\omega_{n-m_{1}+m_{2}})t}$,
and $g=4\pi\hbar^{2}a/M$ with $a$ being the scattering length. In addition,
$g_{0}=|D_{12}|^{2}|E_{L}|^{2}/(\hbar^{2}|\Delta|^{2})$, where
$\Delta=\delta+i\Gamma$ with $\delta$ and $\Gamma$ being the one-photon laser
detuning to the upper electronic excited state and the spontaneous emission
rate of the upper state, respectively. The normalized field is defined as
$\epsilon^{(\pm)}=E_{G}^{(\pm)}(\rho,z,t)/E_{L}^{(\pm)}$, with
$E_{L,G}^{(-)}=E_{L,G}^{(+)\ast}$. The trapping potential
$V_{T}=M\Omega_{T}^{2}\rho^{2}/2$ with trapping frequency $\Omega_{T}$.
$\hbar\omega_{m}=(m2\hbar k)^{2}/2M$ is the $m-$th order recoil energy with
$k=k_{L}$ and $M$ being the pump laser wave vector and the mass of the atom,
respectively.
In the slowly varying envelope approximation the Maxwell equation for the
generated field is given by
$\displaystyle-i\frac{\partial\epsilon^{(+)}}{\partial z}$ $\displaystyle+$
$\displaystyle i\frac{1}{c}\frac{\partial\epsilon^{(+)}}{\partial
t}+\frac{1}{2k_{G}}\nabla^{2}_{\bot}\epsilon^{(+)}=\frac{\kappa_{0}}{\Delta}|\psi_{0}|^{2}\epsilon^{(+)}$
(2) $\displaystyle+$
$\displaystyle\frac{\kappa_{0}}{\Delta}\sum_{n}\psi_{n}\psi_{n+1}^{\ast}e^{i2(n+1)4\omega_{R}t-i\Delta_{L}t},$
where the second term on the right is the polarization source term that drives
the generation of the new field. In deriving Eq. (2) we have only kept the
lowest scattering order, i.e. we neglect $n>1$ terms. Furthermore, we also
neglect $n<0$ terms since it has already been shown that for long pulse
excitation the bandwidth of the laser is sufficiently narrow that $n<0$
scattering orders do not occur.
To investigate the scattered optical field self-focusing effect Eq. (2) must
be solved simultaneously with the atomic response Eq. (1) to third order in
the generated field. We apply a perturbation expansion scheme
$\displaystyle\psi_{0}=\psi_{0}^{(0)}+\lambda^{2}\psi_{0}^{(2)},\,\psi_{1}=\lambda\psi_{1}^{(1)}+\lambda^{3}\psi_{1}^{(3)},\epsilon^{(+)}=\lambda\epsilon^{(+)}.$
(3)
These are well-known multi-scale pertubation schemes that have been widely
used in soliton theories where small ground state population corrections must
be included in the mathmetical theory soliton1 . Inserting Eq. (3) into Eq.
(1) we obtain
$\displaystyle\frac{\partial\psi_{0}^{(0)}}{\partial t}$ $\displaystyle=$
$\displaystyle
i\frac{\hbar}{2M}\nabla_{\bot}^{2}\psi_{n}^{(0)}-iV_{T}\psi_{n}^{(0)}-ig|\psi_{0}^{(0)}|^{2}\psi_{0}^{(0)},$
(4a) $\displaystyle\frac{\partial\psi_{0}^{(2)}}{\partial t}$ $\displaystyle=$
$\displaystyle-\gamma_{0}\psi_{0}^{(2)}+i\frac{\hbar}{2M}\nabla_{\bot}^{2}\psi_{0}^{(2)}-iV_{T}\psi_{0}^{(2)}$
(4b) $\displaystyle-$ $\displaystyle
ig_{0}\delta|\epsilon^{(+)}|^{2}\psi_{0}^{(0)}-2ig|\psi_{+1}^{(1)}|^{2}\psi_{0}^{(0)}-ig|\psi_{0}^{(0)}|^{2}\psi_{0}^{(2)}$
$\displaystyle-$ $\displaystyle
ig_{0}\delta\epsilon^{(+)}\psi_{+1}^{(1)}e^{-i\omega_{1}t+i\Delta_{L}t},$
$\displaystyle\frac{\partial\psi_{+1}^{(1)}}{\partial t}$ $\displaystyle=$
$\displaystyle-\gamma_{1}\psi_{+1}^{(1)}+i\frac{\hbar}{2M}\nabla_{\bot}^{2}\psi_{+1}^{(1)}-iV_{T}\psi_{+1}^{(1)}$
(4c) $\displaystyle-$ $\displaystyle
2ig|\psi_{0}^{(0)}|^{2}\psi_{+1}^{(1)}-ig_{0}\delta\epsilon^{(-)}\psi_{0}^{(0)}e^{i\omega_{1}t-i\Delta_{L}t},$
$\displaystyle\frac{\partial\psi_{+1}^{(3)}}{\partial t}$ $\displaystyle=$
$\displaystyle-\gamma_{1}\psi_{+1}^{(3)}+i\frac{\hbar}{2M}\nabla_{\bot}^{2}\psi_{+1}^{(3)}-iV_{T}\psi_{+1}^{(3)}$
(4d) $\displaystyle-$ $\displaystyle
2ig|\psi_{0}^{(0)}|^{2}\psi_{+1}^{(3)}-ig|\psi_{+1}^{(1)}|^{2}\psi_{+1}^{(1)}$
$\displaystyle-$ $\displaystyle
ig\psi_{+1}^{(1)}\psi_{0}^{(0)}\,{}^{*}\psi_{0}^{(2)}-ig\psi_{+1}^{(1)}\psi_{0}^{(2)}\,{}^{*}\psi_{0}^{(0)}$
$\displaystyle-$ $\displaystyle
ig_{0}\delta\epsilon^{(-)}\psi_{0}^{(2)}e^{i\omega_{1}t-i\Delta_{L}t}.$
It is clear that Eq. (4a), which is the zero-order equation for $n=0$ mean
field wave fucntion $\psi_{0}^{(0)}$, is just the Gross-Pitaevskii equation in
the absence of the external electric field note1a . In our calculation Eq.
(4a) is solved numerically by directly numerical integration. In the
derivation of Eq. (4b-4d) we have introduced decay constants $\gamma_{0}$ and
$\gamma_{1}$ to characterize the loss of coherence of the atomic center-of-
motion states due to the interaction with the pump light field. In general,
the total system population conservation in such a simple two-level model
implies $\gamma_{0}^{(2)}\approx-\gamma_{1}^{(1)}$. This has been verified
numerically. Finaly, we neglected a constant Stark shift/dipole potential due
to the pump field that can be removed by a trivial phase transformation
without affecting the polarization source term in Eq. (2). Enforcing the
first-order Bragg scattering condition $\omega_{1}-\omega_{0}=4\omega_{\rm
R}=\Delta_{\rm L}$, and consistently keeping all terms up to the third order
in the generated field, the Maxwell equation for the generated field now
becomes
$\displaystyle\frac{\partial\epsilon^{(+)}}{\partial z}$ $\displaystyle+$
$\displaystyle\frac{i}{2k_{\rm
G}}\nabla^{2}_{\bot}\epsilon^{(+)}=i\frac{\kappa_{0}}{\Delta}\left(|\psi_{0}^{(0)}|^{2}\epsilon^{(+)}+\psi_{0}^{(0)}\psi_{+1}^{(1)\ast}\right)$
(5) $\displaystyle+i\frac{\kappa_{0}}{\Delta}\left[2{\rm
Re}\left(\psi_{0}^{(0)}\psi_{0}^{(2)\ast}\right)+|\psi_{+1}^{(1)}|^{2}\right]\epsilon^{(+)}$
$\displaystyle+i\frac{\kappa_{0}}{\Delta}\left(\psi_{0}^{(0)}\psi_{+1}^{(3)\ast}+\psi_{0}^{(2)}\psi_{+1}^{(1)\ast}\right).$
Here, we have neglected the $(1/c)\left(\partial\epsilon/\partial t\right)$
term because the dominant propagation velocity comes from the polarization
term Deng2010A . Under the steady state approximation analytical expressions
of $\psi_{0}$ and $\psi_{+1}$ can be obtained. The first-order solution of the
scattered component becomes
$\psi_{+1}^{(1)}=-i\frac{\delta
g_{0}\psi_{0}^{(0)}}{\gamma_{1}+ig|\psi_{0}^{(0)}|^{2}}\epsilon^{(-)}.$ (6)
Using Eq. (6), we obtain
$\psi_{0}^{(2)}=-i\delta g_{0}\psi_{0}^{(0)}\alpha|\epsilon^{(+)}|^{2},$ (7)
where
$\alpha\\!=\\!\frac{1}{\gamma_{0}+ib}\left[1\\!+\\!\frac{\delta
g_{0}g|\psi_{0}^{(0)}|^{2}}{\gamma_{1}^{2}\\!+\\!g^{2}|\psi_{0}^{(0)}|^{4}}-i\frac{\delta
g_{0}\gamma_{1}}{\gamma_{1}^{2}+g^{2}|\psi_{0}^{(0)}|^{4}}\right].$ (8)
Here, we have abrivated the second term on the right of Eq. (4b) as $\hbar
b\equiv\hbar^{2}k_{\bot}^{2}/2M$. Physically, it is a small transverse kinetic
energy of atoms in the zeroth-order condensate due to transverse light force
compression. The third order correct $\psi_{+1}^{(3)}$ is given by
$\displaystyle\psi_{+1}^{(3)}=-\frac{\delta^{2}g_{0}^{2}\psi_{0}^{(0)}}{\gamma_{1}+ig|\psi_{0}^{(0)}|^{2}}|\epsilon^{(+)}|^{2}\epsilon^{(-)}$
$\displaystyle\times\left\\{\alpha+\frac{g|\psi_{0}^{(0)}|^{2}}{\gamma_{1}+ig|\psi_{0}^{(0)}|^{2}}\left[2{\rm
Im}(\alpha)+\frac{\delta
g_{0}}{\gamma_{1}^{2}+g^{2}|\psi_{0}^{(0)}|^{4}}\right]\right\\}.\quad\;$ (9)
We now explain the rationale for the above outlined perturbation scheme where
only the $\psi_{+1}$ order is considered. Our calculations are aimed at
providing a trackable derivation with an analytical solution that can capture
the key physics. It is for this reason that we limit our treatment to a pump
light scattering rate of $R<80$ Hz. In this regime only first-order scattering
has been observed experimentally. Although the $\psi_{+2}^{(2)}$ term, which
is the leading contribution from the $\psi_{+2}$ term, is on the order of
$|\epsilon^{(+)}|^{2}$ (similar to that of $\psi_{+1}^{(3)}$), we have
neglected it in the above calculation because the residual multi-photon
Doppler shift affects the scattering efficiency of a four-photon process (the
$\psi_{+2}$ term) much more strongly than a two-photon process (the
$\psi_{+1}$ term) for a given laser band width. In fact, this energy mismatch
due to a residual Doppler shift is the primary reason why even at higher pump
powers the scattering orders higher than four are difficult to observe under
long-pulse excitation ref29 . We emphasize, however, that we have carried out
directly numerical integration of Eqs. (4a)$-$(4c) and (5) without further
approximation and the results agree well with the above steady state
treatment.
Substituting Eqs. (6)-(Strong optical self-focusing effect in coherent light
scattering with condensates) into Eq. (5) we arrive at a third-order wave
equation analogus to a (2+1)-D nonlinear Schrödinger (NLS) equation where the
3rd-order nonlinear contribution can effectively balance the beam loss due to
diffraction due to the condensate size effect, and result in an optical field
self-focusing phenomenon. In our case, this (2+1)-D NLS equation can be
written as
$i\frac{\partial\epsilon^{(+)}}{\partial z}-\frac{1}{2k_{\rm
G}}\nabla^{2}_{\bot}\epsilon^{(+)}+W|\epsilon^{(+)}|^{2}\epsilon^{(+)}=-\beta\epsilon^{(+)}.$
(10)
Here the linear absorption/gain term is given by
$\displaystyle\beta\approx\frac{\kappa_{0}n}{\delta}\left(1-\frac{\delta
g_{0}gn}{\gamma_{1}^{2}}+i\frac{\delta g_{0}}{\gamma_{1}}\right)$ (11a)
$\displaystyle W\approx\frac{\kappa_{0}\delta
g_{0}^{2}n}{\gamma_{1}^{2}}\left(3-\frac{5\delta
g_{0}gn}{\gamma_{1}^{2}}\right)+2i\frac{\kappa_{0}\delta
g_{0}^{2}n}{\gamma_{1}^{3}},\quad$ (11b)
where $n=|\psi_{0}^{(0)}|^{2}$ is the initial transverse density profile. In
deriving Eqs. (11a, 11b) we have assumed $b\ll|\gamma_{0}|$ for mathematics
simplicity. This assumption has been verified by direct numerical evaluation
of the transverse kinetic energy $\hbar b$. It has been shown previously
agarwal ; shen ; Cumberbatch1970 ; Mourou2006 that the sign of Re$[W]$ given
in Eq. (11b) leads to self-focusing/self-defocusing effects. Indeed, Eq. (11b)
predicts that: (i) For red detunings (i.e. $\delta<0$) Re$[W]$ is always
negative for typical experimental parameters (see below), and this will result
in a reduction of the transverse dimension of the generated field. Thus, one
expects to see reduced diffraction, and possibly a self-focusing effect. (ii)
For blue detunings (i.e. $\delta>0$) Re$[W]$ is also negative for typical
experimental parameters and therefore one also expects a self-focusing effect
note except the strength of the self-focusing effect is considerably weaker
(that is, for typical experimental parameters we always find that
$|\rm{Re}[W_{red}]|>|\rm{Re}[W_{blue}]|$). Finally, for typical experimental
parameters Im$[\beta]$ and Im$[W]$ are always positive for both red and blue
detunings, indicating linear and nonlinear gains.
Figure 2: Third-order nonlinearity $W$ as function of $\eta=\rho/\rho_{0}$.
Dashed line: Re$[W]_{\rm red}$, dotted line: Im$[W]_{\rm red}$ with red
detunings $\delta/2\pi=-2$ GHz. Solid line: Re$[W]_{\rm blue}$, dash-dotted
line: Im$[W]_{\rm blue}$ with blue detunings $\delta/2\pi=+2$ GHz.
Numerical calculation. $-$ To verify the above analysis we performed full
numerical simulations using Eqs. (10) and (11a,b). Other parameters are s
similiar to those reported in literature. Specifically, we consider a rubidium
condensate with $2\times 10^{6}$ atoms, $L=200\ \mu$m, and $\rho_{0}=10\ \mu$m
(peak density about $n_{0}=3.2\times 10^{19}\ $m-3). $\Gamma/2\pi=6\ $MHz,
$\gamma_{1}/2\pi=2$ kHz, $\gamma_{0}/2\pi=-2$ kHz, $\kappa_{0}=2.76\times
10^{-6}\ {\rm m}^{2}{\rm s}^{-1}$, $b=240$ Hz, $g/\hbar=4.85\times 10^{-17}\
{\rm m}^{3}{\rm s}^{-1}$ corresponding to the scattering length $a_{\rm
s}=100a_{0}$ (Bohr radius $a_{0}=5.29\times 10^{-9}$ cm), $\delta/2\pi=\pm 2$
GHz, $k_{\rm G}\approx 8\times 10^{6}$ m-1. In accord with our approximations
we chose $g_{0}=2.5\times 10^{-5}$, which corresponds to $R\approx$ 60 Hz. In
Fig. (2) we plot the values of Re$[W]$ and Im$[W]$ for these parameters. It
can be seen that indeed $|\rm{Re}[W]_{red}|>|\rm{Re}[W]_{blue}|$, and yet both
contribute to a field self-focusing effect note .
Figure 3: Macroscopic atomic mean field distribution as a function of
dimensionless radius $\eta=\rho/\rho_{0}$ at $z=L$ (dashed curve) and at $z=0$
(solid curve). Note $z=L$ is the starting position of $E_{G}$. At this point
$E_{G}$ is negligible and the density distribution is just the original
condensate distribution. The field $E_{G}$ travels backward and it reaches its
maximum value at $z=0$, causing the greatest atomic density change near the
center of the condensate $\eta=0$.
One important consequence of the light field self-focusing effect is its
tendency to compress/decompress the spatial density distribution of the
condensate. This effect uniquely affects a gaseous phase medium where
collective recoil motion is a prominent feature. Indeed, such a density
modification effect due to the light field intensity change is not important
in a solid medium where the atoms are strongly bounded to their lattice sites.
Nor is this important for a normal gas where the collective CM recoil motion
is completely negligible when compared to its intrinsic thermal motion. In the
case of red-detunings in a condensate, the self-focusing effect results in a
rapid field intensity increase which further compresses the condensate. This
process further enhances the local field generation, resulting in positive
feedback and a run away gain effect. For blue-detunings, however, the atoms
are expelled from the region of strong fields, resulting in a reduced density
distribution which reduces the field generation efficiency. In Fig. 3 we plot
the atomic density distribution $|\psi(\rho,z)|^{2}$ as a function of the
normalized radius $\eta$. We emphasize that the significant change in the
local density distribution for red detunings shown in Fig. 3 further enhances
the generation efficiency of the scattered light field, which further
compresses the condensate.
Figure 4: (Color online) Plot of $|\epsilon^{(+)}|^{2}$ as a function of the
propagation distance $z$ and the dimensionless radius $\eta$. Left column:
each plot is normalized to its own peak at $\eta=0$. Right column: all plots
are normalized with respect to the peak of Fig. 4e at $\eta=0$ . Figs. 4a and
4d ($\delta/2\pi=-2$ GHz): The Kerr nonlinearity is neglected . Figs. 4b
($\delta/2\pi=-2$ GHz), 4c ($\delta/2\pi=+2$ GHz), 4e ($\delta/2\pi=-2$ GHz),
and 4f ($\delta/2\pi=+2$ GHz): The Kerr nonlinearity $W$ is included.
This dramatic light field self-focusing effect is shown in Fig. 4 where the
intensity profile of the generated light field is presented with, and without,
the Kerr term for red and blue detunings. Fig. 4a shows the field profile
without the Kerr term ($\delta/2\pi=-2$ GHz). Figures 4b and 4c show the field
distributions with the nonlinear term included. Here, all three plots are
normalized to unity to show the effective transverse field distribution
(width). Clearly, in the case of red-detuned pumps (Fig. 4b) the scattered
field intensity has a cross section that is more than a factor of 2 smaller
when compared to blue-detuned pumps (Fig. 4c), representing a factor of 4
fermion intensity difference. In Figs. 4d-4f we show the same numerical
results but with all three plots normalized with respect to Fig. 4b. This
gives a sense of the relative strengths of the fields in Fig. 4a and 4c when
compared to Fig. 4b.
Conclusion. $-$ In conclusion, we have studied numerically the dynamic light
field self-focusing effect in light scattering in a Bose condensate. By
including the condensate transverse density profile we derived a 3-dimensional
atomic CM Maxwell equation describing the generation and propagation of a new
field, and a set of Gross-Pitaevskii equations for scattered atoms. Using a
standard perturbation expansion, we recast the field equation into a (2+1)-D
NLS equation which reveals the light field self-focusing phenomenon. Numerical
simulations revealed a significant reduction of the transverse profile of a
red-detuned internally generated field as it propagates through the
condensate. With red detunings the rapid increase in field intensity and the
accompanying compression effect further feed back on themselves, leading to a
significant condensate density change and a highly efficient field generation
and scattering process. In the case of blue-detuned pumps, numerical
calculations have shown that the field generation is considerably weaker. Our
study, which provides the first theoretical evidence of nonlinear optical
processes in light scattering in a condensate, has clearly shown that these
higher-order processes play very important roles in light scattering in
quantum gases.
Acknowledgments: Chengjie Zhu acknowledges supported by NSF-China under Grant
Nos. 10874043 and 11174080, and by the Chinese Education Ministry Reward for
Excellent Doctors in Academics under Grant No. MXRZZ2010007.
## References
* (1) G.P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2006).
* (2) Y.R. Shen, The Principle of Nonlinear Optics,(John Wiley & Sons, New York, 1984).
* (3) Cumberbatch, E, J. Inst. Maths Applics 6, 250 (1970).
* (4) Mourou, G. A. et al., Rev. Mod. Phys. 78, 309 (2006).
* (5) S. Inouye, et al, Science 285, 571 (1999).
* (6) D. Schneble, et al, Science 300, 475 (2003).
* (7) D. Schneble et al., Phys. Rev. A 69, 041601(R)(2004).
* (8) Y. Yoshikawa et al., Phys. Rev. A 69, 041603(R)(2004).
* (9) S. Inouye et al., Nature (London) 402, 641 (1999).
* (10) M. Kozuma et al., Science 286, 2309 (1999).
* (11) M.G. Moore and P. Meystre, Phys. Rev. Lett. 83, 5202 (1999).
* (12) $\ddot{\text{O}}$.E. M$\ddot{\text{u}}$stecaplio$\check{\text{g}}$lu and L. You, Phys. Rev. A 62, 063615 (2000).
* (13) N. Piovella et al., Opt. Commun. 187, 165 (2001).
* (14) H. Pu, W. Zhang, and P. Meystre, Phys. Rev. Lett. 91, 150407 (2003).
* (15) R. Bonifacio et al., Opt. Commun. 233, 155 (2004).
* (16) L. Fallani et al., Phys. Rev. A 71, 033612 (2005).
* (17) L. De Sarlo et al., J. Eurp. Phys. D 32, 167 (2005).
* (18) Yu. A. Avetisyan and E. D. Trifonov, Laser Phys. Lett. 1, 373 (2004).
* (19) H. Uys and P. Meystre, Phys. Rev. A 75, 033805 (2007).
* (20) C. Benedek and M. G. Benedikt, J. Opt. B: Quantum Semiclass. Opt. 6, S111 (2004).
* (21) G.R.M. Robb, N. Piovella, and R. Bonifacio, J. Opt. B: Quantum Semiclass. Opt. 7, 93 (2005).
* (22) W. Ketterle and S. Inouye, C.R. Acad. Sci. Paris, IV, 339 (2001).
* (23) O. Zobay and G. M. Nikolopoulos, Phys. Rev. A 73, 013620 (2006).
* (24) E.D. Trifonov, Optics and Spectroscopy 92, 577 (2002).
* (25) M.W. Sorenson and A.S. Sorenson, Phys. Rev. A77, 013826 (2008).
* (26) L. Deng, M.G. Payne, and E. W. Hagley, Phy. Rev. Lett. 104, 050402 (2010).
* (27) L. Deng, E. W. Hagley, Phy. Rev. A. 82, 053613 (2010).
* (28) L.F. Buchmann et al., Phys. Rev A 82, 023608 (2010).
* (29) R. MacKenzie, M.B. Paranjape, and W.J. Zakrzewski, Solitons, Springer-Verlag, New York 2000.
* (30) The ground state chemical potential $\mu$ neither enters Eq. (4) nor leads to a significant oscillation at this pump pulse time scale for a typical condensate.
* (31) Higher-order scatterings occur in a sequential manner, implying that most of the key physics should be revealed by studying first-order scattering.
* (32) While a mild self-focusing effect is predicted for blue-detuned pumps, a much stronger loss mechanism of molecular origin occurs simultaneously when the pump laser is blue detuned. See, L. Deng et al., Phys. Rev. Lett. 105, 220404 (2010); N.S. Kampel et al., ibid. 108, 090401 (2012); Xinyu Luo et al., Phys. Rev. A 86, 043603 (2012).
* (33) In the case of a fermionic gas [see P.J. Wang et al., Phys. Rev. Lett. 106, 210401 (2011)] the difference between $\pm\delta$ is very small because of the low coherence of the gas.
|
arxiv-papers
| 2013-07-24T17:23:45 |
2024-09-04T02:49:48.395271
|
{
"license": "Public Domain",
"authors": "Chengjie Zhu, L. Deng, E.W. Hagley, G.X. Huang",
"submitter": "Chengjie Zhu",
"url": "https://arxiv.org/abs/1307.6500"
}
|
1307.6571
|
# The 1st Fermi LAT SNR Catalog: Constraining the Cosmic Ray Contribution
###### Abstract
Despite tantalizing evidence that supernova remnants (SNRs) are the source of
Galactic cosmic rays (CRs), including the recent detection of a spectral
signature of hadronic $\gamma$-ray emission from two SNRs, their origin in
aggregate remains elusive. We address the long-standing question of Galactic
CR nuclei origins using our statistically significant GeV SNR sample to
estimate the contribution of SNRs to directly observed CRs. Interactions
between CRs and ambient gas near the SNRs emit photons via pion decay at GeV
energies, providing an in situ tracer for CRs otherwise measured directly with
balloon-borne and satellite experiments near the Earth. To date, the Fermi LAT
SNR Catalog has detected more than 50 SNRs and potential associations in
classes with a variety of properties, yet all remain possible accelerators. We
investigate the GeV and multiwavelength (MW) emission from SNRs to constrain
their maximal contribution to observed Galactic CRs. Our work demonstrates the
need for improvements to previously sufficient simple models describing the
GeV and MW emission from these objects.
## 1 Introduction
Direct measurements of cosmic ray (CR) energetics and composition combined
with our understanding of high energy accelerators in the Galaxy have long
suggested that supernova remnants (SNRs) are likely the source of Galactic
CRs. Yet proof has for a long while remained elusive. With the advent of
$\gamma$-ray telescopes with degree-scale spatial resolution in addition to
good spectral resolution, we have made significant strides towards more firmly
associating SNRs with CR acceleration. These have included individual SNR
spectra tending towards being dominated by hadronic emission, where CRs
interacting with the local medium emit $\gamma$-rays via $\pi^{0}$ decay (e.g.
[1]), as well as recent detection of proton acceleration through the low-
energy $\pi^{0}$ cutoff [2].
Yet while such individual results are necessary, they are not sufficient, for
the direct data we would ultimately like to compare to is comprised of CRs
from sources throughout the Galaxy. Thus, it is also necessary to show that
the aggregate contribution from all sources can produce the observed CRs. In
order to do so, we have leveraged several years’ worth of Fermi Large Area
Telescope (LAT) survey data to study systematically all known Galactic SNRs,
the majority of which are detected in the radio and compiled in [3], as well
as a few identifications from other wavelengths. Details of the analysis
procedure are laid out in [4], along with a discussion of the implications of
a radio-GeV flux correlation. As interstellar $\gamma$-ray emission is quite
prevalent along the galactic plane, where the majority of SNRs lie, we also
explore systematics related to the choice of interstellar emission model in
[5, 6] to ensure that we will have the most robust results possible. These
results point to a separation of SNRs into classes, notably those which are
young and those which are interacting, often with molecular clouds [4].
## 2 Particle Populations
In [4], we showed that the synchrotron radio emission from high energy leptons
tends to be correlated for interacting SNRs, suggesting a physical link,
whereas the young SNRs showed more scatter.
### 2.1 Emission Mechanisms
If radio and GeV emission arise from the same particle population(s), e.g.
leptons and hadrons accelerated at the SNR shock front, under simple
assumptions, the GeV and radio indices should be correlated. For inverse
Compton (IC) emitting leptons, the GeV and radio photon indices ($\Gamma$ and
$\alpha$ respectively) can be related as $\Gamma=\alpha+1$ whereas for
$\pi^{0}$ decay and e± bremsstrahlung, $\Gamma=2\alpha+1$. Figure 1 shows
that, contrary to our radio/GeV flux observations, young SNRs seem consistent
with expectations from these simple models. Several of the known, young SNRs
are more consistent with the IC relation (dashed line), suggesting that they
may be lepton-dominated and emitting via IC in the GeV regime. SNRs emitting
via a combination of mechanisms in this scenario have indices falling between
the two index relations, that is, the region spanned by the
$\pi^{0}$/bremsstrahlung (solid) and IC (dashed) lines. The young SNR RX
J1713-3946 is one example which bears out this case [7]. Other SNRs, including
those observed to be interacting with molecular clouds, are softer than
expected, and in most cases are not even consistent with combinations of
emission mechanisms.
The apparent lack of correlation between the indices and emission mechanisms
for the majority of observed SNRs suggests that the data are now able to
challenge model assumptions for these SNRs. These assumptions include that:
* •
the underlying leptonic and hadronic populations may have different power law
indices;
* •
the emitting particle population(s) may not follow a power law but may instead
have break(s);
* •
or there may be different zones with different properties dominating the
emission at different wavelengths.
Figure 1: GeV-Radio index and expected slope correlation for: $\pi^{0}$ decay
or e± bremsstrahlung (solid line) and inverse Compton (dashed line). As
described in more detail in [4], the colors correspond to different types of
SNRs and SNR candidates, namely, young SNRs are blue; those identified,
interacting SNRs are red; the green points correspond to newly identified
SNRs; and the grey points to point-like candidate SNRs (dark grey) and point-
like candidate pulsars (light grey). This scheme is maintained throughout the
paper.
### 2.2 Spectral Break?
With SNRs studied in TeV, we have the opportunity to explore the second of the
model assumptions: that the emitting particle population(s) may have breaks.
Such a break in the underlying particle population(s) can also cause a break
in the observed spectrum. As TeV emission may arise from the same mechanisms
as the Fermi-observed GeV emission, we might expect to see such a break
reflected in the spectrum combining Fermi data with observations from Imaging
Air Cherenkov Telescopes (IACTs) such as H.E.S.S., VERITAS, and MAGIC. In
Figure 2 we plot the GeV index versus TeV index for all SNRs observed with
both Fermi and an IACT. Several SNRs’ TeV indices are lower than their GeV
index, and few lie above the line of equal index. This suggests a break either
at or between GeV and TeV energies. The former has been observed for e.g.
IC443 [8] and the latter in, e.g. the young SNR RX J1713-3946 [7]. Such a
break would be a tantalizing clue, likely reflecting a break in the underlying
particle spectrum, as it does for IC443 and RX J1713-3946, of many SNRs. We
note however that, as the TeV sources are not uniformly surveyed, inferring
population statements from this observation requires a careful understanding
of the non-TeV observed SNR subsample.
Figure 2: GeV-TeV index. The line shows equal indices. SNRs lying below the
line suggest that their spectra have breaks, potentially reflecting a break in
the underlying particle population(s’) index or indices.
We note that the GeV-TeV index plot (Fig. 2) also shows a distinct separation
between young and interacting, often older SNRs, suggesting an evolution in
index with age, from harder when younger to softer when older. We explore this
further by explicitly investigating the evolution of the GeV index with age in
the next section.
## 3 Evolution or Environment?
As we saw a division between young SNRs having harder indices and interacting
SNRs’ tending to softer ones, we explicitly examine the evolution of the GeV
index with age of the SNR. In Figure 3 we see a clear separation of known
young SNRs having lower, harder GeV indices than interacting SNRs, where ages
were drawn from the literature. This separation could be due to decreasing
shock speed, decreasing the maximum acceleration energy as SNRs age. Particles
will be more readily able to escape a slower shock, thereby reducing the
$\gamma$-ray flux. Further, a less energetic shock will no longer be able to
accelerate particles to the highest energy, thereby reducing emission at the
highest energies.
Figure 3: Age versus GeV index. The young (blue) SNRs are separated in GeV
index from the identified interacting SNRs (red).
In [4], we observed that SNRs known to be interacting, in particular with
large molecular clouds, appear to be more luminous in GeV $\gamma$-rays than
young SNRs. The apparent difference in indices and luminosities for the young
and interacting SNRs may also be caused by differing environments: older SNRs
may interact with denser surroundings not yet reached by younger SNRs. Using
MW information such as ambient density estimated from thermal X-rays in the
catalog context will help disentangle the effects of evolution and
environment.
## 4 Constraining CR Acceleration
Recent work examining GeV $\gamma$-ray data at the $\pi^{0}$ rest mass [2] has
added another piece to the accumulating evidence (e.g. [1, 9, 10]) that SNRs
accelerate hadrons: at least two SNRs, IC443 and W44, show evidence of the
$\pi^{0}$ low energy break ($E<100$ MeV), demonstrating that they accelerate
protons. To this necessary evidence that SNRs accelerate hadrons, we must also
add an understanding of the Galactic SNR population’s ability to accelerate
the appropriate composition of hadrons to the energies observed by direct-
detection experiments.
For a SNR at a given distance $d$ interacting with a density $n$ and
accelerating cosmic rays to a maximum energy $E_{CR,max}\gtrsim 200$ GeV with
index $\Gamma_{CR}\approx 2.5$, we can relate the $\gamma$-ray flux above $1$
GeV to the SNR’s energy, $E_{SN}$, and CR acceleration efficiency,
$\epsilon_{CR}\equiv\frac{E_{CR}}{E_{SN}}$, as:
$\begin{split}F(>1\,\textrm{GeV})\approx
10^{-8}&\times\frac{\epsilon_{CR}}{0.1}\times\frac{E_{SN}}{10^{51}\,\textrm{erg}}\\\
&\times\frac{n}{1\,\textrm{cm${}^{-3}$}}\times\left(\frac{d}{1\,\textrm{kpc}}\right)^{-2}\,\textrm{cm${}^{-2}$
s${}^{-1}$}\end{split}$ (1)
which is consistent with e.g. [11]. It is useful to note that this derivation
includes the approximation that the majority of the transfer of SN explosion
energy to hadrons occurs during the Sedov(-like) phase, and that the
efficiency remains roughly constant during this period (see [11] for further
discussion).
Alternatively, we can allow the CR index and maximal energy to vary. Fixing
the acceleration efficiency to a reasonable $\epsilon_{CR}=1\%$, we can plot
the $\gamma$-ray flux as a function of $\Gamma_{CR}$ and $E_{CR,max}$, as seen
in Figure 4.
Figure 4: Under standard assumptions, a SNR’s $\gamma$-ray flux above $1$ GeV
can be related to the accelerated CRs’ maximal energy and index for a given
acceleration efficiency ($0.01$), effective density ($1$ cm-3), and distance
to the SNR ($1$ kpc).
Figure 4 shows that if we know the CR index and maximal energy for a SNR with
a given $\gamma$-ray flux, distance, and density, we can determine its CR
acceleration efficiency and thereby, the amount of energy going into CRs. [12]
compare theoretically calculated efficiencies, including time evolution of the
SNR-interstellar material system, to measured GeV luminosities for several
detected SNRs. With the $1^{st}$ Fermi SNR Catalog, we have measured fluxes or
upper limits for all SNRs in the energy range $1-100$ GeV [4]. Moreover, we
can fix the CR index to, for instance, the GeV $\gamma$-ray index, self-
consistently measured with the flux. Finally we can, for instance, constrain
$E_{CR,max}$ by relating it to either a SNR’s measured break energy or to the
maximum energy inferred from CRs interacting with the interstellar medium and
creating the diffuse Galactic $\gamma$-ray background. [13] and [14]
illustrate the extraction of CR parameters from the diffuse Galactic
$\gamma$-ray background. Combining the flux (upper limit), inferred index, and
upper limit on CRs’ maximum energy, we can place an upper limit on the energy
transferred from a given supernova explosion to its CRs. Doing so for all
known SNRs yields the total energy being transferred to CRs. If this is less
than the observed total CR energy content, within the limits of our
assumptions, another source must contribute to accelerating particles to CR
energies.
We can examine this explicitly for SNRs with GeV flux upper limits and
distances. Assuming that they are merely faint rather than less energetic, we
can use an index of $2.5$, about average for those observed so far. With a
maximum CR energy of $E_{CR,max}\gtrsim 200$ GeV, we find that the GeV flux is
nearly independent of the energy (for indices $\Gamma\gtrsim 2.0$), and scales
equation (1) by $1.5$. Solving for the efficiency,
$\begin{split}\frac{\epsilon_{CR}}{0.1}&\times\frac{n}{1\,\textrm{cm${}^{-3}$}}\\\
&\approx\frac{F(1-100\,\textrm{GeV})}{1.5\times 10^{-8}\,\textrm{cm${}^{2}$
s}}\times\left(\frac{d}{1\,\textrm{kpc}}\right)^{2}\end{split}$ (2)
for SNRs with a canonical energy of $10^{51}$ ergs. We note that if we use an
observed flux, this efficiency is that for particles accelerated up to and
including the moment of observation.
As it is also necessary to know the density of the medium with which the SNR
interacts as well as the distance to the SNR, for our preliminary study, we
turn to the $\sim 175$ SNRs detected in X-rays [15], in search of those with
thermal emission. The thermal X-ray emission from the shock-heated inter-
and/or circumstellar mediums places reasonable constraints on the density.
Likewise, most of these SNRs have a distance estimate. Densities may
subsequently also be obtained from measurements such as IR emission from
collisionally heated dust and hydrodynamics.
We will explore methods and implications for constraining Galactic SNRs’
contribution to the observed CRs by studying the efficiency, including using
flux upper limits under the assumption of entirely hadronic processes.
## 5 Conclusions
By examining correlations between SNRs’ radio and GeV indexes, we observed
that, while several known young SNRs tend to follow the expected radio-GeV
index correlation for standard emission mechanisms, the majority do not. This
challenges the previously sufficient models assumptions. In particular, we
explored the hypothesis that the underlying particle population(s) may have a
spectral break, correlating to a break in their emission spectrum. The GeV-TeV
index correlation does in fact show that several SNRs have spectral breaks at
or between GeV and TeV energies.
The GeV index’s evolution with SNR age suggests that the decreasing shock
speed or decreasing maximum acceleration energy may cause the GeV index to
tend to soften. Combining this with the observed luminosity differences also
allows a scenario where the fainter, harder young SNRs are interacting with
less dense material, while older, interacting SNRs have eg reached nearby
local overdensities, such as molecular clouds. Multiwavelength information in
the context of the $1^{st}$ Fermi SNR Catalog will help disentangle the
effects of evolution and environment.
We also explored a method for constraining SNR’s contribution to the observed
Galactic CR flux using the flux and index measured in the $1^{st}$ Fermi SNR
Catalog, constraining the maximum CR energy using the diffuse Galactic
$\gamma$-ray flux, for SNRs with known distances. Such a constraint may allow
us to infer if another source population may be contributing to the measured
Galactic CR flux or alternatively, if we need to refine the assumptions made.
Acknowledgment: The Fermi LAT Collaboration acknowledges generous ongoing
support from a number of agencies and institutes that have supported both the
development and the operation of the LAT as well as scientific data analysis.
These include the National Aeronautics and Space Administration and the
Department of Energy in the United States, the Commissariat à l’Energie
Atomique and the Centre National de la Recherche Scientifique / Institut
National de Physique Nucléaire et de Physique des Particules in France, the
Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in
Italy, the Ministry of Education, Culture, Sports, Science and Technology
(MEXT), High Energy Accelerator Research Organization (KEK) and Japan
Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg
Foundation, the Swedish Research Council and the Swedish National Space Board
in Sweden.
Additional support for science analysis during the operations phase is
gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy
and the Centre National d’Études Spatiales in France.
## References
* [1] D. J. Thompson, L. Baldini, and Y. Uchiyama, Astroparticle Physics 39 (2012) 22-32.
* [2] M. Ackermann, et. al., Science 339 (2013) 807-811.
* [3] D. A. Green, The Astrophysical Journal, Bulletin of the Astronomical Society of India 37 (2009) 45-61.
* [4] J. W. Hewitt, et. al., ICRC 2013 Proceedings.
* [5] T. J. Brandt, et. al., ICRC 2013 Proceedings.
* [6] F. de Palma, T. J. Brandt, G. Johannesson, and L. Tibaldo, for the Fermi LAT collaboration, Proceedings of the $4^{th}$ Fermi Symposium (2013).
* [7] A. A. Abdo, et. al., The Astrophysical Journal 734 (2011) 28.
* [8] A. A. Abdo, et. al., The Astrophysical Journal 712 (2010) 459-468.
* [9] T. J. Brandt, et. al., Advances in Space Research 51 (2013) 247-252.
* [10] D. Castro, P. Slane, A. Carlton, and E. Figueroa-Feliciano, sub. ApJ (2013)
* [11] L. O. Drury, F. A. Aharonian, and H. J. Voelk, Astronomy & Astrophysics 287 (1994) 959-971.
* [12] C. D. Dermer and G. Powale, Astronomy & Astrophysics 553 (2013) A34.
* [13] C. D. Dermer, J. D. Finke, R. J. Murphy, et. al., Proceedings of the $4^{th}$ Fermi Symposium (2013).
* [14] C. D. Dermer, A. W. Strong, E. Orlando, and L. Tibaldo, for the Fermi LAT collaboration, ICRC 2013 Proceedings.
* [15] G. Ferrand and S. Safi-Harb, Advances in Space Research 49 (2012) 1313-1319.
|
arxiv-papers
| 2013-07-24T20:18:46 |
2024-09-04T02:49:48.411783
|
{
"license": "Public Domain",
"authors": "T. J. Brandt, F. Acero, F. de Palma, J. W. Hewitt, M. Renaud (for the\n Fermi LAT Collaboration)",
"submitter": "Theresa Brandt",
"url": "https://arxiv.org/abs/1307.6571"
}
|
1307.6572
|
# The 1st Fermi LAT SNR Catalog: the Impact of Interstellar Emission Modeling
###### Abstract
Galactic interstellar emission contributes substantially to Fermi LAT
observations in the Galactic plane, the location of the majority of supernova
remnants (SNRs). To explore some systematic effects on SNRs’ properties caused
by interstellar emission modeling, we have developed a method comparing the
official LAT interstellar emission model results to eight alternative models.
We created the eight alternative Galactic interstellar models by varying a few
input parameters to GALPROP, namely the height of the cosmic ray propagation
halo, cosmic ray source distribution in the Galaxy, and atomic hydrogen spin
temperature. We have analyzed eight representative SNRs chosen to encompass a
range of Galactic locations, extensions, and spectral properties using the
eight different interstellar emission models. We will present the results and
method in detail and discuss the implications for studies such as the 1st
Fermi LAT SNR Catalog.
## 1 Introduction
Galactic interstellar $\gamma$-ray emission is produced through interactions
of high-energy cosmic ray (CR) hadrons and leptons with interstellar gas via
nucleon-nucleon inelastic collisions and electron Bremsstrahlung, and with
low-energy radiation fields, via inverse Compton (IC) scattering. Such
interstellar emission accounts for more than $60\%$ of the photons detected by
the Fermi Large Area Telescope (LAT) and is particularly bright toward the
Galactic disk.
In this paper, we present our ongoing effort to explore the systematic
uncertainties due to the modeling of Galactic interstellar emission in the
analysis of Fermi LAT sources, with particular emphasis on its application to
the $1^{st}$ Fermi LAT Supernova Remnant (SNR) Catalog. We compare the results
of analyzing sources with eight alternative interstellar emission models
(IEMs), described in Section 2, to the source parameters obtained with the
standard model in Section 3. In Section 4 we discuss the future application of
this method to the SNR Catalog.
## 2 Interstellar Emission Models
To estimate the systematic uncertainty inherent in the choice of standard
interstellar emission model (IEM) in analyzing a source, we have developed
eight alternative IEMs. By comparing the results of the source analysis using
these eight alternative models to the standard model, we can approximate the
systematic uncertainty therefrom.
For the standard model, we assume that the Galactic interstellar $\gamma$-ray
intensities can be modeled as a linear combination of gas column
densities111Gas column densities are determined from emission lines of atomic
hydrogen ($\mathrm{H\,\scriptstyle{I}}$, extracted from the radio data using a
uniform value for the spin temperature ($200$ K)) and CO, a surrogate tracer
of molecular hydrogen, and from dust optical depth maps used to account for
gas not traced by the lines. and an inverse Compton (IC) intensity map as a
function of energy. For further details on the construction of the standard
Fermi LAT IEM, see [1].
We generated the eight alternative IEMs to probe key sources of systematic
uncertainties by:
* •
adopting a different model building strategy from the standard IEM, resulting
in different gas emissivities, or equivalently CO-to-H2 and dust-to-gas
ratios, and including a different approach for dealing with the remaining
extended residuals;
* •
varying a few important input parameters for building the alternative IEMs:
atomic hydrogen spin temperature ($150$ K and optically thin), CR source
distribution (SNRs and pulsars), and CR propagation halo heights (4 kpc and 10
kpc);
* •
and allowing more freedom in the fit by separately scaling the inverse Compton
emission and $\mathrm{H\,\scriptstyle{I}}$ and CO emission in 4 Galactocentric
rings.
The work in [2], using the GALPROP CR propagation and interaction code222The
GALPROP code has been developed over several years, starting with, e.g. [3]
and [4]., was used as a starting point for our model building strategy. The
GALPROP output intensity maps associated with $\mathrm{H\,\scriptstyle{I}}$,
CO, the gas column densities, determined from and IC are then fit
simultaneously with an isotropic component and 2FGL sources to 2 years of
Fermi LAT data in order to minimize bias in the a priori assumptions on the CR
injection spectra and the proton CR source distribution. The intensity maps
associated with gas were binned into four Galactocentric annuli ($0-4$ kpc,
$4-8$ kpc, $8-10$ kpc and $10-30$ kpc). The spectra of all intensity maps were
individually fit with log parabolas to the data, to allow for possible CR
spectral variations between the annuli for all $\mathrm{H\,\scriptstyle{I}}$
and CO maps while the IC fit accounts for spectral variations in the electron
distribution. We also included in the fit an isotropic template and templates
for Loop I [5] and the Fermi bubbles [6]. The template for Loop I is based on
the geometrical model of [7] while the bubbles are assumed to be uniform with
edges defined in spherical coordinates by $R=R_{0}|\sin\theta|$, where
$\theta$ is the polar angle.
Ackermann, et. al. [2] explored some systematic uncertainties by varying input
parameters. The $\mathrm{H\,\scriptstyle{I}}$ spin temperature, CR source
distribution, and CR propagation halo height were found to be among those
parameters which have the largest impact on the $\gamma$-ray intensity. The
values adopted in this study to generate the eight alternative IEMs were
chosen to be reasonably extreme; we note that they do not reflect the full
uncertainty in the input parameters. Separately scaling the
$\mathrm{H\,\scriptstyle{I}}$ and CO emission in rings and the IC emission
permits the alternative IEMs to better adapt to local structure when analyzing
particular source regions. Figure 1 shows the relative difference between the
standard model and one of the alternative models (Lorimer CR source
distribution with a $4$ kpc halo height, and $150$ K
$\mathrm{H\,\scriptstyle{I}}$ spin temperature). Differences are particularly
large along the Galactic plane, where SNRs are located.
Finally, we note that this strategy for estimating systematic uncertainty from
interstellar emission modeling does not represent the complete range of
systematics involved. In particular, we have tested only one alternative
method for building the IEM, and the input parameters do not encompass their
full uncertainties. Further, as the alternative method differs from that used
to create the standard IEM, the resulting uncertainties will not bracket the
results using the standard model. The estimated uncertainty does not contain
other possibly important sources of systematic error, including uncertainties
in the ISRF model, simplifications to Galaxy’s geometry, small scale non-
uniformities in the CO-to-H2 and dust-to-gas ratios and
$\mathrm{H\,\scriptstyle{I}}$ spin temperature non-uniformities, and
underlying uncertainties in the input gas and dust maps. While the resulting
uncertainty should be considered a limited estimate of the systematic
uncertainty due to interstellar emission modeling, rather than a full
determination, it is critical for interpreting the data, and this work
represents our most complete and systematic effort to date.
Figure 1: The position of the eight candidate SNRs used in this analysis are
overlaid on a map of relative difference between the standard IEM and one of
the alternative models. The alternative model selected for this image has a
Lorimer source distribution, a halo height of $4$ kpc and a spin temperature
of $150$ K. We plot the difference between the models’ predicted counts
divided by the square of the sum of the predicted counts so the map is in
units of sigma. The hardness of the SNRs’ spectra is in two categories: hard
(purple) and soft (black). SNRs detected as extended with Fermi are shown as
circles while point-like are shown as crosses.
## 3 ESTIMATING IEM SYSTEMATICS
### 3.1 Analysis Method
We developed this method for estimating the systematics from the interstellar
emission model using eight candidate SNRs chosen to represent the range of
spectral and spatial SNR characteristics in high and low IEM intensity
regions. Figure 1 shows the candidate SNRs’ location on the sky, illustrating
their range of Galactic longitude. The color indicates those candidates with a
hard or soft index and the shape of the extension (pointlike or extended). The
SNR candidates are overlaid on a map of the relative difference between the
standard IEM and one of the alternative IEMs described in Section 2.
We use the same analysis strategy to obtain all SNR candidates’ Fermi LAT
parameter values with both the standard and all eight alternative IEMs on $3$
years of P7_V6SOURCE data [8] in the energy range $1-100$ GeV. We applied the
standard binned likelihood method333The standard Fermi LAT analysis
description and tools can be found here:
http://fermi.gsfc.nasa.gov/ssc/data/analysis/ ., treating sources as follows.
For each of the eight candidate SNRs, an extended source initially of the
radio size and with a power law (PL) spectral model either replaces the
closest non-pulsar 2FGL source [9] within the radio size or is positioned as a
new source at the location determined from radio observations [10]. All other
2FGL sources within the radio size which are not pulsars are removed from the
source model. We fit the centroid and extension of the SNR candidate disk as
well as the normalization and PL index for the source of interest and the five
closest background sources within $5^{\circ}$ with a significance of $\gtrsim
4\,\sigma$ in order to balance the number of degrees of freedom with
convergence and computation time requirements.
(a) Flux for the eight candidate SNRs’ from $1-100$ GeV.
(b) Index for the eight SNR candidates from $1-100$ GeV.
Figure 2: Results for each candidate SNR, averaging over the eight alternative
IEMs separately for split (red) and summed (green) component models compared
to the standard model solution (black). The error bars for results using the
alternative IEMs show the maximal range of the values given by the $1\,\sigma$
statistical errors.
To generate results for the source of interest with each diffuse model, we fit
the sources’ model to the data with either the standard model or one of the
eight alternative IEMs. In the case of the standard Fermi LAT IEM, we allow
the normalization to vary and fix the accompanying standard isotropic model’s
normalization. For each of the eight alternative models, we use the
corresponding isotropic model fixed to its value resulting from the fit to the
all-sky data (see Section 2). To better understand the effect of allowing
freedom in the $\mathrm{H\,\scriptstyle{I}}$ and CO rings, we fit the
alternative models in two ways: either with the rings’ normalizations free
(“split” models) or with the rings summed together, as given by the all-sky
fit (see Section 2), and only the total normalization free (“summed” models).
The summed alternative IEMs are thus closer to the standard IEM. For the split
alternative IEMs, not all rings are crossed by all lines of sight. We thus fit
only the two innermost $\mathrm{H\,\scriptstyle{I}}$ and CO rings crossed by
the line of sight to our region of interest. The IC template is also free to
vary while the isotropic component remains fixed.
### 3.2 Results for SNRs’ IEM Systematics
We compare the results obtained using the eight alternative IEMs with the
standard model results by averaging each parameter’s eight values from the
alternative IEMs. Figure 2 shows the values for the flux and index from
fitting the data with the alternative IEMs with the rings either split or
summed. These are then plotted along with the standard model results for all
eight SNR candidates studied. We conservatively represent the allowed
parameter range with error bars showing the maximal range for the alternative
IEMs $1\,\sigma$ statistical errors.
Figure 2 shows that the variation in value of the best fit parameters obtained
with the alternative IEMs is larger than the $1\,\sigma$ statistical
uncertainty. The impact of changing the IEM on the source’s parameters depends
strongly on the source’s properties and location. As expected, the parameter
values for the source of interest are generally closer to the standard model
results for the alternative IEMs with components summed rather than split. In
many cases, the allowed parameter range represented by the $1\,\sigma$
statistical errors for each of the alternative IEMs is larger with the
components split than summed. Also as noted earlier, the alternative IEM
results do not as a rule bracket the standard model solution. We observe that
some of the largest differences between the standard and alternative IEM
results for a single source are frequently associated with sources coincident
with templates accounting for remaining residual emission in the standard IEM
(Section 2).
SNR G347.3-0.5 proves to be an interesting source for understanding the impact
nearby source(s) can have on this type of analysis. In particular, our
automated analysis finds a softer index and a much larger flux for SNR
G347.3-0.5 than that obtained in a dedicated analysis [11]. Since the best fit
radius ($0.8^{\circ}$) is larger than that the X-ray data indicates
($0.55^{\circ}$), the automated analysis’s disk encompasses nearby sources
that are only used in the [11] model. Including this additional emission also
affects the spectrum, making it softer in this case than that found in the
dedicated analysis. Given Fermi LAT’s both increasing point spread function
and number of sources with decreasing energy as well as the predominance of
diffuse emission at lower energies, we note that nearby sources may play a
greater role if extending this method below the $1$ GeV minimum energy
examined here.
### 3.3 IEM Input Parameter Comparison
To identify which, if any, of the three IEM input parameters
($\mathrm{H\,\scriptstyle{I}}$ spin temperature, CR source distribution, and
CR propagation halo height) has the largest impact on the fitted source
parameters, we marginalize over the other parameters and examine the relative
ratio of the averaged input parameter values to the values’ dispersion. For a
fitted source parameter $a$, such as flux and a GALPROP input parameter set
$P=\\{i,j\\}$, e.g. spin temperature $Ts\leavevmode\nobreak\
=\leavevmode\nobreak\ \\{150$ K$,10^{5}$ K$\\}$, this becomes:
$\frac{|<a_{i}>-<a_{j}>|}{max(\sigma_{a,i},\sigma_{a,j})}$ (1)
where $\sigma_{a}$ is the rms of the parameter $a$ for a given input parameter
value $P$. A ratio $\geq 1$ implies that changing the selected input parameter
has a greater effect on the flux than all combinations of the other input
parameters.
Figure 3: The impact on the candidate SNRs’ flux of each of the alternative
IEM input parameters, marginalized over the other GALPROP input parameters, is
shown relative to the figure of merit for the other input parameters (source
distribution, halo height, and spin temperature). We calculate the figure of
merit (Eq 1) separately for the alternate IEM components fit separately (left)
and summed (right). The large open cross represents the average figure of
merit over all SNR candidates. As no alternative IEM input parameter has a
figure of merit significantly larger than $1$, no input parameter dominates
the fitted source parameter sufficiently to justify neglecting the others.
In Figure 3 we plot this ratio for each of the alternative IEM’s input
parameters for each of the eight SNR candidates, along with the average over
the SNR candidates, separately for the split and summed components. While the
spin temperature has the largest effect for the split alternative IEMs, the CR
source distribution also becomes relevant with the summed alternative models.
In light of this and as none of the parameters shows a ratio significantly
greater than $1$ for all the sources tested, we conclude that none of the
input parameters has a sufficiently large impact on the fitted source
parameter to justify neglecting the others.
## 4 FUTURE APPLICATIONS
In this work we explored the effect of using alternative interstellar emission
models on the analysis of LAT sources. As the Galactic interstellar emission
contributes substantially to Fermi LAT observations in the Galactic plane, the
choice of IEM can have a significant impact on the parameters determined for a
given source of interest, as demonstrated with eight SNR candidates. To
estimate the reported error we currently use only the most conservative
extreme variation of the source of interest’s output parameters. We are
finalizing our definition of the systematic error using this method, including
through comparison of the present estimate with previous methods’ estimates,
typically found by varying the standard IEM’s normalization by a fraction
estimated from neighboring regions. Although our current method represents the
uncertainty due to a limited range of IEMs, it plays a critical role in
interpreting the data and represents the most complete and systematic attempt
at quantifying the systematic error due to the choice of IEM to date.
As the majority of SNRs lie in the Galactic plane, coincident with the
majority of the Galactic interstellar emission, this method is particularly
pertinent to analyses such as that underway for the $1^{st}$ Fermi LAT SNR
Catalog. Figure 2 shows that the flux and index can vary greatly for our eight
representative SNR candidates, depending on the source and local background’s
specific characteristics. Given these differences, we plan to use this method
to estimate the systematic uncertainty associated with the choice of IEM on
the full set of SNR candidates in the catalog. Such error estimates will allow
us to, among other things, more accurately determine underlying source
characteristics such as the inferred composition (leptonic or hadronic) and
particle spectrum.
Other classes of objects such as pulsar wind nebulae and binary star systems
also lie primarily in the plane and are likely to be strongly affected by the
choice of IEM. We are thus generalizing this method in order to be able to
apply it to the study of Galactic plane sources generally. Another possible
extension to this method is extending it to energies $<1$ GeV, where the
interplay between the Galactic interstellar emission model and background
sources must be carefully examined. By more faithfully accounting for the
systematic uncertainty of our model components we will be better equipped to
draw less biased conclusions from our data.
Acknowledgment: The Fermi LAT Collaboration acknowledges generous ongoing
support from a number of agencies and institutes that have supported both the
development and the operation of the LAT as well as scientific data analysis.
These include the National Aeronautics and Space Administration and the
Department of Energy in the United States, the Commissariat à l’Energie
Atomique and the Centre National de la Recherche Scientifique / Institut
National de Physique Nucléaire et de Physique des Particules in France, the
Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in
Italy, the Ministry of Education, Culture, Sports, Science and Technology
(MEXT), High Energy Accelerator Research Organization (KEK) and Japan
Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg
Foundation, the Swedish Research Council and the Swedish National Space Board
in Sweden.
Additional support for science analysis during the operations phase is
gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy
and the Centre National d’Études Spatiales in France.
## References
* [1] F. de Palma, T. J. Brandt, G. Johannesson, L. Tibaldo, for the Fermi LAT collaboration, $4^{th}$ Fermi Symp. Proc. (2013).
* [2] M. Ackermann, et. al., ApJ 750 (2012) 3.
* [3] A. W. Strong, I. V. Moskalenko, ApJ 493 (1998) 694.
* [4] A. W. Strong, I. V. Moskalenko, ApJ 509 (1998) 212-228.
* [5] J.-M. Casandjian and I. Grenier for the Fermi Large Area Telescope Collaboration, (2009) arXiv:0912.3478.
* [6] M. Su, T. R. Slatyer, and D. P. Finkbeiner ApJ 724 (2010) 1044-1082.
* [7] M. Wolleben, ApJ 664 (2007) 349-356.
* [8] M. Ackermann, et. al., ApJS 203 (2012) 31.
* [9] P. L. Nolan, et. al., ApJ 199 (2012) 31.
* [10] D. A. Green, ApJ BASI (2009) 45-61.
* [11] A. A. Abdo, et. al., ApJ 734 (2011) 28.
|
arxiv-papers
| 2013-07-24T20:19:14 |
2024-09-04T02:49:48.419072
|
{
"license": "Public Domain",
"authors": "T. J. Brandt, J. Ballet, F. de Palma, G. Johannesson, L. Tibaldo (for\n the Fermi-LAT Collaboration)",
"submitter": "Theresa Brandt",
"url": "https://arxiv.org/abs/1307.6572"
}
|
1307.6616
|
# Does generalization performance of $l^{q}$ regularization learning depend on
$q$? A negative example ††thanks: The research was supported by the National
973 Programming (2013CB329404), the Key Program of National Natural Science
Foundation of China (Grant No. 11131006), and the National Natural Science
Foundations of China (Grants No. 61075054).
Shaobo Lin1 Chen Xu2 Jinshan Zeng1 Jian Fang1
###### Abstract
$l^{q}$-regularization has been demonstrated to be an attractive technique in
machine learning and statistical modeling. It attempts to improve the
generalization (prediction) capability of a machine (model) through
appropriately shrinking its coefficients. The shape of a $l^{q}$ estimator
differs in varying choices of the regularization order $q$. In particular,
$l^{1}$ leads to the LASSO estimate, while $l^{2}$ corresponds to the smooth
ridge regression. This makes the order $q$ a potential tuning parameter in
applications. To facilitate the use of $l^{q}$-regularization, we intend to
seek for a modeling strategy where an elaborative selection on $q$ is
avoidable. In this spirit, we place our investigation within a general
framework of $l^{q}$-regularized kernel learning under a sample dependent
hypothesis space (SDHS). For a designated class of kernel functions, we show
that all $l^{q}$ estimators for $0<q<\infty$ attain similar generalization
error bounds. These estimated bounds are almost optimal in the sense that up
to a logarithmic factor, the upper and lower bounds are asymptotically
identical. This finding tentatively reveals that, in some modeling contexts,
the choice of $q$ might not have a strong impact in terms of the
generalization capability. From this perspective, $q$ can be arbitrarily
specified, or specified merely by other no generalization criteria like
smoothness, computational complexity, sparsity, etc..
Keywords: Learning theory, $l^{q}$ regularization learning, sample dependent
hypothesis space, learning rate
MSC 2000: 68T05, 62G07.
1\. Institute for Information and System Sciences, School of Mathematics and
Statistics, Xi’an Jiaotong University Xi’an 710049, P R China
2\. The Methodology Center, The Pennsylvania State University, Department of
Statistics, 204 E. Calder Way, Suite 400, State College, PA 16801, USA
## 1 Introduction
Contemporary scientific investigations frequently encounter a common issue of
exploring the relationship between a response and a number of covariates. In
machine learning research, the subject is typically addressed through learning
a underling rule from the data that accurately predicates future values of the
response. For instance, in banking industry, financial analysts are interested
in building a system that helps to judge the risk of a loan request. Such a
system is often trained based on the risk assessments from previous loan
applications together with the empirical experiences. An incoming loan request
is then viewed as a new input, upon which the corresponding potential risk
(response) is to be predicted. In such applications, the predictive accuracy
of a trained rule is of the key importance.
In the past decade, various strategies have been developed to improve the
prediction (generalization) capability of a learning process, which include
$l^{q}$ regularization as an well-known example [33]. The $l^{q}$
regularization learning prevents over-fitting by shrinking the model
coefficients and thereby attains a higher predictive value. To be specific,
suppose that the data ${\bf z}=\\{x_{i},y_{i}\\}$ for $i=1,\ldots,m$ are
collected independently and identically according to an unknown but definite
distribution, where $y_{i}$ is a response of $i$th unit and $x_{i}$ is the
corresponding $d$-dimensional covariates. Let
$\mathcal{H}_{K,{\bf
z}}:=\left\\{\sum_{i=1}^{m}a_{i}K_{x_{i}}:a_{i}\in\mathbf{R}\right\\}$
be a sample dependent space (SDHS) with $K_{t}(\cdot)=K(\cdot,t)$ and
$K(\cdot,\cdot)$ being a positive definite kernel function. The coefficient-
based $l^{q}$ regularization strategy ($l^{q}$ regularizer) takes the form of
$f_{{\bf z},\lambda,q}=\arg\min_{f\in\mathcal{H}_{K,{\bf
z}}}\left\\{\frac{1}{m}\sum_{i=1}^{m}(f(x_{i})-y_{i})^{2}+\lambda\Omega^{q}_{\bf
z}(f)\right\\},$ (1)
where $\lambda=\lambda(m,q)>0$ is a regularization parameter and $\Omega_{\bf
z}^{q}(f)$ $(0<q<\infty)$ is defined by
$\Omega^{q}_{\bf z}(f)=\sum_{i=1}^{m}|a_{i}|^{q}\ \mbox{when}\
f=\sum_{i=1}^{m}a_{i}K_{x_{i}}\in\mathcal{H}_{K,{\bf z}}.$
With different choices of order $q$, (1) leads to various specific forms of
the $l_{q}$ regularizer. In particular, when $q=2$, $f_{{\bf z},\lambda,q}$
corresponds to the ridge regressor [23], which smoothly shrinks the
coefficients toward zero. When $q=1$, $f_{{\bf z},\lambda,q}$ leads to the
LASSO [29], which set small coefficients exactly at zero and thereby also
serves as a variable selection operator. When $0<q<1$, $f_{{\bf z},\lambda,q}$
coincides with the bridge estimator [8], which tends to produce highly sparse
estimates through a non-continuous shrinkage.
The varying forms and properties of $f_{{\bf z},\lambda,q}$ make the choice of
order $q$ crucial in applications. Apparently, an optimal $q$ may depend on
many factors such as the learning algorithms, the purposes of studies and so
forth. These factors make a simple answer to this question infeasible in
general. To facilitate the use of $l^{q}$-regularization, alteratively, we
intend to seek for a modeling strategy where an elaborative selection on $q$
is avoidable. Specifically, we attempt to reveal some insights for the role of
$q$ in $l^{q}$-learning via answering the following question:
Problem 1. Are there any kernels such that the generalization capability of
(1) is independent of $q$?
In this paper, we provides a positive answer to Problem 1 under the framework
of statistical learning theory. Specifically, we provide a featured class of
positive definite kernels, under which the $l_{q}$ estimators for $0<q<\infty$
attain similar generalization error bounds. We then show that these estimated
bounds are almost essential in the sense that up to a logarithmic factor the
upper and lower bounds are asymptotically identical. In the proposed modeling
context, the choice of $q$ does not have a strong impact in terms of the
generalization capability. From this perspective, $q$ can be arbitrarily
specified, or specified merely by other no generalization criteria like
smoothness, computational complexity, sparsity, etc..
The reminder of the paper is organized as follows. In Section 2, we provide a
literature review and explain our motivation of the research. In Section 3, we
present some preliminaries including spherical harmonics, Gegenbauer
polynomials and so on. In Section 4, we introduce a class of well-localized
needlet type kernels of Petrushev and Xu [22] and show some crucial properties
of them which will play important roles in our analysis. In Section 5, we then
study the generalization capabilities of $l^{q}$-regularizer associated with
the constructed kernels for different $q$. In Section 6, we provide the proof
of the main results. We conclude the paper with some useful remarks in the
last section.
## 2 Motivation and related work
### 2.1 Motivation
In practice, the choice of $q$ in (1) is critical, since it embodies certain
potential attributions of the anticipated solutions such as sparsity,
smoothness, computational complexity, memory requirement and generalization
capability of course. The following simple simulation illustrates that
different choice of $q$ can lead to different sparsity of the solutions.
The samples are identically and independently drawn according to the uniform
distribution from the two dimensional Sinc function pulsing a Gaussian noise
$N(0,\delta^{2})$ with $\delta^{2}=0.1$. There are totally 256 training
samples and 256 test samples. In Fig. 1, we show that different choice of $q$
may deduce different sparsity of the estimator for the kernel
$K_{0.1}(x):=\exp\left\\{-\|x-y\|^{2}/0.1\right\\}$. It can be found that
$l^{q}$ $(0<q\leq 1)$ regularizers can deduce sparse estimator, while it
impossible for $l^{2}$ regularizer.
Figure 1: Sparsity for $l^{q}$ learning schemes
Therefore, for a given learning task, how to choose $q$ is an important and
crucial problem for $l^{q}$ regularization learning. In other words, which
standards should be adopted to measure the quality of $l^{q}$ regularizers
deserves study. As the most important standard of statistical learning theory,
the generalization capability of $l^{q}$ regularization scheme (1) may depend
on the choice of kernel, the size of samples $m$, the regularization parameter
$\lambda$, the behavior of priors, and, of course, the choice of $q$. If we
take the generalization capability of $l^{q}$ regularization learning as a
function of $q$, we then automatically wonder how this function behaves when
$q$ changes for a fixed kernel. If the generalization capabilities depends
heavily on $q$, then it is natural to choose the $q$ such that the
generalization capability of the corresponding $l^{q}$ regularizer is the
smallest. If the generalization capabilities is independent of $q$, then $q$
can be arbitrarily specified, or specified merely by other no generalization
criteria like smoothness, computational complexity, sparsity.
However, the relation between the generalization capability and $q$ depends
heavily on the kernel selection. To show this, we compare the generalization
capabilities of $l^{2}$, $l^{1}$, $l^{1/2}$ and $l^{2/3}$ regularization
schemes for two kernels: $\exp\left\\{-\|x-y\|^{2}/0.1\right\\}$ and
$\exp\left\\{-\|x-y\|/10\right\\}$ in the simulation. The one case shows that
the generalization capabilities of $l^{q}$ regularization schemes may be
independent of $q$ and the other case shows that the generalization capability
of (1) depends heavily on $q$. In the left of Fig. 2, we report the relation
between the test error and regularization parameter for the kernel
$\exp\left\\{-\|x-y\|^{2}/0.1\right\\}$. It is shown that when the
regularization parameters are appropriately tuned, all of the aforementioned
regularization schemes may possess the similar generalization capabilities. In
the right of Fig. 2, for the kernel $\exp\left\\{-\|x-y\|/10\right\\}$, we see
that the generalization capability of $l^{q}$ regularization depends heavily
on the choice of $q$.
Figure 2: Comparisons of test error for $l^{q}$ regularization schemes with
different $q$.
From these simulations, we see that finding kernels such that the
generalization capability of (1) is independent of $q$ is of special
importance in theoretical and practical applications. In particular, if such
kernels exist, with such kernels, $q$ can be solely chosen on the basis of
algorithmic and practical considerations for $l^{q}$ regularization. Here we
emphasize that all these conclusions can, of course only be made in the
premise that the obtained generalization capabilities of all $l^{q}$
regularizers are (almost) optimal.
### 2.2 related work
There have been several papers that focus on the generalization capability
analysis of the $l^{q}$ regularization scheme (1). Wu and Zhou [33] were the
first, to the best of our knowledge, to show a mathematical foundation of
learning algorithms in SDHS. They claimed that the data dependent nature of
the algorithm leads to an extra error term called hypothesis error, which is
essentially different form regularization schemes with sample independent
hypothesis spaces (SIHSs). Based on this, the authors proposed a coefficient-
based regularization strategy and conducted a theoretical analysis of the
strategy by dividing the generalization error into approximation error, sample
error and hypothesis error. Following their work, Xiao and Zhou [34] derived a
learning rate of $l^{1}$ regularizer via bounding the regularization error,
sample error and hypothesis error, respectively. Their result was improved in
[24] by adopting a concentration inequality technique with $l^{2}$ empirical
covering numbers to tackle the sample error. On the other hand, for $l^{q}$
$(1\leq q\leq 2)$ regularizers, Tong et al. [30] deduced an upper bound for
generalization error by using a different method to cope with the hypothesis
error. Later, the learning rate of [30] was improved further in [11] by giving
a sharper estimation of the sample error.
In all those researches, some sharp restrictions on the probability
distributions (priors) have been imposed, say, both spectrum assumption of the
regression function and concentration property of the marginal distribution
should be satisfied. Noting this, for $l^{2}$ regularizer, Sun and Wu [28]
conducted a generalization capability analysis for $l^{2}$ regularizer by
using the spectrum assumption to the regression function only. For $l^{1}$
regularizer, by using a sophisticated functional analysis method, Zhang et al.
[36] and Song et al. [25] built the regularized least square algorithm on the
reproducing kernel Banach space (RKBS), and they proved that the regularized
least square algorithm in RKBS is equivalent to $l^{1}$ regularizer if the
kernel satisfies some restricted conditions. Following this method, Song and
Zhang [26] deduced a similar learning rate for the $l^{1}$ regularizer and
eliminated the concentration property assumption on the marginal distribution
.
Limiting $q$ within $[1,2]$ is certainly incomplete to judge whether the
generalization capability of $l^{q}$ regularization depends on the choice of
$q$. Moreover, in the context of learning theory, to intrinsically
characterize the generalization capability of a learning strategy, the
essential generalization bound [10] rather than the upper bound is required,
that is, we must deduce a lower and an upper bound simultaneously for the
learning strategy and prove that the upper and lower bounds can be
asymptotically identical. We notice, however, that most of the previously
known estimations on generalization capability of learning schemes (1) are
only concerned with the upper bound estimation. Thus, their results can not
serve the answer to Problem 1. Different from the pervious work, the essential
bound estimation of generalization error for $l^{q}$ regularization schemes
(1) with $0<q<\infty$ will be presented in the present paper. As a
consequence, we provide an affirmative answer to Problem 1.
## 3 Preliminaries
In this section, we introduce some preliminaries on spherical harmonics,
Gegenbauer polynomial and orthonormal basis construction., which will be used
in the construction of the positive definite needlet kernel.
### 3.1 Gegenbauer polynomial
The Gegenbauer polynomials are defined by the generating function [31]
$(1-2tz+z^{2})^{-\mu}=\sum_{n=0}^{\infty}G_{n}^{\mu}(t)z^{n},$
where $|z|<1,|t|\leq 1,$ and $\mu>0$. The coefficients $G_{n}^{\mu}(t)$ are
algebraic polynomials of degree $n$ which are called the Gegenbauer
polynomials associated with $\mu$. It is known that the family of polynomials
$\\{G_{n}^{\mu}\\}_{n=0}^{\infty}$ is a complete orthogonal system in the
weighted space $L^{2}(I,w)$, $I:=[-1,1]$,
$w_{\mu}(t):=(1-t^{2})^{\mu-\frac{1}{2}}$ and there holds
$\int_{I}G_{m}^{\mu}(t)G_{n}^{\mu}(t)w_{\mu}(t)dt=\left\\{\begin{array}[]{cc}0,&m\neq
n\\\ h_{n,\mu},&m=n\end{array}\right.\ \mbox{with}\
h_{n,\mu}=\frac{\pi^{1/2}(2\mu)_{n}\Gamma(\mu+\frac{1}{2})}{(n+\mu)n!\Gamma(\mu)},$
where
$(a)_{0}:=0,(a)_{n}:=a(a+1)\dots(a+n-1)=\frac{\Gamma(a+n)}{\Gamma(a)}.$
Define
$U_{n}:=(h_{n,d/2})^{-1/2}G_{n}^{d/2},\ n=0,1,\dots.$ (2)
Then it is easy to see that $\\{U_{n}\\}_{n=0}^{\infty}$ is a complete
orthonormal system for the weighted $L^{2}$ space $L^{2}(I,w)$, where
$w(t):=(1-t^{2})^{\frac{d-1}{2}}$. Let $\mathbf{B}^{d}$ be the unit ball in
$\mathbf{R}^{d}$, $\mathbf{S}^{d-1}$ be the unit sphere in $\mathbf{R}^{d}$
and $\mathcal{P}_{n}$ be the set of algebraic polynomials of degree not larger
than $n$ defined on $\mathbf{B}^{d}$. Denote by $d\omega_{d-1}$ the aero
element of $\mathbf{S}^{d-1}$. Then
$\Omega_{d-1}:=\int_{\mathbf{S}^{d-1}}d\omega_{d-1}=\frac{2\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2})}.$
The following important properties of $U_{n}$ are established in [21].
###### Lemma 1.
Let $U_{n}$ be defined as above. Then for each $\xi,\eta\in\mathbf{S}^{d-1}$
we have
$\int_{\mathbf{B}^{d}}U_{n}(\xi\cdot x)P(x)dx=0\ \mbox{for}\
P\in\mathcal{P}_{n-1},$ (3) $\int_{\mathbf{B}^{d}}U_{n}(\xi\cdot
x)U_{n}(\eta\cdot x)dx=\frac{U_{n}(\xi\cdot\eta)}{U_{n}(1)},$ (4)
$K_{n}^{*}+K_{n-2}^{*}+\dots+K_{\varepsilon_{n}}^{*}=\frac{v_{n}^{2}}{U_{n}(1)}U_{n},$
(5)
and
$\int_{\mathbf{S}^{d-1}}U_{n}(\xi\cdot
x)U_{n}(\xi\cdot\eta)d\omega_{d-1}(\xi)=\frac{U_{n}(1)}{v_{n}^{2}}U_{n}(\eta\cdot
x),$ (6)
where $v_{n}:=\left(\frac{(n+1)_{d-1}}{2(2\pi)^{d-1}}\right)^{\frac{1}{2}}$,
and
$K_{n}^{*}:=\frac{2k+d-2}{(d-2)\Omega_{d-1}}G_{k}^{\frac{d-2}{2}}(\xi\cdot\eta)$.
### 3.2 Spherical harmonics
For any integer $k\geq 0$, the restriction to $\mathbf{S}^{d-1}$ of a
homogeneous harmonic polynomial with degree $k$ is called a spherical harmonic
of degree $k$. The class of all spherical harmonics with degree $k$ is denoted
by $\mathbf{H}^{d-1}_{k}$, and the class of all spherical polynomials with
total degrees $k\leq n$ is denoted by $\Pi_{n}^{d-1}$. It is obvious that
$\Pi_{n}^{d-1}=\bigoplus_{k=0}^{n}\mathbf{H}^{d-1}_{k}$. The dimension of
$\mathbf{H}^{d-1}_{k}$ is given by
$D_{k}^{d-1}:=\mbox{dim
}\mathbf{H}^{d-1}_{k}:=\left\\{\begin{array}[]{ll}\frac{2k+d-2}{k+d-2}{{k+d-2}\choose{k}},&k\geq
1;\\\ 1,&k=0,\end{array}\right.$
and that of $\Pi_{n}^{d-1}$ is $\sum_{k=0}^{n}D^{d-1}_{k}=D_{n}^{d}\sim
n^{d-1},$ where $A\sim B$ denotes that there exist absolute constants $C_{1}$
and $C_{2}$ such that $C_{1}A\leq B\leq C_{2}A$.
The well known addition formula is given by (see [20] and [31])
$\sum_{l=1}^{D_{k}^{d-1}}Y_{k,l}(\xi)Y_{k,l}(\eta)=\frac{2k+d-2}{(d-2)\Omega_{d-1}}G_{k}^{\frac{d-2}{2}}(\xi\cdot\eta)=K_{n}^{*}(\xi\cdot\eta),$
(7)
where $\\{Y_{k,l}:l=1,\dots,D_{k}^{d-1}\\}$ is arbitrary orthonormal basis of
$\mathbf{H}_{k}^{d-1}$.
For $r>0$ and $a\geq 1$, we say that a finite subset
$\Lambda\subset\mathbf{S}^{d-1}$ is an $(r,a)$-covering of $\mathbf{S}^{d-1}$
if
$\mathbf{S}^{d-1}\subset\bigcup_{\xi\in\Lambda}D(\xi,r)\ \ \mbox{and}\ \
\max_{\xi\in\Lambda}\left|\Lambda\bigcap D(\xi,r)\right|\leq a,$
where $|A|$ denotes the cardinality of the set $A$ and
$D(\xi,r)\subset\mathbf{S}^{d-1}$ denotes the spherical cap with the center
$\xi$ and the angle $r$. The following positive cubature formula can be found
in [2].
###### Lemma 2.
There exists a constant $\gamma>0$ depending only on $d$ such that for any
positive integer $n$ and any $(\delta/n,a)$-covering of $\mathbf{S}^{d-1}$
satisfying $0<\delta<a^{-1}\gamma$. There exists a set of numbers
$\\{\eta_{\xi}\\}_{\xi\in\Lambda}$ such that
$\int_{\mathbf{S}^{d-1}}Q(\zeta)d\omega_{d-1}(\zeta)=\sum_{\xi\in\Lambda}\eta_{\zeta}Q(\zeta)\
\ \mbox{for}\ \mbox{any \ \ }Q\in\Pi_{4n}^{d-1}.$
### 3.3 Basis and reproducing kernel for $\mathcal{P}_{n}$
Define
$P_{k,j,i}(x)=v_{k}\int_{\mathbf{S}^{d-1}}Y_{j,i}(\xi)U_{k}(x\cdot\xi)d\omega(\xi).$
(8)
Then it follows from [15] (or [21]) that
$\\{P_{k,j,i}:k=0,\dots,n,j=k,k-2,\dots,\varepsilon_{k},i=1,2,\dots,D_{j}^{d-1}\\}$
consists an orthonormal basis for $\mathcal{P}_{n}$, where
$\varepsilon_{k}:=\left\\{\begin{array}[]{cc}0,&k\ \mbox{even},\\\ 1,&k\
\mbox{odd}\end{array}\right..$
Of course,
$\\{P_{k,j,i}:k=0,1,\dots,j=k,k-2,\dots,\varepsilon_{k},i=1,2,\dots,D_{j}^{d-1}\\}$
is an orthonormal basis for $L^{2}(\mathbf{B}^{d})$. The following Lemma 3
defines a reproducing kernel of $\mathcal{P}_{n}$, whose proof will be
presented in Appendix A.
###### Lemma 3.
The space
$(\mathcal{P}_{n},\langle\cdot,\cdot\rangle_{L^{2}(\mathbf{B}^{d})})$ is a
reproducing kernel Hilbert space. The unique reproducing kernel of this space
is
$K_{n}(x,y):=\sum_{k=0}^{n}v_{k}^{2}\int_{\mathbf{S}^{d-1}}U_{k}(\xi\cdot
x)U_{k}(\xi\cdot y)d\omega(\xi).$ (9)
## 4 The needlet kernel: Construction and Properties
In this section, we construct a concrete positive definite needlet kernel [22]
and show its properties. A function $\eta$ is said to be admissible if
$\eta\in C^{\infty}[0,\infty),$ $\eta(t)\geq 0$, and $\eta$ satisfies the
following condition [22]:
$\mbox{supp}\eta\subset[0,2],\eta(t)=1\ \mbox{on}\ [0,1],\ \mbox{and}\
0\leq\eta(t)\leq 1\ \mbox{on}\ [1,2].$
Such a function can be easily constructed out of an orthogonal wavelet mask
[7]. We define a kernel $L_{2n}(\cdot,\cdot)$ as the following
$L_{2n}(x,y):=\sum_{k=0}^{\infty}\eta\left(\frac{k}{n}\right)v_{k}^{2}\int_{\mathbf{S}^{d-1}}U_{k}(x\cdot\xi)U_{k}(y\cdot\xi)d\omega(\xi).$
(10)
As $\eta(\cdot)$ is admissible, the constructed kernel $L_{2n}(x,y),$ called
the needlet kernel (or localized polynomial kernel) [22] henceforth, is
positive definite. We will show that so defined kernel function $L_{2n}(x,y)$,
deduces the $l^{q}$ regularization learning whose learning rate is independent
of the choice of $q$. To this end, we first show several useful properties of
the needlet kernel.
The following Proposition 1 which can be deduced directly from Lemma 3 and the
definition of $\eta(\cdot)$ reveals that $L_{2n}$ possesses reproducing
property for $\mathcal{P}_{n}$.
###### Proposition 1.
Let $L_{2n}$ be defined as in (10). For arbitrary $P\in\mathcal{P}_{n}$, there
holds
$P(x)=\int_{\mathbf{B}^{d}}L_{2n}(x,y)P(y)dy.$ (11)
Since $\eta(\cdot)$ is an admissible function by definition, it follows that
$L_{2n}(x,\cdot)$ is an algebraic polynomial of degree not larger than $2n$
for any fixed $x\in\mathbf{B}^{d}$. At the first glance, as a polynomial
kernel, it may have good frequency localization property while have bad space
localization property. The following Proposition 2, which can be found in [22,
Theorem 4.2], however, advocates that $L_{2n}$ is actually a polynomial kernel
possessing very good spacial localized properties. This makes it widely
applicable in approximation theory and signal processing [12, 22].
###### Proposition 2.
Let $L_{2n}$ be defined as in (10). For arbitrary $l\in\mathbf{N}$, there
exists a constant $c_{l}$ depending only on $l$, $d$ and $\eta$ such that
$\max_{x,y\in\mathbf{B}^{d}}|L_{2n}(x,y)|\leq
c_{l}\frac{n^{d}}{(\sqrt{1-|x|^{2}}+n^{-1})(\sqrt{1-|y|^{2}}+n^{-1})(1+d(x,y))^{l}}.$
(12)
Let
$E_{n}(f)_{p}:=\inf_{P\in\mathcal{P}_{n}}\|f-P\|_{L^{p}(\mathbf{B}^{d})}$
be the best approximation error of $\mathcal{P}_{n}$. Define
$(\mathcal{L}_{2n}f)(x):=\int_{\mathbf{B}^{d}}L_{2n}(x,y)f(y)dy.$ (13)
It has been shown in [22, Remak 4.8] that the integral operator
$\mathcal{L}_{2n}f$ possesses the following compressive property:
###### Proposition 3.
If $\mathcal{L}_{2n}f$ is defined as in (13), then, for arbitrary $f\in
L^{p}(\mathbf{B}^{d})$, there exists a constant $C$ depending only on $d$ and
$p$ such that
$\|\mathcal{L}_{2n}f\|_{L^{p}(\mathbf{B}^{d})}\leq
C\|f\|_{L^{p}(\mathbf{B}^{d})}.$
By Propositions 1, 2 and 3, a standard method in approximation theory [9]
yields the following best approximation property of $\mathcal{L}_{2n}f$.
###### Proposition 4.
Let $1\leq p\leq\infty,$ and $\mathcal{L}_{2n}$ be defined in (13), then for
arbitrary $f\in L^{p}(\mathbf{B}^{d})$, there exists a constant $C$ depending
only on $d$ and $p$ such that
$\|f-\mathcal{L}_{2n}f\|_{L^{p}(\mathbf{B}^{d})}\leq CE_{n}(f)_{p}.$ (14)
## 5 Almost essential learning rate
In this section, we conduct a detailed generalization capability analysis of
the $l^{q}$ regularization scheme (1) when the kernel function $K$ is
specified as $L_{2n}(x,y)$. Our aim is to derive an almost essential learning
rate of $l^{q}$ regularization strategy (1). We first present a quick review
of learning theory. Then, we given the main result of this paper, where a
$q$-independent learning rate of $l^{q}$ regularization schemes (1) is
deduced. At last, we present some remarks on the main result.
### 5.1 Statistical learning theory
Let $X\subseteq\mathbf{B}^{d}$ be an input space and $Y\subseteq\mathbf{R}$ an
output space. Assume that there exists a unknown but definite relationship
between $x\in X$ and $y\in Y$, which is modeled by a probability distribution
$\rho$ on $Z:=X\times Y$. It is assumed that $\rho$ admits the decomposition
$\rho(x,y)=\rho_{X}(x)\rho(y|x).$
Let ${\bf z}=(x_{i},y_{i})_{i=1}^{m}$ be a set of finite random samples of
size $m$, $m\in\mathbf{N}$, drawn identically, independently according to
$\rho$ from $Z$. The set of examples ${\bf z}$ is called a training set.
Without loss of generality, we assume that $|y_{i}|\leq M$ almost everywhere.
The aim of learning is to learn from a training set a function $f:X\rightarrow
Y$ such that $f(x)$ is an effective estimate of $y$ when $x$ is given. One
natural measurement of the error incurred by using $f$ of this purpose is the
generalization error,
$\mathcal{E}(f):=\int_{Z}(f(x)-y)^{2}d\rho,$
which is minimized by the regression function [3, 4] defined by
$f_{\rho}(x):=\int_{Y}yd\rho(y|x).$
We do not know this ideal minimizer $f_{\rho}$, since $\rho$ is unknown, but
we have access to random examples from $X\times Y$ sampled according to
$\rho$.
Let $L^{2}_{\rho_{{}_{X}}}$ be the Hilbert space of $\rho_{X}$ square
integrable functions on $X$, with norm $\|\cdot\|_{\rho}.$ In the setting of
$f_{\rho}\in L^{2}_{\rho_{{}_{X}}}$, it is well known that, for every $f\in
L^{2}_{\rho_{X}}$, there holds
$\mathcal{E}(f)-\mathcal{E}(f_{\rho})=\|f-f_{\rho}\|^{2}_{\rho}.$ (15)
The goal of learning is then to construct a function $f_{\bf z}$ that
approximates $f_{\rho}$, in the norm $\|\cdot\|_{\rho}$, using the finite
sample ${\bf z}$.
One of the main points of this paper is to formulate the learning problem in
terms of probability estimates rather than expectation estimates. To this end,
we present a formal way to measure the performance of learning schemes in
probability. Let $\Theta\subset L_{\rho_{X}}^{2}$ and $\mathcal{M}(\Theta)$ be
the class of all Borel measures $\rho$ on $Z$ such that $f_{\rho}\in\Theta$.
For each $\varepsilon>0$, we enter into a competition over all estimators
established in the hypothesis space $\mathcal{H}$,
$\Psi_{m}:Z^{m}\rightarrow\mathcal{H},{\bf z}\mapsto f_{\bf z},$ and we define
the accuracy confidence function by [10]
${\bf AC}_{m}(\Theta,\mathcal{H},\varepsilon):=\inf_{f_{\bf
z}\in\Psi_{m}}\sup_{\rho\in\mathcal{M}(\Theta)}P^{m}\\{{\bf
z}:\|f_{\rho}-f_{\bf z}\|_{\rho}^{2}>\varepsilon\\}.$
Furthermore, we define the accuracy confidence function for all possible
estimators based on $m$ samples $\Phi_{m}:{\bf z}\mapsto f_{\bf z}$ by
${\bf AC}_{m}(\Theta,\varepsilon):=\inf_{f_{\bf
z}\in\Phi_{m}}\sup_{\rho\in\mathcal{M}(\Theta)}P^{m}\\{{\bf
z}:\|f_{\rho}-f{\bf z}\|_{\rho}^{2}>\varepsilon\\}.$
From these definitions, it is obvious that
${\bf AC}_{m}(\Theta,\varepsilon)\leq{\bf
AC}_{m}(\Theta,\mathcal{H},\varepsilon)$
for all $\mathcal{H}$.
### 5.2 $q$-independent learning rate
The sample dependent hypothesis space (SDHS) associated with
$L_{2n}(\cdot,\cdot)$ is then defined by
$\mathcal{H}_{L,{\bf
z}}:=\left\\{\sum_{i=1}^{m}a_{i}L_{2n}(x_{i},\cdot):a_{i}\in\mathbf{R}\right\\}$
and the corresponding $l^{q}$ regularization scheme is defined by
$f_{{\bf z},\lambda,q}=\arg\min_{f\in\mathcal{H}_{L,{\bf
z}}}\left\\{\frac{1}{m}\sum_{i=1}^{m}(f(x_{i})-y_{i})^{2}+\Omega_{\bf
z}^{q}(f_{{\bf z},\lambda,q})\right\\},$ (16)
where
$\Omega_{\bf z}^{q}(f):=\lambda\sum_{i=1}^{m}|a_{i}|^{q},\mbox{for}\
f=\sum_{i=1}^{m}a_{i}L_{2n}(x_{i},\cdot).$
The projection operator $\pi_{M}$ from the space of measurable functions
$f:X\rightarrow\mathbf{R}$ to $[-M,M]$ is defined by
$\pi_{M}(f)(x):=\left\\{\begin{array}[]{ll}M,&\mbox{if }\ f(x)>M,\\\
f(x),&\mbox{if }\ -M\leq f(x)\leq M,\\\ -M,&\mbox{if}\
f(x)\leq-M.\end{array}\right.$
As $y\in[-M,M]$ by assumption, it is easy to check [37] that
$\|\pi_{M}f_{{\bf z},\lambda,q}-f_{\rho}\|_{\rho}\leq\|f_{{\bf
z},\lambda,q}-f_{\rho}\|_{\rho}.$
Also, for arbitrary $H\subset L^{2}(\mathbf{B}^{d})$, we denote
$\pi_{M}H:=\\{\pi_{M}f:f\in H\\}$.
We also need to introduce the class of priors. For any $f\in
L^{2}(\mathbf{B}^{d})$, denote by $\mathcal{F}(f)$ or $\hat{f}$ the Fourier
transformation of $f$,
$\hat{f}(u):=(2\pi)^{d/2}\int_{\mathbf{R}^{d}}f(x)e^{iu\cdot x}dx,$
where $u\in\mathbf{B}^{d}$. The inverse Fourier transformation will be denoted
by $\mathcal{F}^{-1}$. In the space $L^{2}(\mathbf{B}^{d})$, the derivative of
$f$ with order $\alpha$ is defined as
$D^{\alpha}f:=\mathcal{F}^{-1}\\{|u|^{\alpha}\mathcal{F}(u)\\},$
where $|u|:=\sqrt{u_{1}^{2}+\cdot+u_{d}^{2}}$. Here, Fourier transformation
and derivatives are all taken sense in distribution. Let $r$ be any positive
number. We consider the Sobolev class of functions
$W_{2}^{r}:=\left\\{f:\max_{0\leq\alpha\leq
r}\|D^{\alpha}f\|_{L^{2}(\mathbf{B}^{d})}<\infty\right\\}.$
It follows from the well known Sobolev embedding theorem that
$W_{2}^{r}\subset C(\mathbf{B}^{d})$ provided $r>\frac{d}{2}.$
Now, we state the main result of this paper, whose proof will be given in the
next section.
###### Theorem 1.
Let $f_{\rho}\in W_{2}^{r}$ with $r>d/2$, $m\in\mathbf{N}$, $\varepsilon>0$ be
any numbers, and $n\sim\varepsilon^{-r/d}$. If $f_{{\bf z},\lambda,q}$ is
defined as in (16) with $\lambda=m^{-1}\varepsilon$ and $0<q<\infty$, then
there exist positive constants $C_{i},$ $i=1,\dots,4,$ depending only on $M$,
$\rho$, $q$ and $d$, $\varepsilon_{0}>0$ and
$\varepsilon_{m}^{-},\varepsilon_{m}^{+}$ satisfying
$C_{1}m^{-2r/(2r+d)}\leq\varepsilon_{m}^{-}\leq\varepsilon_{m}^{+}\leq
C_{2}(m/\log m)^{-2r/(2r+d)},$ (17)
such that for any $\varepsilon<\varepsilon_{m}^{-}$,
$\sup_{f_{\rho}\in W_{2}^{r}}P^{m}\\{{\bf z}:\|f_{\rho}-\pi_{M}f_{{\bf
z},\lambda,q}\|_{\rho}^{2}>\varepsilon\\}\geq{\bf
AC}_{m}(W_{2}^{r},\pi_{M}\mathcal{H}_{L,{\bf z}},\varepsilon)\geq{\bf
AC}_{m}(W_{2}^{r},\varepsilon)\geq\varepsilon_{0},$ (18)
and for any $\varepsilon\geq\varepsilon_{m}^{+}$,
$\displaystyle e^{-C_{3}m\varepsilon}$ $\displaystyle\leq$ $\displaystyle{\bf
AC}_{m}(W_{2}^{r},\varepsilon)\leq{\bf
AC}_{m}(W_{2}^{r},\pi_{M}\mathcal{H}_{L,{\bf z}},\varepsilon)$ (19)
$\displaystyle\leq$ $\displaystyle\sup_{f_{\rho}\in W_{2}^{r}}P^{m}\\{{\bf
z}:\|f_{\rho}-\pi_{M}f_{{\bf z},\lambda,q}\|_{\rho}^{2}>\varepsilon\\}\leq
e^{-C_{4}m\varepsilon}.$
### 5.3 Remarks
We explain Theorem 1 below in more detail. At first, we explain why the
accuracy function is used to characterize the generalization capability of the
$l^{q}$ regularization schemes (16). In applications, we are often faced with
the following problem: There are $m$ data available, and we are asked to
product an estimator with tolerance at most $\varepsilon$ by using these $m$
data only. In such circumstance, we have to know the probability of success.
It is obvious that such probability depends on $m$ and $\varepsilon$. For
example, if $m$ is too small, we can not construct an estimator within small
tolerance. This fact is quantitatively verified by Theorem 1. More
specifically, (18) shows that if there are $m$ data available and $f_{\rho}\in
W_{2}^{r}$ with $r>d/2$, then $l^{q}$ $(0<q<\infty)$ regularization scheme
(16) is impossible to yield an estimator with tolerance error smaller than
$\varepsilon_{m}^{-}$. This is not a negative result, since we can see in (18)
also that the main reason of impossibility is the lack of data rather than
inappropriateness of the learning scheme (16). More importantly, Theorem 1
reveals a quantitive relation between the probability of success and the
tolerance error based on $m$ samples. It says in (19) that if the tolerance
error $\varepsilon$ is relaxed to $\varepsilon_{m}^{+}$ or larger, then the
probability of success of $l^{q}$ regularization is at least
$1-e^{-{C_{4}m\varepsilon}}$. The first inequality (lower bound) of (19)
implies that such confidence can not be improved further. That is, we have
presented an optimal confidence estimation for $l^{q}$ regularization scheme
(16) with $0<q<\infty$. Thus, Theorem 1 basically concludes the following
thing: If $\varepsilon<\varepsilon_{m}^{-}$, then every estimator deduced from
$m$ samples by $l^{q}$ regularization can not approximate the regression
function with tolerance smaller than $\varepsilon$, while if
$\varepsilon\geq\varepsilon_{m}^{+}$, then the $l^{q}$ regularization schemes
with any $0<q<\infty$ can definitely yield the estimators that approximate the
regression function with tolerance $\varepsilon$.
The values $\varepsilon_{m}^{-}$ and $\varepsilon_{m}^{+}$ thus are critical
for indicating the generalization error of a learning scheme. Indeed, the
upper bound of generalization error of a learning scheme depends heavily on
$\varepsilon_{m}^{+}$, while the lower bound of generalization error is
relative to $\varepsilon_{m}^{-}$. Thus, in order to have a tight
generalization error estimate of a learning scheme, we naturally wish to make
the interval $[\varepsilon_{m}^{-},\varepsilon_{m}^{+}]$ as short as possible.
Theorem 1 shows that, for $l^{q}$ regularization scheme (16),
$\varepsilon_{m}^{-}\geq C_{1}m^{-2r/(2r+d)}$, and $\varepsilon_{m}^{+}\leq
C_{2}(m/\log m)^{-2r/(2r+d)}$, which shows that the interval
$[\varepsilon_{m}^{-},\varepsilon_{m}^{+}]$ is almost the shortest one in the
sense that up to a logarithmic factor, the upper bound and lower bound are
asymptotical identical. Noting that the learning rate established in Theorem 1
is independent of $q$, we thus can conclude that the generalization capability
of $l^{q}$ regularization does not depend on the choice of $q$. This gives an
affirmative answer to Problem 1.
The other advantage of using the accuracy confidence function to measure the
generalization capability is that it allows to expose some phenomenon that can
not be founded if the classical expectation standard is utilized. For example,
Theorem 1 shows a sharp phase transition phenomenon of $l^{q}$ regularization
learning, that is, the behavior of the accuracy confidence function changes
dramatically within the critical interval
$[\varepsilon_{m}^{-},\varepsilon_{m}^{+}]$. It drops from a constant
$\varepsilon_{0}$ to an exponentially small quantity. We might call
$[\varepsilon_{m}^{-},\varepsilon_{m}^{+}]$ the interval of phase transition
for a corresponding learning scheme. To make this more intuitive, let us
conduct a simulation on the phase transition of the confidence function below.
Without loss of generality, we implemented the $l^{2}$ regularization strategy
(16) associated with the kernel (10) for $d=1$ and $n=8$ to yield the
estimator. The regularization parameter $\lambda$ was chosen as
$\varepsilon/m$. The training samples were drawn independently and identically
according to the uniform distribution from the well known $Sinc$ function,
that is $f(x):=sinx/x$. The number of the training samples $m$ was chosen from
$1$ to $100$ and the tolerance $\varepsilon$ was chosen from $10^{-4}$ to $1$
with step-length $10^{-4}$. Then, there were totally 1000 test data
$(s_{i},t_{i})_{i=1}^{1000}$ drawn i. i. d according to the uniform
distribution from $sinC$. The test error was defined as
$\delta_{test}:=\sqrt{\frac{1}{100}\sum_{i=1}^{100}(f_{{\bf
z},\lambda,2}(s_{i})-t_{i})^{2}}.$ We repeated 100 times simulations at each
point, and labeled its value as $1$ if $\delta_{test}$ is smaller than the
tolerance error and $0$ otherwise. Simulation result is shown in Fig.3. We can
see from Fig.3 that in the upper right part, the colors of all points are red,
which means that in those setting, the probability that $\delta_{test}$ is
smaller than the tolerance is approximately $0$. Thus, if the number of
samples is small, then $l^{2}$ regularization schemes can not provide an
estimation with very small tolerance. In the lower left area, the colors of
all points are blue, which means that the probability of $\delta_{test}$
smaller than the tolerance is approximately $1$. Between these two areas,
there exists a band, that could be called the phase transition area, in which
the colors of points vary from red to blue dramatically. It is seen that the
length of phase transition interval monotonously decreases with $m$. All these
coincide with the theoretical assertions of Theorem 1.
Figure 3: The phase transition phenomenon of generalization with $l^{2}$
regularization
For comparison, we also present a generalization error bound result in terms
of expectation error. Corollary 1 below can be directly deduced from Theorem 1
and [10, Chapter 3], if we notice the identity:
$E_{\rho^{m}}(\mathcal{E}(f_{\rho})-\mathcal{E}(f_{{\bf
z},\lambda,q}))=\int_{0}^{\infty}P^{m}\\{\mathcal{E}(f_{\rho})-\mathcal{E}(f_{{\bf
z},\lambda,q})>\varepsilon\\}d\varepsilon.$
###### Corollary 1.
Let $f_{\rho}\in W_{2}^{r}$ with $r>d/2$, $q_{0}>0$, $m\in\mathbf{N}$, and
$n\sim\varepsilon^{-r/d}$. If $f_{{\bf z},\lambda,q}$ is defined as in (16)
with $\lambda\sim\frac{m^{-2r/(2r+d)}}{m+1}$ and $0<q<\infty$, then there
exist constants $C_{5}$ and $C_{6}$ depending only on $M$, $d$, $q$ and $\rho$
such that
$\displaystyle C_{5}m^{-2r/(2r+d)}\leq\inf_{f_{\bf
z}\in\Phi_{m}}\sup_{\rho\in\mathcal{M}(W_{2}^{r})}E_{\rho^{m}}\\{\|f_{\rho}-f_{\bf
z}\|_{\rho}^{2}\\}$ $\displaystyle\leq$ $\displaystyle\inf_{f_{\bf
z}\in\pi\mathcal{H}_{L,{\bf
z}}}\sup_{\rho\in\mathcal{M}(W_{2}^{r})}E_{\rho^{m}}\\{\|f_{\rho}-f_{\bf
z}\|_{\rho}^{2}\\}$ $\displaystyle\leq$ $\displaystyle\sup_{f_{\rho}\in
W_{2}^{r}}E_{\rho^{m}}\left\\{\|f_{\rho}-f_{{\bf
z},\lambda,q}\|^{2}\right\\}\leq C_{6}(m/\log m)^{-2r/(2r+d)},$
where $\Phi_{m}$ is the set of all possible estimators based on $m$ samples.
It is noted that the representation theorem in learning theory [27] implies
that the generalization capability of an optimal learning algorithm in SDHS is
not worse than that of learning in RKHS with convex loss function. Corollary 1
then shows that if $f_{\rho}\in W_{2}^{r}$, then the generalization capability
of an optimal learning scheme in SDHS associated with $L_{2n}$ is not worse
than that of any optimal learning algorithms in the corresponding RKHS. More
specifically, (1) shows that as far as the learning rate is concerned, all
$l^{q}$ regularization schemes (16) for $0<q<\infty$ can realize the same
almost optimal theoretical rate. That is to say, the choice of $q$ has no
influence on the generalization capability of the learning schemes (16). This
also gives an affirmative answer to Problem 1 in the sense of expectation.
Here, we emphasize that the independence of generalization of $l^{q}$
regularization on $q$ is based on the understanding of attaining the same
almost optimal generalization error. Thus, in application, $q$ can be
arbitrarily specified, or specified merely by other no generalization criteria
(like complexity, sparsity, etc.).
## 6 Proof of Theorem 1
### 6.1 Methodology
The methodology we adopted in the proof of Theorem 1 seems of novelty.
Traditionally, the generalization error of learning schemes in SDHS is divided
into the approximation, hypothesis and sample errors (three terms) [33]. All
of the aforementioned results about coefficient regularization in SDHS falled
into this style. According to [33], the hypothesis error has been regarded as
the reflection of nature of data dependence of SDHS (sample dependent
hypothesis space), and an indispensable part attributed to an essential
characteristic of learning algorithms in SDHS, compared with the learning in
SIHS (sample independent hypothesis space). With the specific kernel function
$L_{2n}$, we will divide the generalization error of $l^{q}$ regularization in
this paper into the approximation and sample errors (two terms) only. Both of
these two terms are dependent of the samples. The success in this paper then
reveals that for at least some kernels, the hypothesis error is negligible, or
can be avoided in estimation when $l^{q}$ regularization learning are analyzed
in SDHS. We show that such new methodology can bring an important benefit of
yielding an almost optimal generalization error bound for a large types of
priors. Such benefit may reasonably be expected to beyond the $l^{q}$
regularization.
We sketch the methodology to be used as follows. Due to the sample dependent
property, any estimators constructed in SDHS may be a random approximant. To
bound the approximation error, we first deduce a probabilistic cubature
formula for algebraic polynomial. Then we can discretize the near-best
approximation operator $\mathcal{L}_{2n}f$ based on the probabilistic cubature
formula. Thus, the well known Jackson-type error estimate [9] can be applied
to derive the approximation error. To bound the sample error, we will use a
different method from the tranditional approaches [3, 32]. Since the
constructed approximant in SDHS is a random approximant, the concentration
inequality such as Bernstein inequality [4] can not be available. In our
approach, based on the prominent property of the constructed approximant, we
will bound the sample error by using the concentration inequality established
in [3] twice. Then the relation between the so-called Pseudo-dimension and
covering number [18] yields the sample error estimate for $l^{q}$
regularization schemes (16) with arbitrary $o<q<\infty$. Hence, we divide the
proof into four subsections. The first subsection is devoted to establish the
probabilistic cubature formula. The second subsection is to construct the
random approximant and study the approximation error. The third subsection is
to deduce the sample error and the last subsectionis to derive the final
learning rate. We present the details one by one below.
### 6.2 A probabilistic cubature formula
In this subsection, we establish a probabilistic cubature formula. At first,
we need several lemmas. The weighted $L^{p}$ norm on the $d+1$-dimensional
unit sphere $\mathbf{S}^{d}$ is defined as follows. Let
$\alpha=(\alpha_{(1)},\dots,\alpha_{(d+1)})\in\mathbf{S}^{d}$ and
$w_{\alpha}=|\alpha_{(d+1)}|$. Define
$\|f\|_{p,w_{\alpha}}:=\left\\{\begin{array}[]{cc}\left(\int_{\mathbf{S}^{d}}|f(\alpha)|^{p}w_{\alpha}d\omega_{d}(\alpha)\right)^{1/p},&1\leq
p\leq\infty,\\\
\max_{\alpha\in\mathbf{S}^{d}}|f(\alpha)|w_{\alpha},&p=\infty.\end{array}\right.$
The following [6, Lemma 2.3] gives a weighted Nikolskii inequality for
spherical polynomial.
###### Lemma 4.
Let $1\leq p\leq q\leq\infty$. Then for any $Q\in\Pi_{n}^{d}$,
$\|Q\|_{q,w_{\alpha}}\leq Cn^{d(1/p-1/q)}\|Q\|_{p,w_{\alpha}},$
where $C$ is a positive constant depending only on $d,p$ and $q$.
Lemma 5 establishes a relation between cubature formula on the unit sphere and
cubature formula on the unit ball, which can be found in [35, Theorem 4.2].
###### Lemma 5.
If there is a cubature formula of degree $n$ on $\mathbf{S}^{d}$ given by
$\int_{\mathbf{S}^{d}}f(\alpha)w_{\alpha}d\omega_{d}(\alpha)=\sum_{i=1}^{m}a_{i}f(\alpha_{i}),$
whose nodes are all located on $\mathbf{S}^{d}$, then there exists a cubature
formula of degree $n$ on $\mathbf{B}^{d}$, that is,
$2\int_{\mathbf{B}^{d}}f(x)dx=\sum_{i=1}^{m}a_{i}f(x_{i}),$
where $x_{i}\in\mathbf{B}^{d}$ are the first $d$ components of $\alpha_{i}$.
The following Lemma 6 is known as the Bernstein inequality for random
variables, which can be found in [3].
###### Lemma 6.
Let $\xi$ be a random variable on a probability space $Z$ with mean $E(\xi)$,
variance $\sigma^{2}(\xi)=\sigma^{2}$. If $|\xi(z)-E(\xi)|\leq M_{\xi}$ for
almost all ${\bf z}\in Z$. then, for all $\varepsilon>0$,
$\mbox{Prob}_{{\bf z}\in
Z^{m}}\left\\{\left|\frac{1}{m}\sum_{i=1}^{m}\xi(z_{i})-E(\xi)\right|\geq\varepsilon\right\\}\leq
2\exp\left\\{-\frac{m\varepsilon^{2}}{2\left(\sigma^{2}+\frac{1}{3}M_{\xi}\varepsilon\right)}\right\\}.$
We also need a lemma showing that if
$\Xi:=\\{\alpha_{i}\\}_{i=1}^{m}\subset\mathbf{S}^{d}$ is a set of independent
random variables drawn identically according to a distribution $\mu$, then
with high confidence the cubature formula holds.
###### Lemma 7.
Let $0<\varepsilon<1$, and $1\leq p\leq\infty$. If
$\\{\alpha_{i}\\}_{i=1}^{m}$ are i.i.d. random variables drawn according to
arbitrary distribution $\mu$ on $\mathbf{S}^{d}$, then there exits a set of
real numbers $\\{a_{i}\\}_{i=1}^{m}$ such that
$\int_{\mathbf{S}^{d}}Q_{n}(\alpha)w_{\alpha}d\omega_{d}(\xi)=\sum_{i=1}^{m}a_{i}Q_{n}(\alpha_{i})$
holds with confidence at least
$1-2\exp\left\\{-\frac{Cm\varepsilon^{2}}{n^{d}(1+\varepsilon)}\right\\},$
subject to
$\sum_{i=1}^{m}|a_{i}|^{p}\leq\frac{\Omega_{d}}{1-\varepsilon}m^{1-p}.$
Proof. For the sake of brevity, we write $w=w_{\alpha}$ in the following.
Since the sampling set $\Xi$ consists of a sequence of i.i.d. random variables
on $\mathbf{S}^{d}$, the sampling points are a sequence of functions
$\alpha_{j}=\alpha_{j}(\omega)$ on some probability space
$(\Omega,\mathbf{P})$. Without loss of generality, we assume
$\|Q_{n}\|_{p,w}=1$ for arbitrary fixed $p$. If we set
$\xi_{j}^{p}(Q_{n})=|Q_{n}(\alpha_{j})|^{p}w$, then we have
$\frac{1}{m}\sum_{i=1}^{m}|Q_{n}(\alpha_{i})|^{p}w(\alpha_{i})-E\xi_{j}^{p}=\frac{1}{m}\sum_{i=1}^{m}|Q_{n}(\alpha_{i})|^{p}w(\alpha_{i})-\|Q_{n}\|_{p,w}^{p},$
where we have used the equality
$E\xi_{j}^{p}=\int_{\Omega}|Q_{n}(\alpha(\omega_{j}))|^{p}wd\omega_{j}=\int_{S}|Q_{n}(\alpha)|^{p}w(\alpha)d\omega_{d}(\alpha)=\|Q_{n}\|_{p,w}^{p}=1.$
Furthermore,
$|\xi_{j}^{p}-E\xi_{j}^{p}|\leq\sup_{\omega\in\Omega}\left||Q_{n}(\alpha(\omega))|^{p}w(\omega)-\|Q_{n}\|_{p,w}^{p}\right|\leq\|Q_{n}\|_{\infty,w}^{p}-\|Q_{n}\|_{p,w}^{p}.$
It follows from Lemma 4 that
$\|Q_{n}\|_{\infty,w}\leq Cn^{\frac{d}{p}}\|Q_{n}\|_{p,w}=Cn^{\frac{d}{p}}.$
Hence
$|\xi_{j}^{p}-E\xi_{j}^{p}|\leq Cn^{d}-1.$
On the other hand, we have
$\displaystyle\sigma^{2}$ $\displaystyle=$ $\displaystyle
E((\xi_{j}^{p})^{2})-(E(\xi_{j}^{p}))^{2}\leq\int_{\Omega}|Q_{n}(\alpha(\omega))|^{2p}w(\alpha)d\omega-\left(\int_{\Omega}|Q_{n}(\alpha(\omega))|^{p}w(x)d\omega\right)^{2}$
$\displaystyle=$ $\displaystyle\|Q_{n}\|_{2p,w}^{2p}-\|Q_{n}\|_{p,w}^{2p}.$
Then using Lemma 4 again, there holds
$\sigma^{2}\leq
Cn^{2dp(\frac{1}{p}-\frac{1}{2p})}\|Q_{n}\|_{p,w}^{2p}-\|Q_{n}\|_{p,w}^{2p}=Cn^{d}-1.$
Thus it follows from Lemma 6 that with confidence at least
$1-2\mbox{exp}\left\\{-\frac{m\varepsilon^{2}}{2\left(\sigma^{2}+\frac{1}{3}M_{\xi}\varepsilon\right)}\right\\}\geq
1-2\mbox{exp}\left\\{-\frac{m\varepsilon^{2}}{2\left((Cn^{d}-1)+\frac{1}{3}(Cn^{d}-1)\varepsilon\right)}\right\\},$
there holds
$\left|\frac{1}{m}\sum_{i=1}^{m}|Q_{n}(\alpha_{i})|^{p}w(\alpha_{i})-\|Q_{n}\|_{p,w}^{p}\right|\leq\varepsilon.$
This means that if $\Xi$ is a sequence of i.i.d. random variables, then the
Marcinkiewicz-Zygmund inequality
$(1-\varepsilon)\|Q_{n}\|_{p,w}^{p}\leq\frac{1}{m}\sum_{i=1}^{m}|Q_{n}(\alpha_{i})|^{p}w(x)\leq(1+\varepsilon)\|Q_{n}\|_{p,w}^{p}\quad\forall
Q_{n}\in\Pi_{n}^{d}$ (21)
holds with probability at least
$1-2\mbox{exp}\left\\{-\frac{Cm\varepsilon^{2}}{n^{d-1}(1+\varepsilon)}\right\\}.$
Then, almost same argument as that in [19, Theorem 4.1] or [5, Theorem 4.2]
implies Lemma 7. $\Box$
By virtue of the above lemmas, we can prove the following Proposition 5.
###### Proposition 5.
Let $1\leq p\leq\infty$ and ${\bf x}:=(x_{i})_{i=1}^{m}\subset\mathbf{B}^{d}$
be a set of random variables independently and identically drawn according to
arbitrary distribution $\mu$. Then there exits a set of real numbers
$\\{a_{i}\\}_{i=1}^{m}$ and a constant $C$ depending only on $d$ such that the
equality
$\int_{\mathbf{B}^{d}}P_{n}(x)dx=\sum_{i=1}^{m}a_{i}P_{n}(x_{i}),\quad
P_{n}\in\mathcal{P}_{n}$
holds with confidence at least
$1-2\mbox{exp}\left\\{-\frac{Cm}{n^{d}}\right\\},$
subject to
$\sum_{i=1}^{m}|a_{i}|^{p}\leq Cm^{1-p}.$
### 6.3 Error decomposition and an approximation error estimate
To estimate the upper bound of
$\mathcal{E}(\pi_{M}f_{{\bf z},\lambda,q})-\mathcal{E}(f_{\rho}),$
we first introduce an error decomposition strategy. It follows from the
definition of $f_{{\bf z},\lambda,q}$ that, for arbitrary
$f\in\mathcal{H}_{L,{\bf z}}$,
$\displaystyle\mathcal{E}(\pi_{M}f_{{\bf z},\lambda,q})-\mathcal{E}(f_{\rho})$
$\displaystyle\leq$ $\displaystyle\mathcal{E}(\pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}(f_{\rho})+\lambda\Omega_{\bf z}^{q}(f_{{\bf
z},\lambda,q})$ $\displaystyle\leq$ $\displaystyle\mathcal{E}(\pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}_{\bf z}(\pi_{M}f_{{\bf
z},\lambda,q})+\mathcal{E}_{\bf z}(f)-\mathcal{E}(f)$ $\displaystyle+$
$\displaystyle\mathcal{E}_{\bf z}(\pi_{M}f_{{\bf
z},\lambda,q})+\lambda\Omega_{\bf z}^{q}(f_{{\bf
z},\lambda,q})-\mathcal{E}_{\bf z}(f)-\lambda\Omega_{\bf z}^{q}(f)$
$\displaystyle+$
$\displaystyle\mathcal{E}(f)-\mathcal{E}(f_{\rho})+\lambda\Omega_{\bf
z}^{q}(f)$ $\displaystyle\leq$ $\displaystyle\mathcal{E}(\pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}_{\bf z}(\pi_{M}f_{{\bf
z},\lambda,q})+\mathcal{E}_{\bf z}(f)-\mathcal{E}(f)$ $\displaystyle+$
$\displaystyle\mathcal{E}(f)-\mathcal{E}(f_{\rho})+\lambda\Omega_{\bf
z}^{q}(f).$
Since $f_{\rho}\in W_{2}^{r}$ with $r>\frac{d}{2}$, it follows from the
Sobolev embedding theorem that $f_{\rho}\in C(\mathbf{B}^{d})$. Thus, it can
be deduced from Proposition 3 and Proposition 4 that there exists a
$P_{\rho}\in\mathcal{P}_{n}$ such that
$\|P_{\rho}\|\leq c\|f_{\rho}\|\quad\mbox{and}\quad\|f_{\rho}-P_{\rho}\|\leq
CE_{[n/2]}(f_{\rho}),$ (22)
where $[t]$ denotes the largest integer not larger than $t$ and $\|\cdot\|$
denotes the uniform norm on $\mathbf{B}^{d}$. The above inequalities together
with the well known Jackson inequality [9] imply that there exists a
$P_{\rho}\in\mathcal{P}_{n}$ such that for all $f_{\rho}\in W_{2}^{r}$ with
$r>\frac{d}{2}$, there holds
$\|P_{\rho}\|\leq
c\|f_{\rho}\|\quad\mbox{and}\quad\|f_{\rho}-P_{\rho}\|^{2}\leq Cn^{-2r}.$ (23)
Let $\mathcal{H}_{L,{\bf z}}^{*}:=\left\\{f\in\mathcal{H}_{L,{\bf
z}}:\|f\|\leq cM\right\\}$, where $c$ is defined as in (22). Define
$f_{\bf z}^{*}:=\arg\min_{f\in\mathcal{H}_{L,{\bf
z}}^{*}}\|f-f_{\rho}\|_{\rho}^{2}+\lambda\Omega_{\bf z}^{q}(f).$ (24)
Then we have
$\displaystyle\mathcal{E}(\pi_{M}f_{{\bf z},\lambda,q})-\mathcal{E}(f_{\rho})$
$\displaystyle\leq$ $\displaystyle\left\\{\mathcal{E}(f_{\bf
z}^{*})-\mathcal{E}(f_{\rho})+\lambda\Omega_{\bf z}^{q}(f_{\bf
z}^{*})\right\\}$ $\displaystyle+$
$\displaystyle\left\\{\mathcal{E}(\Pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}_{\bf z}(\Pi_{M}f_{{\bf
z},\lambda,q})+\mathcal{E}_{\bf z}(f_{\bf z}^{*})-\mathcal{E}(f_{\bf
z}^{*})\right\\}$ $\displaystyle=:$ $\displaystyle\mathcal{D}({\bf
z},\lambda,q)+\mathcal{S}({\bf z},\lambda,q),$
where $\mathcal{D}({\bf z},\lambda,q)$ and $\mathcal{S}({\bf z},\lambda,q)$ is
called the approximation error and sample error, respectively.
###### Proposition 6.
Let $m,n\in\mathbf{N}$, $r>d/2$ and $f_{\rho}\in W_{2}^{r}$. Then, with
confidence at least $1-2\exp\\{{-cm/n^{d}}\\},$ there holds
$\mathcal{D}({\bf z},\lambda,q)\leq C\left(n^{-2r}+\lambda m\right),$ (25)
where $C$ and $c$ are constants depending only on $d$ and $r$.
Proof. From Proposition 1, it is easy to deduce that
$P_{\rho}(x)=\int_{\mathbf{B}^{d}}P_{\rho}(y)L_{2n}(x,y)dy.$
Thus, Lemma 5 with $\varepsilon=\frac{1}{2}$ yields that with confidence at
least $1-2\exp\\{{-cm/n^{d}}\\},$ there exists a set of real numbers
$\\{a_{i}\\}_{i=1}^{m}$ satisfying $\sum_{i=1}^{m}|a_{i}|^{q}\leq
2\Omega_{d}m^{1-q}$ for $q\geq 1$ such that
$P_{\rho}(x)=\sum_{i=1}^{m}a_{i}P_{\rho}(x_{i})L_{2n}(x_{i},x).$
The above observation together with (23) implies that with confidence at least
$1-2\exp\\{{-cm/n^{d}}\\},$ there exists a
$g^{*}(x):=\sum_{i=1}^{m}a_{i}P_{\rho}(x_{i})L_{2n}(x_{i},x)\in\mathcal{H}_{L,{\bf
z}}^{*}$
such that for arbitrary $f_{\rho}\in W_{2}^{r}$, there holds
$\|g^{*}-f_{\rho}\|_{\rho}^{2}\leq\|g^{*}-f_{\rho}\|^{2}\leq Cn^{-2r},$
and
$\Omega_{\bf
z}^{q}(g^{*})=\sum_{i=1}^{m}|a_{i}P_{\rho}(x_{i})|^{q}\leq(cM)^{q}\sum_{i=1}^{m}|a_{i}|^{q}\leq
Cm,$
where $C$ is a constant depending only on $d$ and $M$. Indeed, if $q\geq 1$,
we have $\sum_{i=1}^{m}|a_{i}|^{q}\leq 2\Omega_{d}m^{1-q}$. Without loss of
generality, we assume $m\geq cM$. Then there holds
$\sum_{i=1}^{m}|a_{i}P_{\rho}(x_{i})|^{q}\leq(cM)^{q}2\Omega_{d}m^{1-q}\leq
2\Omega_{d-1}m.$
If $0<q<1$, it follows from the Hölder inequality that
$\sum_{i=1}^{m}|a_{i}|^{q}\leq\left(\sum_{i=1}^{m}|a_{i}|\right)^{q}\left(\sum_{i=1}^{m}1\right)^{1-q}\leq
m^{1-q}(2\Omega_{d})^{q}\leq 2\Omega_{d}m.$
Thus, for all $q_{0}\leq q\leq\infty$, there holds
$\sum_{i=1}^{m}|a_{i}P_{\rho}(x_{i})|^{q}\leq 2cM\Omega_{d-1}m.$
It thus follows from the definition of $f_{\bf z}^{*}$ that the inequalities
$\mathcal{D}({\bf
z},\lambda,q)\leq\|g^{*}-f_{\rho}\|_{\rho}^{2}+\lambda\Omega_{\bf
z}^{q}(g^{*})\leq C\left(n^{-2r}+\lambda m\right)$ (26)
holds with confidence at least $1-2\exp\\{{-cm/n^{d}}\\}.$ $\Box$
### 6.4 A sample error estimate
For further use, we also need introducing some quantities to measure the
complexity of a space [14, 16]. Let $B$ be a Banach space and $V$ a compact
set in $B$. The quantity $H_{\varepsilon}(V,B)=\log_{2}N_{\varepsilon}(V,B)$,
where $N_{\varepsilon}(V,B)$ is the number of elements in least
$\varepsilon$-net of $V$, is called $\varepsilon$-entropy of $V$ in $B$. The
quantity $N_{\varepsilon}(V,B)$ is called the $\varepsilon$-covering number of
$V$. For any $t\in\mathbf{R}$, define
$\mbox{sgn}(t):=\left\\{\begin{array}[]{cc}1,&\mbox{if}\ t\geq 0,\\\
-1,&\mbox{if}\ t<0.\end{array}\right.$
If a vector ${\bf t}=(t_{1},\dots,t_{n})$ belongs to $\mathbf{R}^{n}$, then we
denote by $\mbox{sgn}({\bf t})$ the vector $(\mbox{sgn}(t_{1}),$
$\dots$,$\mbox{sgn}(t_{n}))$. The VC dimension of a set $V$ over
$\mathbf{B}^{d}$, denoted as $VCdim(V,\mathbf{B}^{d})$, is the maximal natural
number $m$ such that there exists a collection $(\mu_{1},\dots,\mu_{m})$ in
$\mathbf{B}^{d}$ such that the cardinality of the sgn-vectors set
$S=\\{(\mbox{sgn}(v(\mu_{1})),\dots,\mbox{sgn}(v(\mu_{m}))):v\in V\\}$
equals to $2^{m}$, that is, the set $S$ coincides with the set of all vertexes
of unit cube in $\mathbf{R}^{m}$. The quantity
$Pdim(V,\mathbf{B}^{d}):=\max_{g}VCdim(V+g,\mathbf{B}^{d}),$
is called pseudo-dimension of the set $V$ over $\mathbf{B}^{d}$, where $g$
runs all functions defined on $\mathbf{B}^{d}$ and $V+g=\\{v+g:v\in V\\}$ .
Mendelson and Vershinin [18] (see also [16]) has established the following
important relation between Pseudo-dimension and $\varepsilon$-entropy.
###### Lemma 8.
Let $V(\mathbf{B}^{d})$ be a class of functions which consists of all
functions $f\in V$ satisfying $|f(x)|\leq R$ for all $x\in\mathbf{B}^{d}$.
Then,
$H_{\varepsilon}(V(\mathbf{B}^{d}),L^{2}(\mathbf{B}^{d}))\leq
c{Pdim}(V,\mathbf{B}^{d})\log_{2}\frac{R}{\varepsilon},$
where $c$ is an absolute positive constant.
The following Lemma 9 [13] further shows that the pseudo-dimension of
arbitrary $m$-dimensional vector space is $m$.
###### Lemma 9.
Let $\mathcal{H}$ be an $m$-dimensional vector space of functions from
$\mathbf{B}^{d}$ into $\mathbf{R}$. Then
${Pdim}(\mathcal{H},\mathbf{B}^{d})=m$.
We also need to apply the following concentration inequality [3].
###### Lemma 10.
Let $\mathcal{G}$ be a set of functions on $Z$ such that, for some $c\geq 0$,
$|g-E(g)|\leq B$ almost everywhere and $E(g^{2})\leq cE(g)$ for each
$g\in\mathcal{G}$. Then, for every $\varepsilon>0$,
$\mbox{Prob}_{z\in
Z^{m}}\left\\{\sup_{f\in\mathcal{G}}\frac{E(g)-\frac{1}{m}\sum_{i=1}^{m}g(z_{i})}{\sqrt{E(g)+\varepsilon}}\leq\sqrt{\varepsilon}\right\\}\leq\mathcal{N}_{\varepsilon}(\mathcal{G},C(\mathbf{B}^{d}))\mbox{exp}\left\\{-\frac{m\varepsilon}{2c+\frac{2B}{3}}\right\\}.$
The following Proposition 7 give an upper bound of sample error.
###### Proposition 7.
Let $m,n\in\mathbf{N}$, $\varepsilon>0$, and $f_{\bf z,\lambda,q}$ be defined
as in (16). Then with confidence at least
$\displaystyle
1-\exp\left\\{cn^{d}\log\frac{Cn^{d+2}\max\\{m^{1-1/q},1\\}M^{\frac{2}{q}+1}}{\lambda^{1/q}\varepsilon}-\frac{3m\varepsilon}{128M^{2}}\right\\}$
$\displaystyle-\exp\left\\{Cn^{d}\log\left(\frac{32M+32cM}{\varepsilon}\right)-\frac{3m\varepsilon}{16(c+3)^{2}M^{2}}\right\\}$
there holds
$\mathcal{S}({\bf z},\lambda,q)\leq\frac{1}{2}(\mathcal{E}(\pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}(f_{\rho}))+\frac{1}{2}\mathcal{D}({\bf
z},\lambda,q)+2\varepsilon.$
Proof. If we set $\xi_{1}:=(\pi_{M}f_{{\bf
z},\lambda,q}(x)-y)^{2}-(f_{\rho}(x)-y)^{2},$ and $\xi_{2}:=(f_{\bf
z}^{*}(x)-y)^{2}-(f_{\rho}(x)-y)^{2},$ then
$E(\xi_{1})=\mathcal{E}(\pi_{M}f_{{\bf z},\lambda,q})-\mathcal{E}(f_{\rho}),\
\mbox{and}\ E(\xi_{2})=\mathcal{E}(f_{\bf z}^{*})-\mathcal{E}(f_{\rho}),$
both of which are random variables. Hence, we can rewrite the sample error as
$S({\bf
z},\lambda,q)=\left\\{E(\xi_{1})-\frac{1}{m}\sum_{i=1}^{m}\xi_{1}(z_{i})\right\\}+\left\\{\frac{1}{m}\sum_{i=1}^{m}\xi_{2}(z_{i})-E(\xi_{2})\right\\}=:\mathcal{S}_{1}+\mathcal{S}_{2}.$
Define
$\mathcal{B}_{R}^{q}:=\left\\{f=\sum_{i=1}^{m}a_{i}L_{2n}(x_{i},x):\sum_{i=1}^{m}|a_{i}|^{q}\leq
R\right\\}.$
As $f_{{\bf z},\lambda,q}:=\sum_{i=1}^{m}b_{i}L_{2n}(x_{i},x)$, it follows
from (16) that
$\lambda\sum_{i=1}^{m}|b_{i}|^{q}\leq\frac{1}{m}\sum_{i=1}^{m}(0-y_{i})^{2}+0\leq
M^{2},$
which implies $f_{{\bf z},\lambda,q}\in\mathcal{B}^{q}_{M^{2}/\lambda}$. Let
$\mathcal{F}_{\lambda}:=\left\\{g=(\pi_{M}f(x)-y)^{2}-(f_{\rho}(x)-y)^{2}:f\in\mathcal{B}^{q}_{M^{2}/\lambda}\right\\}.$
Then, for any fixed $g\in\mathcal{F}_{\lambda},$ there exists
$f\in\mathcal{B}^{q}_{M^{2}/\lambda}$ such that
$g(z)=(\pi_{M}f(x)-y)^{2}-(f_{\rho}(x)-y)^{2}$. It is easy to deduce that
$E(g)=\mathcal{E}(\pi_{M}f)-\mathcal{E}(f_{\rho})\geq 0,$
$\frac{1}{m}\sum_{i=1}^{m}g(z_{i})=\mathcal{E}_{\bf
z}(\pi_{M}f)-\mathcal{E}_{\bf z}(f_{\rho}),$
and
$g({z})=\left(\pi_{M}f(x)-f_{\rho}(x)\right)\left[\left(\pi_{M}f(x)-y\right)+\left(f_{\rho}(x)-y\right)\right].$
Since $|y|\leq M$ and $|f_{\rho}(x)|\leq M$ almost everywhere, we find that
$|g({z})|\leq(M+M)(M+3M)\leq 8M^{2}.$
Of course, we have
$|g({z})-E(g)|\leq B:=16M^{2}$
almost everywhere and
$\displaystyle E(g^{2})$ $\displaystyle=$ $\displaystyle
E\left[\left(\pi_{M}f(x)-f_{\rho}(x)\right)^{2}\left\\{\left(\pi_{M}f(x)-y\right)+\left(f_{\rho}(x)-y\right)\right\\}^{2}\right]$
$\displaystyle\leq$ $\displaystyle
16M^{2}\|\pi_{M}f-f_{\rho}\|_{\rho}^{2}=16M^{2}E(g),$
Therefore, we can apply Lemma 10 to the set of functions
$\mathcal{F}_{\lambda}$ with $B=c=16M^{2}$, yielding
$\sup_{f\in\mathcal{B}^{q}_{M^{2}/\lambda}}\frac{\mathcal{E}(\pi_{M}f)-\mathcal{E}(f_{\rho})-\left(\mathcal{E}_{\bf
z}(\pi_{M}f)-\mathcal{E}_{\bf
z}(f_{\rho})\right)}{\sqrt{\mathcal{E}(\pi_{M}f)-\mathcal{E}(f_{\rho})+\varepsilon}}\leq\sqrt{\varepsilon}$
(27)
with confidence at least
$1-\mathcal{N}_{\varepsilon/4}\left(\mathcal{F}_{\lambda},C(\mathbf{B}^{d})\right)\exp\left\\{-\frac{m\varepsilon}{2B+\frac{2}{3}c}\right\\}\geq
1-\mathcal{N}_{\varepsilon/4}\left(\mathcal{F}_{\lambda},C(\mathbf{B}^{d})\right)\exp\left\\{-\frac{3m\varepsilon}{128M^{2}}\right\\}.$
For every $f_{1},f_{2}\in\mathcal{B}^{q}_{M^{2}/\lambda}$ , we have
$\left|(\pi_{M}f_{1}(x)-y)^{2}-(\pi_{M}f_{2}(x)-y)^{2}\right|\leq
4M\|f_{1}-f_{2}\|.$
Thus, a $\left(\frac{\varepsilon}{4M}\right)$-covering of
$\mathcal{B}^{q}_{M^{2}/\lambda}$ provides an $\varepsilon$-covering of
$\mathcal{F}_{\lambda}$ for any $\varepsilon>0$. This implies
$\mathcal{N}_{\varepsilon/4}\left(\mathcal{F}_{\lambda},C(\mathbf{B}^{d})\right)\leq\mathcal{N}_{\varepsilon/(16M)}\left(\mathcal{B}^{q}_{M^{2}/\lambda},C(\mathbf{B}^{d})\right)\leq\mathcal{N}_{\varepsilon/(16M)}\left(\mathcal{B}^{q}_{M^{2}/\lambda},L^{2}(\mathbf{B}^{d})\right).$
It is also needed to derive an upper bound estimation for
$\mathcal{N}_{\varepsilon/(16M)}\left(\mathcal{B}^{q}_{M^{2}/\lambda},L^{2}(\mathbf{B}^{d})\right)$.
For $q\geq 1$, and $f\in\mathcal{B}_{M^{2}/\lambda}^{q}$, it follows from
Proposition 2 and the Hölder inequality that
$\left|\sum_{i=1}^{m}b_{i}L_{2n}(x_{i},x)\right|\leq\max_{x,y\in\mathbf{B}^{d}}L_{2n}(x,y)\sum_{i=1}^{m}|b_{i}|\leq
Cn^{2+d}(M^{2}/\lambda)^{\frac{1}{q}}m^{1-1/q}.$
For $0<q<1$, and $f\in\mathcal{B}_{M^{2}/\lambda}^{q}$, using (12) again we
can obtain
$\left|\sum_{i=1}^{m}b_{i}L_{2n}(x_{i},x)\right|\leq\max_{x,y\in\mathbf{B}^{d}}L_{2n}(x,y)\sum_{i=1}^{m}|b_{i}|\leq
Cn^{2+d}(M^{2}/\lambda)^{\frac{1}{q}}.$
Consequently, for arbitrary $f\in\mathcal{B}_{M^{2}/\lambda}^{q}$ and
arbitrary $0<q<\infty$, there holds
$\left|\sum_{i=1}^{m}b_{i}L_{2n}(x_{i},x)\right|\leq
Cn^{2+d}\max\\{m^{1-1/q},1\\}(M^{2}/\lambda)^{\frac{1}{q_{0}}}.$
Noting that $\mathcal{H}_{L,{\bf z}}$ is a finite dimensional linear space
with its dimension not larger than $cn^{d}$, it follows from Lemma 9 and Lemma
8 that
$\log\mathcal{N}_{\varepsilon/(16M)}\left(\mathcal{B}^{q}_{M^{2}/\lambda},L^{2}(\mathbf{B}^{d})\right)\leq
cn^{d}\log\frac{Cn^{d+2}\max\\{m^{1-1/q},1\\}M^{\frac{2}{q}+1}}{\lambda^{1/q}\varepsilon}.$
Accordingly,
$\mathcal{N}_{\varepsilon/4}\left(\mathcal{F}_{\lambda},C(\mathbf{B}^{d})\right)\leq\exp\left\\{cn^{d}\log\frac{Cn^{d+2}\max\\{m^{1-1/q},1\\}M^{\frac{2}{q}+1}}{\lambda^{1/q}\varepsilon}\right\\},$
which together with (27) further yields
$\mathcal{S}_{1}\leq\frac{1}{2}(\mathcal{E}(\pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}(f_{\rho}))+\varepsilon$ (28)
with confidence at least
$1-\exp\left\\{cn^{d}\log\frac{Cn^{d+2}\max\\{m^{1-1/q},1\\}M^{\frac{2}{q}+1}}{\lambda^{1/q}\varepsilon}-\frac{3m\varepsilon}{128M^{2}}\right\\}.$
Now, we turn to estimate $\mathcal{S}_{2}$. By definition of $f_{\bf z}^{*}$,
we have $\|f_{\bf z}^{*}\|\leq cM$. Let
$\mathcal{G}:=\left\\{g=(f(x)-y)^{2}-(f_{\rho}(x)-y)^{2}:f\in\mathcal{H}^{*}_{L,{\bf
z}}\right\\}.$
Then for any fixed $g\in\mathcal{G},$ there exists an
$f\in\mathcal{\mathcal{missing}}H_{L,{\bf z}}^{*}$ such that
$g(z)=(f(x)-y)^{2}-(f_{\rho}(x)-y)^{2}$. Similarly, we have
$E(g)=\mathcal{E}(f)-\mathcal{E}(f_{\rho})\geq
0\quad\mbox{and}\quad\frac{1}{m}\sum_{i=1}^{m}g(z_{i})=\mathcal{E}_{\bf
z}(f)-\mathcal{E}_{\bf z}(f_{\rho}).$
Since $|y|\leq M$, $|f_{\rho}(x)|\leq M$ and $\|f\|\leq cM$ almost everywhere,
we get
$|g({z})|\leq(c+3)^{2}M^{2}\quad\mbox{and}\quad|g({z})-E(g)|\leq
B:=2(c+3)^{2}M^{2}$
almost everywhere. Furthermore,
$E(g^{2})\leq 2(c+3)^{2}M^{2}\|f-f_{\rho}\|_{\rho}^{2}=2(c+3)^{2}M^{2}E(g).$
Then we apply Lemma 10 again to the set of functions $\mathcal{G}$ with
$B=c=2(c+3)^{2}M^{2}$ and obtain
$\sup_{f\in\mathcal{H}_{L,{\bf
z}}}\frac{\mathcal{E}(f)-\mathcal{E}(f_{\rho})-\left(\mathcal{E}_{\bf
z}(f)-\mathcal{E}_{\bf
z}(f_{\rho})\right)}{\sqrt{\mathcal{E}(f)-\mathcal{E}(f_{\rho})+\varepsilon}}\leq\sqrt{\varepsilon}$
(29)
with confidence at least
$1-\mathcal{N}_{\varepsilon/4}\left(\mathcal{G},C(\mathbf{B}^{d})\right)\exp\left\\{-\frac{m\varepsilon}{2B+\frac{2}{3}c}\right\\}\geq
1-\mathcal{N}_{\varepsilon/4}\left(\mathcal{G},C(\mathbf{B}^{d})\right)\exp\left\\{-\frac{3m\varepsilon}{16(c+3)^{2}M^{2}}\right\\}.$
For every $f_{1},f_{2}\in\mathcal{H}_{L,{\bf z}}^{*}$ , we have
$\left|(f_{1}(x)-y)^{2}-(f_{2}(x)-y)^{2}\right|\leq(2c+2)M\|f_{1}-f_{2}\|.$
Thus, for any $\varepsilon>0$, a
$\left(\frac{\varepsilon}{2cM+2M}\right)$-covering of $\mathcal{H}^{*}_{L,{\bf
z}}$ provides an $\varepsilon$-covering of $\mathcal{G}$. This means
$\mathcal{N}_{\varepsilon/4}\left(\mathcal{G},C(\mathbf{B}^{d})\right)\leq\mathcal{N}_{\varepsilon/(8M+8cM)}\left(\mathcal{H}_{L,{\bf
z}}^{*},C(\mathbf{B}^{d})\right)$
By definition of $\mathcal{H}_{L,{\bf z}}^{*}$, we then deduce from [4,
Theorem 5.3] that
$\log\mathcal{N}_{\varepsilon/(8M+8cM)}\left(\mathcal{H}_{L,{\bf
z}}^{*},C(\mathbf{B}^{d})\right)\leq
Cn^{d}\log\left(\frac{32M+32cM}{\varepsilon}\right).$
Hence,
$\mathcal{N}_{\varepsilon/4}\left(\mathcal{G},C(\mathbf{B}^{d})\right)\leq\exp\left\\{Cn^{d}\log\left(\frac{32M+32cM}{\varepsilon}\right)\right\\},$
which together with (29) yields
$\mathcal{S}_{2}\leq\frac{1}{2}(\mathcal{E}(f_{\bf
z}^{*})-\mathcal{E}(f_{\rho}))+\varepsilon$ (30)
with confidence at least
$1-\exp\left\\{Cn^{d}\log\left(\frac{32M+32cM}{\varepsilon}\right)-\frac{3m\varepsilon}{16(c+3)^{2}M^{2}}\right\\}.$
This finishes the proof of Proposition 7. $\Box$
### 6.5 Learning rate analysis
Now we are in a position to deduce the final learning rate of $l^{q}$
regularization schemes (16). Firstly, it follows from Propositions 6 and 7
that
$\displaystyle\mathcal{E}(\pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}(f_{\rho}))$ $\displaystyle\leq$
$\displaystyle\mathcal{D}({\bf
z},\lambda,q)+\mathcal{S}_{1}+\mathcal{S}_{2}\leq C\left(n^{-2r}+\lambda
m\right)$ $\displaystyle+$
$\displaystyle\frac{1}{2}(\mathcal{E}(\pi_{M}f_{{\bf
z},\lambda,q})-\mathcal{E}(f_{\rho}))+\varepsilon+\frac{1}{2}(\mathcal{E}(f_{\bf
z}^{*})-\mathcal{E}(f_{\rho}))+\varepsilon$
holds with confidence at least
$\displaystyle
1-2\exp\\{{-cm/n^{d}}\\}-\exp\left\\{cn^{d}\log\frac{Cn^{d+2}\max\\{m^{1-1/q},1\\}M^{\frac{2}{q_{0}}+1}}{\lambda^{1/q_{0}}\varepsilon}-\frac{3m\varepsilon}{128M^{2}}\right\\}$
$\displaystyle-$
$\displaystyle\exp\left\\{Cn^{d}\log\left(\frac{32M+32cM}{\varepsilon}\right)-\frac{3m\varepsilon}{16(c+3)^{2}M^{2}}\right\\}.$
Then, by setting $\varepsilon\geq\varepsilon_{m}^{+}\geq C(m/\log
m)^{-2r/(2r+d)}$, $n=\left[c_{0}\varepsilon^{-1/(2r)}\right]$ and
$\lambda=m^{-1}\varepsilon$, it follows from $r>d/2$ that
$\displaystyle
1-2\exp\\{-Cm\varepsilon^{d/(2r)}\\}-\exp\left\\{C\varepsilon^{-d/(2r)}\log\frac{1}{\varepsilon}-3m\varepsilon/(16(c+3)^{2}M^{2})\right\\}$
$\displaystyle-$ $\displaystyle\exp\left\\{C\varepsilon^{-d/(2r)}\left(\log
1/\varepsilon+\log\lambda^{-1/q_{0}}\right)-3m\varepsilon/(128M^{2})\right\\}$
$\displaystyle\geq$ $\displaystyle
1-2\exp\\{-Cm\varepsilon\\}-\exp\left\\{C\varepsilon^{-d/(2r)}\log
m-3m\varepsilon/(16(c+3)^{2}M^{2})\right\\}$ $\displaystyle-$
$\displaystyle\exp\left\\{C\varepsilon^{-d/(2r)}\log
m-3m\varepsilon/(128M^{2})\right\\}$ $\displaystyle\geq$ $\displaystyle
1-\exp\\{-Cm\varepsilon\\}.$
That is, for $\varepsilon\geq\varepsilon^{+}$
$\mathcal{E}(\pi_{M}f_{{\bf z},\lambda,q})-\mathcal{E}(f_{\rho})\leq
6\varepsilon$
holds with confidence at least $1-\exp\\{-Cm\varepsilon\\}$.
The lower bound can be more easily deduced. Actually, it follows from [10,
Equation (3.27)] (see also [17]) that for any estimator $f_{\bf
z}\in\Phi_{m}$, there holds
$\sup_{f_{\rho}\in W_{2}^{r}}P_{m}\\{{\bf z}:\|f_{\bf
z}-f_{\rho}\|_{\rho}^{2}\geq\varepsilon\\}\geq\left\\{\begin{array}[]{cc}\varepsilon_{0},&\varepsilon<\varepsilon^{-},\\\
e^{-cm\varepsilon},&\varepsilon\geq\varepsilon^{-},\end{array}\right.$
where $\varepsilon_{0}=\frac{1}{2}$ and $\varepsilon^{-}=cm^{-2r/(2r+d)}$ for
some universal constant $c$. With this, the proof of Theorem 1 is completed.
## 7 Further discussion and conclusion
In studies and applications, regularization is a fundamental skill to improve
on performance of a learning machine. The $l^{q}$ regularization schemes (1)
with $0<q<\infty$ are well known to be central in use. In this paper, we have
studied the dependency problem of the generalization capability of $l^{q}$
regularization with the choice of $q$. Through formulating a new methodology
of estimation of generalization error, we have shown that there is at least a
positive definite kernel, say, $L_{2n}$, such that associated with such a
kernel, the learning rate of the $l^{q}$ regularization schemes is independent
of the choice of $q$. (To be more precise, we verified that with the kernel
$L_{2n}$, all $l^{q}$ regularization schemes (1) can attain the same almost
optimal learning rate in the following sense: up to a logarithmic factor, the
upper and lower bounds of generalization error of the $l^{q}$ regularization
schemes are asymptotically identical). This implies that for some kernels, the
generalization capability of $l^{q}$ regularization may not depend on $q$.
Therefore, as far as the generalization capability is concerned, for those
kernels, the choice of $q$ is not important, which then relaxes the model
selection difficulty in applications. The problem is, however, far
complicated. We have also illustrated in Section 2 that there exists a kernel
with which the generalization capability of $l^{q}$ regularization heavily
depends on the choice of $q$. Thus, answering completely whether or not the
choice of $q$ affects the generalization of $l^{q}$ regularization is by no
means easy and completed.
Though we have constructed a concrete kernel example, the localized polynomial
kernel $L_{2n}$, with which implementing the $l^{q}$ regularization in SDHS
can realize the almost optimal learning rate, and this is independence of the
choice of $q$, we have not provided a practically feasible algorithm to
implement the learning with the almost optimal generalization capability. This
is because the kernel $L_{2n}$ we have constructed is not easily computed in
practice, even though we can use the cubature formula (Lemma 2) to discretize
it. Thus, seeking the kernels that possesses the similar property as that of
$L_{2n}$ and can be implemented easily deserve study. This is under our
current investigation.
## Appendix A: Proof of Lemma 3
To prove Lemma 3, we need the following Aronszajn Theorem (see [1]).
###### Lemma 11.
Let $\mathcal{H}$ be a separable Hilbert space of functions over $X$ with
orthonormal basis $\\{\phi_{k}\\}_{k=0}^{\infty}$. $\mathcal{H}$ is a
reproducing kernel Hilbert space if and only if
$\sum_{k=0}^{\infty}|\phi_{k}(x)|^{2}<\infty$
for all $x\in X$. The unique reproducing kernel $K$ is defined by
$K(x,y):=\sum_{k=0}^{\infty}\phi_{k}(x)\phi_{k}(y).$
Proof of Lemma 3. Since
$\\{P_{k,j,i}:k=0,\dots,n,j=k,k-2,\dots,\varepsilon_{k},i=1,2,\dots,D_{j}^{d-1}\\}$
is an orthonormal basis for $\mathcal{P}_{n}$, for arbitrary
$P\in\mathcal{P}_{n}$, there exists a set of real numbers $a_{k,j,i}$ such
that
$P(x)=\sum_{k=0}^{n}\sum_{j}\sum_{i=1}^{D_{j}^{d-1}}a_{k,j,i}P_{k,j,i}(x),$
where the summation concerning the index $j$ is $k,k-2,\dots,\varepsilon_{k}$.
On the other hand, it follows from (8) that
$\displaystyle\sum_{j}\sum_{i=1}^{D_{j}^{d-1}}P_{k,j,i}(x)P_{k,j,i}(y)$
$\displaystyle=$
$\displaystyle\sum_{j}\sum_{i=1}^{D_{j}^{d-1}}v_{k}^{2}\int_{\mathbf{S}^{d-1}}Y_{j,i}(\xi)U_{k}(x\cdot\xi)d\omega_{d-1}(\xi)\int_{\mathbf{S}^{d-1}}Y_{j,i}(\eta)U_{k}(y\cdot\eta)d\omega_{d-1}(\eta)$
$\displaystyle=$ $\displaystyle
v_{k}^{2}\sum_{j}\int_{\mathbf{S}^{d-1}}U_{k}(x\cdot\xi)\int_{\mathbf{S}^{d-1}}U_{k}(y\cdot\eta)\sum_{i=1}^{D_{j}^{d-1}}Y_{j,i}(\xi)Y_{j,i}(\eta)d\omega(\xi)d\omega_{d-1}(\eta).$
Thus, the addition formula (7) yields
$\sum_{j}\sum_{i=1}^{D_{j}^{d-1}}P_{k,j,i}(x)P_{k,j,i}(y)=v_{k}^{2}\sum_{j}\int_{\mathbf{S}^{d-1}}U_{k}(x\cdot\xi)\int_{\mathbf{S}^{d-1}}U_{k}(y\cdot\eta)K^{*}_{j}(\xi\cdot\eta)d\omega_{d-1}(\xi)d\omega_{d-1}(\eta).$
The above equality together with (5) and (6) implies
$\displaystyle\sum_{j}\sum_{i=1}^{D_{j}^{d-1}}P_{k,j,i}(x)P_{k,j,i}(y)$
$\displaystyle=$ $\displaystyle
v_{k}^{2}\int_{\mathbf{S}^{d-1}}U_{k}(x\cdot\xi)\int_{\mathbf{S}^{d-1}}U_{k}(y\cdot\eta)\sum_{j}K_{j}^{*}(\xi\cdot\eta)d\omega_{d-1}(\xi)d\omega_{d-1}(\eta)$
$\displaystyle=$
$\displaystyle\frac{v_{k}^{4}}{U_{k}(1)}\int_{\mathbf{S}^{d-1}}U_{k}(x\cdot\xi)\int_{\mathbf{S}^{d-1}}U_{k}(y\cdot\eta)U_{k}(\xi\cdot\eta)d\omega_{d-1}(\xi)d\omega_{d-1}(\eta)$
$\displaystyle=$ $\displaystyle v_{k}^{2}\int_{\mathbf{S}^{d-1}}U_{k}(\xi\cdot
x)U_{k}(\xi\cdot y)d\omega_{d-1}(\xi).$
Therefore, there holds
$K_{n}(x,y)=\sum_{k=0}^{\infty}\sum_{j}\sum_{i=1}^{D_{j}^{d-1}}P_{k,j,i}(x)P_{k,j,i}(y).$
The above equality together with Lemma 11 yields Lemma 3.
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|
arxiv-papers
| 2013-07-25T00:48:04 |
2024-09-04T02:49:48.431616
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaobo Lin, Chen Xu, Jingshan Zeng, Jian Fang",
"submitter": "Shao-Bo Lin",
"url": "https://arxiv.org/abs/1307.6616"
}
|
1307.6726
|
# Information content versus word length in natural language: A reply to
Ferrer-i-Cancho and Moscoso del Prado Martin (2011)
Steven T. Piantadosi, Harry Tily, Edward Gibson
###### Abstract
Recently, ? (?) argued that an observed linear relationship between word
length and average surprisal (?, ?) is not evidence for communicative
efficiency in human language. We discuss several shortcomings of their
approach and critique: their model critically rests on inaccurate assumptions,
is incapable of explaining key surprisal patterns in language, and is
incompatible with recent behavioral results. More generally, we argue that
statistical models must not critically rely on assumptions that are
incompatible with the real system under study.
## 1 Introduction
One of the most famous properties of language, first studied by ? (?), is that
frequent words are typically also short. Zipf offered a communicative theory
for this property, under which lexicons have evolved to be efficient: words
which must be re-used repeatedly should be short to minimize the effort of
language users. Recently, we (?, ?, henceforth, PT&G) demonstrated an
improvement on Zipf’s ideas. Under a more elaborate notion of communicative
efficiency, word length should depend not on frequency, but on the typical
_amount_ of information conveyed by a word. Efficient languages will convey
information close to the channel capacity (?, ?) of human perceptual and
cognitive systems. In this case, one should observe a linear relationship
between a word’s negative log probability (surprisal) and its length, in an
attempt to keep the number of bits communicated per unit time roughly
constant. Such a prediction is a lexical version of a now popular idea that
choices made in language production also attempt to maintain a roughly uniform
rate of information transmission (?, ?, ?, ?, ?, ?). PT&G demonstrated across
$10/11$ languages for which corpora were readily available that information
content does predict word length better than frequency, both in total
correlations and partial correlations.
Recently, ? (?, henceforth F&M) argued that the roughly linear relation
observed in PT&G was not necessarily evidence of communicative efficiency.
They prove that a model in which language is generated by choosing characters
independently also shows a linear relationship between average information
content and word length111Because in their model predictability reduces to
frequency, their work replicates ? (?) (e.g. Equation 2) and extends it to the
case of unequal letter probabilities.. In such a system, words are generated
by randomly typing characters, and occasionally hitting the “space” character
to create a word boundary. It is intuitively not surprising that such a system
would show the required linear relationship, since the probability of
achieving a word of a given length $l$ will decrease geometrically in $l$, so
the log probability scales linearly in $l$. F&M furnish a mathematical proof
for a more general version where all words must be longer than some minimum
length $l_{0}$ and individual letters occur with arbitrary probabilities.
Because random typing does not consider communicative efficiency, they argue
that it is possible to achieve a linear relationship without any notion of
communicative optimization222F&M do not specify what they mean by
communicative efficiency. For PT&G and other UID work before them, language
would be efficient if it tended to communicative bits of information at the
channel capacity of human cognitive systems. Under this definition, F&M’s
random typing model actually could be efficient.. Though they do not say so
explicitly, their paper implies that such simple statistical models should be
treated as baselines, where any properties of language that hold in them are
not expected to be the result of any interesting causal processes. F&M’s
random typing model is one exemplar of a long history of _monkey models_ in
psycholinguistics, so-called because they capture the Borel’s process of “a
million monkeys typing on a million typewriters.” Such models were first
articulated in the study of language by ? (?) and ? (?) in an attempt to
account of the power law distribution of word frequencies (?, ?).
F&M’s model is essentially a toy model of language and does indeed exhibit a
linear relationship between word length and information content. However, we
argue that their model is, in some sense, too simplified for the strong claims
they make. Here, we discuss some of the limitations of F&M’s model. After
clarifying the main finding of PT&G, we argue that the model’s assumptions
make it inherently unlike real human language, and therefore a poor choice for
statistical comparison. We then review recent behavioral work indicating that
PT&G’s results are not statistical artifacts.
### Random typing cannot explain the primary finding of PT&G
First—and perhaps most importantly—F&M do not address the primary data point
reported by PT&G. Our main finding was _not_ a linear relationship. Instead,
we focused on reporting that a word’s average in-context surprisal predicted
word length _better than frequency predicts word length_. This pattern held in
general for several different corpora, ways of measuring word length, and ways
of estimating surprisal, and in partial correlations (e.g. surprisal
partialing out frequency vs. frequency partialing out surprisal). Thus, PT&G’s
measure of the average amount of information conveyed by a word was a more
important determinant of word length than frequency—so much so that the
partial effect of frequency was near zero in some languages. Indeed, our
primary correlations reported were not even linear correlations, but
nonparametric (although both give very similar results).
In the random typing setup used by F&M, frequency and our information measure
are mathematically identical—a fact used in their derivations—so random typing
could never find that information content was a better predictor of length
than frequency. This both illustrates a limitation of F&M’s model, and
provides evidence that it is a poor description of the statistical patterns in
natural language.
### Independent data argues against random typing
Beyond the fact that F&M’s analysis cannot address the reported differences
between frequency and information content, it is worth considering its
limitations when viewed as a statistical model of language. The model’s
generative process assumes that words are created by just happening to
randomly sample their component pieces. This probabilistic scheme is what
assigns words of varying lengths their varying probabilities.
But language generation does not work that way333? (?) presented a similar
critique to Miller’s random typing model for deriving the Zipfian distribution
of word frequencies: “If Zipf’s law indeed referred to the writings of ‘random
monkeys,’ Miller’s argument would be unassailable, for the assumption he bases
it upon are appropriate to the behavior of those conjectural creatures. But to
justify his conclusion that people also obey Zipf’s law for the same reason,
Miller must perforce establish that the same assumptions are also appropriate
to human language. In fact, as we shall see, they are directly contradicted by
well-known and obvious properties of languages.”. Instead, speakers _know
whole words_. This fact is not hard to establish psychologically or
statistically. For instance, psychologically, speakers know word meanings and
produce them in the correct context—they don’t just happen to say words by
randomly saying syllables or phonemes. Statistically, idealized models over
sequences of characters infer not only the presence of words but the correct
words themselves (?, ?, ?). Though such models were proposed as language
acquisition models, they equally serve as idealized data analysis models,
demonstrating that the evidence provided by statistical dependencies between
characters in natural language strongly favors the existence of words.
The existence of words as psychological units undermines F&M’s primary point,
that “a linear correlation between information content and word length may
simply arise internally, from the units making a word (e.g., letters) and not
necessarily from the interplay between words and their context … .” If humans
generate language by remembering entire words, rather than by sampling their
component parts, then there is no necessary relationship between word length,
frequency, and information content. In the real psychological system wordforms
are memorized sequences, making their probability of generation is not longer
intrinsically tied to their length. Indeed, it is hard to see what the
behavior of models that critically rely on randomly generating word
_components_ could tell us about a psychological system that does not work
that way.
### Random typing is not even a good statistical model
It is still worth considering that even though the generative assumptions of
random typing models are limiting, they still may provide a useful
_statistical_ description of language. Unfortunately, a considerable amount of
evidence has amassed that such models are poor statistical theories. ? (?,
pg106-107) analyzes the predictions of a random typing model with respect to
coarse properties of lexical systems, including word frequency distributions,
frequency/length relationships, and neighborhood density. He finds that while
a random typing model provides qualitative trends in the right directions, its
quantitative fit is not very good and is eclipsed by the fit of other models
such as a Yule-Simon model. Indeed, a considerable amount of work—confusingly,
much of it by the first author of F&M—has detailed the ways in which the
output of random typing models are _unlike_ those found in natural languages,
especially with respect to Zipf’s law (?, ?, ?, ?, ?, ?). Other statistical
properties of random texts have been found to be divergent from real language.
For instance, ? (?) shows that random texts, but not natural texts, follow the
statistics of Heaps’ law, a growth pattern relating types and tokens to text
sample size (?, ?). ? (?) compares entropy-based measures on natural and
random typing model texts, with the goal of finding metrics that best
distinguish these texts. Our other work has detailed ways in which lexical
systems are not only non-random, but specifically structured for communicative
efficiency, in terms of ambiguity (?, ?) or lexical properties such as stress
(?, ?). Additionally, not all short, phonotactically possible words are used
in language (?, ?), contrary to the predictions of a random typing model, but
consistent with communicative theories based on entropy rate (PT&G) or
possibly the avoidance of confusable code words. In short, it is clear that
random typing models don’t even produce the correct detailed _statistical_
properties of language, although they may appear qualitatively similar with
respect to some coarse-grained properties.
### The importance of a model’s key assumptions
The implication of F&M (and before them, ? (?)) is that even though random
typing models are implausible descriptions of the generative process of
language and poor statistical theories, they still provide a null hypothesis
which should be considered in the course of scientific theorizing. Properties
of language that are also exhibited by random typing models should be looked
on cautiously, as phenomena which likely have a trivial and uninteresting
cause. In contrast, we believe that it is a fallacy to think that the fact a
random typing model exhibits a linguistic property should cast any doubt on
alternative theories, such as those proposed by PT&G and Zipf. The hypothesis
of random typing—and all models like them—have already been disproven by other
sources of evidence like the statistical and psychological existence of words
as memorized units.
We find this point interesting because it raises a difficult issue for
modeling. All models are inaccurate in that they do not exactly mirror the
“real” process happening in the world. Most models do or should attempt to
make their key assumption analogous to the key causal process at play in the
world. Thus, changing “good” models to make them more realistic will not break
their core predictions and properties. But F&M’s model is different: it cannot
be given knowledge of words—like people have—without destroying the behavior
F&M aim to explain. The key assumption of generating words by happening to
choose their components make the model critically _unlike_ people in terms of
representation, processing, and knowledge of language.
### Independent evidence for PT&G’s optimization process
Aside from discussions of the plausibility of various models, there are good
independent reasons for rejecting F&M’s assertion that the relationship
observed by PT&G is a statistical artifact. PT&G posited that the observed
relationships might result from lexicalization of phonetic reduction (e.g. ?,
?, ?). It is well-known that speakers shorten or reduce syllables in
predictable locations and PT&G’s findings plausibly result from these speech
production factors being integrated into lexical representations. If a word is
used in predictable locations, it will be reduced, and eventually might be
learned to be its shorter form, giving a relationship between word length and
predictability.
Second, there are independent behavioral studies showing that speakers
actually _do_ prefer short forms of words in predictable contexts, exactly as
PT&G’s theory would predict. ? (?) gave people a choice between two synonymous
pairs (e.g. “chimp”/“chimpanzee”) in either predictive or non-predictive
contexts. They found that people preferred the the short form when the word
was predictable and the long form was it was not. These kind of behavioral
tendencies have also been examined in corpus research by ? (?), who showed
that contracted forms (“do not” / “don’t”) occur more frequently in predictive
contexts. Such behavior is predicted by PT&G, but not explainable with F&M’s
view that the relationship between information content and word length is a
statistical artifact.
### Conclusion
Random typing models provide an interesting case study for considering what
modelers should want from models. Good models do not simply exhibit the
correct surface statistics; good models capture the right core assumptions of
the system under study, and show how the observed properties of the system
result from those properties. It is not informative to show that other
assumptions could also lead to the observed behavior, _if_ those other
assumptions are demonstrably not at play.
This means that one is not free to study _any_ conceivable statistical process
and conclude that it is relevant for understanding how language works. Models
under consideration must respect what is independently known about the system
under study. Since words are actively _chosen_ by language users to convey a
meaning, there is no point to studying models for which the uttered word is
generated according to some statistical properties of the wordform itself—that
is the wrong causal direction. As such, results about random typing models
only apply to systems that are critically unlike human language in terms of
the structure of language, knowledge of words, and the transmission of
meaningful information.
## References
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* BaayenBaayen Baayen, R. (2001). _Word frequency distributions_ (Vol. 1). Kluwer Academic Publishers.
* Bernhardsson, Baek, MinnhagenBernhardsson et al. Bernhardsson, S., Baek, S., Minnhagen, P. (2011). A paradoxical property of the monkey book. _Journal of Statistical Mechanics: Theory and Experiment_ , _2011_ , P07013.
* CohenCohen Cohen, A. (2006). Why ambiguity? In H.-M. Gaertner, S. Beck, R. Eckardt, R. Musan, B. Stiebels (Eds.), _Between 40 and 60 Puzzles for Manfred Krifka._
* Cohen, Mantegna, HavlinCohen et al. Cohen, A., Mantegna, R., Havlin, S. (1997). Numerical analysis of word frequencies in artificial and natural language texts. _Fractals-an Interdisciplinary Journal on the Complex Geometry_ , _5_(1), 95–104.
* Ferrer i Cancho ElvevågFerrer i Cancho Elvevåg Ferrer i Cancho, R., Elvevåg, B. (2010). Random Texts Do Not Exhibit the Real Zipf’s Law-Like Rank Distribution. _PLoS ONE_ , _5_(3).
* Ferrer i Cancho Moscoso del Prado MartínFerrer i Cancho Moscoso del Prado Martín Ferrer i Cancho, R., Moscoso del Prado Martín, F. (2011). Information content versus word length in random typing. _Journal of Statistical Mechanics: Theory and Experiment_ , _2011_ , L12002.
* Ferrer i Cancho SoléFerrer i Cancho Solé Ferrer i Cancho, R., Solé, R. (2002). Zipf’s law and random texts. _Advances in Complex Systems_ , _5_(1), 1–6.
* Frank JaegerFrank Jaeger Frank, A., Jaeger, T. (2008). Speaking rationally: Uniform information density as an optimal strategy for language production. In _Proceedings of the Cognitive Science Society._
* Genzel CharniakGenzel Charniak Genzel, D., Charniak, E. (2002). Entropy rate constancy in text. In _Proceedings of the 40th Annual Meeting on Association for Computational Linguistics_ (pp. 199–206).
* Genzel CharniakGenzel Charniak Genzel, D., Charniak, E. (2003). Variation of entropy and parse trees of sentences as a function of the sentence number. In _Proceedings of empirical methods in natural language processing_ (pp. 65–72).
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* JaegerJaeger Jaeger, T. (2010). Redundancy and reduction: Speakers manage syntactic information density. _Cognitive Psychology_ , _61_ , 23–62.
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* LiebermanLieberman Lieberman, P. (1963). Some effects of semantic and grammatical context on the production and perception of speech. _Language and Speech_ , _6_ , 172–187.
* Mahowald, Fedorenko, Piantadosi, GibsonMahowald et al. Mahowald, K., Fedorenko, E., Piantadosi, S., Gibson, E. (in press). Info/information theory: speakers actively choose shorter words in predictable contexts. _Cognition_.
* MandelbrotMandelbrot Mandelbrot, B. (1953). An informational theory of the statistical structure of language. _Communication theory_ , 486–502.
* MillerMiller Miller, G. (1957). Some effects of intermittent silence. _The American Journal of Psychology_ , 311–314.
* Miller ChomskyMiller Chomsky Miller, G., Chomsky, N. (1963). Finitary models of language users. _Handbook of mathematical psychology_ , _2_ , 419–491.
* MontemurroMontemurro Montemurro, M. (2001). Beyond the Zipf–Mandelbrot law in quantitative linguistics. _Physica A: Statistical Mechanics and its Applications_ , _300_(3), 567–578.
* Pearl, Goldwater, SteyversPearl et al. Pearl, L., Goldwater, S., Steyvers, M. (2011). Online learning mechanisms for bayesian models of word segmentation. _Research on Language & Computation_, 1–26.
* Piantadosi, Tily, GibsonPiantadosi et al. Piantadosi, S., Tily, H., Gibson, E. (2009). The communicative lexicon hypothesis. In _The 31st annual meeting of the Cognitive Science Society (CogSci09)_ (pp. 2582–2587).
* Piantadosi, Tily, GibsonPiantadosi et al. Piantadosi, S., Tily, H., Gibson, E. (2011). Word lengths are optimized for efficient communication. _Proceedings of the National Academy of Sciences_.
* Piantadosi, Tily, GibsonPiantadosi et al. Piantadosi, S., Tily, H., Gibson, E. (2012). The communicative function of ambiguity in language. _Cognition_ , _122_ , 280–291.
* ShannonShannon Shannon, C. (1948). _The mathematical theory of communication_. Urbana, IL: University of Illinois Press.
* Tripp FeitelsonTripp Feitelson Tripp, O., Feitelson, D. (1982). Zipf’s law re-visited. _Studies on Zipf’s law_ , 1–28.
* ZipfZipf Zipf, G. (1936). _The Psychobiology of Language_. London: Routledge.
|
arxiv-papers
| 2013-07-25T12:53:54 |
2024-09-04T02:49:48.446649
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Steven T. Piantadosi and Harry Tily and Edward Gibson",
"submitter": "Steven Piantadosi",
"url": "https://arxiv.org/abs/1307.6726"
}
|
1307.6731
|
# Evolution of the tangent vectors and localization
of the stable and unstable manifolds of hyperbolic orbits
by Fast Lyapunov Indicators
Massimiliano Guzzo
Dipartimento di Matematica
Via Trieste, 63 - 35121 Padova, Italy
[email protected]
Elena Lega
Université de Nice Sophia Antipolis, CNRS UMR 7293
Observatoire de la Côte d’Azur
Bv. de l’Observatoire, B.P. 4229, 06304 Nice cedex 4, France
[email protected]
###### Abstract
The Fast Lyapunov Indicators are functions defined on the tangent fiber of the
phase–space of a discrete (or continuous) dynamical system, by using a finite
number of iterations of the dynamics. In the last decade, they have been
largely used in numerical computations to localize the resonances in the
phase–space and, more recently, also the stable and unstable manifolds of
normally hyperbolic invariant manifolds. In this paper, we provide an analytic
description of the growth of tangent vectors for orbits with initial
conditions which are close to the stable-unstable manifolds of a hyperbolic
saddle point of an area–preserving map. The representation explains why the
Fast Lyapunov Indicator detects the stable-unstable manifolds of all fixed
points which satisfy a certain condition. If the condition is not satisfied, a
suitably modified Fast Lyapunov Indicator can be still used to detect the
stable-unstable manifolds. The new method allows for a detection of the
manifolds with a number of precision digits which increases linearly with
respect to the integration time. We illustrate the method on the critical
problem of detection of the so–called tube manifolds of the Lyapunov orbits of
$L_{1},L_{2}$ in the circular restricted three–body problem.
## 1 Introduction
Since the first detection of chaotic motions in 1964 (Henon–Heiles [17]),
several indicators have been largely used to characterize the different
dynamics of dynamical systems. Many dynamical indicators, such as the Lyapunov
characteristic exponents and the more recently introduced finite–time chaos
indicators (such as the Finite Time Lyapunov Exponent–FTLE [31], Fast Lyapunov
Indicator–FLI [7], Mean Exponential Growth of Nearby Orbits–MEGNO [4]), are
defined by the local divergence of nearby initial conditions, that is by the
variational dynamics. For example, for a discrete dynamical system defined by
the map
$\displaystyle\Phi:M$ $\displaystyle\longrightarrow$ $\displaystyle M$ (1)
$\displaystyle z$ $\displaystyle\longmapsto$ $\displaystyle\Phi(z),$ (2)
with $M\subseteq{\mathbb{R}}^{n}$ open invariant, by denoting with $D\Phi_{z}$
the tangent map of $\Phi$ at $z$:
$\displaystyle D\Phi_{z}:{\mathbb{R}}^{n}$ $\displaystyle\longrightarrow$
$\displaystyle{\mathbb{R}}^{n}$ (3) $\displaystyle v$
$\displaystyle\longmapsto$ $\displaystyle D\Phi_{z}v,$ (4)
the characteristic Lyapunov exponent of a point $z\in M$ and a vector
$v\in{\mathbb{R}}^{n}\backslash 0$ is defined by the limit
$\lambda(z,v)=\lim_{T\rightarrow+\infty}{1\over
T}\log{\left\|D\Phi^{T}_{z}v\right\|\over\left\|v\right\|},$ (5)
and the largest Lyapunov exponent of $z$ is the maximum of $\lambda(z,v)$ for
$v\neq 0$. As a matter of fact, the numerical estimation of the characteristic
Lyapunov exponents (see [2]) relies on extrapolation of finite time
computations, since computers cannot integrate on infinite time intervals. The
so–called finite–time chaos indicators (such as the FTLE, the FLI and the
MEGNO) have been afterwards introduced as surrogate indicators of the largest
Lyapunov exponent, with the aim to discriminate between regular orbits and
chaotic orbits using time intervals which are significantly smaller than the
time interval required for a reliable estimation of the largest characteristic
Lyapunov exponent ([7], [4]). For example, the function Fast Lyapunov
Indicator of $z$ and $v$ is simply defined by
$l_{T}(z,v)=\log{\left\|D\Phi^{T}_{z}v\right\|\over\left\|v\right\|},$ (6)
and depends parametrically on the integer $T>0$, as well as on the choice of a
norm on ${\mathbb{R}}^{n}$. The definition of finite time chaos indicators was
justified by the possibility of their systematic numerical computation over
large grids of initial conditions in the phase–space in a reasonable
computational time. We remark that, specifically in Celestial Mechanics, the
numerical detection of the resonances of a system using dynamical indicators,
both formulated using the Lyapunov exponent theory or alternatively the
Fourier analysis (such as the frequency analysis [19, 21, 20]), is one of the
major tools for studying its long–term instability (for recent examples, see
[27, 28, 26, 25, 8, 9, 33]). The papers [5],[11], focused and proved
properties of the finite time chaos indicators, specifically the FLI, which
are lost by taking the limit of $l_{T}(z,v)/T$, thus differentiating the use
of these indicators from the parent largest Lyapunov characteristic exponent.
Specifically, since [5],[11], the FLI has been used to discriminate regular
motions of different nature: for example the motions which are regular because
are supported by a KAM torus from the regular motions in the resonances of a
system. This property of the FLI improved a lot the precision in the numerical
localization of different types of resonant motions, the so–called Arnold web,
and provided the technical tool for the first numerical computations of
diffusion along the resonances of quasi–integrable systems in exponentially
long times [22, 12, 6, 14, 16], as depicted in the celebrate Arnold’s paper
[1].
More recently, the FLI has been successfully used to compute the stable and
unstable manifolds of normally hyperbolic invariant manifolds of the standard
map and its generalizations [10, 13], and of the three–body–problem [32, 23,
15]. In these cases it happens that, depending on the choice of the parameter
$T$, finite pieces of the stable and unstable manifolds appear as sharp local
maxima of the FLI. As a matter of fact, the possibility of sharp detection of
the stable and unstable manifolds of a fixed point, or periodic orbit, with a
FLI computation is not general and turns out to be a property of the
manifolds. A model example is represented by the stable and unstable manifold
of the fixed point $(0,0)$ of the symplectic map
$\Phi(\varphi,I)=\left(\varphi+I\ ,\
I+{\sin(\varphi+I)\over(\sigma\cos(\varphi+I)+2)^{2}}\right),$ (7)
where $(\varphi,I)\in M=(2\pi{\mathbb{S}}^{1})\times{\mathbb{R}}$ are the
phase–space variables, $\sigma=\pm 1$ is a parameter: for $\sigma=-1$ the FLI
may be used for excellent detection of the manifolds; for $\sigma=1$ the FLI
does not provide any detection.
To explain this fact, in this paper we provide a representation for the growth
of tangent vectors for orbits with initial conditions close to the stable
manifold of a saddle fixed point. To better illustrate the theory, we consider
a two dimensional area–preserving map with a saddle fixed point $z_{*}$, but
the techniques which we use (the local stable manifold theorem and Lipschitz
estimates) can be used also in the higher dimensional cases. The two
dimensional case allows us to treat also Poincaré sections of the circular
restricted three body problem.
Let us denote by $z_{*}$ the saddle point of the map, and by $W_{s},W_{u}$ its
stable and unstable manifold. We consider a point $z_{s}\in W_{s}$, a tangent
vector $v\in{\mathbb{R}}^{2}$, and we provide estimates about the norm of the
tangent vector $D\Phi^{T}_{z}v$, for points $z\notin W_{s}$ which are close to
$z_{s}$. As it is usual, the same arguments applied to the inverse map
$\Phi^{-1}$, allow to reformulate the result by exchanging the role of the
stable manifold with that of the unstable manifold. For the points $z$ which
are the suitably close to $z_{s}\in W_{s}$, the orbit $\Phi^{k}(z)$ follows
closely the orbit $\Phi^{k}(z_{s})$ for any $k\leq T$, and
$\left\|D\Phi^{k}_{z}v\right\|$ remains close to
$\left\|D\Phi^{k}_{z_{s}}v\right\|$ as well. The most interesting situation
happens for the points $z$ which are little more distant from the stable
manifold: their orbit (i) follows closely the orbit $\Phi^{k}(z_{s})$ only for
$k$ smaller than some $K_{0}<T$; (ii) then remains close to the hyperbolic
fixed point (for a number of iterations which increases logarithmically with
respect to some distance between $z$ and $z_{s}$, see Section 2), (iii) then
follows closely the orbit of a point on the unstable manifold $W_{u}$ in the
remaining iterations. It is during the process (iii) that the growth of the
tangent vector $\left\|D\Phi^{k}_{z}v\right\|$ can be significantly different
from the growth of $\left\|D\Phi^{k}_{z_{s}}v\right\|$, and the difference may
be possibly used to characterize the distance of $z$ from the stable manifold.
As a matter of fact, with evidence any difference may exist only due to the
non–linearity of the map $\Phi$. In Section 2 we provide a representation for
such a difference, and we discuss a condition which guarantees the desired
scaling of the FLI with respect to the distance of $z$ from the stable
manifold. If this condition is satisfied, the computation of the FLI on a grid
of initial conditions provides a sharp detection of the stable and unstable
manifolds (see Section 3): typically, the time $T$ used for the FLI
computation, which is the time needed by the orbits with initial condition $z$
to approach the fixed point $z_{*}$, turns out to be proportional to the
number of precision digits of the detection.
At the light of the representation provided in Section 2, we propose a
generalization of the FLI which weakens a lot the condition for the detection
of the stable and unstable manifold. For any smooth and positive function
$u:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}^{+}$
we define the modified FLI indicator of $z\in M$, $v\in{\mathbb{R}}^{2}$ at
time $T>0$, as the $T$–th element of the sequence
$l_{1}=\ln\left\|v\right\|\ \ ,\ \
l_{j+1}=l_{j}+u({z_{j}})\ln{\left\|D\Phi_{z_{j}}v_{j}\right\|\over\left\|v_{j}\right\|},$
(8)
where $z_{j}:=\Phi^{j}(z)$ and $v_{j}:=D\Phi^{j}_{z_{j}}v$. The traditional
FLI is obtained with the choice $u(z)=1$ for any $z\in M$. We consider the
alternative case of functions $u(z)$ which are test functions of some
neighbourhood ${\cal B}\subseteq M$ of the fixed point, and precisely with
$u(z)=1$ for $z\in{\overline{\cal B}}$, and $u(z)=0$ for $z$ outside a given
open set $V\supseteq{\overline{\cal B}}$. When the diameter of the set ${\cal
B}$ is small, but not necessarily extremely small, the computation of the
modified FLI indicator allows to refine the localization of the fixed point by
many orders of magnitude. Therefore, at variance with the traditional FLI
indicator, the modified indicators are proposed as a general tool for the
numerical detection of the stable and unstable manifolds. An illustration of
the potentialities of these indicators is given in Section 3, where we provide
computations of the stable and unstable manifolds and their heteroclinic
intersections, of the Lyapunov orbits around $L_{1}$, $L_{2}$ of the circular
restricted three–body problem. The application is particularly critical, since
these manifolds are located in a region of the phase–space close to the
singularity due to the secondary mass.
The paper is structured as follows. In Section 2 we provide the representation
for the evolution $D\Phi^{T}_{z}v$ of the norm of tangent vector $v$ for
points $z\notin W_{s}$ which are suitably close to the stable manifold, and we
also discuss a sufficient condition for the FLI to detect sharply the stable
and unstable manifolds of the map. In Section 3 we provide an illustration of
the method for the computation of the stable and unstable manifolds of the
Lyapunov orbits around $L_{1}$, $L_{2}$ of the circular restricted three body
problem; in Section 4 we provide the proof of Proposition 1. In Section 5 we
formulate and prove two technical lemmas. Finally, Conclusions are provided in
Section 6.
## 2 Evolution of the tangent vectors close to the stable manifolds of the
saddle points of two dimensional area–preserving maps
We consider a smooth two–dimensional area–preserving map:
$\Phi(z)=Az+f(z),$ (9)
where $A$ is a $2\times 2$ diagonal matrix with $A_{11}=\lambda_{u}>1$,
$A_{22}=1/\lambda_{u}$ and $f$ is at least quadratic in $z_{1},z_{2}$, that is
$f_{i}(0,0)=0$ and ${\partial f_{i}\over\partial z_{j}}(0,0)=0$, for any
$i,j=1,2$. Therefore, the origin is a saddle fixed point.
We need to introduce some constants which characterize the analytic properties
of $\Phi$. We denote by $\lambda_{\Phi},\lambda_{\Phi^{-1}},\lambda_{D\Phi}$
the Lipschitz constants of $\Phi,\Phi^{-1},D\Phi$ respectively defined with
respect to the norm
$\left\|u\right\|:=\max\\{\left|u_{1}\right|,\left|u_{2}\right|\\}$, in the
set $B(R)=\\{z:\ \left\|z\right\|\leq R\\}$. Also, we set $\eta$ such that,
for any $z\in B(R)$, we have
$\left\|f(z)\right\|\leq\eta\left\|z\right\|^{2}\ \ ,\ \
\left\|Df_{z}\right\|\leq\eta\left\|z\right\|\ \ ,\ \
\left\|D^{2}f_{z}\right\|\leq\eta$
$\left\|f(z^{\prime})-f(z^{\prime\prime})\right\|\leq\eta\max\\{\left\|z^{\prime}\right\|,\left\|z^{\prime\prime}\right\|\\}\left\|z^{\prime}-z^{\prime\prime}\right\|,$
where $D^{2}f_{z}$ denotes the Hessian matrix of $f$ at the point $z$ and, by
denoting with $\Phi^{-1}(z)=A^{-1}z+{\tilde{f}}(z)$ the inverse map, we also
have
$\left\|\tilde{f}(z)\right\|\leq\eta\left\|z\right\|^{2}\ \ ,\ \
\left\|D{\tilde{f}}_{z}\right\|\leq\eta\left\|z\right\|\ \ ,\ \
\left\|D^{2}{\tilde{f}}_{z}\right\|\leq\eta$
$\left\|{\tilde{f}}(z^{\prime})-{\tilde{f}}(z^{\prime\prime})\right\|\leq\eta\max\\{\left\|z^{\prime}\right\|,\left\|z^{\prime\prime}\right\|\\}\left\|z^{\prime}-z^{\prime\prime}\right\|.$
Moreover, since $\Phi$ is a diffeomorphism, we have
$\sigma=\min_{z\in B(R)}\min_{\left\|v\right\|=1}\left\|D\Phi_{z}v\right\|>0.$
(10)
By the local stable manifold theorem, we consider e neighbourhood $B(r_{*})$
of the origin where the local stable and unstable manifolds
$W^{l}_{s},W^{l}_{u}$ are Cartesian graphs over the $z_{2}$ and $z_{1}$ axes
respectively, that is
$W^{l}_{s}=\left\\{z:\left|z_{2}\right|\leq r_{*}\ \ ,\ \
z_{1}=w_{s}(z_{2})\right\\}$ $W^{l}_{u}=\left\\{z:\left|z_{1}\right|\leq
r_{*}\ \ ,\ \ z_{2}=w_{u}(z_{1})\right\\}$
with $w_{s}(0)=w_{u}(0)=0$, $w^{\prime}_{s}(0)=w^{\prime}_{u}(0)=0$ and, by
possibly increasing $\eta$,
$\left|w_{s}(z_{2})\right|\leq\eta\left|z_{2}\right|^{2}\ \ ,\ \
\left|w_{u}(z_{1})\right|\leq\eta\left|z_{1}\right|^{2}$
and
$\left|w_{s}(\xi^{\prime})-w_{s}(\xi^{\prime\prime})\right|\leq\lambda_{w}\max\\{\left|\xi^{\prime}\right|,\left|\xi^{\prime\prime}\right|\\}\left|\xi^{\prime}-\xi^{\prime\prime}\right|$
$\left|w_{u}(\xi^{\prime})-w_{u}(\xi^{\prime\prime})\right|\leq\lambda_{w}\max\\{\left|\xi^{\prime}\right|,\left|\xi^{\prime\prime}\right|\\}\left|\xi^{\prime}-\xi^{\prime\prime}\right|.$
We denote by $W_{s},W_{u}$ the stable and unstable manifolds of the origin. We
consider a point $z_{s}\in W_{s}$, a tangent vector $v\in{\mathbb{R}}^{2}$,
and we provide estimates about the norm of the tangent vector
$D\Phi^{T}_{z}v$, for points $z\notin W_{s}$ which are suitably close to
$z_{s}$, precisely in a curve $z_{\varepsilon}$, with $z_{0}=z_{s}$ and
$\left\|z-z_{\varepsilon}\right\|=\varepsilon$.
Figure 1: Illustration of $z_{s},z_{\varepsilon}$; of
$\Phi^{T_{s}}(z_{\varepsilon})$ and its parallel projection
$\pi_{\varepsilon}$ on the local stable manifold; of
$\Phi^{T_{s}+T_{\varepsilon}}(z_{\varepsilon})$ and its parallel projection
$\zeta_{\varepsilon}$ on the local unstable manifold.
Let us consider a small $\delta:={\delta_{0}\over T}$, with $\delta_{0}$
satisfying
$\delta_{0}\leq\min\left({1\over
16\max(1,\eta)^{2}e^{3}\lambda_{u}^{2}}\left(1-{1\over\lambda_{u}}\right),{r_{*}\over
2}\right).$
Then, we consider the minimum $T_{s}:=T_{s}(\delta)$ such that
$\Phi^{T_{s}}(z_{s})\in B(\delta-2\delta^{2})$. Typically, one has
$T_{s}\sim\ln(1/\delta)$. For all $\varepsilon$, we have (see Lemma 5.2):
$\left\|\Phi^{T_{s}}(z_{\varepsilon})-\Phi^{T_{s}}(z_{s})\right\|\leq\lambda_{\Phi}^{T_{s}}\varepsilon$
(11)
$\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v-D\Phi^{T_{s}}_{z_{s}}v\right\|\leq\left\|D\Phi^{T_{s}}_{z_{s}}v\right\|\lambda^{T_{s}}\varepsilon,$
(12)
where
$\lambda=\max(\lambda_{\Phi},(\left\|D\Phi\right\|+\lambda_{D\Phi})/\sigma)$.
We consider only the small $\varepsilon$ satisfying
$\lambda^{T_{s}}\varepsilon<\delta^{2}$, so that
$\Phi^{T_{s}}(z_{\varepsilon})\in B(\delta-\delta^{2})$, are close to
$\Phi^{T_{s}}(z_{s})$ and $\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v\right\|$
are close to $\left\|D\Phi^{T_{s}}_{z_{s}}v\right\|$. We rename the vector
$D\Phi^{T_{s}}_{z_{s}}v$ as follows:
$w=w_{s}+w_{u}=D\Phi^{T_{s}}_{z_{s}}v,$
where $w_{s},w_{u}$ are the orthogonal projections of $w$ over the stable and
unstable spaces of the matrix $A$, i.e. the $z_{2}$ and $z_{1}$ axes,
respectively. We need a condition which ensures that $v$ is not close to some
special contracting direction. Precisely, we assume that the initial vector
$v$ is such that
$\left\|w_{s}\right\|\leq\left\|w_{u}\right\|=\left\|w\right\|.$
In particular, for any $k\geq 0$, we have
$\left\|A^{k}w\right\|=\lambda_{u}^{k}\left\|w_{u}\right\|$. Let us denote by
$\pi_{\varepsilon}=\Big{(}w_{s}(\Phi^{T_{s}}_{2}(z_{\varepsilon})),\Phi^{T_{s}}_{2}(z_{\varepsilon})\Big{)}\in
W^{l}_{s}$
the parallel projection of $\Phi^{T_{s}}(z_{\varepsilon})$ on the local stable
manifold (see figure 1), that is the point on $W^{l}_{s}$ with
$z_{2}=\Phi^{T_{s}}_{2}(z_{\varepsilon})$, and by
$\Delta_{\varepsilon}=\left|\Phi^{T_{s}}_{1}(z_{\varepsilon})-w_{s}(\Phi^{T_{s}}_{2}(z_{\varepsilon}))\right|$
the distance between $\Phi^{T_{s}}(z_{\varepsilon})$ and the point
$\pi_{\varepsilon}$. Since $\Delta_{\varepsilon}$ depends continuously on
$\varepsilon$, $\Delta_{0}=0$, and the local stable manifold is invariant,
there exists $\varepsilon_{1}$ such that $\Delta_{\varepsilon}$ is strictly
monotone increasing function of $\varepsilon\in[0,\varepsilon_{1}]$. We have
also (see Section 4):
$\Delta_{\varepsilon}\leq(1+\lambda_{w})\lambda_{\Phi}^{T_{s}}\varepsilon,$
(13)
so that if $(1+\lambda_{w})\lambda^{T_{s}}\varepsilon<\delta^{2}$ we have
$\pi_{\varepsilon}\in B(\delta)$. We use $\Delta_{\varepsilon}$ to
parameterize the distance of $z_{\varepsilon}$ from the stable manifold
$W_{s}$, and we introduce the time
$T_{\varepsilon}=\left[{1\over\ln\lambda_{u}}{\ln{e\delta\over\Delta_{\varepsilon}}}\right]$
(14)
which, as we will prove (see Lemma 4.1), is required by the orbit with initial
condition $\Phi^{T_{s}}(z_{\varepsilon})$ to exit from $B(\delta)$. We also
denote by
$\zeta_{\varepsilon}=\Big{(}\Phi^{T_{s}+T_{\varepsilon}}_{1}(z_{\varepsilon}),w_{u}(\Phi^{T_{s}+T_{\varepsilon}}_{1}(z_{\varepsilon}))\Big{)}\in
W^{l}_{u}$
the parallel projection of $\Phi^{T_{s}+T_{\varepsilon}}_{1}(z_{\varepsilon})$
over the local unstable manifold.
###### Proposition 1
Let us consider any large $T$ satisfying
$\displaystyle e\delta\lambda_{u}^{-\alpha(T-T_{s})}$ $\displaystyle\leq$
$\displaystyle\Delta_{\varepsilon_{1}}$ (15) $\displaystyle
e\lambda_{u}^{-\alpha(T-T_{s})}$ $\displaystyle\leq$
$\displaystyle{\sigma^{T_{s}}\delta_{0}\over\max(1,\eta)(1+\lambda_{w})\lambda^{T_{s}}T^{2}}$
(16) $\displaystyle T$ $\displaystyle>$ $\displaystyle T_{s}+{1\over
1-\alpha}$ (17)
with
$\alpha={\ln\lambda\over\ln\lambda+\ln\lambda_{u}}.$
By denoting with $\varepsilon_{0}$ the constant such that
$\Delta_{\varepsilon_{0}}=e\delta\lambda_{u}^{-\alpha(T-T_{s})},$ (18)
then, for any $\varepsilon\leq\varepsilon_{0}$, if $T_{\varepsilon}\geq
T-T_{s}$ we have
$\left\|D\Phi^{T}_{z_{\varepsilon}}v-A^{T-T_{s}}w\right\|\leq\lambda_{u}^{T-T_{s}}{\left\|w_{u}\right\|\over
T}\ \ ,\ \ w=D\Phi^{T_{s}}_{z_{s}}v,$ (19)
if $\alpha(T-T_{s})\leq T_{\varepsilon}<T-T_{s}$ we have
${\left\|D\Phi^{T}_{z_{\varepsilon}}v\right\|\over\left\|D\Phi^{T}_{z_{s}}v\right\|}\leq\left(1+{1\over
T}\right){\left\|D\Phi^{j}_{\zeta_{\varepsilon}}\right\|\over\lambda_{u}^{j}}\
\ ,\ \ j=T-T_{s}-T_{\varepsilon}.$ (20)
The proof is reported in Section 4.
Remark. Conditions (15), (16) and (17) may be all satisfied by times $T$ which
are suitably large, but not necessarily extremely large, because of the
presence of the exponentials in (15) and (16), and because of the typical
dependence $T_{s}(\delta)\sim\ln(1/\delta)\sim\ln T$. Therefore, the
proposition is meaningful also for $\varepsilon_{0}$ which are small, but not
necessarily extremely small. Moreover, from the definition of
$\varepsilon_{0}$, apart from a small difference due to the use of the integer
part in the definition of $T_{\varepsilon}$, we have
$T_{\varepsilon_{0}}\sim\alpha(T-T_{s})$, and $T-T_{s}-T_{\varepsilon}\leq
T_{u}:=(T-T_{s})(1-\alpha)$. $\Box$
For $z_{s}\in W_{s}$, and for all the points $z_{\varepsilon}$ which are so
close to the stable manifold that $T_{\varepsilon}\geq T-T_{s}$, the FLI is
approximated by
$\ln\left\|A^{T-T_{s}}w\right\|=(T-T_{s})\ln\lambda_{u}+\ln\left\|w_{u}\right\|.$
Therefore, the only possibility for the FLI to strongly decrease by increasing
$\varepsilon$ is that, for $\alpha(T-T_{s})\leq T_{\varepsilon}<T-T_{s}$, we
have an exponential decrement of
$\left\|D\Phi^{j}_{\zeta_{\varepsilon}}\right\|/\lambda_{u}^{j}$ with respect
to $j$. The assumption which guarantees a desired scaling of the FLI with
respect to $\varepsilon$ is
$\sup_{\varepsilon:\alpha(T-T_{s})\leq T_{\varepsilon}\leq
T-T_{s}}{\left\|D\Phi^{T-T_{s}-T_{\varepsilon}}_{\zeta_{\varepsilon}}\right\|\over(C\lambda_{u})^{T-T_{s}-T_{\varepsilon}}}\leq
1$ (21)
with some $C<1$, so that we have
$\ln\left\|D\Phi^{T}_{z_{\varepsilon}}v\right\|\leq\ln\left\|D\Phi^{T}_{z_{s}}v\right\|-(T-T_{s}-T_{\varepsilon})\left|\ln
C\right|+\ln\left(1+{1\over T}\right).$
From the definition of $T_{\varepsilon}$, we have therefore a linear decrement
of the FLI with respect to $\ln\Delta_{\varepsilon}$, up to the maximum value
of $T-T_{s}-T_{\varepsilon}\leq(1-\alpha)(T-T_{s})$. Therefore, at the
exponentially small distance from the manifold (18) the FLI has decreased of a
quantity which is proportional to integration time $T$, and conversely, the
differences of units in the FLI value typically determines a proportional
number of precision digits in the localization of the stable manifold.
With evidence, condition (21) may be satisfied if $\left\|D\Phi_{z}\right\|$
has an absolute maximum for $z\in\cup_{k\leq T_{u}}\Phi^{-k}(W_{u}^{l})$. For
example, the condition may be satisfied for the map (7) with $\sigma=-1$,
since the origin is a local strict maximum for $\left\|D\Phi_{z}\right\|$,
$z\in W_{u}$, while it is not satisfied for $\sigma=1$, since in this case the
origin is a local strict minimum for $\left\|D\Phi_{z}\right\|$, $z\in W_{u}$.
In any case, it is not practical to verify if condition (21) is satisfied by a
certain choices of the parameters. Therefore, at the light of the above
analysis, we consider a generalization of the FLI indicators which depend on a
function
$u:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}^{+}$
as follows: let us consider $z\in M$, $v\in{T_{z}M}$, and $T>0$. Then, we
consider $l_{T}(z,v)$ defined as the $T$–th element of the sequence
$l_{1}=\ln\left\|v\right\|\ \ ,\ \
l_{j+1}=l_{j}+u({z_{j}})\ln{\left\|D\Phi_{z_{j}}v_{j}\right\|\over\left\|v_{j}\right\|},$
(22)
where $z_{j}:=\Phi^{j}(z)$ and $v_{j}:=D\Phi^{j}_{z_{j}}v$. The usual FLI is
obtained by $u(z)=1$ for any $z\in M$. We consider the alternative case of
functions $u(z)$ which are test functions of some neighbourhood ${\cal
B}\subseteq M$ of the fixed point, and precisely with $u(z)=1$ for
$z\in{\overline{\cal B}}$, and $u(z)=0$ for $z$ outside a given open set
$V\supseteq{\overline{\cal B}}$. We remark that the set ${\cal B}$ needs to be
small, but not necessarily extremely small. For example, if ${\cal B}\subseteq
B(\delta)$, we only need, in $V\backslash{\cal B}$,
$\left\|D\Phi_{z}\right\|^{u(z)}\leq C\lambda_{u}$
for some $C<1$. The function $u$ described above depends on a specific
hyperbolic fixed point. If one is interested in the stable or unstable
manifolds of more fixed points (or hyperbolic periodic orbits), with the same
numerical integration of the variational equations, forward and backward in
time, one may compute the FLI indicators related to the different fixed points
without increasing significantly the computational time, and use the results
to find, for example, homoclinic and heteroclinic intersections between the
different manifolds. If instead, one is interested in determining with a
single numerical integration the largest number of manifolds in some finite
domain $B$, one can divide the domain $B$ in many small sets ${\cal B}_{j}$,
$j\leq N$, and compute the $N$ indicators FLIj adapted to the sets ${\cal
B}_{j}$. This procedure increases the computational time only logarithmically
with $N$, since the time required for the numerical localization of a point in
one of the sets ${\cal B}_{j}$ increases logarithmically with $N$. Then, the
portrait of all the manifolds is obtained by representing, for any initial
condition, the maximum between all the FLIj. Therefore, at variance with the
traditional FLI indicator, the modified indicators are proposed as a general
tool for the numerical detection of the stable and unstable manifolds.
## 3 A numerical example: the tube manifolds of $L_{1}$ and $L_{2}$ in the
planar circular restricted three body problem
The circular restricted three-body problem describes the motion of a massless
body $P$ in the gravitation field of two massive bodies $P_{1}$ and $P_{2}$,
called primary and secondary body respectively, which rotate uniformly around
their common center of mass. In a rotating frame $xOy$, the equations of
motion of $P$ are:
$\left\\{\begin{array}[]{rcl}\ddot{x}&=&2\dot{y}+x-(1-\mu)\frac{x+\mu}{r_{1}^{3}}-\mu\frac{x-1+\mu}{r_{2}^{3}}\\\
\ddot{y}&=&-2\dot{x}+y-(1-\mu)\frac{y}{r_{1}^{3}}-\mu\frac{y}{r_{2}^{3}}\\\
\end{array}\right.$ (23)
where the units of masses, lengths and time have been chosen so that the
masses of $P_{1}$ and $P_{2}$ are $1-\mu$ and $\mu$ ($\mu\leq 1/2$)
respectively, their coordinates are $(-\mu,0)$ and $(1-\mu,0)$ and their
revolution period is $2\pi$. We denoted by $r_{1}^{2}=(x+\mu)^{2}+y^{2}$ and
by $r_{2}^{2}=(x-1+\mu)^{2}+y^{2}$. As it is well known, equations (23) have
an integral of motion, the so–called Jacobi constant, defined by:
${\cal
C}(x,y,\dot{x},\dot{y})=x^{2}+y^{2}+2\frac{1-\mu}{r_{1}}+2\frac{\mu}{r_{2}}-\dot{x}^{2}-\dot{y}^{2},$
(24)
and five equilibria usually denoted by $L_{1},\ldots,L_{5}$. Here we consider
$\mu=0.0009537$, which corresponds to the Jupiter–Sun mass ratio value, and a
value of the Jacobi constant slightly smaller than ${\cal
C}(x_{L_{2}},0,0,0):=C_{2}$. As it is extensively explained in [18], in these
conditions, one may find particularly interesting dynamics, which we briefly
summarize. The equilibrium points $L_{1},L_{2}$ are partially hyperbolic, and
their center manifolds $W^{c}_{L_{1}},W^{c}_{L_{2}}$ are two–dimensional, and
foliated near $L_{1},L_{2}$ respectively by periodic orbits called Lyapunov
orbits. For values $C$ of the Jacobi constant slightly smaller than $C_{2}$,
there exist one Lyapunov orbit related to $L_{1}$ and one Lyapunov orbit
related to $L_{2}$ respectively with Jacobi constant equal to $C$ (see figure
2).
Figure 2: Projection on the plane x-y of the Lyapunov orbits related to the
points $L_{1}$ and $L_{2}$, for the value
$C=3.03685733643946038606918461928938$ of the Jacobi constant. The shaded area
represents a region of the orbit plane which is forbidden for this value of
the Jacobi constant.
The Lyapunov orbits are hyperbolic, and transverse intersections of their
stable and unstable manifolds–usually called tube manifolds– produce the
complicate dynamics related to the heteroclinic chaos. The numerical
computation of the tube manifolds has been afforded in several papers, and has
important implications also for modern space mission design (see [29], [18]).
In this Section we analyze the FLI method for the detection of the tube
manifolds introduced in [24, 15] at the light of the theoretical analysis
performed in Section 2, and we show that the method allows for a detection of
the manifolds with a number of precision digits which increase linearly with
respect to the integration time. Moreover, the modified FLI allows us to
compute the manifolds with a precision limited only by the round–off of the
numerical computations.
We report here three numerical experiments. In the first one we illustrate the
numerical precision of the FLI method in the determination of the stable tube
manifold of a Lyapunov periodic orbit around $L_{1}$; in the second one, we
provide some snapshots of the stable tube manifold of the Lyapunov periodic
orbit around $L_{2}$ and the unstable tube manifold of the Lyapunov periodic
orbit around $L_{1}$, obtained by extending the integration time; in the third
one we illustrate the numerical precision of the FLI method for the
localization of a heteroclinic intersection between these two manifolds. We
remark that these computations are particularly critical since the tube
manifolds are located in a region of the phase space close to the singularity
at $(x,y)=(1-\mu,0)$. In these circumstances, the numerical computation of
both equations of motions (23) and their variational equations becomes
critical, and several approaches have been introduced (see [32, 23, 3, 15]).
For the computation of the tube manifolds, we find particularly useful to
define the variational equation in the space of the variables obtained by
regularizing equations (23) with respect to the secondary mass, as in [3, 15].
Precisely, we consider the Levi–Civita regularization defined by the space
transformation
$\left\\{\begin{array}[]{lll}x-(1-\mu)&=&u_{1}^{2}-u_{2}^{2}\\\
y&=&2u_{1}u_{2}\\\ \end{array}\right.$ (25)
and by the fictitious time $s$ related to $t$ by $dt=r_{2}ds$. The equations
of motion in the variables $u_{1},u_{2}$, and fictitious time $s$ are (see for
example [30]):
$\left\\{\begin{array}[]{lll}u_{1}^{\prime\prime}&=&{1\over
4}[(a+b)u_{1}+cu_{2}]\\\ u_{2}^{\prime\prime}&=&{1\over
4}[(a-b)u_{2}+cu_{1}]\\\ \end{array}\right.$ (26)
with:
$\left\\{\begin{array}[]{lll}a&=&{\frac{2(1-\mu)}{r_{1}}}-C+x^{2}+y^{2}\\\
b&=&4y^{\prime}+2r_{2}x-{\frac{2(1-\mu)r_{2}(x-1+\mu)}{r_{1}^{3}}}\\\
c&=&2r_{2}y-4x^{\prime}-{\frac{2(1-\mu)r_{2}y}{r_{1}^{3}}}\end{array}\right.$
(27)
where $C$ denotes the value of the Jacobi constant, and the primed derivatives
denote derivatives with respect the fictitious time $s$. To define the FLI, we
first write (26) as a system of first order differential equations:
$\left\\{\begin{array}[]{lll}u^{\prime}_{1}&=&v_{1}\\\
u^{\prime}_{2}&=&v_{2}\\\ v_{1}^{\prime}&=&{1\over 4}[(a+b)u_{1}+cu_{2}]\\\
v_{2}^{\prime}&=&{1\over 4}[(a-b)u_{2}+cu_{1}]\\\ \end{array}\right.$ (28)
and we introduce its compact form:
$\xi^{\prime}=F(\xi)\\\ $ (29)
with $\xi=(u_{1},u_{2},v_{1},v_{2})$. The variational equations of (29) are
therefore:
$\left\\{\begin{array}[]{lcr}&\xi^{\prime}=F(\xi)&\cr&w^{\prime}={\partial
F\over\partial\xi}(\xi)w&,\end{array}\right.$ (30)
where $w\in\mathbb{R}^{4}$ represents a tangent vector. Following [15], we
here consider the regularized FLI indicator defined by
$FLI(\xi(0),w(0),T)=\log\left\|w({s(T)})\right\|$ (31)
where $\xi(s),w(s)$ denotes the solution of the variational equations (30)
with initial condition $\xi(0),w(0)$ and $s(T)$ is the fictitious time which
corresponds to the physical time $T$ for that orbit. The indicator (31) will
be computed also for negative times $T<0$.
FLI detection of the tube manifolds. In order to test the precision of the FLI
method in the localization of the tube manifolds, we consider a point
$z_{s}=(x_{s},y_{s},\dot{x}_{s},\dot{y}_{s})\in W^{s}_{L_{1}}$ in the stable
tube manifold of the Lyapunov orbit around $L_{1}$ (see Figure 3), and we
compute the traditional and modified FLIs for a set of many initial
conditions. with $(x(0),y(0))=(x_{s},\dot{x}_{s})$ (see Fig.3),
$\log\left|y(0)-y_{s}\right|$ in the interval $[-25,-1]$ and $y(0)$ obtained
from the value of the Jacobi constant $C=3.03685733643946038606918461928938$.
The integration times are respectively $T=15$ and $T=25$. We appreciate a
localization of the manifold determined by a linear decrement of the FLI with
respect to $\log\left|y(0)-y_{s}\right|$. The time $T=15$ allows us to
localize the manifold with a precision of order $10^{-15}$, which is greatly
improved by using $T=25$. We obtain a good localization of the manifold
already with the traditional FLI, see Figure 4, although the irregularities in
the FLI curve limit the precision of the localization to $10^{-22}$, higher
than the numerical round–off precision.
Then, we considered a modified FLI defined by equations (8) with function
$u(z)$ which is a test function of a neighbourhood of the Lyapunov orbit
$\gamma_{1}$ around $L_{1}$. Precisely, we use a test function defined by:
$u(z)=\left\\{\begin{array}[]{lcr}&1&\ {\rm if}\ \
\left|z-\gamma_{1}\right|\leq{r_{1}\over 2}\\\ &{1\over
2}[{\cos(({\left|z-\gamma_{1}\right|\over r_{1}}-{1\over 2})\pi)+1}]&{\rm if}\
\ {r_{1}\over 2}<\left|z-\gamma_{1}\right|\leq{3r_{1}\over 2}\\\ &0&\ {\rm
if}\ \ \left|z-\gamma_{1}\right|>{3r_{1}\over 2}\end{array}\right.$ (32)
where $\left|z-\gamma_{1}\right|$ denotes the distance between $z$ and the
Lyapunov orbit $\gamma_{1}$ (we set $r_{1}=10^{-3}$ in the following
computations). Also in this case the time $T=15$ allows us to localize the
manifold with a precision of order $10^{-15}$, while the time $T=25$ allows us
to localize the manifold more precisely than $10^{-25}$. The use of the
modified FLI has eliminated the irregularities in the curves of Figure 4, and
improved the precision of the localization. As a matter of fact, the precision
of the localization is reduced to the round–off used for the numerical
computation.
Figure 3: Projection on the plane $(x,y)$ of an orbit with initial condition
$z_{s}=(x_{s},y_{s},\dot{x}_{s},\dot{y}_{s})\in W^{s}_{L_{1}}$, with
$x^{s}=0.687020836763335598413507147121355$,
$y^{s}=-0.227669455733293321520979535995733$,
$\dot{x}^{s}=0.331597964276881596512604348842892$, and $\dot{y}^{s}$ obtained
from the Jacobi constant $C=3.03685733643946038606918461928938$. The shaded
area represents a region of the orbit plane which is forbidden for the value
$C$ of the Jacobi constant. Figure 4: Values of the traditional FLI computed
on a set of 960 initial conditions with $(x(0),y(0))=(x_{s},\dot{x}_{s})$ (see
Fig.3), $\log\left|y(0)-y_{s}\right|$ in the interval $[-25,-1]$ and
$\dot{y}(0)$ obtained from the Jacobi constant
$C=3.03685733643946038606918461928938$. The integration times are respectively
$T=15$ and $T=25$, (the negative values correspond to initial conditions with
$y(0)<y^{s}$). We appreciate a localization of the manifold determined by a
linear decrement of the FLI with respect to $\log\left|y(0)-y_{s}\right|$. The
time $T=15$ allows us to localize the manifold with a precision of order
$10^{-15}$, while the time $T=25$ allows us to localize the manifold more
precisely than $10^{-22}$. Figure 5: Values of the modified FLI defined by
equations (8) with function $u(z)$ which is a test function of a neighbourhood
of the Lyapunov orbit around $L_{1}$. The initial conditions are the same 960
initial conditions considered in Figure 4, that is
$(x(0),y(0))=(x_{s},\dot{x}_{s})$ (see Fig.3), $\log\left|y(0)-y_{s}\right|$
in the interval $[-25,-1]$ and $y(0)$ obtained from the Jacobi constant
$C=3.03685733643946038606918461928938$. The integration times are respectively
$T=15$ and $T=25$, (the negative values correspond to initial conditions with
$y(0)<y_{s}$). We appreciate a localization of the manifold determined by a
linear decrement of the FLI with respect to $\log\left|y(0)-y_{s}\right|$. The
time $T=15$ allows us to localize the manifold with a precision of order
$10^{-15}$, while the time $T=25$ allows us to localize the manifold more
precisely than $10^{-25}$. The use of the modified FLI has eliminated the
irregularities in the curves of Figure 4, and improved he precision of the
localization. As a matter of fact, the precision of the localization is
reduced to the round–off used for the numerical computation.
Snapshots of tube manifolds of $W^{u}_{L_{1}}$ and $W^{s}_{L_{2}}$. Motivated
by these results, we obtained sharp representations of the intersections
$W^{s}_{L_{2}}\cap\Sigma\ \ ,\ \ W^{u}_{L_{1}}\cap\Sigma$
of the stable tube manifold $W^{s}_{L_{2}}$ of the Lyapunov orbit $\gamma_{2}$
around $L_{2}$ and of the unstable tube manifold $W^{u}_{L_{1}}$ of the
Lyapunov orbit $\gamma_{1}$ around $L_{1}$ with the two–dimensional section of
the phase–space defined by
$\Sigma=\\{(x,y,\dot{x},\dot{y}):\ \ y=0\ \ ,\ \ \dot{y}\geq 0:\ \ {\cal
C}(x,0,\dot{x},\dot{y})=C\\}.$ (33)
Any point $z\in\Sigma$ is parameterized and identified by its two components
$(x,\dot{x})$. The representation of the manifolds are obtained by computing
the modified FLIs on refined grids of initial conditions $(x,\dot{x})$ on
$\Sigma$ for different integration times $T$. The stable manifold
$W^{s}_{L_{2}}$ is obtained by computing the modified FLI on a time $T_{2}$,
using a test function defined by
$u(z)=\left\\{\begin{array}[]{lcr}&1&\ {\rm if}\ \
\left|z-\gamma_{2}\right|\leq{r_{2}\over 2}\\\ &{1\over
2}[{\cos(({\left|z-\gamma_{2}\right|\over r_{2}}-{1\over 2})\pi)+1}]&{\rm if}\
\ {r_{2}\over 2}<\left|z-\gamma_{2}\right|\leq{3r_{2}\over 2}\\\ &0&\ {\rm
if}\ \ \left|z-\gamma_{2}\right|>{3r_{2}\over 2}\end{array}\right.$ (34)
where $\left|z-\gamma_{2}\right|$ denotes the distance between $z$ and the
Lyapunov orbit $\gamma_{2}$ and $r_{2}=5\,10^{-4}$. The unstable manifold
$W^{u}_{L_{1}}$ is obtained by computing the modified FLI on a negative time
$-T_{1}$, using a test function defined by
$u(z)=\left\\{\begin{array}[]{lcr}&1&\ {\rm if}\ \
\left|z-\gamma_{1}\right|\leq{r_{1}\over 2}\\\ &{1\over
2}[{\cos(({\left|z-\gamma_{1}\right|\over r_{1}}-{1\over 2})\pi)+1}]&{\rm if}\
\ {r_{1}\over 2}<\left|z-\gamma_{1}\right|\leq{3r_{1}\over 2}\\\ &0&\ {\rm
if}\ \ \left|z-\gamma_{1}\right|>{3r_{1}\over 2}\end{array}\right.$ (35)
where $\left|z-\gamma_{1}\right|$ denotes the distance between $z$ and the
Lyapunov orbit $\gamma_{1}$ and $r_{1}=10^{-3}$. In such a way, for any
$x,\dot{x}$ we compute the modified FLIs: FLI1, FLI2. The representation of
both manifolds on the same picture is obtained by representing with a color
scale a weighted average of the two indicators:
${w{\rm FLI}_{1}+{\rm FLI}_{2}\over(w+1)}.$ (36)
The results are represented in Figures 6 and 7 for $T=5$ and $T=100$
respectively. We clearly appreciate different lobes of both manifolds already
for the shorter integration time $T=5$. The longer time $T=100$ allows us to
appreciate additional lobes, which contain initial condition approaching the
manifolds only after several revolution periods of Jupiter.
Figure 6: Representation of the modified FLIs computed on a grid of
$4000\times 4000$ initial conditions regularly spaced on $(x,\dot{x})$ (the
axes on the picture–the other initial conditions are $y=0$ and $\dot{y}$ is
computed from the Jacobi constant $C=3.0368573364394607$), computed with
integration time $T=5$. In order to represent both manifolds on the same
picture, we represent with a color scale the weighted average (36) of the two
indicators ${\rm FLI}_{1}$, ${\rm FLI}_{2}$ with weight $w=100$. The yellow
curves on the picture correspond to different lobes of the manifolds. Figure
7: Representation of the modified FLIs computed on a grid of $4000\times 4000$
initial conditions regularly spaced on $(x,\dot{x})$ (the axes on the
picture–the other initial conditions are $y=0$ and $\dot{y}$ is computed from
the Jacobi constant $C=3.0368573364394607$), computed with an integration time
$T=100$. In order to represent both manifolds on the same picture, we
represent with a color scale the weighted average (36) of the two indicators
${\rm FLI}_{1}$, ${\rm FLI}_{2}$ with weight $w=500$. The yellow curves on the
picture correspond to different lobes of the manifolds. Due to the integration
time which is much longer than the time used in Figure 6, many additional
lobes of the tube manifolds of both $\gamma_{1}$ and $\gamma_{2}$ appear on
this figure. Their corresponding initial conditions approach the manifolds
only after several revolution periods of Jupiter.
Localization of heteroclinic intersections. The detection of both manifolds
$W^{u}_{L_{1}}$ and $W^{s}_{L_{2}}$ on the same picture (see Figure 6 and
Figure 7) allows us to obtain a precise localization of the heteroclinic
intersections points, which we denote by $z_{he}$. Precisely, the intersection
between the two yellow curves in the box of Fig.6 corresponds to an
intersection point $z_{he}$. Of course, accordingly to the resolution of the
computation, at first we are only able to determine a point $z_{he,1}$ in the
box which is close $z_{he}$. To improve the localization of $z_{he}$ we
compute again the modified FLIs on a refined grid of points in the box of
Fig.6, and we obtain a new point $z_{he,2}$ (the point with the maximum value
of the averaged FLI (36)) closer to the intersection point. The procedure is
iterated by computing again the FLIs on zoomed out grids of initial conditions
centered on $z_{he,j}$ with $j=2,...15$, with increasing integration times to
increase the number of precision digits in the localization of the
heteroclinic point.
In Fig.8 we plot the FLI values computed on a grid of $500\times 500$ initial
conditions centered on the point $z_{he,15}$, using the integration time
$T=18$. The maximum value of the FLI in this picture provides a new refined
initial condition that we used to compute the heteroclinic orbit shown in
Fig.9. The convergence of the forward (backward) integration towards the
Lyapunov orbit related to $L_{2}$ ($L_{1}$) clearly shows the validity of the
method for the precise localization of heteroclinic orbits.
Figure 8: Computation of the averaged FLI (36) on a grid of $500\times 500$
initial conditions centered on the point $z_{he,15}$ of coordinates:
$x_{he,15}=1.041239777351473900$, $y_{he,15}=0$,
$\dot{x}_{he,15}=0.0460865533656582000$. The velocity $\dot{y}_{he,15}$ is
obtained from the Jacobi constant $C=3.0368573364394607$. The integration time
is $T=18$. The values of the FLI are provided as the average between FLI1 and
FLI2. A sharp detection of both manifolds appears thanks to the
differentiation of the FLI values on this refined grid. The maximum value of
the FLI in this picture provides a new refined initial condition for the orbit
plotted in Fig.9. Figure 9: Projection on the plane $(x,y)$ of the
heteroclinic orbit found through the maximum of the FLI (see text). The
conditions are : $x_{he}(0)=1.041239777351473912$, $y_{he}(0)=0$,
$\dot{x}_{he}(0)=0.046086553365658360$ and $\dot{y}_{he}(0)$ obtained from the
Jacobi constant $C=3.0368573364394607$. Blue points: forward integration, the
orbit converges to the Lyapunov orbit related to $L_{2}$. Red points: backward
integration, the orbit converges to the Lyapunov orbit related to $L_{1}$.
## 4 Proofs
Proof of Proposition 1. We first remark that (15) implies
$\varepsilon_{0}\leq\varepsilon_{1}$, and condition (16) implies
$\varepsilon_{0}\leq{\delta_{0}^{2}\over\max(1,\eta)(1+\lambda_{w})\lambda^{T_{s}}T^{2}}.$
The proof of Proposition 1 is a consequence of the following:
###### Lemma 4.1
For any $\varepsilon,\delta$ satisfying
$\displaystyle\max(1,\eta)(1+\lambda_{w})\lambda^{T_{s}}\varepsilon\leq\delta^{2}$
(37) $\displaystyle\delta\leq\min\left({1\over 2}\ ,\ {r_{*}\over 2}\ ,\
{1\over\eta}\ ,\ {1\over
4e^{2}\lambda_{u}\eta}\Big{(}1-{1\over\lambda_{u}}\Big{)}\ ,\ {1\over
2e^{3}\lambda_{u}^{2}\eta}{1\over T_{\varepsilon}}\right)$ (38)
we have:
$\delta-\eta\Delta_{\varepsilon}\leq\left|\Phi^{T_{s}+T_{\varepsilon}}_{1}(z_{\varepsilon})\right|\leq
e^{2}\lambda_{u}\delta+\eta\Delta_{\varepsilon}$ (39)
$\displaystyle\left|\Phi^{T_{s}+T_{\varepsilon}}_{2}(z_{\varepsilon})\right|\leq
e^{2}\lambda_{u}\delta+2\eta\Delta_{\varepsilon}$ (40)
$\displaystyle\left|\Phi^{T_{s}+T_{\varepsilon}}_{2}(z_{\varepsilon})-(\zeta_{\varepsilon})_{2}\right|\leq{1\over\lambda_{u}^{T_{\varepsilon}}}4e^{3}\lambda_{u}\delta$
, (41)
and the tangent vector $D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v$
satisfies
$\left\|D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v-A^{T_{\varepsilon}}(w_{u}+w_{s})\right\|\leq\lambda_{u}^{T_{\varepsilon}}\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\left\|w_{u}\right\|.$
(42)
First, as we anticipated in Section 2, the time $T_{\varepsilon}$ can be
identified as the time required by the orbit with initial condition
$\Phi^{T_{s}}(z_{\varepsilon})$ to exit from $B(\delta)$ and to arrive at the
small distance $(4e^{3}\lambda_{u}\delta)/\lambda_{u}^{T_{\varepsilon}}$ from
the local unstable manifold. If $T_{\varepsilon}\geq T-T_{s}$, we can repeat
the proof of Lemma 4.1 by limiting all the estimates to the time interval
$[0,T]$, and obtaining
$\left\|D\Phi^{T}_{z_{\varepsilon}}v-A^{T-T_{s}}w\right\|\leq\lambda_{u}^{T-T_{s}}\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\left\|w_{u}\right\|,$
(43)
so that (19) is proved. If $T_{\varepsilon}<T-T_{s}$, we need an estimate of
the growth of the tangent vectors in the remaining time interval
$[T_{s}+T_{\varepsilon},T]$, and we obtain it by comparison with the growth of
the tangent vectors of the orbits with initial condition in the point
$\zeta_{\varepsilon}$ on the unstable manifold. We will provide estimates of
the FLI for $T_{\varepsilon}$ in the interval:
$(T-T_{s}(\delta)){\ln\lambda\over\ln\lambda+\ln\lambda_{u}}\leq
T_{\varepsilon}<T-T_{s}(\delta).$ (44)
Let us consider
$j=T-T_{s}-T_{\varepsilon}\in\\{1,(T-T_{s}(\delta)){\ln\lambda_{u}\over\ln\lambda+\ln\lambda_{u}}\\}$.
First, we have
$\left\|D\Phi^{T}_{z_{\varepsilon}}v-D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|\leq
4e^{3}\lambda_{u}{\lambda^{j}\over\lambda_{u}^{T_{\varepsilon}}}\delta\left\|D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|,$
(45)
In fact, since
$D\Phi^{T}_{z_{\varepsilon}}v=D\Phi^{j}_{\Phi^{T_{s}+T_{\varepsilon}}(z_{\varepsilon})}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v=D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v+\Big{(}D\Phi^{j}_{\Phi^{T_{s}+T_{\varepsilon}}(z_{\varepsilon})}-D\Phi^{j}_{\zeta_{\varepsilon}}\Big{)}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v,$
using Lemmas 4.1 and 5.2 we obtain
$\left\|D\Phi^{T}_{z_{\varepsilon}}v-D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|\leq\lambda^{j}\left\|\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}-\zeta_{\varepsilon}\right\|\left\|D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|\leq
4e^{3}\lambda_{u}{\lambda^{j}\over\lambda_{u}^{T_{\varepsilon}}}\delta\left\|D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|.$
Therefore, we have
$\left\|D\Phi^{T}_{z_{\varepsilon}}v\right\|\leq\left\|D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|\left(1+4e^{3}\lambda_{u}{\lambda^{j}\over\lambda_{u}^{T_{\varepsilon}}}\delta\right).$
We now analyze and compare the FLI for initial conditions at different
distances from the stable manifold. We have
${\left\|D\Phi^{T}_{z_{\varepsilon}}v\right\|\over\left\|D\Phi^{T}_{z_{s}}v\right\|}\leq{\left\|D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|\over\left\|D\Phi^{T}_{z_{s}}v\right\|}\left(1+4e^{3}\lambda_{u}{\lambda^{j}\over\lambda_{u}^{T_{\varepsilon}}}\delta\right).$
Using inequalities (43) and (42), we obtain
${\left\|D\Phi^{T}_{z_{\varepsilon}}v\right\|\over\left\|D\Phi^{T}_{z_{s}}v\right\|}\leq{\left\|D\Phi^{j}_{\zeta_{\varepsilon}}\right\|\over\lambda_{u}^{j}}\
{1+\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\over
1-\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)}\left(1+4e^{3}\lambda_{u}{\lambda^{j}\over\lambda_{u}^{T_{\varepsilon}}}\delta\right).$
(46)
In fact, from (43) and
$\left\|A^{T-T_{s}}(w_{u}+w_{s})\right\|=\lambda_{u}^{T-T_{s}}\left\|w_{u}\right\|$,
we obtain that for all $\varepsilon$ with $T_{\varepsilon}\geq T-T_{s}$,
including $z_{0}=z_{s}$, we have
$\left\|D\Phi^{T}_{z_{\varepsilon}}v\right\|\geq\lambda_{u}^{T-T_{s}}\left(1-\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\right)\left\|w_{u}\right\|.$
(47)
From (42), for all $\varepsilon$ with $T_{\varepsilon}\leq T-T_{s}$, we have:
$\left\|D\Phi^{j}_{\zeta_{\varepsilon}}D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|\leq\left\|D\Phi^{j}_{\zeta_{\varepsilon}}\right\|\left\|D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v\right\|$
$\leq\left\|D\Phi^{j}_{\zeta_{\varepsilon}}\right\|\left(\left\|A^{T_{\varepsilon}}w\right\|+\lambda_{u}^{T_{\varepsilon}}\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\left\|w\right\|\right)$
$\leq\left\|D\Phi^{j}_{\zeta_{\varepsilon}}\right\|\lambda_{u}^{T_{\varepsilon}}\left\|w_{u}\right\|\left(1+\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\right).$
(48)
Since for all $\varepsilon$ with
$T_{\varepsilon}\geq(T-T_{s}){\ln\lambda\over\ln\lambda+\ln\lambda_{u}},$
we have
${\lambda^{j}\over\lambda_{u}^{T_{\varepsilon}}}={\lambda^{T-T_{s}-T_{\varepsilon}}\over\lambda_{u}^{T_{\varepsilon}}}\leq
1,$
using also
$\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\leq{1\over
2eT}\ ,\ 4e^{3}\lambda_{u}\delta<{1\over 4T}$
we have
${1+\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\over
1-\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)}\left(1+4e^{3}\lambda_{u}{\lambda^{j}\over\lambda_{u}^{T_{\varepsilon}}}\delta\right)\leq\left(1+{1\over
T}\right),$
so that, from (46), we immediately obtain (20).
Proof of (13). We have:
$\Delta_{\varepsilon}=\left|\Phi^{T_{s}}_{1}(z_{\varepsilon})-w_{s}(\Phi^{T_{s}}_{2}(z_{\varepsilon}))\right|\leq\left|\Phi^{T_{s}}_{1}(z_{\varepsilon})-\Phi^{T_{s}}_{1}(z_{s})\right|+\left|w_{s}(\Phi^{T_{s}}_{2}(z_{s}))-w_{s}(\Phi^{T_{s}}_{2}(z_{\varepsilon}))\right|$
$\leq\lambda_{\Phi}^{T_{s}}\varepsilon+\lambda_{w}\lambda_{\Phi}^{T_{s}}\varepsilon\leq(1+\lambda_{w})\lambda_{\Phi}^{T_{s}}\varepsilon.$
Proof of Lemma 4.1. We consider the segments which join
$\Phi^{k}(\pi_{\varepsilon})$ and $\Phi^{k}(\Phi^{T_{s}}(z_{\varepsilon}))$,
and define
$\Delta^{k}_{1}=\left|\Phi^{k}_{1}(\Phi^{T_{s}}(z_{\varepsilon}))-\Phi^{k}_{1}(\pi_{\varepsilon})\right|$
$\Delta^{k}_{2}=\left|\Phi^{k}_{2}(\Phi^{T_{s}}(z_{\varepsilon}))-\Phi^{k}_{2}(\pi_{\varepsilon})\right|.$
We prove that, for all the $k$ such that
$\Phi^{k}(\pi_{\varepsilon}),\Phi^{k}(\Phi^{T_{s}}(z_{\varepsilon}))\leq
B(A\delta)$, for $A>1$, we have
$\Delta^{k}_{2}<\Delta^{k}_{1}.$
In fact, we have $\Delta^{0}_{1}=\Delta_{\varepsilon}$, $\Delta^{0}_{2}=0$;
then, if $\Delta^{k-1}_{2}<\Delta^{k-1}_{1}$, we have
$\Delta^{k}_{2}=\left|\Phi^{k}_{2}(\Phi^{T_{s}}(z_{\varepsilon}))-\Phi^{k}_{2}(\pi_{\varepsilon})\right|=\left|\Phi_{2}(\Phi^{k-1}(\Phi^{T_{s}}(z_{\varepsilon})))-\Phi_{2}(\Phi^{k-1}(\pi_{\varepsilon}))\right|$
$\leq{1\over\lambda_{u}}\Delta^{k-1}_{2}+\left|f_{2}(\Phi^{k-1}(\Phi^{T_{s}}(z_{\varepsilon})))-f_{2}(\Phi^{k-1}(\pi_{\varepsilon}))\right|$
$\leq{1\over\lambda_{u}}\Delta^{k-1}_{2}+A\eta\delta(\Delta^{k-1}_{1}+\Delta^{k-1}_{2})\leq\Big{(}{1\over\lambda_{u}}+2A\eta\delta\Big{)}\Delta^{k-1}_{1}<\Delta^{k-1}_{1}$
as soon as
${1\over\lambda_{u}}+2A\eta\delta<1.$
Therefore, we have
$\Delta^{k}_{1}=\left|\Phi^{k}_{1}(\Phi^{T_{s}}(z_{\varepsilon}))-\Phi^{k}_{1}(\pi_{\varepsilon})\right|\leq\lambda_{u}\Delta^{k-1}_{1}+A\eta\delta(\Delta^{k-1}_{1}+\Delta^{k-1}_{2})$
$\leq(\lambda_{u}+2\eta A\delta)\Delta^{k-1}_{1}\leq(\lambda_{u}+2\eta
A\delta)^{k}\Delta^{0}_{1}=\lambda_{u}^{k}\Big{(}1+{2\eta
A\delta\over\lambda_{u}}\Big{)}^{k}\Delta_{\varepsilon}\leq
e\lambda_{u}^{k}\Delta_{\varepsilon}$
and
$\Delta^{k}_{1}\geq\lambda_{u}\Delta^{k-1}_{1}-A\eta(\Delta^{k-1}_{1}+\Delta^{k-1}_{2})\geq(\lambda_{u}-2A\eta\delta)\Delta^{k-1}_{1}$
$\geq\lambda_{u}^{k}\Big{(}1-{2\eta
A\delta\over\lambda_{u}}\Big{)}^{k}\Delta_{\varepsilon}\geq{1\over
e}\lambda_{u}^{k}\Delta_{\varepsilon}$
as soon as $k\leq T$ and
${2\eta A\delta\over\lambda_{u}}\leq{1\over 2k}.$
We obtained
${1\over e}\lambda_{u}^{k}\Delta_{\varepsilon}\leq\Delta^{k}_{1}\leq
e\lambda_{u}^{k}\Delta_{\varepsilon}.$
We now provide an estimate of $\Phi^{k}_{1}(\pi_{\varepsilon})$ and
$\Phi^{k}_{2}(\pi_{\varepsilon})$. We consider the segment which joins the
origin $(0,0)$ and $\Phi^{k}(\pi_{\varepsilon})$ and define
$\delta_{1}^{k}=\left|\Phi^{k}_{1}(\pi_{\varepsilon})\right|\ \ ,\ \
\delta_{2}^{k}=\left|\Phi^{k}_{2}(\pi_{\varepsilon})\right|.$
We have $\delta_{1}^{k}<\delta_{2}^{k}$ for any $k$. In fact, since
$\pi_{\varepsilon}\in B(\delta)$ and $\pi_{\varepsilon}\in W^{l}_{s}$, then
$\Phi^{k}(\pi_{\varepsilon})\in B(\delta)$ for any $k$ and we have
$\delta_{1}^{k}=\left|\Phi^{k}_{1}(\pi_{\varepsilon})\right|=\left|w_{s}(\Phi^{k}_{2}(\pi_{\varepsilon}))\right|\leq\eta\left|\Phi^{k}_{2}(\pi_{\varepsilon})\right|^{2}\leq\eta\delta\delta^{k}_{2}<\delta^{k}_{2}$
as soon as
$\eta\delta<1.$
For $k=0$ we have
$\delta_{1}^{0}=\left|(\pi_{\varepsilon})_{1}\right|=\left|w_{s}((\pi_{\varepsilon})_{2})\right|\leq\eta\left|\Phi^{T_{s}}_{2}(z_{s})\right|^{2}\leq\eta\delta\left|\Phi^{T_{s}}_{2}(z_{s})\right|\leq\eta\delta^{2}\
\ ,\ \ \delta_{2}^{0}=\left|\Phi^{T_{s}}_{2}(z_{s})\right|\leq\delta$
Then, we have
$\delta_{2}^{k}=\left|\Phi^{k}_{2}(\pi_{\varepsilon})\right|=\left|\Phi_{2}(\Phi^{k-1}(\pi_{\varepsilon}))\right|\leq{1\over\lambda_{u}}\left|\Phi^{k-1}_{2}(\pi_{\varepsilon})\right|+\eta\left\|\Phi^{k-1}(\pi_{\varepsilon})\right\|^{2}\leq{1\over\lambda_{u}}\delta_{2}^{k-1}+\eta(\delta_{2}^{k-1})^{2}$
$\leq\Big{(}{1\over\lambda_{u}}+\eta\delta\Big{)}\delta_{2}^{k-1}\leq\Big{(}{1\over\lambda_{u}}+\eta\delta\Big{)}^{k}\delta_{2}^{0}\leq{1\over\lambda_{u}^{k}}\Big{(}1+\eta\lambda_{u}\delta\Big{)}^{k}\delta\leq{1\over\lambda_{u}^{k}}e\delta$
as soon as
$\eta\lambda_{u}\delta\leq{1\over ek},$
and
$\delta_{1}^{k}=\left|w_{s}(\Phi^{k}_{2}(\pi_{\varepsilon})\right|\leq\eta\left|\Phi^{k}_{2}(\pi_{\varepsilon})\right|^{2}=\eta(\delta_{2}^{k})^{2}\leq\eta
e{1\over\lambda_{u}^{k}}\delta.$
Therefore, from
${1\over
e}\lambda_{u}^{k}\Delta_{\varepsilon}\leq\left|\Phi^{k}_{1}(\Phi^{T_{s}}(z_{\varepsilon}))-\Phi^{k}_{1}(\pi_{\varepsilon})\right|\leq
e\lambda_{u}^{k}\Delta_{\varepsilon}$
we have
${1\over e}\lambda_{u}^{k}\Delta_{\varepsilon}-\eta
e{1\over\lambda_{u}^{k}}\delta\leq\left|\Phi^{k}_{1}(\Phi^{T_{s}}(z_{\varepsilon}))\right|\leq
e\lambda_{u}^{k}\Delta_{\varepsilon}+\eta e{1\over\lambda_{u}^{k}}\delta.$
Finally, from $\Delta^{k}_{2}<\Delta^{k}_{1}$ we have
$\left|\Phi^{k}_{2}(\Phi^{T_{s}}(z_{\varepsilon}))\right|\leq\Delta^{k}_{1}+\left|\Phi^{k}_{2}(\pi_{\varepsilon})\right|\leq
e\lambda_{u}^{k}\Delta_{\varepsilon}+{1\over\lambda_{u}^{k}}e\delta.$
From the definition of $T_{\varepsilon}$, we have
${e\delta\over\Delta_{\varepsilon}}\leq\lambda_{u}^{T_{\varepsilon}}<{\lambda_{u}e\delta\over\Delta_{\varepsilon}},$
and therefore we have
$\delta-\eta\Delta_{\varepsilon}\leq\delta-\eta
e{\delta\over\lambda_{u}^{T_{\varepsilon}}}\leq\left|\Phi^{k}_{1}(\Phi^{T_{s}}(z_{\varepsilon}))\right|\leq
e^{2}\lambda_{u}\delta+\eta\Delta_{\varepsilon}$
$\left|\Phi^{k}_{2}(\Phi^{T_{s}}(z_{\varepsilon}))\right|\leq
e^{2}\lambda_{u}\delta+2\eta\Delta_{\varepsilon}.$
Therefore, since
$\Delta_{\varepsilon}\leq(1+\lambda_{w})\lambda_{\Phi}^{T_{s}}\varepsilon$, as
soon as
$\max(1,\eta)(1+\lambda_{w})\lambda^{T_{s}}\varepsilon<\delta^{2}$
we have
$\left\|\Phi^{k}(\Phi^{T_{s}}(z_{\varepsilon}))\right\|\leq
e^{2}\lambda_{u}\delta+2\eta\Delta_{\varepsilon}\leq
e^{2}\lambda_{u}\delta+2\eta(1+\lambda_{w})\lambda_{\Phi}^{T_{s}}\varepsilon<e^{2}\lambda_{u}\delta+\delta^{2}<2e^{2}\lambda_{u}\delta=A\delta$
for $A=2e^{2}\lambda_{u}$. The thresholds conditions on $\delta$ become
$\delta\leq{1\over 4e^{2}\lambda_{u}\eta}\Big{(}1-{1\over\lambda_{u}}\Big{)}\
\ ,\ \ \delta\leq{1\over 8e^{2}\eta}{1\over T_{\varepsilon}}\ \ ,\ \
\delta\leq{1\over 2e^{3}\lambda_{u}^{2}\eta}{1\over T_{\varepsilon}}.$
We now consider the point
$\zeta_{\varepsilon}=\Big{(}\Phi^{T_{s}+T_{\varepsilon}}_{1}(z_{\varepsilon}),w_{u}(\Phi^{T_{s}+T_{\varepsilon}}_{1}(z_{\varepsilon}))\Big{)}$
and the segments which join $\Phi^{-k}(\zeta_{\varepsilon})$ and
$\Phi^{-k}(\Phi^{T_{s}+T_{\varepsilon}}(z_{\varepsilon}))$, for $k\leq
T_{\varepsilon}$. We already know that
$\Phi^{-k}(\Phi^{T_{s}+T_{\varepsilon}}(z_{\varepsilon}))\in B(A\delta)$, and
$\left\|\zeta_{\varepsilon}\right\|=\left\|\Big{(}\Phi^{T_{s}+T_{\varepsilon}}(z_{\varepsilon}),w_{u}(\Phi^{T_{s}+T_{\varepsilon}}(z_{\varepsilon}))\Big{)}\right\|\leq\max(A\delta,\eta
A\delta^{2})\leq A\delta$
as soon as $\eta\delta\leq 1$. By definition of local unstable manifold, we
have $\Phi^{-k}(\zeta_{\varepsilon})\in B(A\delta)$. We define
$\Delta^{-k}_{1}=\left|\Phi^{-k+T_{s}+T_{\varepsilon}}_{1}(z_{\varepsilon})-\Phi^{-k}_{1}(\zeta_{\varepsilon})\right|$
$\Delta^{-k}_{2}=\left|\Phi^{-k+T_{s}+T_{\varepsilon}}_{2}(z_{\varepsilon}))-\Phi^{-k}_{2}(\zeta_{\varepsilon})\right|,$
in particular we have
$\Delta^{0}_{1}=0\ \ ,\ \ \Delta^{0}_{2}:=\Delta^{\varepsilon}.$
By repeating the above arguments using the inverse map $\Phi^{-1}(x)$, we have
$\Delta^{-k}_{1}<\Delta^{-k}_{2}$ for any $k$ and:
${1\over e}\lambda_{u}^{k}\Delta^{\varepsilon}\leq\Delta_{2}^{-k}\leq
e\lambda_{u}^{k}\Delta^{\varepsilon}$
$\Delta^{\varepsilon}=\left|\Phi^{T_{s}+T_{\varepsilon}}_{2}(z_{\varepsilon})-(\zeta_{\varepsilon})_{2}\right|\leq{e\over\lambda_{u}^{T_{\varepsilon}}}\Delta_{2}^{-T_{\varepsilon}}\leq{e\over\lambda_{u}^{T_{\varepsilon}}}2A\delta.$
It remains to prove (42). For any $k\leq T_{\varepsilon}$, we have:
$\left\|\Phi^{k}(\Phi^{T_{s}}(z_{\varepsilon}))\right\|\leq
e\lambda_{u}^{k}\Delta_{\varepsilon}+\max(1,\eta)e{1\over\lambda_{u}^{k}}\delta\leq
e^{2}{\lambda_{u}\over\lambda_{u}^{T_{\varepsilon}-k}}\delta+\max(1,\eta)e{1\over\lambda_{u}^{k}}\delta$
and
$\sum_{k=0}^{T_{\varepsilon}-1}\left\|\Phi^{k}(\Phi^{T_{s}}(z_{\varepsilon}))\right\|\leq\sum_{k=0}^{T_{\varepsilon}-1}\left(e^{2}{\lambda_{u}\over\lambda_{u}^{T_{\varepsilon}-k}}+\max(1,\eta)e{1\over\lambda_{u}^{k}}\right)\delta$
$\leq
2e\lambda_{u}\max(e,\eta)\delta\sum_{k=0}^{T_{\varepsilon}}{1\over\lambda_{u}^{k}}\leq
2e\lambda_{u}\max(e,\eta){\lambda_{u}\over\lambda_{u}-1}\delta.$
so that, by using also lemma 5.1, we have:
$\left\|D\Phi^{T_{\varepsilon}}_{\Phi^{T_{s}}(z_{\varepsilon})}D\Phi^{T_{s}}_{z_{\varepsilon}}v-A^{T_{\varepsilon}}D\Phi^{T_{s}}_{z_{\varepsilon}}v\right\|\leq\left\|D\Phi^{T_{\varepsilon}}_{\Phi^{T_{s}}(z_{\varepsilon})}-A^{T_{\varepsilon}}\right\|\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v\right\|$
$\leq\eta\lambda_{u}^{T_{\varepsilon}}\left(1+2\eta
e^{2}\delta\right)^{T_{\varepsilon}-1}\sum_{k=0}^{T_{\varepsilon}-1}\left\|\Phi^{k}(\Phi^{T_{s}}(z_{\varepsilon}))\right\|\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v\right\|$
$\leq\eta\max(e,\eta){2e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}\
\lambda_{u}^{T_{\varepsilon}}\delta\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v\right\|.$
We have
$\left\|D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v-A^{T_{\varepsilon}}w\right\|\leq\left\|D\Phi^{T_{\varepsilon}}_{\Phi^{T_{s}}(z_{\varepsilon})}D\Phi^{T_{s}}_{z_{\varepsilon}}v-A^{T_{\varepsilon}}D\Phi^{T_{s}}_{z_{\varepsilon}}v\right\|+\left\|A^{T_{\varepsilon}}(D\Phi^{T_{s}}_{z_{\varepsilon}}v-w)\right\|$
$\leq\eta\max(e,\eta){2e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}\
\lambda_{u}^{T_{\varepsilon}}\delta\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v\right\|+\left\|A\right\|^{T_{\varepsilon}}\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v-w\right\|$
$\leq\eta\max(e,\eta){2e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}\
\lambda_{u}^{T_{\varepsilon}}\delta\Big{(}\left\|w\right\|+\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v-w\right\|\Big{)}+{\lambda_{u}}^{T_{\varepsilon}}\left\|D\Phi^{T_{s}}_{z_{\varepsilon}}v-w\right\|$
and using (12) and $\left\|w\right\|=\left\|w_{u}\right\|$ we obtain
$\left\|D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v-A^{T_{\varepsilon}}w\right\|\leq\lambda_{u}^{T_{\varepsilon}}\left(\eta\max(e,\eta){2e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}\
\delta(1+\lambda^{T_{s}}\varepsilon)+\lambda^{T_{s}}\varepsilon\right)\left\|w_{u}\right\|$
and, since $\lambda^{T_{s}}\varepsilon\leq\delta^{2}\leq\delta\leq 1$:
$\left\|D\Phi^{T_{s}+T_{\varepsilon}}_{z_{\varepsilon}}v-A^{T_{\varepsilon}}(w_{u}+w_{s})\right\|\leq\lambda_{u}^{T_{\varepsilon}}\delta\left(\eta\max(e,\eta){4e^{2}\lambda_{u}^{2}\over\lambda_{u}-1}+1\right)\left\|w_{u}\right\|.$
(49)
## 5 Two Technical Lemmas
In this Section we prove two technical Lemmas which we obtain by using
Lipschitz inequalities for $\Phi$ and $D\Phi$.
###### Lemma 5.1
Let $U\subseteq{\mathbb{R}}^{n}$ be a neighbourhood of $0$, and
$\Phi:U\rightarrow{\mathbb{R}}^{n}$ be a smooth map:
$\Phi(z)=Az+f(z)$
with $f_{i}(0,\ldots,0)=0$, ${\partial f_{i}\over\partial
z_{j}}(0,\ldots,0)=0$ for any $i,j$ and, for any $z\in B(R)$, satisfying
$\left\|f(z)\right\|\leq\eta\left\|z\right\|^{2}\ \ ,\ \
\left\|Df_{z}\right\|\leq\eta\left\|z\right\|\ \ ,\ \
\left\|D\Phi_{z}\right\|\leq l.$
Then, for any $z,K$ such that $\Phi^{k}(z)\in B(R)$ for any $k=0,\ldots,K$, we
have
$\left\|D\Phi^{k}_{z}-A^{k}\right\|\leq\eta\sum_{j=0}^{k-1}\lambda_{u}^{j}\left\|\Phi^{j}(z)\right\|\left(\lambda_{u}+\eta\left\|\Phi^{j+1}(z)\right\|\right)\ldots\left(\lambda_{u}+\eta\left\|\Phi^{k-1}(z)\right\|\right)$
(50)
and
$\left\|D\Phi^{k}_{z}-A^{k}\right\|\leq\eta\Big{(}\lambda_{u}+\eta\max_{j\leq
k-1}\left\|\Phi^{j}(z)\right\|\Big{)}^{k-1}\sum_{j=0}^{k-1}\left\|\Phi^{j}(z)\right\|.$
(51)
Proof of Lemma 5.1. For $k=1$ we have
$\left\|D\Phi_{z}-A\right\|=\left\|Df_{z}\right\|\leq\eta\left\|z\right\|$.
For generic $k\leq K$, since $\left\|A\right\|=\lambda_{u}$, we have
$\left\|D\Phi^{k}_{z}-A^{k}\right\|\leq\left\|D\Phi_{\Phi^{k-1}(z)}D\Phi^{k-1}_{z}-A^{k}\right\|=\left\|D\Phi_{\Phi^{k-1}(z)}(D\Phi^{k-1}_{z}-A^{k-1})+(D\Phi_{\Phi^{k-1}(z)}-A)A^{k-1}\right\|$
$\leq\left\|D\Phi_{\Phi^{k-1}(z)}\right\|\left\|D\Phi^{k-1}_{z}-A^{k-1}\right\|+\left\|D\Phi_{\Phi^{k-1}(z)}-A\right\|\left\|A\right\|^{k-1}$
$=\left\|A+Df_{\Phi^{k-1}(z)}\right\|\left\|D\Phi^{k-1}_{z}-A^{k-1}\right\|+\left\|D\Phi_{\Phi^{k-1}(z)}-A\right\|\left\|A\right\|^{k-1}$
$\leq\left(\lambda_{u}+\eta\left\|\Phi^{k-1}(z)\right\|\right)\left\|D\Phi^{k-1}_{z}-A^{k-1}\right\|+\eta\left\|\Phi^{k-1}(z)\right\|\lambda_{u}^{k-1}.$
Assuming that (50) is valid for $k-1$, we have
$\left\|D\Phi^{k}_{z}-A^{k}\right\|\leq\left(\lambda_{u}+\eta\left\|\Phi^{k-1}(z)\right\|\right)\eta\sum_{j=0}^{k-2}\lambda_{u}^{j}\left\|\Phi^{j}(z)\right\|\left(\lambda_{u}+\eta\left\|\Phi^{j+1}(z)\right\|\right)\ldots\left(\lambda_{u}+\eta\left\|\Phi^{k-2}(z)\right\|\right)$
$+\eta\left\|\Phi^{k-1}(z)\right\|\lambda_{u}^{k-1}=\eta\sum_{j=0}^{k-2}\lambda_{u}^{j}\left\|\Phi^{j}(z)\right\|\left(\lambda_{u}+\eta\left\|\Phi^{j+1}(z)\right\|\right)\ldots\left(\lambda_{u}+\eta\left\|\Phi^{k-1}(z)\right\|\right)+\eta\left\|\Phi^{k-1}(z)\right\|\lambda_{u}^{k-1}$
$=\eta\sum_{j=0}^{k-1}\lambda_{u}^{j}\left\|\Phi^{j}(z)\right\|\left(\lambda_{u}+\eta\left\|\Phi^{j+1}(z)\right\|\right)\ldots\left(\lambda_{u}+\eta\left\|\Phi^{k-1}(z)\right\|\right).$
From (50) we immediately obtain (51).
###### Lemma 5.2
Let $U\subseteq{\mathbb{R}}^{n}$ be a neighbourhood of $0$, and
$\Phi:U\rightarrow{\mathbb{R}}^{n}$ be a smooth map with finite Lipschitz
constants $\lambda_{\Phi}$, $\lambda_{D\Phi}$ for $\Phi$ and $D\Phi$
respectively. For any initial conditions
$z^{\prime}_{0},z^{\prime\prime}_{0}$, their time–evolutions
$z^{\prime}_{k}=\Phi^{k}(z^{\prime}_{0})$,
$z^{\prime\prime}_{k}=\Phi^{k}(z^{\prime\prime}_{0})$ satisfy
$\left\|z^{\prime}_{T}-z^{\prime\prime}_{T}\right\|\leq\lambda_{\Phi}^{T}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|$
(52)
and for any $v\neq 0$, the time–evolution of the tangent vectors
$v^{\prime}_{T}=D\Phi^{T}_{z^{\prime}_{0}}v\ \ ,\ \
v^{\prime\prime}_{T}=D\Phi^{T}_{z^{\prime\prime}_{0}}v$
satisfies
${\left\|v^{\prime}_{T}-v^{\prime\prime}_{T}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}\leq\lambda^{T}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|.$
(53)
with
$\lambda=\max\left(\lambda_{\Phi},{\left\|D\Phi\right\|+\lambda_{D\Phi}\over\sigma}\right)$
where $\sigma=\min_{z\in
U}\min_{\left\|v\right\|=1}\left\|D\Phi_{z}v\right\|$.
Proof of Lemma 5.2. We prove (52) by induction on $T$. If $T=1$ we have
$\left\|z^{\prime}_{1}-z^{\prime\prime}_{1}\right\|=\left\|\Phi(z^{\prime}_{0})-\Phi(z^{\prime\prime}_{0})\right\|\leq\lambda_{\Phi}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|.$
Le us assume
$\left\|z^{\prime}_{T-1}-z^{\prime\prime}_{T-1}\right\|\leq\lambda_{\Phi}^{T-1}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|.$
Then, we have
$\left\|z^{\prime}_{T}-z^{\prime\prime}_{T}\right\|=\left\|\Phi(z^{\prime}_{T-1})-\Phi(z^{\prime\prime}_{T-1})\right\|\leq\lambda_{\Phi}\left\|z^{\prime}_{T-1}-z^{\prime\prime}_{T-1}\right\|\leq\lambda_{\Phi}^{T}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|.$
Then, let us prove (53) by induction on $T$. If $T=1$, we have
${\left\|v^{\prime}_{1}-v^{\prime\prime}_{1}\right\|\over\left\|v^{\prime\prime}_{1}\right\|}={\left\|(D\Phi_{z^{\prime}_{0}}-D\Phi_{z^{\prime\prime}_{0}})v\right\|\over\left\|v^{\prime\prime}_{1}\right\|}={\left\|(D\Phi_{z^{\prime}_{0}}-D\Phi_{z^{\prime\prime}_{0}})v\right\|\over\left\|v\right\|}{\left\|v\right\|\over\left\|v^{\prime\prime}_{1}\right\|}={\left\|(D\Phi_{z^{\prime}_{0}}-D\Phi_{z^{\prime\prime}_{0}})v\right\|\over\left\|v\right\|}{\left\|v\right\|\over\left\|D\Phi_{z^{\prime\prime}_{0}}v\right\|}.$
By Lipschitz estimate and inequality (10) we have:
${\left\|(D\Phi_{z^{\prime}_{0}}-D\Phi_{z^{\prime\prime}_{0}})v\right\|\over\left\|v\right\|}\leq\left\|D\Phi_{z^{\prime}_{0}}-D\Phi_{z^{\prime\prime}_{0}}\right\|\leq\lambda_{D\Phi}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|$
${\left\|D\Phi_{z^{\prime\prime}_{0}}v\right\|\over\left\|v\right\|}\geq\min_{\left\|v\right\|=1}\left\|D\Phi_{z^{\prime\prime}_{0}}v\right\|=\sigma>0,$
and therefore we obtain
${\left\|v^{\prime}_{1}-v^{\prime\prime}_{1}\right\|\over\left\|v^{\prime\prime}_{1}\right\|}\leq{\lambda_{D\Phi}\over\sigma}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|\leq\lambda\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|.$
We now assume that (53) is satisfied for $T-1$, that is:
${\left\|v^{\prime}_{T-1}-v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T-1}\right\|}\leq\lambda^{T-1}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|.$
(54)
Then, let us consider
${\left\|v^{\prime}_{T}-v^{\prime\prime}_{T}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}={\left\|D\Phi_{z^{\prime}_{T-1}}v^{\prime}_{T-1}-D\Phi_{z^{\prime\prime}_{T-1}}v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}$
$\leq{\left\|D\Phi_{z^{\prime}_{T-1}}(v^{\prime}_{T-1}-v^{\prime\prime}_{T-1})\right\|\over\left\|v^{\prime\prime}_{T}\right\|}+{\left\|(D\Phi_{z^{\prime}_{T-1}}-D\Phi_{z^{\prime\prime}_{T-1}})v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}$
$\leq\Big{(}\sup_{z}\left\|D\Phi_{z}\right\|\Big{)}{\left\|v^{\prime}_{T-1}-v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}+\left\|D\Phi_{z^{\prime}_{T-1}}-D\Phi_{z^{\prime\prime}_{T-1}}\right\|{\left\|v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}$
$=\Big{(}\sup_{z}\left\|D\Phi_{z}\right\|\Big{)}{\left\|v^{\prime}_{T-1}-v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T-1}\right\|}{\left\|v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}+\lambda_{D\Phi}\left\|z^{\prime}_{T-1}-z^{\prime\prime}_{T-1}\right\|{\left\|v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}.$
Using (10):
${\left\|v^{\prime\prime}_{T-1}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}={\left\|v^{\prime\prime}_{T-1}\right\|\over\left\|D\Phi_{z^{\prime\prime}_{T-1}}v^{\prime\prime}_{T-1}\right\|}\leq{1\over\sigma}$
and (52), (54), we obtain
${\left\|v^{\prime}_{T}-v^{\prime\prime}_{T}\right\|\over\left\|v^{\prime\prime}_{T}\right\|}\leq{\Big{(}\sup_{z}\left\|D\Phi_{z}\right\|\Big{)}\over\sigma}\lambda^{T-1}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|+{\lambda_{D\Phi}\over\sigma}\lambda_{\Phi}^{T-1}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|$
$={\Big{(}\sup_{z}\left\|D\Phi_{z}\right\|\Big{)}\lambda^{T-1}+\lambda_{D\Phi}\lambda_{\Phi}^{T-1}\over\sigma}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|\leq\lambda^{T}\left\|z^{\prime}_{0}-z^{\prime\prime}_{0}\right\|.$
$\Box$
## 6 Conclusions
In this paper we have explained why the FLI indicators, suitably modified by
the introduction of test functions, may be used for high precision
computations of the stable and unstable manifolds of dynamical systems,
including the critical computations of the so called tube manifolds of the
restricted three–body problem. An advantage of the FLI method is that it does
not requires a preliminary high precision localization of the hyperbolic fixed
points or periodic orbits to provide high precision computations of their
stable and unstable manifolds. This is particularly useful for practical
applications, since additional perturbations can be easily included in the
numerical computations.
## Acknowledgments
Part of the computations have been done on the “Mesocentre SIGAMM” machine,
hosted by the Observatoire de la Cote d’Azur.
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|
arxiv-papers
| 2013-07-25T13:14:29 |
2024-09-04T02:49:48.454734
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Massimiliano Guzzo and Elena Lega",
"submitter": "Lega Elena",
"url": "https://arxiv.org/abs/1307.6731"
}
|
1307.6794
|
# On FK’ Conjecture
Zhengtang Tan, Shouchuan Zhang, Weicai Wu
Department of Mathematics, Hunan University
Changsha 410082, P.R. China, Emails: [email protected]
###### Abstract
We give the relationship between FK’ Conjecture and Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ of transposition
over symmetry group by means of quiver Hopf algebras. That is, if
$\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)=\infty$, then
so is $\dim\mathcal{E}_{n}$.
2000 Mathematics Subject Classification: 16W30, 16G10
keywords: Conjecture, Hopf algebra, Weyl group.
## 0 Introduction
S. Fomin and A.N. Kirillov [FK97, Conjecture 2.2] point out a conjecture “
${\mathcal{E}}_{n}$ is finite dimensional ” to study the cohomology ring of
the flag manifold. In [FK] it is shown that ${\mathcal{E}}_{3}=12$ and
${\mathcal{E}}_{4}=24^{2}$ (see [AS02, Section 3.4]). Many papers ( for
example, [MS, GHV, Ba06]) refer to this conjecture.
In this paper we give the relationship between FK’ Conjecture and Nichols
algebra $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ of
transposition over symmetry group by means of quiver Hopf algebras. That is,
if $\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)=\infty$,
then so is $\dim\mathcal{E}_{n}$.
## Preliminaries And Conventions
Let ${k}$ be the complex field, A quiver $Q=(Q_{0},Q_{1},s,t)$ is an oriented
graph, where $Q_{0}$ and $Q_{1}$ are the sets of vertices and arrows,
respectively; $s$ and $t$ are two maps from $Q_{1}$ to $Q_{0}$. For any arrow
$a\in Q_{1}$, $s(a)$ and $t(a)$ are called its start vertex and end vertex,
respectively, and $a$ is called an arrow from $s(a)$ to $t(a)$. For any $n\geq
0$, an $n$-path or a path of length $n$ in the quiver $Q$ is an ordered
sequence of arrows $p=a_{n}a_{n-1}\cdots a_{1}$ with $t(a_{i})=s(a_{i+1})$ for
all $1\leq i\leq n-1$. Note that a 0-path is exactly a vertex and a 1-path is
exactly an arrow. In this case, we define $s(p)=s(a_{1})$, the start vertex of
$p$, and $t(p)=t(a_{n})$, the end vertex of $p$. For a 0-path $x$, we have
$s(x)=t(x)=x$. Let $Q_{n}$ be the set of $n$-paths. Let ${}^{y}Q_{n}^{x}$
denote the set of all $n$-paths from $x$ to $y$, $x,y\in Q_{0}$. That is,
${}^{y}Q_{n}^{x}=\\{p\in Q_{n}\mid s(p)=x,t(p)=y\\}$.
A quiver $Q$ is finite if $Q_{0}$ and $Q_{1}$ are finite sets. A quiver $Q$ is
locally finite if ${}^{y}Q_{1}^{x}$ is a finite set for any $x,y\in Q_{0}$.
Let ${\mathcal{K}}(G)$ denote the set of conjugacy classes in $G$. A formal
sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugacy classes of $G$ with
cardinal number coefficients is called a ramification (or ramification data )
of $G$, i.e. for any $C\in{\mathcal{K}}(G)$, $r_{C}$ is a cardinal number. In
particular, a formal sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugacy
classes of $G$ with non-negative integer coefficients is a ramification of
$G$.
For any ramification $r$ and $C\in{\mathcal{K}}(G)$, since $r_{C}$ is a
cardinal number, we can choose a set $I_{C}(r)$ such that its cardinal number
is $r_{C}$ without loss of generality. Let
${\mathcal{K}}_{r}(G):=\\{C\in{\mathcal{K}}(G)\mid
r_{C}\not=0\\}=\\{C\in{\mathcal{K}}(G)\mid I_{C}(r)\not=\emptyset\\}$. If
there exists a ramification $r$ of $G$ such that the cardinal number of
${}^{y}Q_{1}^{x}$ is equal to $r_{C}$ for any $x,y\in G$ with $x^{-1}y\in
C\in{\mathcal{K}}(G)$, then $Q$ is called a Hopf quiver with respect to the
ramification data $r$. In this case, there is a bijection from $I_{C}(r)$ to
${}^{y}Q_{1}^{x}$, and hence we write ${\ }^{y}Q_{1}^{x}=\\{a_{y,x}^{(i)}\mid
i\in I_{C}(r)\\}$ for any $x,y\in G$ with $x^{-1}y\in C\in{\mathcal{K}}(G)$.
$(G,r,\overrightarrow{\rho},u)$ is called a ramification system with
irreducible representations (or RSR in short ), if $r$ is a ramification of
$G$; $u$ is a map from ${\mathcal{K}}(G)$ to $G$ with $u(C)\in C$ for any
$C\in{\mathcal{K}}(G)$; $I_{C}(r,u)$ and $J_{C}(i)$ are sets with
$\mid\\!J_{C}(i)\\!\mid$ = ${\rm deg}(\rho_{C}^{(i)})$ and
$I_{C}(r)=\\{(i,j)\mid i\in I_{C}(r,u),j\in J_{C}(i)\\}$ for any
$C\in{\mathcal{K}}_{r}(G)$, $i\in I_{C}(r,u)$;
$\overrightarrow{\rho}=\\{\rho_{C}^{(i)}\\}_{i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)}\
\in\prod_{C\in{\mathcal{K}}_{r}(G)}(\widehat{{G^{u(C)}}})^{\mid
I_{C}(r,u)\mid}$ with $\rho_{C}^{(i)}\in\widehat{{G^{u(C)}}}$ for any $i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)$. In this paper we always assume that
$I_{C}(r,u)$ is a finite set for any $C\in{\mathcal{K}}_{r}(G).$ Furthermore,
if $\rho_{C}^{(i)}$ is a one dimensional representation for any
$C\in{\mathcal{K}}_{r}(G)$, then $(G,r,\overrightarrow{\rho},u)$ is called a
ramification system with characters (or RSC $(G,r,\overrightarrow{\rho},u)$
in short ) (see [ZZC04, Definition 1.8]). In this case, $a_{y,x}^{(i,j)}$ is
written as $a_{y,x}^{(i)}$ in short since $J_{C}(i)$ has only one element.
For ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, let $\chi_{C}^{(i)}$ denote the
character of $\rho_{C}^{(i)}$ for any $i\in I_{C}(r,u)$,
$C\in{\mathcal{K}}_{r}(C)$. If ramification $r=r_{C}C$ and
$I_{C}(r,u)=\\{i\\}$ then we say that ${\rm RSR}(G,r,\overrightarrow{\rho},u)$
is bi-one, written as ${\rm RSR}(G,{\mathcal{O}}_{s},\rho)$ with $s=u(C)$ and
$\rho=\rho_{C}^{(i)}$ in short, since $r$ only has one conjugacy class $C$ and
$\mid\\!I_{C}(r,u)\\!\mid=1$. Quiver Hopf algebras, Nichols algebras and
Yetter-Drinfeld modules, corresponding to a bi-one ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$, are said to be bi-one.
If $(G,r,\overrightarrow{\rho},u)$ is an ${\rm RSR}$, then it is clear that
${\rm RSR}(G,{\mathcal{O}}_{u(C)},\rho_{C}^{(i)})$ is bi-one for any
$C\in{\mathcal{K}}$ and $i\in I_{C}(r,u)$, which is called a bi-one sub-${\rm
RSR}$ of ${\rm RSR}(G,r,\overrightarrow{\rho},u)$,
For $s\in G$ and $(\rho,V)\in\widehat{G^{s}}$, here is a precise description
of the YD module $M({\mathcal{O}}_{s},\rho)$, introduced in [Gr00, AZ07]. Let
$t_{1}=s$, …, $t_{m}$ be a numeration of ${\mathcal{O}}_{s}$, which is a
conjugacy class containing $s$, and let $g_{i}\in G$ such that $g_{i}\rhd
s:=g_{i}sg_{i}^{-1}=t_{i}$ for all $1\leq i\leq m$. Then
$M({\mathcal{O}}_{s},\rho)=\oplus_{1\leq i\leq m}g_{i}\otimes V$. Let
$g_{i}v:=g_{i}\otimes v\in M({\mathcal{O}}_{s},\rho)$, $1\leq i\leq m$, $v\in
V$. If $v\in V$ and $1\leq i\leq m$, then the action of $h\in G$ and the
coaction are given by
$\displaystyle\delta(g_{i}v)=t_{i}\otimes g_{i}v,\qquad
h\cdot(g_{i}v)=g_{j}(\gamma\cdot v),$ (0.1)
where $hg_{i}=g_{j}\gamma$, for some $1\leq j\leq m$ and $\gamma\in G^{s}$.
The explicit formula for the braiding is then given by
$c(g_{i}v\otimes g_{j}w)=t_{i}\cdot(g_{j}w)\otimes
g_{i}v=g_{j^{\prime}}(\gamma\cdot w)\otimes g_{i}v$ (0.2)
for any $1\leq i,j\leq m$, $v,w\in V$, where $t_{i}g_{j}=g_{j^{\prime}}\gamma$
for unique $j^{\prime}$, $1\leq j^{\prime}\leq m$ and $\gamma\in G^{s}$. Let
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ denote
$\mathfrak{B}(M({\mathcal{O}}_{s},\rho))$. $M({\mathcal{O}}_{s},\rho)$ is a
simple YD module (see [AZ07, Section 1.2 ]).
## 1 Relation between bi-one arrow Nichols algebras and
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ (see [WZT13])
In this section it is shown that bi-one arrow Nichols algebras and
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ introduced in [Gr00, AZ07, AFZ] are the
same up to isomorphisms.
For any ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, we can construct an arrow
Nichols algebra $\mathfrak{B}(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},$ $u))$
( see [ZCZ, Pro. 2.4]), written as $\mathfrak{B}(G,r,\overrightarrow{\rho},$
$u)$ in short. Let us recall the precise description of arrow YD module. For
an ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and a $kG$-Hopf bimodule
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ with the module operations
$\alpha^{-}$ and $\alpha^{+}$, define a new left $kG$-action on $kQ_{1}$ by
$g\rhd x:=g\cdot x\cdot g^{-1},\ g\in G,x\in kQ_{1},$
where $g\cdot x=\alpha^{-}(g\otimes x)$ and $x\cdot g=\alpha^{+}(x\otimes g)$
for any $g\in G$ and $x\in kQ_{1}$. With this left $kG$-action and the
original left (arrow) $kG$-coaction $\delta^{-}$, $kQ_{1}$ is a Yetter-
Drinfeld $kG$-module. Let $Q_{1}^{1}:=\\{a\in Q_{1}\mid s(a)=1\\}$, the set of
all arrows with starting vertex $1$. It is clear that $kQ_{1}^{1}$ is a
Yetter-Drinfeld $kG$-submodule of $kQ_{1}$, denoted by
$(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$, called the arrow YD module.
###### Lemma 1.1.
For any $s\in G$ and $\rho\in\widehat{G^{s}}$, there exists a bi-one arrow
Nichols algebra $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ such that
$\mathfrak{B}({\mathcal{O}}_{s},\rho)\cong\mathfrak{B}(G,r,\overrightarrow{\rho},u)$
as graded braided Hopf algebras in ${}^{kG}_{kG}\\!{\mathcal{Y}D}$.
Proof. Assume that $V$ is the representation space of $\rho$ with
$\rho(g)(v)=g\cdot v$ for any $g\in G,v\in V$. Let $C={\mathcal{O}_{s}}$,
$r=r_{C}C$, $r_{C}={\rm deg}\rho$, $u(C)=s$, $I_{C}(r,u)=\\{1\\}$ and
$(v)\rho_{C}^{(1)}(h)=\rho(h^{-1})(v)$ for any $h\in G$, $v\in V$. We get a
bi-one arrow Nichols algebra $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$.
We now only need to show that
$M({\mathcal{O}}_{s},\rho)\cong(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$
in ${}^{kG}_{kG}\\!{\mathcal{Y}D}$. We recall the notation in [ZCZ,
Proposition 1.2]. Assume $J_{C}(1)=\\{1,2,\cdots,n\\}$ and $X_{C}^{(1)}=V$
with basis $\\{x_{C}^{(1,j)}\mid j=1,2,\cdots,n\\}$ without loss of
generality. Let $v_{j}$ denote $x_{C}^{(1,j)}$ for convenience. In fact, the
left and right coset decompositions of $G^{s}$ in $G$ are
$\displaystyle G=\bigcup_{i=1}^{m}g_{i}G^{s}\ \ \hbox{and }\ \ G$
$\displaystyle=$ $\displaystyle\bigcup_{i=1}^{m}G^{s}g_{i}^{-1}\ \ ,$ (1.1)
respectively.
Let $\psi$ be a map from $M({\mathcal{O}}_{s},\rho)$ to $(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))$ by sending $g_{i}v_{j}$ to
$a_{t_{i},1}^{(1,j)}$ for any $1\leq i\leq m,1\leq j\leq n$. Since the
dimension is $mn$, $\psi$ is a bijective. See
$\displaystyle\delta^{-}(\psi(g_{i}v_{j}))$ $\displaystyle=$
$\displaystyle\delta^{-}(a_{t_{i},1}^{(1,j)})$ $\displaystyle=$ $\displaystyle
t_{i}\otimes a_{t_{i},1}^{(1,j)}=(id\otimes\psi)\delta^{-}(g_{i}v_{j}).$
Thus $\psi$ is a $kG$-comodule homomorphism. For any $h\in G$, assume
$hg_{i}=g_{i^{\prime}}\gamma$ with $\gamma\in G^{s}$. Thus
$g_{i}^{-1}h^{-1}=\gamma^{-1}g_{i^{\prime}}^{-1}$, i.e.
$\zeta_{i}(h^{-1})=\gamma^{-1}$, where $\zeta_{i}$ was defined in [ZZC04,
(0.3)]. Since $\gamma\cdot x^{(1,j)}\in V$, there exist
$k_{C,h^{-1}}^{(1,j,p)}\in k$, $1\leq p\leq n$, such that $\gamma\cdot
x^{(1,j)}=\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}x^{(1,p)}$. Therefore
$\displaystyle x^{(1,j)}\cdot\zeta_{i}(h^{-1})$ $\displaystyle=$
$\displaystyle\gamma\cdot x^{(1,j)}\ \ (\hbox{by definition of
}\rho_{C}^{(1)})$ (1.2) $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}x^{(1,p)}.$
See
$\displaystyle\psi(h\cdot g_{i}v_{j})$ $\displaystyle=$
$\displaystyle\psi(g_{i^{\prime}}(\gamma v_{j}))$ $\displaystyle=$
$\displaystyle\psi(g_{i^{\prime}}(\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}v_{p}))$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}a_{t_{i^{\prime}},1}^{(1,p)}$
and
$\displaystyle h\rhd(\psi(g_{i}v_{j}))$ $\displaystyle=$ $\displaystyle
h\rhd(a_{t_{i},1}^{(1,j)})$ $\displaystyle=$ $\displaystyle
a_{ht_{i},h}^{(1,j)}\cdot h^{-1}$ $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}a_{t_{i^{\prime}},1}^{(1,p)}\
\ (\hbox{by \cite[cite]{[\@@bibref{}{ZCZ08}{}{}, Pro.1.2]} and
}(\ref{e1.11})).$
Therefore $\psi$ is a $kG$-module homomorphism. $\Box$
Therefore we also say that $\mathfrak{B}({\mathcal{O}}_{s},\rho)$ is a bi-one
Nichols Hopf algebra.
###### Remark 1.2.
The representation $\rho$ in $\mathfrak{B}({\mathcal{O}}_{s},\rho)$ introduced
in [Gr00, AZ07] and $\rho_{C}^{(i)}$ in RSR are different. $\rho(g)$ acts on
its representation space from the left and $\rho_{C}^{(i)}(g)$ acts on its
representation space from the right.
Otherwise, when $\rho=\chi$ is a one dimensional representation, then
$(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$ is PM (see [ZZC04, Def. 1.1]).
Thus the formulae are available in [ZZC04, Lemma 1.9]. That is, $g\cdot
a_{t}=a_{gt_{i},g}$, $a_{t_{i}}\cdot g=\chi(\zeta_{i}(g))a_{t_{i}g,g}$.
## 2 Transposition and FK’ Conjecture
In this section we give the relationship between FK’ Conjecture and Nichols
algebra $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ of
transposition over symmetry group by means of quiver Hopf algebras.
We first consider Nichols algebra $\mathfrak{B}({\mathcal{O}}_{(12)},\rho)$ of
transposition $\sigma$ over symmetry groups, where $\rho=sgn\otimes sgn$ or
$\rho=\epsilon\otimes sgn.$
Let $\sigma=(12)\in S_{n}:=G$, $\mathcal{O}_{\sigma}=\\{(ij)|1\leq i,j\leq
n\\}$, $G^{\sigma}=\\{g\in G\mid g\sigma=\sigma g\\}$.
$G=\bigcup\limits_{1\leq i<j\leq n}G^{\sigma}g_{i,j}$.
Let $g_{kj}:=\left\\{\begin{array}[]{lll}id\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
k=1,j=2\\\ (2j)\ \ \ \ \ \ \ \ \ \ \ \ \ \ k=1,j>2\\\ (1j)\ \ \ \ \ \ \ \ \ \
\ \ \ \ k=2,j>2\\\ (1k)(2j)\ \ \ \ \ \ \ \ \ k>2,j>k\\\ \end{array}\right.$
and $t_{kj}:=(k\ j).$
###### Lemma 2.1.
(see [WZT13]) In The following equations in $\mathbb{S}_{m}$ hold.
$(12)id=id(12)$
$(12)(2j)=(2j1)=(1j)(12)$
$(12)(1j)=(1j2)=(2j)(12)$
$(12)(1k)(2j)=(1k2)(2j)=(1k)(2j)(kj)$
$(1j)id=(1j)id$
$(1j)(2j)=(j21)=(2j)(12)$
$(1j)(2j_{1})=(1j)(2j_{1})id\ \ j<j_{1}$
$(1j)(2j_{1})=(1j_{1})(2j)(jj_{1})(12)\ \ j>j_{1}$
$(1j)(1j)=id$
$(1j)(1j_{1})=(1j_{1}j)=(1j_{1})(jj_{1})$
$(1j)(1k)(2j)=(1kj)(2j)=(2k)(12)(kj)$
$(1j)(1k)(2j_{1})=(2j_{1})id\ \ \ j=k$
$(1j)(1k)(2j_{1})=(1k)(kj)(2j_{1})=(1k)(2j_{1})(kj)\ \ j\neq j_{1}$
$(2j)id=(2j)id$
$(2j)(2j)=id$
$(2j)(2j_{1})=(2j_{1}j)=(2j_{1})(jj_{1})$
$(2j)(1j)=(j12)=(1j)(12)$
$(2j)(1j_{1})=(1j)(2j_{1})(jj_{1})(12)\ \ j<j_{1}$
$(2j)(1j_{1})=(1j_{1})(2j)id\ \ j>j_{1}$
$(2j)(1k)(2j)=(1k)=(1k)id$
$(2j)(1k)(2j_{1})=(1k)(12)(2j_{1})=(1k)(1j_{1})(12)=(1j_{1})(12)(kj_{1})\ \
j=k$
$(2j)(1k)(2j_{1})=(2j)(2j_{1})(1k)=(2j_{1})(j_{1}j)(1k)=(2j_{1})(1k)(j_{1}j)\
\ j\neq j_{1},j\neq k$
$(kj)id=id(kj)$
$(kj)(2j)=(2k)(kj)$
$(kj)(2k)=(2j)(kj)$
$(kj)(2j_{1})=(2j_{1})(kj)\ \ k\neq j_{1},j\neq j_{1}$
$(kj)(1j)=(1k)(kj)$
$(kj)(1k)=(1j)(kj)$
$(kj)(1j_{1})=(1j_{1})(kj)\ \ k\neq j_{1},j\neq j_{1}$
$(kj)(1k)(2j)=(1k)(2j)(12)$
$(kj)(1k_{1})(2j_{1})=(1k)(2j_{1})(kj)\ \ k_{1}=j$
$(kj)(1k_{1})(2j_{1})=(1k_{1})(2k)(kj)\ \ k_{1}<k,j_{1}=j$
$(kj)(1k_{1})(2j_{1})=(1k)(2k_{1})(12)(kjk_{1})\ \ k_{1}>k,j_{1}=j$
$(kj)(1k_{1})(2j_{1})=(1j)(2j_{1})(kj)\ \ j_{1}>j,k_{1}=k$
$(kj)(1k_{1})(2j_{1})=(1j_{1})(2j)(12)(jkj_{1})\ \ j_{1}<j,k_{1}=k$
$(kj)(1k_{1})(2j_{1})=(1k_{1})(2j)(kj)\ \ k_{1}\neq j,j_{1}=k$
$(kj)(1k_{1})(2j_{1})=(1k_{1})(2j_{1})(kj)\ \ k_{1}\neq k,k_{1}\neq
j,j_{1}\neq j,j_{1}\neq k$.
Remark: By Lemma above, we can obtain $\zeta_{st}(t_{ij})\in G^{\sigma}$ such
that $g_{st}t_{ij}=\zeta_{st}(t_{ij})g_{s^{\prime}t^{\prime}}$ for any $1\leq
i,j,s,t\leq n.$ Let $a_{ij}$ denote the arrow $a_{t_{ij},1}$ from $1$ to
$t_{ij}$ in short. By Lemma 1.1, the algebra generated by $\\{a_{ij}\mid 1\leq
i<j\leq n\\}$ in its co-path Hopf algebra is isomorphic to Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho).$
###### Definition 2.2.
(See [FK97, Def. 2.1]) algebra ${\mathcal{E}}_{n}$ is generated by
$\\{x_{ij}\mid 1\leq i<j\leq n\\}$ with definition relations:
(i) $x_{ij}^{2}=0$ for $i<j.$
(ii) $x_{ij}x_{jk}=x_{jk}x_{ik}+x_{ik}x_{ij}$ and
$x_{jk}x_{ij}=x_{ik}x_{jk}+x_{ij}x_{ik},$ for $i<j<k.$
(iii) $x_{ij}x_{kl}=x_{kl}x_{ij}$ for any distinct $i,j,k$ and $l,$ $i<j,k<l.$
###### Theorem 2.3.
Assume $n>3$.
(i) $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$ is an image of
$\mathcal{E}_{n}$.
(ii) If $\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes
sgn)=\infty$, then so is $\dim\mathcal{E}_{n}$.
Proof. Let $b_{ij}=-a_{ij}$ when $i=2$ and $j>2$; $b_{ij}=a_{ij}$ otherwise.
By [WZT13, Table 1, Lemma 5.2(i)(ii)(iii)], there exists $\alpha_{i,j,k}$,
$\beta_{i,j,k}\in\\{1,-1\\}$ such that
$\displaystyle
a_{ij}a_{jk}+\alpha_{ijk}a_{jk}a_{ki}+\beta_{ijk}a_{ki}a_{ij}=0,$ (2.1)
for any distinct $i,j$ and $k$. It follows from [WZT13, formula (5.2) ] that
case $\alpha_{ijk}$ $\beta_{ijk}$ $2<i<j<k$ $-1$ $-1$ $i=1,j=2<k$ $-1$ $1$
$i=1,2<j<k$ $-1$ $-1$ $i=2<j<k$ $-1$ $1$ $2<i<k<j$ $-1$ $1$ $i=1,k=2<j$ $-1$
$-1$ $i=1,2<k<j$ $-1$ $1$ $i=2<k<j$ $-1$ $-1$ .
$\hbox{Table }1$
Using the Table 1 we can obtain that $b_{ij}^{2}=0$ for $i<j;$
$b_{ij}b_{jk}=b_{jk}b_{ik}+b_{ik}b_{ij}$ and
$b_{jk}b_{ij}=b_{ik}b_{jk}+b_{ij}b_{ik},$ for $i<j<k.$ By [WZT13, Lemma
5.2(iv)], $b_{ij}b_{kl}=b_{kl}b_{ij}$ for any distinct $i,j,k$ and $l,$
$i<j,k<l.$ Consequently, Part (i) holds since $\\{b_{ij}\mid 1\leq i<j\leq
n\\}$ generates $\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)$.
$\Box$
Notation: N. Andruskiewitsch and H.-J. Schneider point out that
${\mathcal{E}}_{5}$ is finite dimensional by using a computer program (see
[AS02, Section 3.4]). A. Kirillov told me that ${\mathcal{E}}_{5}$ has
dimension $4^{4}5^{2}6^{4}$ and the real problem is to compute the Hilbert
series of the algebra ${\mathcal{E}}_{6}$. Consequently,
$\dim\mathfrak{B}({\mathcal{O}}_{{(1,2)}},\epsilon\otimes sgn)<\infty$ when
$n=5.$
Let $\chi^{\prime}:=sgn\otimes sgn$ and $\chi^{\prime\prime}:=\epsilon\otimes
sgn$. Let $\phi_{1}$ and $\phi_{2}$ be maps from $\mathbb{S}_{n}\times T$ to
${\bf k}\setminus 0$ such that
$\phi_{1}(g,t):=\left\\{\begin{array}[]{ll}1&\hbox{if }g(i)<g(j)\\\
-1&\hbox{if }g(j)<g(i)\\\ \end{array}\right.$ and $\phi_{2}(g,t):=(-1)^{l(g)}$
where $1\leq<j\leq n$, $T:=\\{(u,v)\mid 1\leq u,v\leq n,u\not=v\\}$ and $l(g)$
is the length of $g$, that is, $l(g)$ is the minimal number $q$ such that
$g=g_{1}g_{2}\cdots g_{q}$ with $g_{i}\in T$ , $1\leq i\leq q.$ Let
$\\{x_{ij}\mid(i,j)\in T\\}$ be a basis of $M(\mathbb{S}_{n},T,\phi)$ with
$\phi=\phi_{1}$ or $\phi=\phi_{2}.$ Define module and comodule operations as
follows: $g\rhd x_{ij}=\phi(g,(i,j))x_{g\rhd(i,j)}$,
$\delta^{-}(x_{ij})=(i,j)\otimes x_{ij},$ for any $g\in\mathbb{S}_{n},(i,j)\in
T.$ By [MS, Def. 5.1 and Example 5.3], $M(\mathbb{S}_{n},T,\phi)$ is a ${\rm
YD}$ module over $\mathbb{S}_{n}.$ Its Nichols algebra is written
$\mathfrak{B}(\mathbb{S}_{n},T,\phi)$. By [FK97, Def. 2.1] and [MS, Example
6.2 ], $\mathfrak{B}(\mathbb{S}_{n},T,\phi)$ is an image of $\mathcal{E}_{n}$.
###### Lemma 2.4.
If $n>4$, then $M(\mathbb{S}_{n},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime})$ as YD modules over
$\mathbb{S}_{n}.$ $M(\mathbb{S}_{m},T,\phi_{2})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime\prime})$ as YD modules over
$\mathbb{S}_{n}$.
Proof. If there exists isomorphism $\psi:M(\mathbb{S}_{n},T,\phi)\rightarrow
M({\mathcal{O}}_{{(1,2)}},\chi)$ as YD mosules over $\mathbb{S}_{n}$, where
$\phi=\phi_{1}$ or $\phi=\phi_{2}$, $\chi=\chi^{\prime}$ or
$\chi=\chi^{\prime\prime}$, then there exists $k_{ij}\in{\bf k}$ such that
$\psi(x_{ij})=k_{ij}a_{ij}$ for any $(i,j)\in T$, where $a_{ij}$ denotes arrow
$a_{t_{ij},1}$ in short, since $\psi$ is a comodule isomorphism. By
$\psi(g\rhd x_{ij})=g\rhd\psi(x_{ij})$, we have
$\displaystyle
k_{g\rhd(ij)}\phi(g,(i,j))a_{g\rhd(ij)}=k_{ij}\chi(\zeta_{ij}(g^{-1}))a_{g\rhd(ij)},$
(2.2)
for any $g\in\mathbb{S}_{n}$, $(i,j)\in T.$ For convenience, set
$k_{ij}=k_{ji}.$
Let $2<i<j$, $g=(1,2)$. It is clear $g_{ij}g=(1i)(2j)(12)=(ij)(1i)(2j)$. We
have $g=g^{-1}$ and $\zeta_{ij}(g)=(ij)$. Thus
$\chi^{\prime}(\zeta_{ij}(g))=-1$ and $\chi^{\prime\prime}(\zeta_{ij}(g))=1$;
$\phi_{1}(g,(ij))=1$ and $\phi_{2}(g,(ij))=-1$. Considering (2.2) we have
$M(\mathbb{S}_{n},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime})$ as YD modules over
$\mathbb{S}_{n}.$ $M(\mathbb{S}_{m},T,\phi_{2})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime\prime})$ as YD modules over
$\mathbb{S}_{n}$ since
$k_{g\rhd(ij)}\phi_{1}(g,(i,j))\not=k_{ij}\chi^{\prime}(\zeta_{ij}(g))$ and
$k_{g\rhd(ij)}\phi_{2}(g,(i,j))\not=k_{ij}\chi^{\prime\prime}(\zeta_{ij}(g))$.
$\Box$
###### Theorem 2.5.
If there exists a natural number $n_{0}>5$ such that
$M(\mathbb{S}_{n_{0}},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)}},\chi^{\prime\prime})$ as YD modules over
$\mathbb{S}_{n_{0}}$, then
$\dim\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})=\infty$ and
$\dim\mathcal{E}_{n}=\infty$ for any $n>n_{0}.$.
Proof. $M(\mathbb{S}_{6},T,\phi_{1})$ is not isomorphic to
$M({\mathcal{O}}_{{(1,2)(3,4)(5,6)}},\rho)$ as YD modules over
$\mathbb{S}_{6}$ since they are not isomorphic as comodules over
$\mathbb{S}_{6}$, when $\rho$ is one dimensional representation.
If $\dim\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})<\infty$, then
$\dim\mathfrak{B}(\mathbb{S}_{n_{0}},T,\phi_{1})<\infty$ since
$\mathbb{S}_{n_{0}}$ is a subgroup of $\mathbb{S}_{n}.$
$M(\mathbb{S}_{n},T,\phi_{1})$ is a reducible YD modules over $\mathbb{S}_{n}$
by Lemma 2.4 and [AFGV08, Th. 1.1]. However, every reducible YD modules over
$\mathbb{S}_{n}$ is infinite dimensional by [HS08, Cor. 8.4]. This is a
contradiction. Consequently,
$\dim\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})=\infty.$ By [FK97, Def. 2.1] and
[MS, Example 6.2 ], $\mathfrak{B}(\mathbb{S}_{n},T,\phi_{1})$ is an image of
$\mathcal{E}_{n}$. Therefore, $\dim\mathcal{E}_{n}=\infty$ for any $n>n_{0}.$.
$\Box$
## References
* [AFZ] N. Andruskiewitsch, F. Fantino, S. Zhang, On pointed Hopf algebras associated with the symmetric groups, Manuscripta Math., 128(2009) 3, 359-371.
* [AFGV08] N. Andruskiewitsch, F. Fantino, M. Graña and L.Vendramin, Finite-dimensional pointed Hopf algebras with alternating groups are trivial, preprint arXiv:0812.4628, to appear Aparecer en Ann. Mat. Pura Appl..
* [AS02] N. Andruskiewitsch, H.-J. Schneider, Pointed Hopf algebras, New directions in Hopf algebras, MSRI series Cambridge Univ. Press; 2002, 1–68.
* [AZ07] N. Andruskiewitsch and S. Zhang, On pointed Hopf algebras associated to some conjugacy classes in $\mathbb{S}_{n}$, Proc. Amer. Math. Soc. 135 (2007), 2723-2731.
* [Ba06] Y. Bazlov, Nichols CWoronowicz algebra model for Schubert calculus on Coxeter groups, Journal of Algebra, 297 ( 2006) 2, 372 C399.
* [CR02] C. Cibils and M. Rosso, _Hopf quivers_ , J. Alg., 254 (2002), 241-251.
* [FK97] S. Fomin and A.N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Progress in Geometry, ed. J.-L. Brylinski and R. Brylinski, 1997.
* [GHV] M. Graña, I. Heckenberger, L. Vendramin , Nichols algebras of group type with many quadratic relations, Advances in Mathematics, 227(2011)5, 1956-1989.
* [Gr00] M. Graña, On Nichols algebras of low dimension, Contemp. Math. 267 (2000), 111–134.
* [HS08] I. Heckenberger and H.-J. Schneider, Root systems and Weyl groupoids for Nichols algebras, preprint arXiv:0807.0691, to appear Proc. London Math. Soc..
* [MS] A. Milinski and H-J. Schneider, _Pointed Indecomposable Hopf Algebras over Coxeter Groups_ , Contemp. Math. 267 (2000), 215–236.
* [WZT13] Weicai Wu, Shouchuan Zhang, Zhengtang Tan, Pointed Hopf algebras with classical Weyl groups (II), Preprint arXiv: 1307.8227.
* [ZZC04] Shouchuan Zhang, Y-Z Zhang and H. X. Chen, Classification of PM Quiver Hopf Algebras, J. Alg. and Its Appl. 6 (2007)(6), 919-950. Also see in math. QA/0410150.
* [ZCZ] S. Zhang, H. X. Chen and Y.-Z. Zhang, Classification of quiver Hopf algebras and pointed Hopf algebras of type one, Bull. Aust. Math. Soc. 87 (2013), 216-237. Also in arXiv:0802.3488.
|
arxiv-papers
| 2013-07-25T15:33:11 |
2024-09-04T02:49:48.467989
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Zhengtang Tan, Weicai Wu, Shouchuan Zhang",
"submitter": "Shouchuan Zhang",
"url": "https://arxiv.org/abs/1307.6794"
}
|
1307.6894
|
lrcornerulcorner txfontslrcornertxfontsulcorner
# The operad of temporal wiring diagrams: formalizing a graphical language for
discrete-time processes
Dylan Rupel Department of Mathematics, Northeastern University, Boston, MA
02115 [email protected] and David I. Spivak Department of Mathematics,
Massachusetts Institute of Technology, Cambridge MA 02139
[email protected]
###### Abstract.
We investigate the hierarchical structure of processes using the mathematical
theory of operads. Information or material enters a given process as a stream
of inputs, and the process converts it to a stream of outputs. Output streams
can then be supplied to other processes in an organized manner, and the
resulting system of interconnected processes can itself be considered a macro
process. To model the inherent structure in this kind of system, we define an
operad $\mathcal{W}$ of black boxes and directed wiring diagrams, and we
define a $\mathcal{W}$-algebra $\mathcal{P}$ of processes (which we call
propagators, after [RS]). Previous operadic models of wiring diagrams (e.g.
[Sp2]) use undirected wires without length, useful for modeling static systems
of constraints, whereas we use directed wires with length, useful for modeling
dynamic flows of information. We give multiple examples throughout to ground
the ideas.
Spivak acknowledges support by ONR grant N000141310260.
###### Contents
1. 1 Introduction
2. 2 $\mathcal{W}$, the operad of directed wiring diagrams
3. 3 $\mathcal{P}$, the algebra of propagators on $\mathcal{W}$
4. 4 Future work
## 1\. Introduction
Managing processes is inherently a hierarchical and self-similar affair.
Consider the case of preparing a batch of cookies, or if one prefers, the
structurally similar case of manufacturing a pharmaceutical drug. To make
cookies, one generally follows a recipe, which specifies a process that is
undertaken by subdividing it as a sequence of major steps. These steps can be
performed in series or in parallel. The notion of self-similarity arises when
we realize that each of these major steps can itself be viewed as a process,
and thus it can also be subdivided into smaller steps. For example, procuring
the materials necessary to make cookies involves getting oneself to the
appropriate store, selecting the necessary materials, paying for them, etc.,
and each of these steps is itself a simpler process.
Perhaps every such hierarchy of nesting processes must touch ground at the
level of atomic detail. Hoping that the description of processes within
processes would not continue ad infinitum may have led humanity to investigate
matter and motion at the smallest level possible. This investigation into
atomic and quantum physics has yielded tremendous technological advances, such
as the invention of the microchip.
Working on the smallest possible scale is not always effective, however. It
appears that the planning and execution of processes benefits immensely from
hierarchical chunking. To write a recipe for cookies at the level of atomic
detail would be expensive and useless. Still, when executing our recipe, the
decision to add salt will initiate an unconscious procedure, by which signals
are sent from the brain to the muscles of the arm, on to individual cells, and
so on until actual atoms move through space and “salt has been added”. Every
player in the larger cookie-making endeavor understands the current demand
(e.g. to add salt) as a procedure that makes sense at his own level of
granularity. The decision to add salt is seen as a mundane (low-level) job in
the context of planning to please ones girlfriend by baking cookies; however
this same decision is seen as an abstract (high-level) concept in the context
of its underlying performance as atomic movements.
For designing complex processes, such as those found in manufacturing
automobiles or in large-scale computer programming, the architect and
engineers must be able to change levels of abstraction with ease. In fact,
different engineers working on the same project are often thinking about the
same basic structures, but in different terms. They are most effective when
they can chunk the basic structures as they see fit.
A person who studies a supply chain in terms of the function played by each
chain member should be able to converse coherently with a person who studies
the same supply chain in terms of the contracts and negotiations that exist at
each chain link. These are two radically different viewpoints on the same
system, and it is useful to be able to switch fluidly between them. Similarly,
an engineer designing a system’s hardware must be able to converse with an
engineer working on the system’s software. Otherwise, small perturbations made
by one of them will be unexpected by the other, and this can lead to major
problems.
The same types of issues emerge whether one is concerned with manufacturers in
a supply chain, neurons in a functional brain region, modules in a computer
program, or steps in a recipe. In each case, what we call propagators (after
[RS]) are being arranged into a system that is itself a propagator at a higher
level. The goal of this paper is to provide a mathematical basis for thinking
about this kind of problem. We offer a formalism that describes the
hierarchical and self-similar nature of a certain kind of wiring diagram.
A similar kind of wiring diagram was described in [Sp2], the main difference
being that the present one is built for time-based processes whereas the one
in [Sp2] was built for static relations. In the present work we take the
notion of time (or one may say distance) seriously. We go through considerable
effort to integrate a notion of time and distance into the fundamental
architecture of our description, by emphasizing that communication channels
have a length, i.e. that communication takes time.
Design choices such as these greatly affect the behavior of our model, and
ours was certainly not the only viable choice. We hope that the basic idea we
propose will be a basis upon which future engineers and mathematicians will
improve. For the time being, we may at least say that the set of rules we
propose for our wiring diagrams roughly conform with the IDEF0 standard set by
the National Institute of Standards and Technology [NIST]. The main
differences are that in our formalism,
* •
wires can split but not merge (each merging must occur within a particular
box),
* •
feedback loops are allowed,
* •
the so-called control and mechanism arrows are subsumed into input and output
arrows, and
* •
the rules for and meaning of hierarchical composition is made explicit.
The basic picture to have in mind for our wiring diagrams is the following:
(1)
In this picture we see an exterior box, some interior boxes, and a collection
of directed wires. These directed wires transport some type of product from
the export region of some box to the import region of some box. In (1) we have
a supply chain involving three propagators, one of whom imports flour, sugar,
and salt and exports dry mixture, and another of whom imports eggs and milk
and exports egg yolks and wet mixture. The dry mixture and the wet mixture are
then transported to a third propagator who exports cookie batter. The whole
system itself constitutes a propagator that takes five ingredients and
produces cookie batter and egg yolks.
The formalism we offer in this paper is based on a mathematical structure
called an operad (more precisely, a symmetric colored operad), chosen because
they capture the self-similar nature of wiring diagrams. The rough idea is
that if we have a wiring diagram and we insert wiring diagrams into each of
its interior boxes, the result is a new wiring diagram.
(2)
We will make explicit what constitutes a box, what constitutes a wiring
diagram (WD), and how inserting WDs into a WD constitutes a new WD. Like
Russian dolls, we may have a nesting of WDs inside of WDs inside of WDs, etc.
We will prove an associativity law that guarantees that no matter how deeply
our Russian dolls are nested, the resulting WD is well-defined. Once all this
is done, we will have an operad $\mathcal{W}$.
To make this directed wiring diagrams operad $\mathcal{W}$ useful, we will
take our formalism to the next logical step and provide an algebra on
$\mathcal{W}$. This algebra $\mathcal{P}$ encodes our application to process
management by telling us what fits in the boxes and how to use wiring diagrams
to build more complex systems out of simpler components. More precisely, the
algebra $\mathcal{P}$ makes explicit
* •
the set of things that can go in every box, namely the set of propagators, and
* •
a method for taking a wiring diagram and a propagator for each of its interior
boxes and producing a propagator for the exterior box.
To prove that we have an algebra, we will show that no matter how one decides
to group the various internal propagators, the behavior of the resulting
system is unchanged.
Operads were invented in the 1970s by [May] and [BV] in order to encode the
relationship between various operations they noticed taking place in the
mathematical field of algebraic topology. At the moment we are unconcerned
with topological properties of our operads, but the formalism grounds the
picture we are trying to get across. For more on operads, see [Lei].
### 1.1. Structure of the paper
In Section 2 we discuss operads. In Section 2.1 we give the mathematical
definition of operads and some examples. In Section 2.2 we propose the operad
of interest, namely $\mathcal{W}$, the operad of directed wiring diagrams. We
offer an example wiring diagram in Section 2.3 that will run throughout the
paper and eventually output the Fibonacci sequence. In Section 2.4 we prove
that $\mathcal{W}$ has the required properties so that it is indeed an operad.
In Section 3 we discuss algebras on an operad. In Section 3.1 we give the
mathematical definition of algebras. In Sections 3.2 we discuss some
preliminaries on lists and define our notion of historical propagators, which
we will then use in 3.3 where we propose the $\mathcal{W}$-algebra of
interest, the algebra of propagators. In Section 3.5 we prove that
$\mathcal{P}$ has the required properties so that it is indeed a
$\mathcal{W}$-algebra.
We expect the majority of readers to be most interested in the running
examples sections, Sections 2.3 and 3.4. Readers who want more details, e.g.
those who may wish to write code for propagators, will need to read Sections
2.2, 3.3. The proof that our algebra satisfies the necessary requirements is
technical; we expect only the most dedicated readers to get through it.
Finally, in Section 4 we discuss some possibilities for future work in this
area.
The remainder of the present section is devoted to our notational conventions
(Section 1.2) and our acknowledgments (1.3).
### 1.2. Notation and background
Here we describe our notational conventions. These are only necessary for
readers who want a deep understanding of the underlying mathematics. Such
readers are assumed to know some basic category theory. For mathematicians we
recommend [Awo] or [Mac], for computer scientists we recommend [Awo] or [BW],
and for a general audience we recommend [Sp1].
We will primarily be concerned only with the category of small sets, which we
denote by ${\bf Set}$, and some related categories. We denote by ${\bf
Fin}\subseteq{\bf Set}$ the full subcategory spanned by finite sets. We often
use the symbol $n\in\textnormal{Ob}({\bf Fin})$ to denote a finite set, and
may speak of elements $i\in n$. The cardinality of a finite set is a natural
number, denoted $|n|\in{\mathbb{N}}$. In particular, we consider $0$ to be a
natural number.
Suppose given a finite set $n$ and a function $X\colon
n\rightarrow\textnormal{Ob}({\bf Set})$, and let $\amalg_{i\in I}X(i)$ be the
disjoint union. Then there is a canonical function $\pi_{X}\colon\amalg_{i\in
n}X(i)\longrightarrow n$ which we call the component projection. We use almost
the same symbol in a different context; namely, for any function $s\colon
m\rightarrow n$ we denote the $s$-coordinates projection by
$\pi_{s}\colon\prod_{i\in n}X(i)\longrightarrow\prod_{j\in m}X(s(j)).$
In particular, if $i\in n$ is an element, we consider it as a function
$i\colon\\{*\\}\rightarrow n$ and write $\pi_{i}\colon\prod_{i\in
n}X(i)\rightarrow X(i)$ for the usual $i$th coordinate projection.
A pointed set is a pair $(S,s)$ where $S\in\textnormal{Ob}({\bf Set})$ is a
set and $s\in S$ is a chosen element, called the base point. In particular a
pointed set cannot be empty. Given another pointed set $(T,t)$, a pointed
function from $(S,s)$ to $(T,t)$ consists of a function $f\colon S\rightarrow
T$ such that $f(s)=t$. We denote the category of pointed sets by ${\bf
Set}_{*}$. There is a forgetful functor ${\bf Set}_{*}\rightarrow{\bf Set}$
which forgets the basepoint; it has a left adjoint which adjoins a free
basepoint $X\mapsto X\amalg\\{*\\}$. We often find it convenient not to
mention basepoints; if we speak of a set $X$ as though it is pointed, we are
actually speaking of $X\amalg\\{*\\}$. If $S,S^{\prime}$ are pointed sets then
the product $S\times S^{\prime}$ is also naturally pointed, with basepoint
$(*,*)$, again denoted simply by $*$.
We often speak of functions $n\rightarrow\textnormal{Ob}({\bf Set}_{*})$,
where $n$ is a finite set. Of course, $\textnormal{Ob}({\bf Set}_{*})$ is not
itself a small set, but using the theory of Grothendieck universes [Bou], this
is not a problem. It will be even less of a problem in applications.
### 1.3. Acknowledgements
David Spivak would like to thank Sam Cho as well as the NIST community,
especially Al Jones and Eswaran Subrahmanian. Special thanks go to Nat
Stapleton for many valuable conversations in which substantial progress was
made toward subjects quite similar to the ones we discuss here.
Dylan Rupel would like to thank Jason Isbell and Kiyoshi Igusa for many useful
discussions.
## 2\. $\mathcal{W}$, the operad of directed wiring diagrams
In this section we will define the operad $\mathcal{W}$ of black boxes and
directed wiring diagrams (WDs). It governs the forms that a black box can
take, the rules that a WD must follow, and the formula for how the
substitution of WDs into a WD yields a WD. There is no bound on the depth to
which wiring diagrams can be nested. That is, we prove an associative law
which roughly says that the substitution formula is well-defined for any
degree of nesting, shallow or deep.
We will use the operad $\mathcal{W}$ to discuss the hierarchical nature of
processes. Each box in our operad will be filled with a process, and each
wiring diagram will effectively build a complex process out of simpler ones.
However, this is not strictly a matter of the operad $\mathcal{W}$ but of an
algebra on $\mathcal{W}$. This algebra will be discussed in Section 3.
The present section is organized as follows. First, in Section 2.1 we give the
technical definition of the term operad and a few examples. In Section 2.2 we
propose our operad $\mathcal{W}$ of wiring diagrams. It will include drawings
that should clarify the matter. In Section 2.3 we present an example that will
run throughout the paper and end up producing the Fibonacci sequence. This
section is recommended especially to the more category-theoretically shy
reader. Finally, in Section 2.4 we give a technical proof that our proposal
for $\mathcal{W}$ satisfies the requirements for being a true operad, i.e. we
establish the well-definedness of repeated substitution as discussed above.
### 2.1. Definition and basic examples of operads
Before we begin, we should give a warning about our use of the term “operad”.
###### Warning 2.1.1.
Throughout this paper, we use the word operad to mean what is generally called
a symmetric colored operad or a symmetric multicategory. This abbreviated
nomenclature is not new, for example it is used in [Lur]. Hopefully no
confusion will arise. For a full treatment of operads, multicategories, and
how they fit into a larger mathematical context, see [Lei].
Most of Section 2.1 is recycled material, taken almost verbatim from [Sp2]. We
repeat it here for the convenience of the reader.
###### Definition 2.1.2.
An operad $\mathcal{O}$ is defined as follows: One announces some constituents
(A. objects, B. morphisms, C. identities, D. compositions) and proves that
they satisfy some requirements (1. identity law, 2. associativity law).
Specifically,
1. A.
one announces a collection $\textnormal{Ob}(\mathcal{O})$, each element of
which is called an object of $\mathcal{O}$.
2. B.
for each object $y\in\textnormal{Ob}(\mathcal{O})$, finite set
$n\in\textnormal{Ob}({\bf Fin})$, and $n$-indexed set of objects $x\colon
n\rightarrow\textnormal{Ob}(\mathcal{O})$, one announces a set
$\mathcal{O}_{n}(x;y)\in\textnormal{Ob}({\bf Set})$. Its elements are called
morphisms from $x$ to $y$ in $\mathcal{O}$.
3. C.
for every object $x\in\textnormal{Ob}(\mathcal{O})$, one announces a specified
morphism denoted $\textnormal{id}_{x}\in\mathcal{O}_{1}(x;x)$ called the
identity morphism on $x$.
4. D.
Let $s\colon m\rightarrow n$ be a morphism in ${\bf Fin}$. Let
$z\in\textnormal{Ob}(\mathcal{O})$ be an object, let $y\colon
n\rightarrow\textnormal{Ob}(\mathcal{O})$ be an $n$-indexed set of objects,
and let $x\colon m\rightarrow\textnormal{Ob}(\mathcal{O})$ be an $m$-indexed
set of objects. For each element $i\in n$, write $m_{i}:=s^{-1}(i)$ for the
pre-image of $s$ under $i$, and write $x_{i}=x|_{m_{i}}\colon
m_{i}\rightarrow\textnormal{Ob}(\mathcal{O})$ for the restriction of $x$ to
$m_{i}$. Then one announces a function
(3) $\displaystyle\circ\colon\mathcal{O}_{n}(y;z)\times\prod_{i\in
n}\mathcal{O}_{m_{i}}(x_{i};y(i))\longrightarrow\mathcal{O}_{m}(x;z),$
called the composition formula for $\mathcal{O}$.
Given an $n$-indexed set of objects $x\colon
n\rightarrow\textnormal{Ob}(\mathcal{O})$ and an object
$y\in\textnormal{Ob}(\mathcal{O})$, we sometimes abuse notation and denote the
set of morphisms from $x$ to $y$ by $\mathcal{O}(x_{1},\ldots,x_{n};y)$.
111There are three abuses of notation when writing
$\mathcal{O}(x_{1},\ldots,x_{n};y)$, which we will fix one by one. First, it
confuses the set $n\in\textnormal{Ob}({\bf Fin})$ with its cardinality
$|n|\in{\mathbb{N}}$. But rather than writing
$\mathcal{O}(x_{1},\ldots,x_{|n|};y)$, it would be more consistent to write
$\mathcal{O}(x(1),\ldots,x(|n|);y)$, because we have assigned subscripts
another meaning in D. However, even this notation unfoundedly suggests that
the set $n$ has been endowed with a linear ordering, which it has not. This
may be seen as a more serious abuse, but see Remark 2.1.3. We may write
$\textnormal{Hom}_{\mathcal{O}}(x_{1},\ldots,x_{n};y)$, in place of
$\mathcal{O}(x_{1},\ldots,x_{n};y)$, when convenient. We can denote a morphism
$\phi\in\mathcal{O}_{n}(x;y)$ by $\phi\colon x\rightarrow y$ or by
$\phi\colon(x_{1},\ldots,x_{n})\rightarrow y$; we say that each $x_{i}$ is a
domain object of $\phi$ and that $y$ is the codomain object of $\phi$. We use
infix notation for the composition formula, e.g. writing
$\psi\circ(\phi_{1},\ldots,\phi_{n})$.
These constituents (A,B,C,D) must satisfy the following requirements:
1. 1.
for every $x_{1},\ldots,x_{n},y\in\textnormal{Ob}(\mathcal{O})$ and every
morphism $\phi\colon(x_{1},\ldots,x_{n})\rightarrow y$, we have
$\phi\circ(\textnormal{id}_{x_{1}},\ldots,\textnormal{id}_{x_{n}})=\phi\hskip
21.68121pt\textnormal{and}\hskip 21.68121pt\textnormal{id}_{y}\circ\phi=\phi;$
2. 2.
Let $m\xrightarrow{s}n\xrightarrow{t}p$ be composable morphisms in ${\bf
Fin}$. Let $z\in\textnormal{Ob}(\mathcal{O})$ be an object, let $y\colon
p\rightarrow\textnormal{Ob}(\mathcal{O})$, $x\colon
n\rightarrow\textnormal{Ob}(\mathcal{O})$, and $w\colon
m\rightarrow\textnormal{Ob}(\mathcal{O})$ respectively be a $p$-indexed,
$n$-indexed, and $m$-indexed set of objects. For each $i\in p$, write
$n_{i}=t^{-1}(i)$ for the pre-image and $x_{i}\colon
n_{i}\rightarrow\textnormal{Ob}(\mathcal{O})$ for the restriction. Similarly,
for each $k\in n$ write $m_{k}=s^{-1}(k)$ and $w_{k}\colon
m_{k}\rightarrow\textnormal{Ob}(\mathcal{O})$; for each $i\in p$, write
$m_{i,-}=(t\circ s)^{-1}(i)$ and $w_{i,-}\colon
m_{i,-}\rightarrow\textnormal{Ob}(\mathcal{O})$; for each $j\in n_{i}$, write
$m_{i,j}:=s^{-1}(j)$ and $w_{i,j}\colon
m_{i,j}\rightarrow\textnormal{Ob}(\mathcal{O})$. Then the diagram below
commutes:
---
$\textstyle{{\hskip
65.04256pt\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\prod\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\mathcal{O}_{p}(y;z)\times\prod_{i\in
p}\mathcal{O}_{n_{i}}(x_{i};y(i))\times\prod_{i\in p,\ j\in
n_{i}}\mathcal{O}_{m_{i,j}}(w_{i,j};x_{i}(j))$
---
$\textstyle{{\hskip
72.26999pt\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\prod\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\mathcal{O}_{n}(x;z)\times\prod_{k\in n}\mathcal{O}_{m_{k}}(w_{k};x(k))$
---
$\textstyle{{\hskip
72.26999pt\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\prod\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\mathcal{O}_{p}(y;z)\times\prod_{i\in p}\mathcal{O}_{m_{i,-}}(w_{i,-};y(i))$
---
$\textstyle{\mathcal{O}_{m}(w;z)}$
###### Remark 2.1.3.
In this remark we will discuss the abuse of notation in Definition 2.1.2 and
how it relates to an action of a symmetric group on each morphism set in our
definition of operad. We follow the notation of Definition 2.1.2, especially
following the use of subscripts in the composition formula.
Suppose that $\mathcal{O}$ is an operad, $z\in\textnormal{Ob}(\mathcal{O})$ is
an object, $y\colon n\rightarrow\textnormal{Ob}(\mathcal{O})$ is an
$n$-indexed set of objects, and $\phi\colon y\rightarrow z$ is a morphism. If
we linearly order $n$, enabling us to write
$\phi\colon(y(1),\ldots,y(|n|))\rightarrow z$, then changing the linear
ordering amounts to finding an isomorphism of finite sets $\sigma\colon
m\xrightarrow{\cong}n$, where $|m|=|n|$. Let $x=y\circ\sigma$ and for each
$i\in n$, note that $m_{i}=\sigma^{-1}(\\{i\\})=\\{\sigma^{-1}(i)\\}$, so
$x_{i}=x|_{\sigma^{-1}(i)}=y(i)$. Taking
$\textnormal{id}_{x_{i}}\in\mathcal{O}_{m_{i}}(x_{i};y(i))$ for each $i\in n$,
and using the identity law, we find that the composition formula induces a
bijection $\mathcal{O}_{n}(y;z)\xrightarrow{\cong}\mathcal{O}_{m}(x;z)$, which
we might denote by
$\sigma\colon\mathcal{O}(y(1),y(2),\ldots,y(n);z)\cong\mathcal{O}\big{(}y(\sigma(1)),y(\sigma(2)),\ldots,y(\sigma(n));z\big{)}.$
In other words, there is an induced group action of $\textnormal{Aut}(n)$ on
$\mathcal{O}_{n}(y(1),\ldots,y(n);z)$, where $\textnormal{Aut}(n)$ is the
group of permutations of an $n$-element set.
Throughout this paper, we will permit ourselves to abuse notation and speak of
morphisms $\phi\colon(x_{1},x_{2},\ldots,x_{n})\rightarrow y$ for a natural
number $n\in{\mathbb{N}}$, without mentioning the abuse inherent in choosing
an order, so long as it is clear that permuting the order of indices would not
change anything up to canonical isomorphism.
###### Example 2.1.4.
We define the operad of sets, denoted ${\bf Sets}$, as follows. We put
$\textnormal{Ob}({\bf Sets}):=\textnormal{Ob}({\bf Set})$. Given a natural
number $n\in{\mathbb{N}}$ and objects
$X_{1},\ldots,X_{n},Y\in\textnormal{Ob}({\bf Sets})$, we define
${\bf Sets}(X_{1},X_{2},\ldots,X_{n};Y):=\textnormal{Hom}_{\bf
Set}(X_{1}\times X_{2}\times\cdots\times X_{n},Y).$
For any $X\in\textnormal{Ob}({\bf Sets})$ the identity morphism
$\textnormal{id}_{X}\colon X\rightarrow X$ is the same identity as that in
${\bf Set}$.
The composition formula is as follows. Suppose given a set
$Z\in\textnormal{Ob}({\bf Set})$, a finite set $n\in\textnormal{Ob}({\bf
Fin})$, for each $i\in n$ a set $Y_{i}\in\textnormal{Ob}({\bf Set})$ and a
finite set $m_{i}\in\textnormal{Ob}({\bf Fin})$, and for each $j\in m_{i}$ a
set $X_{i,j}\in\textnormal{Ob}({\bf Set})$. Suppose furthermore that we have
composable morphisms: a function $g\colon\prod_{i\in n}Y_{i}\rightarrow Z$ and
for each $i\in n$ a function $f_{i}\colon\prod_{j\in m_{i}}X_{i,j}\rightarrow
Y_{i}$. Let $m=\amalg_{i}m_{i}$. We need a function $\prod_{j\in
m}X_{j}\rightarrow Z$, which we take to be the composite
$\prod_{i\in n}\prod_{j\in m_{i}}X_{i,j}\xrightarrow{\ \ \prod_{i\in n}f_{i}\
\ }\prod_{i\in n}Y_{i}\xrightarrow{\ \ g\ \ }Z.$
It is not hard to see that this composition formula is associative.
###### Example 2.1.5.
The commutative operad $\mathcal{E}$ has one object, say
$\textnormal{Ob}(\mathcal{E})=\\{{\blacktriangle}\\}$, and for each
$n\in{\mathbb{N}}$ it has a single $n$-ary morphism,
$\mathcal{E}_{n}(\blacktriangle,\ldots,\blacktriangle;\blacktriangle)=\\{\mu_{n}\\}$.
### 2.2. The announced structure of the wiring diagrams operad $\mathcal{W}$
To define our operad $\mathcal{W}$, we need to announce its structure, i.e.
* •
define what constitutes an object of $\mathcal{W}$,
* •
define what constitutes a morphism of $\mathcal{W}$,
* •
define the identity morphisms in $\mathcal{W}$, and
* •
the formula for composing morphisms of $\mathcal{W}$.
For each of these we will first draw and describe a picture to have in mind,
then give a mathematical definition. In Section 2.4 we will prove that the
announced structure has the required properties.
#### Objects are black boxes
Each object $X$ will be drawn as a box with input arrows entering on the left
of the box and output arrows leaving from the right of the box. The arrows
will be called wires. All input and output wires will be drawn across the
corresponding vertical wall of the box.
(4)
Each wire is also assigned a set of values that it can carry, and this set can
be written next to the wire, or the wires may be color coded. See Example
2.2.2 below. As above, we often leave off the values assignment in pictures
for readability reasons.
###### Announcement 2.2.1 (Objects of $\mathcal{W}$).
An object $X\in\textnormal{Ob}(\mathcal{W})$ is called a black box, or box for
short. It consists of a tuple $X:=({\tt in}(X),{\tt out}(X),{\tt vset})$,
where
* •
${\tt in}(X)\in\textnormal{Ob}({\bf Fin})$ is a finite set, called the set of
input wires to $X$,
* •
${\tt out}(X)\in\textnormal{Ob}({\bf Fin})$ is a finite set, called the set of
output wires from $X$, and
* •
${\tt vset}(X)\colon{\tt in}(X)\amalg{\tt
out}(X)\rightarrow\textnormal{Ob}({\bf Set}_{*})$ is a function, called the
values assignment for $X$. For each wire $i\in{\tt in}(X)\amalg{\tt out}(X)$,
we call ${\tt vset}(i)\in\textnormal{Ob}({\bf Set}_{*})$ the set of values
assigned to wire $i$, and we call its basepoint element the default value on
wire $i$.
$\lozenge$
###### Example 2.2.2.
We may take $X=(\\{1\\},\\{2,3\\},{\tt vset})$, where ${\tt
vset}\colon\\{1,2,3\\}\rightarrow\textnormal{Ob}({\bf Set}_{*})$ is given by
${\tt vset}(1)={\mathbb{N}}$, ${\tt vset}(2)={\mathbb{N}}$, and ${\tt
vset}(3)=\\{a,b,c\\}$. 222 The functor ${\tt vset}$ is supposed to assign
pointed sets to each wire, but no base points are specified in the description
above. As discussed in Section 1.2, in this case we really have ${\tt
vset}(1)={\mathbb{N}}\amalg\\{*\\}$, ${\tt
vset}(2)={\mathbb{N}}\amalg\\{*\\}$, and ${\tt
vset}(3)=\\{a,b,c\\}\amalg\\{*\\}$, where $*$ is the default value. We would
draw $X$ as follows.
$X$${\mathbb{N}}$$\\{a,b,c\\}$${\mathbb{N}}$
The input wire carries natural numbers, as does one of the output wires, and
the other output wire carries letters $a,b,c$.
#### Morphisms are directed wiring diagrams
Given black boxes $Y_{1},\ldots,Y_{n}\in\textnormal{Ob}(\mathcal{W})$ and a
black box $Z\in\textnormal{Ob}(\mathcal{W})$, we must define the set
$\mathcal{W}_{n}(Y;Z)$ of wiring diagrams (WDs) of type
$Y_{1},\ldots,Y_{n}\rightarrow Z$. Such a wiring diagram can be taken to
denote a way to wire black boxes $Y_{1},\ldots,Y_{n}$ together to form a
larger black box $Z$. A typical such wiring diagram is shown below:
(5)
Here $n=\underline{3}$, and for example $Y_{1}$ has two input wire and three
outputs wires. Each wire in a WD has a specified directionality. As it travels
a given wire may split into separate wires, but separate wires cannot come
together. The wiring diagram also includes a finite set of delay nodes; in the
above case there are four.
One should think of a wiring diagram $\psi\colon Y_{1},\ldots,Y_{n}\rightarrow
Z$ as a rule for managing material (or information) flow between the
components of an organization. Think of $\psi$ as representing this
organization. The individual components of the organization are the interior
black boxes (the domain objects of $\psi$) and the exterior black box (the
codomain object of $\psi$). Each component supplies material to $\psi$ as well
as demands material from $\psi$. For example component $Z$ supplies material
on the left side of $\psi$ and demands it on the right side of $\psi$. On the
other hand, each $Y_{i}$ supplies material on its right side and demands
material on its left. Like the IDEF0 standard for functional modeling diagrams
[NIST], we always adhere to this directionality.
We insist on one perhaps surprising (though seemingly necessary rule), namely
that the wiring diagram cannot connect an output wire of $Z$ directly to an
input wire of $Z$. Instead, each output wire of $Z$ is supplied either by an
output wire of some $Y(i)$ or by a delay node.
###### Announcement 2.2.3 (Morphisms of $\mathcal{W}$).
Let $n\in\textnormal{Ob}({\bf Fin})$ be a finite set, let $Y\colon
n\rightarrow\textnormal{Ob}(\mathcal{W})$ be an $n$-indexed set of black
boxes, and let $Z\in\textnormal{Ob}(\mathcal{W})$ be another black box. We
write
(6) $\displaystyle{\tt in}(Y)$ $\displaystyle=\amalg_{i\in n}{\tt in}(Y(i)),$
$\displaystyle{\tt out}(Y)$ $\displaystyle=\amalg_{i\in n}{\tt out}(Y(i)).$
We take ${\tt vset}\colon{\tt in}(Y)\amalg{\tt
out}(Y)\rightarrow\textnormal{Ob}({\bf Set}_{*})$ to be the induced map.
A morphism
$\psi\colon Y(1),\ldots,Y(n)\rightarrow Z$
in $\mathcal{W}_{n}(Y;Z)$ is called a temporal wiring diagram, a wiring
diagram, or a WD for short. It consists of a tuple $(DN_{\psi},{\tt
vset},s_{\psi})$ as follows. 333A morphism $\psi\colon Y\rightarrow Z$ is in
fact an isomorphism class of this data. That is, given two tuples
$(DN_{\psi},{\tt vset},s_{\psi})$ and $(DN_{\psi}^{\prime},{\tt
vset}^{\prime},s^{\prime}_{\psi})$ as above, with a bijection $DN_{\psi}\cong
DN_{\psi}^{\prime}$ making all the appropriate diagrams commute, we consider
these two tuples to constitute the same morphism $\psi\colon Y\rightarrow Z$.
* •
$DN_{\psi}\in\textnormal{Ob}({\bf Fin})$ is a finite set, called the set of
delay nodes for $\psi$. At this point we can define the following sets:
${Dm_{\psi}}:={\tt out}(Z)\amalg{\tt in}(Y)\amalg DN_{\psi}$ | the set of demand wires in $\psi$, and
---|---
${Sp_{\psi}}:={\tt in}(Z)\amalg{\tt out}(Y)\amalg DN_{\psi}$ | the set of supply wires in $\psi$.
* •
${\tt vset}\colon DN_{\psi}\rightarrow\textnormal{Ob}({\bf Set})$ is a
function, called the value-set assignment for $\psi$, such that the diagram
$\textstyle{DN_{\psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}_{DN_{\psi}}}$$\scriptstyle{\textnormal{id}_{DN_{\psi}}}$$\scriptstyle{{\tt
vset}}$$\textstyle{{Dm_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tt
vset}}$$\textstyle{{Sp_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tt
vset}}$$\textstyle{{\bf Set}_{*}}$
commutes (meaning that every delay node demands the same value-set that it
supplies).
* •
$s_{\psi}\colon{Dm_{\psi}}\rightarrow{Sp_{\psi}}$ is a function, called the
supplier assignment for $\psi$. The supplier assignment $s_{\psi}$ must
satisfy two requirements:
1. (1)
The following diagram commutes:
$\textstyle{{Dm_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{\psi}}$$\scriptstyle{{\tt
vset}}$$\textstyle{{Sp_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tt
vset}}$$\textstyle{{\bf Set}_{*}}$
meaning that whenever a demand wire is assigned a supplier, the set of values
assigned to these wires must be the same.
2. (2)
If $z\in{\tt out}(Z)$ then $s_{\psi}(z)\not\in{\tt in}(Z)$. Said another way,
$s_{\psi}|_{{\tt out}(Z)}\subseteq{\tt out}(Y)\amalg DN_{\psi},$
meaning that a global output cannot be directly supplied by a global input. We
call this the non-instantaneity requirement.
We have functions ${\tt vset}\colon{\tt in}(Z)\amalg{\tt
out}(Z)\rightarrow{\bf Set}_{*},{\tt vset}\colon{\tt in}(Y(i))\amalg{\tt
out}(Y(i))\rightarrow{\bf Set}_{*},$ and ${\tt vset}\colon
DN_{\psi}\rightarrow{\bf Set}_{*}$. It should not cause confusion if we use
the same symbol to denote the induced functions ${\tt
vset}\colon{Dm_{\psi}}\rightarrow{\bf Set}_{*}$ and ${\tt
vset}\colon{Sp_{\psi}}\rightarrow{\bf Set}_{*}$.
$\lozenge$
###### Remark 2.2.4.
We have taken the perspective that $\mathcal{W}$ is an operad. One might more
naturally think of $\mathcal{W}$ as the underlying operad of a symmetric
monoidal category whose objects are again black boxes and whose morphisms are
again wiring diagrams, though now a morphism connects a single internal domain
black box to the external codomain black box. From this perspective one should
merge the many isolated black boxes occurring in the domain of a multicategory
wiring diagram into a single black box as the domain of the monoidal category
wiring diagram.
Though mathematically equivalent and though we make use of this perspective in
the course of our proofs, it is somewhat unnatural to perform this grouping in
applications. For example, though it makes some sense to view ourselves
writing this paper and you reading this paper as black boxes inside a single
“information conveying” wiring diagram it would be rather strange to
conglomerate all of our collective inputs and outputs so that we become a
single meta-information entity. For reasons of this sort we choose to take the
perspective of the underlying operad rather than of a monoidal category.
On the other hand, the notation of monoidal categories is convenient, so we
introduce it here. Given a finite set $n$ and an $n$-indexed set of objects
$Y\colon n\rightarrow\textnormal{Ob}(\mathcal{W})$, we discussed in (6) what
should be seen as a tensor product
$\bigotimes_{i\in n}Y(i)=(\amalg_{i\in n}{\tt in}(Y(i)),\amalg_{i\in n}{\tt
out}(Y(i)),{\tt vset}),$
which we write simply as $Y=({\tt in}(Y),{\tt out}(Y),{\tt vset})$.
Similarly, given an $n$-indexed set of morphisms $\phi_{i}\colon
X_{i}\rightarrow Y(i)$ in $\mathcal{W}$, we can form their tensor product
$\bigotimes_{i\in n}\phi_{i}\colon\bigotimes_{i\in
n}X_{i}\rightarrow\bigotimes_{i\in n}Y(i),$
which we write simply as $\phi\colon X\rightarrow Y$, in a similar way. That
is, we form a set of delay nodes $DN_{\phi}=\amalg_{i\in n}{DN_{\phi_{i}}}$,
supplies ${Sp_{\phi}}=\amalg_{i\in n}{Sp_{\phi_{i}}}$, demands
${Dm_{\phi}}=\amalg_{i\in n}{Dm_{\phi_{i}}}$, and a supplier assignment
$s_{\phi}=\amalg_{i\in n}s_{\phi_{i}}$, all by taking the obvious disjoint
unions.
###### Example 2.2.5.
In the example below, we see a big box with three little boxes inside, and we
see many wires with arrowheads placed throughout. It is a picture of a wiring
diagram $\phi\colon(X_{1},X_{2},X_{3})\rightarrow Y$. The big box can be
viewed as $Y$, which has some number of input and output wires; however, when
we see the big box as a container of the little boxes wired together, we are
actually seeing the morphism $\phi$.
$\phi\colon(X_{1},X_{2},X_{3})\rightarrow Y$eggsmilksaltsugarflourdry mixwet
mixegg yolkscookie batter$X_{1}$$X_{2}$$X_{3}$
We aim to explain our terminology of demand and supply, terms which interpret
the organization forced on us by the mathematics. Each wire has a demand side
and a supply side; when there are no feedback loops, as in the picture above,
supplies are on the left side of the wire and demands are to the right, but
this is not always the case. Instead, the distinction to make is whether an
arrowhead is entering the big box or leaving it: those that enter the big box
are supplies to $\phi$, and those that are leaving the big box are demands
upon $\phi$. The five left-most arrowheads are entering the big box, so flour,
sugar, etc. are being supplied. But flour, sugar, and salt are demands when
they leave the big box to enter $X_{1}$. Counting, one finds 9 supply wires
and 9 demand wires (though the equality of these numbers is just a coincidence
due to the fact that no wire splits or is wasted).
#### Identity morphisms are identity supplier assignments
Let $Z=({\tt in}(Z),{\tt out}(Z),{\tt vset})$. The identity wiring diagram
$\textnormal{id}_{Z}\colon Z\rightarrow Z$ might be drawn like this:
$Z$$Z$
Even though the interior box is of a different size than the exterior box, the
way they are wired together is as straightforward as possible.
###### Announcement 2.2.6 (Identity morphisms in $\mathcal{W}$).
Let $Z=({\tt in}(Z),{\tt out}(Z),{\tt vset}_{Z})$. The identity wiring diagram
$\textnormal{id}_{Z}\colon Z\rightarrow Z$ has
$DN_{\textnormal{id}_{Z}}=\emptyset$ with the unique function ${\tt
vset}\colon\emptyset\rightarrow\textnormal{Ob}({\bf Set})$, so that
${Dm_{\textnormal{id}_{Z}}}={\tt out}(Z)\amalg{\tt in}(Z)$ and
${Sp_{\textnormal{id}_{Z}}}={\tt in}(Z)\amalg{\tt out}(Z)$. The supplier
assignment
$s_{\textnormal{id}_{Z}}\colon{Sp_{\textnormal{id}_{Z}}}\rightarrow{Dm_{\textnormal{id}_{Z}}}$
is given by the identity function, which satisfies the non-instantaneity
requirement.
$\lozenge$
#### Composition of morphisms is achieved by removing intermediary boxes and
associated arrow-heads
We are interested in substituting a wiring diagram into each black box of a
wiring diagram, to produce a more detailed wiring diagram. The basic picture
to have in mind is the following:
On the top we see a wiring diagram $\psi$ in which each internal box, say
$Y(1)$ and $Y(2)$, has a corresponding wiring diagram $\phi_{1}$ and
$\phi_{2}$ respectively. Dropping them into place and then removing the
intermediary boxes leaves a single wiring diagram $\omega$. One can see that
every input of $Y(i)$ plays a dual role. Indeed, it is a demand from the
perspective of $\psi$, and it is a supply from the perspective of $\phi_{i}$.
Similarly, every output of $Y(i)$ plays a dual role as supply in $\psi$ and
demand in $\phi_{i}$.
In Announcement 2.2.8 we will provide the composition formula for
$\mathcal{W}$. Namely, we will be given morphisms $\phi_{i}\colon
X_{i}\rightarrow Y(i)$ and $\psi\colon Y\rightarrow Z$. Each of these has its
own delay nodes, $DN_{\phi_{i}}$ and $DN_{\psi}$ as well as its own supplier
assignments. Write $\phi=\bigotimes_{i}\phi_{i}\colon X\rightarrow Y$ as in
Remark 2.2.4. For the reader’s convenience, we now summarize the demands and
supplies for each of the given morphisms $\phi_{i}\colon X_{i}\rightarrow
Y(i)$ and $\psi\colon Y\rightarrow Z$, as well as their (not-yet defined)
composition $\omega\colon X\rightarrow Z$. Let $DN_{\omega}=DN_{\phi}\amalg
DN_{\psi}$.
(14)
###### Lemma 2.2.7.
Suppose given morphisms $X\xrightarrow{\phi}Y$ and $Y\xrightarrow{\psi}Z$ in
$\mathcal{W}$, as above. That is, we are given sets of delay nodes,
$DN_{\phi}$ and $DN_{\psi}$, as well as supplier assignments
$s_{\phi}\colon{Dm_{\phi}}\rightarrow{Sp_{\phi}}\hskip
21.68121pt\textnormal{and}\hskip
21.68121pts_{\psi}\colon{Dm_{\psi}}\rightarrow{Sp_{\psi}}$
each of which is subject to a non-instantaneity requirement,
(15) $\displaystyle s_{\phi}\big{|}_{{\tt out}(Y)}\subseteq{\tt out}(X)\amalg
DN_{\phi}\hskip 21.68121pt\textnormal{and}\hskip
21.68121pts_{\psi}\big{|}_{{\tt out}(Z)}\subseteq{\tt out}(Y)\amalg
DN_{\psi}.$
Let $s_{\omega}$ be as in Table 14. It follows that the diagram below is a
pushout
$\textstyle{{Sp_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{Sp_{\omega}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(Y)\amalg{\tt
out}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{e}$$\textstyle{{Sp_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$
where
(16) $\displaystyle e$ $\displaystyle=s_{\psi}\big{|}_{{\tt
in}(Y)}\amalg\textnormal{id}_{{\tt out}(Y)}$ $\displaystyle f$
$\displaystyle=\textnormal{id}_{{\tt in}(Z)}\amalg s_{\phi}\big{|}_{{\tt
out}(Y)}\amalg\textnormal{id}_{DN_{\psi}}$ $\displaystyle g$
$\displaystyle=\textnormal{id}_{{\tt in}(Y)}\amalg s_{\phi}\big{|}_{{\tt
out}(Y)}$ $\displaystyle h$ $\displaystyle=(f\circ s_{\psi})\big{|}_{{\tt
in}(Y)}\amalg\textnormal{id}_{{\tt out}(X)}\amalg\textnormal{id}_{DN_{\phi}}.$
Moreover, each of $e,f,g,$ and $h$ commute with the appropriate functions
${\tt vset}$.
###### Proof.
We first show that the diagram commutes; here are the calculations on each
component:
$\displaystyle f\circ e\big{|}_{{\tt in}(Y)}=f\circ s_{\psi}\big{|}_{{\tt
in}(Y)}=h\circ g\big{|}_{{\tt in}(Y)}$ $\displaystyle f\circ e\big{|}_{{\tt
out}(Y)}=s_{\phi}\big{|}_{{\tt out}(Y)}=h\circ g\big{|}_{{\tt out}(Y)}.$
We now show that the diagram is a pushout. Suppose given a set $Q$ and a
commutative solid-arrow diagram (i.e. with $h^{\prime}\circ g=f^{\prime}\circ
e$):
---
$\textstyle{Q}$$\textstyle{{\tt in}(Y)\amalg{\tt out}(X)\amalg
DN_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\scriptstyle{h}$$\textstyle{{\tt
in}(Z)\amalg{\tt out}(X)\amalg DN_{\phi}\amalg
DN_{\psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\scriptstyle{\alpha}$$\textstyle{{\tt
in}(Y)\amalg{\tt
out}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{e}$$\textstyle{{\tt
in}(Z)\amalg{\tt out}(Y)\amalg
DN_{\psi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$
Looking at components on which $f$ and $h$ are identities, we see that if we
want the equations $\alpha\circ f=f^{\prime}$ and $\alpha\circ h=h^{\prime}$
to hold, there is at most one way to define
$\alpha\colon{Sp_{\omega}}\rightarrow Q$. Namely,
$\alpha:=f^{\prime}\big{|}_{{\tt in}(Z)\amalg DN_{\psi}}\amalg
h^{\prime}\big{|}_{{\tt out}(X)\amalg DN_{\phi}}.$
To see that this definition works, it remains to check that $\alpha\circ
f\big{|}_{{\tt out}(Y)}=f^{\prime}\big{|}_{{\tt out}(Y)}$ and that
$\alpha\circ h\big{|}_{{\tt in}(Y)}=h^{\prime}\big{|}_{{\tt in}(Y)}$. For the
first we use a non-instantaneity requirement (15) to calculate:
$\displaystyle\alpha\circ f\big{|}_{{\tt out}(Y)}=\alpha\circ
s_{\phi}\big{|}_{{\tt out}(Y)}$ $\displaystyle=\alpha\big{|}_{{\tt
out}(X)\amalg DN_{\phi}}\circ s_{\phi}\big{|}_{{\tt out}(Y)}$
$\displaystyle=h^{\prime}\circ s_{\phi}\big{|}_{{\tt out}(Y)}$
$\displaystyle=h^{\prime}\circ g\big{|}_{{\tt out}(Y)}=f^{\prime}\circ
e\big{|}_{{\tt out}(Y)}=f^{\prime}\big{|}_{{\tt out}(Y)}$
Now we have shown that $\alpha\circ f=f^{\prime}$ and the second calculation
follows:
$\displaystyle\alpha\circ h\big{|}_{{\tt in}(Y)}=\alpha\circ f\circ
s_{\psi}\big{|}_{{\tt in}(Y)}=f^{\prime}\circ s_{\psi}\big{|}_{{\tt
in}(Y)}=f^{\prime}\circ e\big{|}_{{\tt in}(Y)}=h^{\prime}\circ g\big{|}_{{\tt
in}(Y)}=h^{\prime}\big{|}_{{\tt in}(Y)}$
Each of $e,f,g,h$ commute with the respective functions ${\tt vset}$ because
each is built solely out of identity functions and supplier assignments. This
completes the proof.
∎
###### Announcement 2.2.8 (Composition formula for $\mathcal{W}$).
Let $m,n\in\textnormal{Ob}({\bf Fin})$ be finite sets and let $t\colon
m\rightarrow n$ be a function. Let $Z\in\textnormal{Ob}(\mathcal{W})$ be a
black box, let $Y\colon n\rightarrow\textnormal{Ob}(\mathcal{O})$ be an
$n$-indexed set of black boxes, and let $X\colon
m\rightarrow\textnormal{Ob}(\mathcal{O})$ be an $m$-indexed set of black
boxes. For each element $i\in n$, write $m_{i}:=t^{-1}(i)$ for the pre-image
of $i$ under $t$, and write $X_{i}=X\big{|}_{m_{i}}\colon
m_{i}\rightarrow\textnormal{Ob}(\mathcal{O})$ for the restriction of $X$ to
$m_{i}$. Then the composition formula
$\circ\colon\mathcal{W}_{n}(Y;Z)\times\prod_{i\in
n}\mathcal{W}_{m_{i}}(X_{i};Y(i))\longrightarrow\mathcal{W}_{m}(X;Z),$
is defined as follows.
Suppose that we are given morphisms $\phi_{i}\colon X_{i}\rightarrow Y(i)$ for
each $i\in n$, which we gather into a morphism
$\phi=\bigotimes_{i}\phi_{i}\colon X\rightarrow Y$ as in Remark 2.2.4, and
that we are also given a morphism $\psi\colon Y\rightarrow Z$. Then we have
finite sets of delay nodes $DN_{\phi}$ and $DN_{\psi}$, and supplier
assignments
$s_{\phi}\colon{Dm_{\phi}}\rightarrow{Sp_{\phi}}\hskip
21.68121pt\textnormal{and}\hskip
21.68121pts_{\psi}\colon{Dm_{\psi}}\rightarrow{Sp_{\psi}}$
as in Announcement 2.2.3.
We are tasked with defining a morphism $\omega:=\psi\circ\phi\colon
X\rightarrow Z$. The set of demand wires and supply wires for $\omega$ are
given in Table (14). Thus our job is to define a set $DN_{\omega}$ and a
supplier assignment $s_{\omega}\colon{Dm_{\omega}}\rightarrow{Sp_{\omega}}$.
We put $DN_{\omega}=DN_{\phi}\amalg DN_{\psi}$. It suffices to find a function
$s_{\omega}\colon{\tt out}(Z)\amalg{\tt in}(X)\amalg
DN_{\omega}\longrightarrow{\tt in}(Z)\amalg{\tt out}(X)\amalg DN_{\omega},$
which satisfies the two requirements of being a supplier assignment. We first
define the function by making use of the following diagram, where the pushout
is as in Lemma 2.2.7:
(23)
Thus we can define a function
(24) $\displaystyle s_{\omega}=h\circ s_{\phi}\big{|}_{{\tt in}(X)\amalg
DN_{\phi}}\amalg f\circ s_{\psi}\big{|}_{{\tt out}(Z)\amalg DN_{\psi}}.$
We need to show that $s_{\omega}$ satisfies the two requirements of being a
supplier assignment (see Announcement 2.2.3).
1. (1)
The fact that $s_{\omega}$ commutes with the appropriate functions ${\tt
vset}$ follows from the fact that $s_{\phi},s_{\psi},f,$ and $h$ do so (by
Lemma 2.2.7).
2. (2)
The fact that the non-instantaneity requirement holds for $s_{\omega}$, i.e.
that $s_{\omega}({\tt out}(Z))\subseteq{\tt out}(X)\amalg DN_{\omega}$,
follows from the fact that it holds for $s_{\psi}$ and $s_{\psi}$ (see (15)),
as follows.
$\displaystyle s_{\omega}({\tt out}(Z))$ $\displaystyle=f\circ s_{\psi}({\tt
out}(Z))$ $\displaystyle\subseteq f({\tt out}(Y)\amalg DN_{\psi})$
$\displaystyle=s_{\phi}({\tt out}(Y))\amalg DN_{\psi}$
$\displaystyle\subseteq{\tt out}(X)\amalg DN_{\phi}\amalg DN_{\psi}={\tt
out}(X)\amalg DN_{\omega}.$
$\lozenge$
### 2.3. Running example to ground ideas and notation regarding $\mathcal{W}$
In this section we will discuss a few objects of $\mathcal{W}$ (i.e. black
boxes), a couple morphisms of $\mathcal{W}$ (i.e. wiring diagrams), and a
composition of morphisms. We showed objects and morphisms in more generality
above (see Examples 2.2.2 and 2.2.5). Here we concentrate on a simple case,
which we will take up again in Section 3.4 and which will eventually result in
a propagator that outputs the Fibonacci sequence. First, we draw three
objects, $X,Y,Z\in\textnormal{Ob}(\mathcal{W})$.
(25)
These objects are not complete until the pointed sets associated to each wire
are specified. Let $N:=({\mathbb{N}},1)$ be the set of natural numbers with
basepoint 1, and put
${\tt vset}(a_{X})={\tt vset}(b_{X})={\tt vset}(c_{X})={\tt vset}(a_{Y})={\tt
vset}(c_{Y})={\tt vset}(c_{Z})=N.$
Now we draw two morphisms, i.e. wiring diagrams, $\phi\colon X\rightarrow Y$
and $\psi\colon Y\rightarrow Z$:
(26)
To clarify the notion of inputs, outputs, supplies, and demands, we provide
two tables that lay out those sets in the case of (26).
Objects shown above
---
Object | ${\tt in}(-)$ | ${\tt out}(-)$
$X$ | $\\{a_{X},b_{X}\\}$ | $\\{c_{X}\\}$
$Y$ | $\\{a_{Y}\\}$ | $\\{c_{Y}\\}$
$Z$ | $\\{\\}$ | $\\{c_{Z}\\}$
| |
Morphisms shown above
---
Morphism | $DN_{-}$ | ${Dm_{-}}$ | ${Sp_{-}}$
$\phi$ | $\\{\\}$ | $\\{c_{Y},a_{X},b_{X}\\}$ | $\\{a_{Y},c_{X}\\}$
$\psi$ | $\\{d_{\psi}\\}$ | $\\{c_{Z},a_{Y},d_{\psi}\\}$ | $\\{c_{Y},d_{\psi}\\}$
| | |
To specify the morphism $\phi\colon X\rightarrow Y$ (respectively $\psi\colon
Y\rightarrow Z$), we are required not only to provide a set of delay nodes
$DN_{\phi}$, which we said was $DN_{\phi}=\emptyset$ (respectively,
$DN_{\psi}=\\{d_{\psi}\\}$), but also a supplier assignment function
$s_{\phi}\colon{Dm_{\phi}}\rightarrow{Sp_{\phi}}$ (resp.,
$s_{\psi}\colon{Dm_{\psi}}\rightarrow{Sp_{\psi}}$). Looking at the picture of
$\phi$ (resp. $\psi$) above, the reader can trace backward to see how every
demand wire is attached to some supply wire. Thus, the supplier assignment
$s_{\phi}$ for $\phi\colon X\rightarrow Y$ is
$c_{Y}\mapsto c_{X},\hskip 21.68121pta_{X}\mapsto a_{Y},\hskip
21.68121ptb_{X}\mapsto c_{X},$
and the supplier assignment $s_{\psi}$ for $\psi\colon Y\rightarrow Z$ is
$c_{Z}\mapsto d_{\psi},\hskip 21.68121pta_{Y}\mapsto d_{\psi},\hskip
21.68121ptd_{\psi}\mapsto c_{Y}.$
We now move on to the composition of $\psi$ and $\phi$. The idea is that we
“plug the $\phi$ diagram into the $Y$-box of the $\psi$ diagram, then erase
the $Y$-box”. We follow this in two steps below: on the left, we shrink down a
copy of $\phi$ and fit it into the $Y$-box of $\psi$. On the right, we erase
the $Y$-box:
$X\xrightarrow{\phi}Y\xrightarrow{\psi}Z$$Z$$Y$$X$$X\xrightarrow{\psi\circ\phi}Z$$Z$$X$
The pushout (23) ensures that wires of $Y$ connect wires inside (i.e. from
$\phi$) to wires outside (i.e. from $\psi$). In other words, when we erase box
$Y$, we do not erase the connections it made for us. We compute the pushout of
the diagram
$\\{a_{Y},c_{X}\\}\xleftarrow{a_{Y}\mapsto a_{Y},\;\;c_{Y}\mapsto
c_{X}}\\{a_{Y},c_{Y}\\}\xrightarrow{a_{Y}\mapsto d_{\psi},\;\;c_{Y}\mapsto
c_{Y}}\\{c_{Y},d_{\psi}\\},$
defining ${Sp_{\omega}}$, to be isomorphic to $\\{d_{\psi},c_{X}\\}.$ The
supplier assignment
$s_{\omega}\colon{Dm_{\omega}}=\\{c_{Z},a_{X},b_{X},d_{\psi}\\}\rightarrow\\{d_{\psi},c_{Z}\\}={Sp_{\omega}}$
is given by
(27) $\displaystyle c_{Z}\mapsto d_{\psi},\hskip 21.68121pta_{X}\mapsto
d_{\psi},\hskip 21.68121ptb_{X}\mapsto c_{X},\hskip 21.68121ptd_{\psi}\mapsto
c_{X}.$
We take this example up again in Section 3.4, where we show that installing a
“plus” function into box $X$ yields the Fibonacci sequence.
### 2.4. Proof that the operad requirements are satisfied by $\mathcal{W}$
We need to show that the announced operad $\mathcal{W}$ satisfies the
requirements set out by Definition 2.1.2. There are two such requirements: the
first says that composing with the identity morphism has no effect, and the
second says that composition is associative.
###### Proposition 2.4.1.
The identity law holds for the announced structure of $\mathcal{W}$.
###### Proof.
Let $X_{1},\ldots,X_{n}$ and $Y$ be black boxes and let $\phi\colon
X_{1},\ldots,X_{n}\rightarrow Y$ be a morphism. We need to show that the
following equations hold:
$\phi\circ(\textnormal{id}_{x_{1}},\ldots,\textnormal{id}_{x_{n}})\stackrel{{\scriptstyle?}}{{=}}\phi\hskip
21.68121pt\textnormal{and}\hskip
21.68121pt\textnormal{id}_{y}\circ\phi\stackrel{{\scriptstyle?}}{{=}}\phi.$
We are given a set $DN_{\phi}$ and a function ${\tt vset}\colon
DN_{\phi}\rightarrow\textnormal{Ob}({\bf Set})$. Let
$\textnormal{id}_{X}=\bigotimes_{i\in n}\textnormal{id}_{X_{i}}$, and form
${\tt in}(X)$ and ${\tt out}(X)$ as in Remark 2.2.4. Thus we have
${Sp_{\phi}}={\tt in}(Y)\amalg{\tt out}(X)\amalg DN_{\phi}\hskip
21.68121pt\textnormal{and}\hskip 21.68121pt{Dm_{\phi}}={\tt out}(Y)\amalg{\tt
in}(X)\amalg DN_{\phi}$
and a supplier assignment $s_{\phi}\colon{Dm_{\phi}}\rightarrow{Sp_{\phi}}$.
For each $i\in n$ we have
${Sp_{\textnormal{id}_{X_{i}}}}={Dm_{\textnormal{id}_{X_{i}}}}$, and the
supplier assignments are the identity, so we have
${Sp_{\textnormal{id}_{X}}}={Dm_{\textnormal{id}_{X}}}={\tt in}(X)\amalg{\tt
out}(X)$
The supplier assignment $s_{\textnormal{id}_{X}}$ is the identity function.
Similarly, ${Sp_{\textnormal{id}_{Y}}}={Dm_{\textnormal{id}_{Y}}}={\tt
in}(Y)\amalg{\tt out}(Y)$, and the supplier assignment
$s_{\textnormal{id}_{Y}}$ is the identity function.
Let $\omega=\phi\circ(\textnormal{id}_{X_{1}},\ldots,\textnormal{id}_{X_{n}})$
and $\omega^{\prime}=\textnormal{id}_{Y}\circ\phi$. Then the relevant pushouts
become
$\textstyle{{\tt
in}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\big{|}_{{\tt
in}(X)}}$$\textstyle{{Sp_{\textnormal{id}_{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{\phi}\big{|}{{\tt
in}(X)}\amalg\textnormal{id}\big{|}_{{\tt
out}(X)}}$$\textstyle{{Sp_{\omega}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(X)\amalg{\tt
out}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{Sp_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\tt
out}(Y)\amalg
DN_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{\phi}\big{|}_{{\tt
out}(Y)\amalg DN_{\phi}}}$ $\textstyle{{\tt in}(X)\amalg
DN_{\phi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{\phi}\big{|}_{{\tt
in}(X)\amalg
DN_{\phi}}}$$\textstyle{{Sp_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{Sp_{\omega^{\prime}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(Y)\amalg{\tt
out}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{Sp_{\textnormal{id}_{Y}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\big{|}_{{\tt
in}(Y)}\amalg s_{\phi}\big{|}_{{\tt out}(Y)}}$$\textstyle{{\tt
out}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\big{|}_{{\tt
out}(Y)}}$
The pushout of an isomorphism is an isomorphism so we have isomorphisms
${Sp_{\phi}}\cong{Sp_{\omega}}$ and ${Sp_{\phi}}\cong{Sp_{\omega^{\prime}}}$.
444Note that a morphism (e.g. $\omega$) in $\mathcal{W}$ are defined only up
to isomorphism class of tuples $(DN_{\omega},{\tt vset},s_{\omega})$, see
Announcement 2.2.3. In both the case of $\omega$ and $\omega^{\prime}$, one
checks using (16) that the induced supplier assignments are also in agreement
(up to isomorphism), $s_{\omega}=s_{\phi}=s_{\omega^{\prime}}.$
∎
###### Proposition 2.4.2.
The associativity law holds for the announced structure of $\mathcal{W}$.
###### Proof.
Suppose we are given morphisms $\tau\colon W\rightarrow X$, $\phi\colon
X\rightarrow Y$ and $\psi\colon Y\rightarrow Z$. We must check that
$(\psi\circ\phi)\circ\tau=\psi\circ(\phi\circ\tau)$. With notation as in Lemma
2.2.7, pushout square defining $\phi\circ\tau$ and then
$\psi\circ(\phi\circ\tau)$ are these:
$\textstyle{{Sp_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\phi,\tau}}$$\textstyle{{Sp_{\phi\circ\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(X)\amalg{\tt
out}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{\phi,\tau}}$$\scriptstyle{e_{\phi,\tau}}$$\textstyle{{Sp_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\phi,\tau}}$
$\textstyle{{Sp_{\phi\circ\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\psi,\phi\circ\tau}}$$\textstyle{{Sp_{\psi\circ(\phi\circ\tau)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(Y)\amalg{\tt
out}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{\psi,\phi\circ\tau}}$$\scriptstyle{g_{\psi,\phi\circ\tau}}$$\textstyle{{Sp_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\psi,\phi\circ\tau}}$
whereas the pushout square defining $\psi\circ\phi$ and then
$(\psi\circ\phi)\circ\tau$ are these:
$\textstyle{{Sp_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\psi,\phi}}$$\textstyle{{Sp_{\psi\circ\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(Y)\amalg{\tt
out}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{\psi,\phi}}$$\scriptstyle{g_{\psi,\phi}}$$\textstyle{{Sp_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\psi,\phi}}$
$\textstyle{{Sp_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\psi\circ\phi,\tau}}$$\textstyle{{Sp_{(\psi\circ\phi)\circ\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(X)\amalg{\tt
out}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{\psi\circ\phi,\tau}}$$\scriptstyle{e_{\psi\circ\phi,\tau}}$$\textstyle{{Sp_{\psi\circ\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\psi\circ\phi,\tau}}$
One checks directly from the formulas (16) that
$e_{\psi\circ\phi,\tau}=h_{\psi,\phi}\circ e_{\phi,\tau}$ as functions ${\tt
in}(X)\amalg{\tt out}(X)\rightarrow{Sp_{\psi\circ\phi}}$, and that
$g_{\psi,\phi\circ\tau}=f_{\phi,\tau}\circ g_{\psi,\phi}$ as functions ${\tt
in}(Y)\amalg{\tt out}(Y)\rightarrow{Sp_{\phi\circ\tau}}$.
We combine them into the following pushout diagram:
$\textstyle{{Sp_{\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\phi,\tau}}$$\textstyle{{Sp_{\phi\circ\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\psi,\phi\circ\tau}}$$\textstyle{\llcorner}$$\textstyle{{Sp_{\psi\circ\phi\circ\tau}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(X)\amalg{\tt
out}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{\phi,\tau}}$$\scriptstyle{e_{\phi,\tau}}$$\textstyle{{Sp_{\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\phi,\tau}}$$\scriptstyle{h_{\psi,\phi}}$$\textstyle{{Sp_{\psi\circ\phi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\psi\circ\phi,\tau}}$$\textstyle{\llcorner}$$\textstyle{{\tt
in}(Y)\amalg{\tt
out}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e_{\psi,\phi}}$$\scriptstyle{g_{\psi,\phi}}$$\textstyle{{Sp_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\psi,\phi}}$
The pasting lemma for pushout squares ensures that the set labeled
${Sp_{\psi\circ\phi\circ\tau}}$ is isomorphic to
${Sp_{\psi\circ(\phi\circ\tau)}}$ and to ${Sp_{(\psi\circ\phi)\circ\tau}}$, so
these are indeed isomorphic to each other. It is also easy to check using the
formulas provided in (24) and (16) that the supplier assignments
${Dm_{\psi\circ\phi\circ\tau}}={\tt out}(Z)\amalg{\tt in}(W)\amalg
DN_{\tau}\amalg DN_{\phi}\amalg
DN_{\psi}\longrightarrow{Sp_{\psi\circ\phi\circ\tau}}$
agree regardless of the order of composition. This proves the result.
∎
## 3\. $\mathcal{P}$, the algebra of propagators on $\mathcal{W}$
In this section we will introduce our algebra of propagators on $\mathcal{W}$.
This is where form meets function: the form called “black box” is a
placeholder for a propagator, i.e. a function, that carries input streams to
output streams, and the form called “wiring diagram” is a placeholder for a
circuit that links propagators together to form a larger propagator.
To formalize these ideas we introduce the mathematical notion of operad
algebra in Section 3.1. In Section 3.2 we discuss some preliminaries on lists
and streams, and define our notion of historical propagator. In Section 3.3 we
announce our algebra of these propagators and in Section 3.4 we ground it in
our running example. Finally in Section 3.5 we prove that the announced
structure really satisfies the requirements of being an algebra.
### 3.1. Definition and basic examples of algebras
In this section we give the formal definition for algebras over an operad.
###### Definition 3.1.1.
Let $\mathcal{O}$ be an operad. An $\mathcal{O}$-algebra, denoted
$F\colon\mathcal{O}\rightarrow{\bf Sets}$, is defined as follows: One
announces some constituents (A. map on objects, B. map on morphisms) and
proves that they satisfy some requirements (1. identity law, 2. composition
law). Specifically,
1. A.
one announces a function
$\textnormal{Ob}(F)\colon\textnormal{Ob}(\mathcal{O})\rightarrow\textnormal{Ob}({\bf
Sets})$.
2. B.
for each object $y\in\textnormal{Ob}(\mathcal{O})$, finite set
$n\in\textnormal{Ob}({\bf Fin})$, and $n$-indexed set of objects $x\colon
n\rightarrow\textnormal{Ob}(\mathcal{O})$, one announces a function
$F_{n}\colon\mathcal{O}_{n}(x;y)\rightarrow\textnormal{Hom}_{\bf
Sets}(Fx;Fy).$
As in B. above, we often denote $\textnormal{Ob}(F)$, and also each $F_{n}$,
simply by $F$.
These constituents (A,B) must satisfy the following requirements:
1. 1.
For each object $x\in\textnormal{Ob}(\mathcal{O})$, the equation
$F(\textnormal{id}_{x})=\textnormal{id}_{Fx}$ holds.
2. 2.
Let $s\colon m\rightarrow n$ be a morphism in ${\bf Fin}$. Let
$z\in\textnormal{Ob}(\mathcal{O})$ be an object, let $y\colon
n\rightarrow\textnormal{Ob}(\mathcal{O})$ be an $n$-indexed set of objects,
and let $x\colon m\rightarrow\textnormal{Ob}(\mathcal{O})$ be an $m$-indexed
set of objects. Then, with notation as in Definition 2.1.2, the following
diagram of sets commutes:
(32)
###### Example 3.1.2.
Let $\mathcal{E}$ be the commutative operad of Example 2.1.5. An
$\mathcal{E}$-algebra $S\colon\mathcal{E}\rightarrow{\bf Sets}$ consists of a
set $M\in\textnormal{Ob}({\bf Set})$, and for each natural number
$n\in{\mathbb{N}}$ a morphism $\mu_{n}\colon M^{n}\rightarrow M$. It is not
hard to see that, together, the morphism $\mu_{2}\colon M\times M\rightarrow
M$ and the element $\mu_{0}\colon{\\{*\\}}\rightarrow M$ give $M$ the
structure of a commutative monoid. Indeed, the associativity and unit axioms
are encoded in the axioms for operads and their morphisms. The commutativity
of multiplication arises by applying the commutative diagram (32) in the case
$s\colon\\{1,2\\}\rightarrow\\{1,2\\}$ is the non-identity bijection, as
discussed in Remark 2.1.3.
### 3.2. Lists, streams, and historical propagators
In this section we discuss some background on lists. We also develop our
notion of historical propagator, which formalizes the idea that a machine’s
output at time $t_{0}$ can depend only on what has happened previously, i.e.
for time $t<t_{0}$. While strictly not necessary for the development of this
paper, we also discuss the relation of historical propagators to streams.
Given a set $S$, an $S$-list is a pair $(t,\ell)$, where $t\in{\mathbb{N}}$ is
a natural number and $\ell\colon\\{1,2,\ldots,t\\}\rightarrow S$ is a
function. We denote the set of $S$-lists by $\textnormal{List}(S)$. We call
$t$ the length of the list; in particular a list may be empty because we may
have $t=0$. Note that there is a canonical bijection
$\textnormal{List}(S)\cong\coprod_{t\in{\mathbb{N}}}S^{t}.$
We sometimes denote a list simply by $\ell$ and write $|\ell|$ to denote its
length; that is we have the component projection
$|\cdot|\colon\textnormal{List}(A)\rightarrow{\mathbb{N}}$. We typically
write-out an $S$-list as $\ell=[\ell(1),\ell(2),\ldots,\ell(t)]$, where each
$\ell(i)\in S$. We denote the empty list by $[\;]$. Given a function $f\colon
S\rightarrow S^{\prime}$, there is an induced function
$\textnormal{List}(f)\colon\textnormal{List}(S)\rightarrow\textnormal{List}(S^{\prime})$
sending $(t,\ell)$ to $(t,f\circ\ell)$; in the parlance of computer science
$\textnormal{List}(f)$ is the function that “maps $f$ over $\ell$”.
Given sets $X_{1},\ldots,X_{k}\in\textnormal{Ob}({\bf Set})$, an element in
$\textnormal{List}(\prod_{1\leq i\leq k}X_{i})$ is a list of $k$-tuples. Given
sets $A$ and $B$ there is a bijection
$\;\raisebox{3.0pt}{${}_{\varcurlyvee}$}\;\colon\textnormal{List}(A)\times_{\mathbb{N}}\textnormal{List}(B)\xrightarrow{\
\ \cong\ \ }\textnormal{List}(A\times B),$
where on the left we have formed the fiber product of the diagram
$\textnormal{List}(A)\xrightarrow{|\cdot|}{\mathbb{N}}\xleftarrow{|\cdot|}\textnormal{List}(B)$.
We call this bijection zipwith, following the terminology from modern
functional programming languages. The idea is that an $A$-list $\ell_{A}$ can
be combined with a $B$-list $\ell_{B}$, as long as they have the same length
$|\ell_{A}|=|\ell_{B}|$; the result will be an $(A\times B)$-list
$\ell_{A}\;\raisebox{3.0pt}{${}_{\varcurlyvee}$}\;\ell_{B}$ again of the same
length. We will usually abuse this distinction and freely identify
$\textnormal{List}(A\times
B)\cong\textnormal{List}(A)\times_{\mathbb{N}}\textnormal{List}(B)$ with its
image in $\textnormal{List}(A)\times\textnormal{List}(B)$. For example, we may
consider the ${\mathbb{N}}\times{\mathbb{N}}$-list
$[(1,2),(3,4),(5,6)]=[1,3,5]\;\raisebox{3.0pt}{${}_{\varcurlyvee}$}\;[2,4,6]$
as an element of
$\textnormal{List}({\mathbb{N}})\times\textnormal{List}({\mathbb{N}})$.
Hopefully this will not cause confusion.
Let $\textnormal{List}_{\geq 1}(S)\subseteq\textnormal{List}(S)$ denote the
set $\amalg_{t\geq 1}S^{t}$. We write
$\partial_{S}\colon\textnormal{List}_{\geq
1}(S)\rightarrow\textnormal{List}(S)$ to denote the function that drops off
the last entry. More precisely, for any integer $t\geq 1$ if we consider
$\ell$ as a function $\ell\colon\\{1,2,\ldots,t\\}\rightarrow S$, then the
list $\partial_{S}\ell$ is given by pre-composition with the subset consisting
of the first $t-1$ elements,
$\\{1,2,\ldots,t-1\\}\hookrightarrow\\{1,2,\ldots,t\\}\xrightarrow{\ell}S.$
For example we have $\partial[0,1,4,9,16]=[0,1,4,9]$.
###### Definition 3.2.1.
Let $R,S$ be pointed sets and let $n\in{\mathbb{N}}$. A $n$-historical
propagator $f$ from $R$ to $S$ is a function
$f\colon\textnormal{List}(R)\rightarrow\textnormal{List}(S)$ satisfying the
following conditions:
1. (1)
If a list $\ell\in\textnormal{List}(R)$ has length $|\ell|=t$, then
$|f(\ell)|=t+n$,
2. (2)
If $\ell\in\textnormal{List}(R)$ is a list of length $t\geq 1$, then
$\partial_{S}f(\ell)=f(\partial_{R}\ell).$
We denote the set of $n$-historical propagators from $R$ to $S$ by
$\textnormal{Hist}^{n}(R,S)$. If $f$ is $n$-historical for some $n\geq 0$ we
say that $f$ is historical.
We usually drop the subscript from the symbol $\partial_{-}$, writing e.g.
$\partial f(\ell)=f(\partial\ell)$.
###### Example 3.2.2.
Let $S$ be a pointed set and let $n\in{\mathbb{N}}$ be a natural number.
Define an $n$-historical propagator $\delta^{n}\in\textnormal{Hist}^{n}(S,S)$
as follows for $\ell\in\textnormal{List}(S)$:
$\delta^{n}(\ell)(i)=\begin{cases}*&\textnormal{ if }1\leq i\leq n\\\
\ell(i-n)&\textnormal{ if }n+1\leq i\leq t+n\end{cases}$
We call $\delta^{n}$ the $n$-moment delay function. For example if
$n=3,S=\\{a,b,c,d\\}\amalg\\{*\\}$, and $\ell=[a,a,b,*,d]\in S^{5}$ then
$\delta^{3}(S)=[*,*,*,a,a,b,*,d]\in S^{8}$.
The following Lemma describes the behavior of historical functions.
###### Lemma 3.2.3.
Let $S,S^{\prime},S^{\prime\prime},T,T^{\prime}\in{\bf Set}_{*}$ be pointed
sets.
1. (1)
Let $f\colon S\rightarrow T$ be a function. The induced function
$\textnormal{List}(f)\colon\textnormal{List}(S)\rightarrow\textnormal{List}(T)$
is $0$-historical.
2. (2)
Given $n$-historical propagators $q\in\textnormal{Hist}^{n}(S,S^{\prime})$ and
$r\in\textnormal{Hist}^{n}(T,T^{\prime})$, there is an induced $n$-historical
propagator $q\times r\in\textnormal{Hist}^{n}(S\times T,S^{\prime}\times
T^{\prime})$.
3. (3)
Given $q\in\textnormal{Hist}^{m}(S,S^{\prime})$ and
$q^{\prime}\in\textnormal{Hist}^{n}(S^{\prime},S^{\prime\prime})$, then
$q^{\prime}\circ
q\colon\textnormal{List}(S)\rightarrow\textnormal{List}(S^{\prime\prime})$ is
$(m+n)$-historical.
4. (4)
If $n\geq 1$ is an integer and $q\in\textnormal{Hist}^{n}(S,S^{\prime})$ is
$n$-historical then $\partial
q\colon\textnormal{List}(S)\rightarrow\textnormal{List}(S^{\prime})$ is
$(n-1)$-historical.
###### Proof.
We show each in turn.
1. (1)
Let $\ell\in\textnormal{List}(S)$ be a list of length $t$. Clearly,
$\textnormal{List}(f)$ sends $\ell$ to a list of length $t$. If $t\geq 1$ then
the fact that
$\partial\textnormal{List}(f)(\ell)=\textnormal{List}(f)(\partial\ell)$
follows by associativity of composition in ${\bf Set}$. That is,
$\textnormal{List}(f)(\ell)$ is the right-hand composition and $\partial\ell$
is the left-hand composition below:
$\\{1,\ldots,t-1\\}\hookrightarrow\\{1,\ldots,t\\}\xrightarrow{\ell}S\xrightarrow{f}T.$
2. (2)
On the length $t$ component we use the function $(S\times T)^{t}=S^{t}\times
T^{t}\xrightarrow{q\times r}S^{t+n}\times T^{t+n}=(S\times T)^{t+n}$. As
necessary, we have
$\partial\circ(q\times r)=\partial q\times\partial r=q\partial\times
r\partial=(q\times r)\circ\partial.$
3. (3)
This is straightforward; for example the second condition is checked
$\partial q^{\prime}(q(\ell))=q^{\prime}(\partial
q(\ell))=q^{\prime}(q(\partial\ell)).$
4. (4)
On lengths we indeed have $|\partial q(\ell)|=|q(\ell)|-1=|\ell|+n-1$. If
$|\ell|=t\geq 1$ then $\partial(\partial q)(\ell)=\partial(\partial
q(\ell))=\partial q(\partial\ell)$ because $q$ is historical.
∎
###### Definition 3.2.4.
Let $S$ be a pointed set. An $S$-stream is a function
$\sigma\colon{\mathbb{N}}_{\geq 1}\rightarrow S$. We denote the set of
$S$-streams by ${\textnormal{Strm}(S)}$.
For any natural number $t\in{\mathbb{N}}$, let
$\sigma\big{|}_{[1,t]}\in\textnormal{List}(S)$ denote the list of length $t$
corresponding to the composite
$\\{1,2,\ldots,t\\}\hookrightarrow{\mathbb{N}}_{\geq 1}\xrightarrow{\sigma}S$
and call it the $t$-restriction of $S$.
###### Lemma 3.2.5.
Let $S$ be a pointed set, let $\\{*\\}$ be a pointed set with one element, and
let $n\in{\mathbb{N}}$ be a natural number. There is a bijection
$\textnormal{Hist}^{n}(\\{*\\},S)\xrightarrow{\cong}{\textnormal{Strm}(S)}.$
###### Proof.
For any natural number $t\in{\mathbb{N}}$, let
${\underline{t}}=\\{1,2,\ldots,t\\}\in\textnormal{Ob}({\bf Set})$. Let
$[{\mathbb{N}}]$ be the poset (considered as a category) with objects
$\\{{\underline{t}}{\;|\;}t\in{\mathbb{N}}\\}$, ordered by inclusion of
subsets. For any $n\in{\mathbb{N}}$ there is a functor
$[{\mathbb{N}}]\rightarrow{\bf Set}$ sending
$\underline{t}\in\textnormal{Ob}([{\mathbb{N}}])$ to
$\\{1,2,\ldots,t+n\\}\in\textnormal{Ob}({\bf Set})$.
For any $n\in{\mathbb{N}}$, there is a bijection
${\mathbb{N}}\cong\mathop{\textnormal{colim}}_{t\in[{\mathbb{N}}]}\\{1,2,\ldots,t+n\\}.$
Thus we have a bijection
${\textnormal{Strm}(S)}=\textnormal{Hom}_{\bf Set}({\mathbb{N}}_{\geq
1},S)\cong\lim_{t\in[{\mathbb{N}}]}\textnormal{Hom}_{\bf
Set}(\\{1,2,\ldots,t+n\\},S).$
On the other hand, an $n$-historical function
$f\colon\textnormal{List}(\\{*\\})\rightarrow\textnormal{List}(S)$ acts as
follows. For each $t\in{\mathbb{N}}$ and list $[*,\ldots,*_{t}]$ of length
$t$, it assigns a list $f([*,\ldots,*_{t}])\in\textnormal{List}(S)$ of length
$t+n$, i.e. a function $\\{1,\ldots,t+n\\}\rightarrow S$, such that
$f([*,\ldots,*_{t-1}])$ is the restriction to the subset
$\\{1,\ldots,t+n-1\\}$.
The fact that these notions agree follows from the construction of limits in
the category ${\bf Set}$.
∎
Below we define an awkward-sounding notion of $n$-historical stream
propagator. The idea is that a function carrying streams to streams is
$n$-historical if, for all $t\in{\mathbb{N}}$, its output up to time $t+n$
depends only on its input up to time $t$. In Proposition 3.2.7 we show that
this notion of historicality for streams is equivalent to the notion for lists
given in Definition 3.2.1.
###### Definition 3.2.6.
Let $S$ and $T$ be pointed sets, and let $n\in{\mathbb{N}}$ be a natural
number. A function
$f\colon{\textnormal{Strm}(S)}\rightarrow{\textnormal{Strm}(T)}$ is called an
$n$-historical stream propagator if, given any natural number
$t\in{\mathbb{N}}$ and any two streams
$\sigma,\sigma^{\prime}\in{\textnormal{Strm}(S)}$, if
$\sigma\big{|}_{[1,t]}=\sigma^{\prime}\big{|}_{[1,t]}$ then
$f(\sigma)\big{|}_{[1,t+n]}=f(\sigma^{\prime})\big{|}_{[1,t+n]}$. Let
$\textnormal{Hist}_{strm}^{n}(S,T)$ denote the set of $n$-historical stream
propagators ${\textnormal{Strm}(S)}\rightarrow{\textnormal{Strm}(T)}$.
###### Proposition 3.2.7.
Let $S$ and $T$ be pointed sets. There is a bijection
$\textnormal{Hist}^{n}(S,T)\xrightarrow{\cong}\textnormal{Hist}^{n}_{strm}(S,T).$
###### Proof.
We construct two functions
$\alpha\colon\textnormal{Hist}^{n}(S,T)\rightarrow\textnormal{Hist}^{n}_{strm}(S,T)$
and
$\beta\colon\textnormal{Hist}^{n}_{strm}(S,T)\rightarrow\textnormal{Hist}^{n}(S,T)$
that are mutually inverse.
Given an $n$-historical function
$f\colon\textnormal{List}(S)\rightarrow\textnormal{List}(T)$ and a stream
$\sigma\in{\textnormal{Strm}(S)}$, define the stream
$\alpha(f)(\sigma)\colon{\mathbb{N}}_{\geq 1}\rightarrow T$ to be the function
whose $(t+n)$-restriction (for any $t\in{\mathbb{N}}$) is given by
$\alpha(f)(\sigma)\big{|}_{[1,t+n]}=f(\sigma\big{|}_{[1,t]}).$
Because $f$ is historical, this construction is well defined.
Given an $n$-historical stream propagator
$F\colon{\textnormal{Strm}(S)}\rightarrow{\textnormal{Strm}(T)}$ and a list
$\ell\in\textnormal{List}(S)$ of length $|\ell|=t$, let
$\ell_{*}\in{\textnormal{Strm}(S)}$ denote the stream ${\mathbb{N}}_{\geq
1}\rightarrow S$ given on $i\in{\mathbb{N}}_{\geq 1}$ by
$\ell_{*}(i)=\begin{cases}\ell(i)&\textnormal{ if }1\leq i\leq t\\\
*&\textnormal{ if }i\geq t+1.\end{cases}$
Now define the list $\beta(F)(\ell)\in\textnormal{List}(T)$ by
$\beta(F)(\ell)=F(\ell_{*})\big{|}_{[1,t+n]}.$
One checks directly that for all $F\in\textnormal{Hist}_{strm}^{n}(S,T)$ we
have $\alpha\circ\beta(F)=F$ and that for all $f\in\textnormal{Hist}^{n}(S,T)$
we have $\beta\circ\alpha(f)=f$.
∎
The above work shows that the notion of historical propagator is the same
whether one considers it as acting on lists or on streams. Throughout the rest
of this paper we work solely with the list version. However, we sometimes say
the word “stream” (e.g. “a propagator takes a stream of inputs and returns a
stream of outputs”) for the image it evokes.
### 3.3. The announced structure of the propagator algebra $\mathcal{P}$
In this section we will announce the structure of our $\mathcal{W}$-algebra of
propagators, which we call $\mathcal{P}$. That is, we must specify
* •
the set $\mathcal{P}(Y)$ of allowable “fillers” for each black box
$Y\in\textnormal{Ob}(\mathcal{W})$,
* •
how a wiring diagram $\psi\colon Y_{1},\ldots,Y_{n}\rightarrow Z$ and a filler
for each $Y_{i}$ serves to produce a filler for $Z$.
In this section we will explain in words and then formally announce
mathematical definitions. In Section 2.4 we will prove that the announced
structure has the required properties.
As mentioned above, the idea is that each black box is a placeholder for (i.e.
can be filled with) those propagators which carry the specified local input
streams to the specified local output streams. Each wiring diagram with
propagators installed in each interior black box will constitute a new
propagator for the exterior black box, which carries the specified global
input streams to the specified global output streams. We now go into more
detail and make these ideas precise.
#### Black boxes are filled by historical propagators
Let $Z=({\tt in}(Z),{\tt out}(Z),{\tt vset})$ be an object in $\mathcal{W}$.
Recall that each element $w\in{\tt in}(Z)$ is called an input wire, which
carries a set ${\tt vset}(w)$ of possible values, and that element
$w^{\prime}\in{\tt out}(Z)$ is called an output wire, which also carries a set
${\tt vset}(w^{\prime})$ of possible values. This terminology is suggestive of
a machine, which we call a historical propagator (or propagator for short),
which takes a list of values on each input wire, processes it somehow, and
emits a list of values on each output wire. The propagator’s output at time
$t_{0}$ can depend on the input it received for time $t<t_{0}$, but not on
input that arrives later.
###### Announcement 3.3.1 ($\mathcal{P}$ on objects).
Let $Z=({\tt in}(Z),{\tt out}(Z),{\tt vset})$ be an object in $\mathcal{W}$.
For any subset $I\subseteq{\tt in}(Z)\amalg{\tt out}(Z)$ we define
${\tt vset}_{I}=\prod_{i\in I}{\tt vset}(i).$
In particular, if $I=\emptyset$ then ${\tt vset}_{I}$ is a one-element set.
We define $\mathcal{P}(Z)\in\textnormal{Ob}({\bf Set})$ to be the set of
1-historical propagators of type $Z$,
$\mathcal{P}(Z):=\textnormal{Hist}^{1}({\tt vset}_{{\tt in}(Z)},{\tt
vset}_{{\tt out}(Z)}).$
$\lozenge$
Consider the propagator below, which has one input wire and one output wire,
say both carrying integers.
$``\Sigma"$
The name $``\Sigma"$ suggests that this propagator takes a list of integers
and returns their running total. But for it to be 1-historical, its input up
to time $t$ determines its output up to time $t+1$. Thus for example it might
send an input list $\ell:=[1,3,5,7,10]$ of length $5$ to the output list
$``\Sigma"(\ell)=[0,1,4,9,16,26]$ of length $6$.
###### Remark 3.3.2.
As in Remark 2.2.4 the following notation is convenient. Given a finite set
$n\in\textnormal{Ob}({\bf Fin})$ and black boxes
$Y_{i}\in\textnormal{Ob}(\mathcal{W})$ for $i\in n$, we can form
$Y=\bigotimes_{i\in n}Y_{i}$, with for example ${\tt in}(Y)=\amalg_{i\in
n}{\tt in}(Y_{i})$. Similarly, given a $1$-historical propagator
$g_{i}\in\mathcal{P}(Y_{i})$ for each $i\in n$ we can form a 1-historical
propagator $g:=\bigotimes_{i\in n}g_{i}\in\textnormal{Hist}^{1}({\tt
vset}_{{\tt in}(Y)},{\tt vset}_{{\tt out}(Y)})$ simply by $g=\prod_{i\in
n}g_{i}$.
#### Wiring diagrams shuttle value streams between propagators
Let $Z\in\textnormal{Ob}(\mathcal{W})$ be a black box, let
$n\in\textnormal{Ob}({\bf Fin})$ be a finite set, and let $Y\colon
n\rightarrow\textnormal{Ob}(\mathcal{W})$ be an $n$-indexed set of black
boxes. A morphism $\psi\colon Y\rightarrow Z$ in $\mathcal{W}$ is little more
than a supplier assignment $s_{\psi}\colon{Dm_{\psi}}\rightarrow{Sp_{\psi}}$.
In other words, it connects each demand wire to a supply wire carrying the
same set of values. Therefore, if a propagator is installed in each black box
$Y(i)$, then $\psi$ tells us how to take each value stream being produced by
some propagator and feed it into the various propagators that it supplies.
###### Announcement 3.3.3 ($\mathcal{P}$ on morphisms).
Let $Z\in\textnormal{Ob}(\mathcal{W})$ be a black box, let
$n\in\textnormal{Ob}({\bf Fin})$ be a finite set, let $Y\colon
n\rightarrow\textnormal{Ob}(\mathcal{W})$ be an $n$-indexed set of black
boxes, and let $\psi\colon Y\rightarrow Z$ be a morphism in $\mathcal{W}$. We
must construct a function
$\mathcal{P}(\psi)\colon\mathcal{P}(Y(1))\times\cdots\times\mathcal{P}(Y(n))\rightarrow\mathcal{P}(Z).$
That is, given a historical propagator $g_{i}\in\textnormal{Hist}^{1}({\tt
vset}_{{\tt in}(Y(i))},{\tt vset}_{{\tt out}(Y(i))})$ for each $i\in n$, we
need to produce a historical propagator
$\mathcal{P}(\psi)(g_{1},\ldots,g_{n})\in\textnormal{Hist}^{1}({\tt
vset}_{{\tt in}(Z)},{\tt vset}_{{\tt out}(Z)}).$ Define
$g\in\textnormal{Hist}^{1}({\tt vset}_{{\tt in}(Y)},{\tt vset}_{{\tt
out}(Y)})$ by $g:=\bigotimes_{i\in n}g_{i}$, as in Remark 3.3.2. Let
${in{Dm_{\psi}}}={\tt in}(Y)\amalg DN_{\psi}$ and ${in{Sp_{\psi}}}={\tt
out}(Y)\amalg DN_{\psi},$ denote the set of internal demands of $\psi$ and the
set of internal supplies of $\psi$, respectively.
We will define $\mathcal{P}(\psi)(g)$ by way of five helper functions:
$\displaystyle S_{\psi}$ $\displaystyle\in\textnormal{Hist}^{0}({\tt
vset}_{{Sp_{\psi}}},{\tt vset}_{{Dm_{\psi}}}),$ $\displaystyle
S^{\prime}_{\psi}$ $\displaystyle\in\textnormal{Hist}^{0}({\tt
vset}_{{Sp_{\psi}}},{\tt vset}_{{in{Dm_{\psi}}}}),$ $\displaystyle
S^{\prime\prime}_{\psi}$ $\displaystyle\in\textnormal{Hist}^{0}({\tt
vset}_{{in{Sp_{\psi}}}},{\tt vset}_{{\tt out}(Z)}),$ $\displaystyle
E_{\psi,g}$ $\displaystyle\in\textnormal{Hist}^{1}({\tt
vset}_{{in{Dm_{\psi}}}},{\tt vset}_{{in{Sp_{\psi}}}}),$ $\displaystyle
C_{\psi,g}$ $\displaystyle\in\textnormal{Hist}^{0}({\tt vset}_{{\tt
in}(Z)},{\tt vset}_{{Sp_{\psi}}}),$
where we will refer to the
$S_{\psi},S^{\prime}_{\psi},S^{\prime\prime}_{\psi}$ as “shuttle”,
$E_{\psi,g}$ as “evaluate”, and $C_{\psi,g}$ as “cascade”. We will abbreviate
by $\overline{{\tt in}(Z)}$ the set $\textnormal{List}({\tt vset}_{{\tt
in}(Z)})$, and similarly for $\overline{{Sp_{\psi}}},$
$\overline{{in{Dm_{\psi}}}},$ etc.
By Announcement 2.2.3, a morphism $\psi\colon Y\rightarrow Z$ in $\mathcal{W}$
is given by a tuple $(DN_{\psi},{\tt vset},s_{\psi})$, where in particular we
remind the reader of a commutative diagram
$\textstyle{{Dm_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{\psi}}$$\scriptstyle{{\tt
vset}}$$\textstyle{{Sp_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tt
vset}}$$\textstyle{{\bf Set}_{*}}$
where we require $s_{\psi}({\tt out}(Z))\subseteq{in{Sp_{\psi}}}$. The
function $s_{\psi}\colon{Dm_{\psi}}\rightarrow{Sp_{\psi}}$ induces the
coordinate projection function $\pi_{s_{\psi}}\colon{\tt
vset}_{{Sp_{\psi}}}\rightarrow{\tt vset}_{{Dm_{\psi}}}$ (see Section 1.2).
Applying the functor List gives a $0$-historical function (see Lemma 3.2.3),
$\textnormal{List}(\pi_{s_{\psi}})$ which we abbreviate as
$S_{\psi}\colon\overline{{Sp_{\psi}}}\rightarrow\overline{{Dm_{\psi}}}.$
This is the function that shuttles a list of tuples from where they are
supplied directly along a wire to where they are demanded. We define a
commonly-used projection,
$S^{\prime}_{\psi}:=\pi_{\overline{{in{Dm_{\psi}}}}}\circ
S_{\psi}\colon\overline{{Sp_{\psi}}}\rightarrow\overline{{in{Dm_{\psi}}}}.$
The purpose of defining the set ${in{Dm_{\psi}}}$ of internal demands above is
that the supplier assignment sends ${\tt out}(Z)$ into it, i.e. we have
$s_{\psi}\big{|}_{{\tt out}(Z)}\colon{\tt out}(Z)\rightarrow{in{Sp_{\psi}}}$
by the non-instantaneity requirement. It induces $\pi_{s_{\psi}\big{|}_{{\tt
out}(Z)}}\colon{\tt vset}_{{in{Sp_{\psi}}}}\rightarrow{\tt vset}_{{\tt
out}(Z)}$. Applying List gives a 0-historical function
$\textnormal{List}(\pi_{s_{\psi}\big{|}_{{\tt out}(Z)}})$ which we abbreviate
as
$S^{\prime\prime}\colon\overline{{in{Sp_{\psi}}}}\rightarrow\overline{{\tt
out}(Z)}.$
Thus $S^{\prime}$ and $S^{\prime\prime}$ first shuttle from supply lines to
all demand lines, and then focus on only a subset of them. Let
$\delta_{\psi}^{1}\in\textnormal{Hist}^{1}({\tt vset}_{DN_{\psi}},{\tt
vset}_{DN_{\psi}})$ be the 1-moment delay. Note that if $DN_{\psi}=\emptyset$
then $\delta_{\psi}^{1}\colon\\{*\\}\rightarrow\\{*\\}$ carries no information
and can safely be ignored.
We now define the remaining helper functions:
(33) $\displaystyle E_{\psi,g}$ $\displaystyle:=(g\times\delta_{\psi}^{1}),$
$\displaystyle C_{\psi,g}(\ell)$
$\displaystyle:=\begin{cases}[\;]&\textnormal{ if }|\ell|=0\\\
(\ell,E_{\psi,g}\circ S^{\prime}_{\psi}\circ
C_{\psi,g}(\partial\ell))&\textnormal{ if }|\ell|\geq 1.\end{cases}$
The last is an inductive definition, which we can rewrite for $|\ell|\geq 1$
as
$C_{\psi,g}=\big{(}\textnormal{id}_{\overline{{\tt
in}(Z)}}\times(E_{\psi,g}\circ S^{\prime}_{\psi}\circ
C_{\psi,g}\circ\partial)\big{)}\circ\Delta,$
where $\Delta:\overline{{\tt in}(Z)}\rightarrow\overline{{\tt
in}(Z)}\times\overline{{\tt in}(Z)}$ is the diagonal map. Intuitively it says
that a list of length $t$ on the input wires will produces a list of length
$t$ on all supply wires. By Lemma 3.2.3 $E_{\psi,g}$ is 1-historical and
$C_{\psi,g}$ is $0$-historical.
We are ready to define the 1-historical function
(34) $\displaystyle\mathcal{P}(\psi)(g)=S^{\prime\prime}_{\psi}\circ
E_{\psi,g}\circ S^{\prime}_{\psi}\circ C_{\psi,g}.$
$\lozenge$
###### Remark 3.3.4.
The definitions of $S^{\prime}_{\psi}$ and $E_{\psi,g}$ above implicitly make
use of the “zipwith” functions
$\;\raisebox{3.0pt}{${}_{\varcurlyvee}$}\;\colon\overline{{\tt
in}(Z)}\times_{\mathbb{N}}\overline{{in{Dm_{\psi}}}}\xrightarrow{\ \ \cong\ \
}\overline{{Dm_{\psi}}}\quad\text{and}\quad\;\raisebox{3.0pt}{${}_{\varcurlyvee}$}\;\colon\overline{{\tt
in}(Y)}\times_{\mathbb{N}}\overline{DN_{\psi}}\xrightarrow{\ \ \cong\ \
}\overline{{in{Dm_{\psi}}}},$
respectively. In section 3.5 we will make similar abuses in the calculations;
however, when commutative diagrams are given, the zipwith is made “explicit”
by writing an equality between products of streams and streams of products
when we mean that ⋎ should be applied to a product of streams.
### 3.4. Running example to ground ideas and notation regarding $\mathcal{P}$
In this section we compose elementary morphisms and apply them to a simple
“addition” propagator to construct a propagator that outputs the Fibonacci
sequence. Let $X,Y,Z\in\textnormal{Ob}(\mathcal{W})$ and $\phi\colon
X\rightarrow Y$ and $\psi\colon Y\rightarrow Z$ be as in (25) and (26). Let
$N=({\mathbb{N}},1)\in{\bf Set}_{*}$ denote the set of natural numbers with
basepoint 1. We recall the shapes of $X,Y$, and $Z$ here, but draw them with
different labels:
$``\\!+\\!"$$a_{X}$$b_{X}$$c_{X}$$``1+\Sigma"$$c_{Y}$$a_{Y}$$``Fib"$$c_{Z}$
We have replaced the symbol $X$ with the symbol $``\\!+\\!"$ because we are
about to define an $X$-shaped propagator $``\\!+\\!"\in\mathcal{P}(X)$. Given
an incoming list of numbers on wire $a_{X}$ and another incoming list of
numbers on wire $b_{X}$, it will create a list of their sums and output that
on $c_{X}$. More precisely, we take $``\\!+\\!"\colon\textnormal{List}(N\times
N)\rightarrow\textnormal{List}(N)$ to be the 1-historical propagator defined
as follows. Suppose given a list $\ell\in\textnormal{List}(N\times N)$ of
length $t$, say
$\ell=\big{[}\ell_{a}(1),\ell_{a}(2),\ldots,\ell_{a}(t),\big{]}\;\raisebox{3.0pt}{${}_{\varcurlyvee}$}\;\big{[}\ell_{b}(1),\ell_{b}(2),\ldots,\ell_{b}(t)\big{]}$
Define $``\\!+\\!"(\ell)\in\textnormal{List}(N)$ to be the list whose $n$th
entry (for $1\leq n\leq t+1$) is
$``\\!+\\!"(\ell)(n)=\begin{cases}1&\textnormal{ if }n=1\\\
\ell_{a}(n-1)+\ell_{b}(n-1)&\textnormal{ if }2\leq n\leq t+1\end{cases}$
So for example
$``+"([4,5,6,7]\;\raisebox{3.0pt}{${}_{\varcurlyvee}$}\;[1,1,3,7])=[1,5,6,9,14]$.
We will use only this $``\\!+\\!"$ propagator to build our Fibonacci sequence
generator. To do so, we will use wiring diagrams $\phi$ and $\psi$, whose
shapes we recall here from (26) above.
$``1+\Sigma"$$a_{Y}$$c_{Y}$$``\\!+\\!"$$a_{X}$$b_{X}$$c_{X}$$``1+\Sigma"=\mathcal{P}(\phi)(``\\!+\\!")$$``Fib"$$c_{Z}$$``1+\Sigma"$$a_{Y}$$c_{Y}$$d_{\psi}$$``Fib"=\mathcal{P}(\psi)(``1+\Sigma")$
The $Y$-shaped propagator
$``1+\Sigma"=\mathcal{P}(\phi)(``\\!+\\!")\in\mathcal{P}(Y)$ will have the
following behavior: given an incoming list of numbers on wire $a_{Y}$, it will
return a list of their running totals, plus 1. More precisely
$``1+\Sigma"\colon\textnormal{List}(N)\rightarrow\textnormal{List}(N)$ is the
1-historical propagator defined as follows. Suppose given a list
$\ell\in\textnormal{List}(N)$ of length $t$, say
$\ell=[\ell_{1},\ell_{2},\ldots,\ell_{t}]$. Then $``1+\Sigma"(\ell)$ will be
the list whose $n$th entry (for $1\leq n\leq t+1$) is
(35) $\displaystyle``1+\Sigma"(\ell)(n)=1+\sum_{i=1}^{n-1}\ell_{i}.$
But this is not by fiat—it is calculated using the formula given in
Announcement 3.3.3. We begin with the following table.
${{{{\displaystyle\small\begin{array}[]{| l || l | l | l |}\cr\vrule\lx@intercol\hrule height=0.79999pt}\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\vrule\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\lx@intercol\lx@intercol\lx@intercol\lx@intercol\lx@intercol\vrule\lx@intercol\lx@intercol\hfil\textnormal{Calculating }``1+\Sigma"\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\\\ \cr\hrule height=0.79999pt}&&&\\\ {\ell\in\overline{a_{Y}}}&C_{\phi,``\\!+\\!"}(\ell)\in\overline{\\{a_{Y},c_{X}\\}}&S^{\prime}_{\phi}C_{\phi,``\\!+\\!"}(\ell)\in\overline{\\{a_{X},b_{X}\\}}&E_{\phi,``\\!+\\!"}S^{\prime}_{\phi}C_{\phi,``\\!+\\!"}(\ell)\in\overline{c_{X}}\\\ \cr\hrule height=0.99998pt}[\;]&[\;]&[\;]&[1]\\\ \hline\cr[\ell_{1}]&[\ell_{1}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1]&[\ell_{1}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1]&[1,1+\ell_{1}]\\\ \hline\cr[\ell_{1},\ell_{2}]&[\ell_{1},\ell_{2}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1+\ell_{1}]&[\ell_{1},\ell_{2}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1+\ell_{1}]&[1,1+\ell_{1},1+\ell_{1}+\ell_{2}]\\\ \hline\cr[\ell_{1},\ell_{2},\ell_{3}]&[\ldots,\ell_{3}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![\ldots,1+\ell_{1}+\ell_{2}]&[\ldots,\ell_{3}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![\ldots,1+\ell_{1}+\ell_{2}]&[\ldots,1+\ell_{1}+\ell_{2}+\ell_{3}]\\\ \hline\cr[\ell_{1},\ldots,\ell_{t}]&[\ldots,\ell_{t}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![\ldots,1+\sum_{i=1}^{t-1}\ell_{i}]&[\ldots,\ell_{t}]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![\ldots,1+\sum_{i=1}^{t-1}\ell_{i}]&[\ldots,1+\sum_{i=1}^{t}\ell_{i}]\\\ \cr\hrule height=0.79999pt}\end{array}$
where the last row can be established by induction. The ellipses ($\ldots$) in
the later boxes indicate that the beginning part of the sequence is repeated
from the row above, which is a consequence of the fact that the formulas in
Announcement 3.3.3 are historical. We need only calculate
$\displaystyle``1+\Sigma"(\ell)=\mathcal{P}(\phi)(``\\!+\\!")(\ell)$
$\displaystyle=S^{\prime\prime}_{\phi}\circ E_{\phi,``\\!+\\!"}\circ
S^{\prime}_{\phi}\circ C_{\phi,``\\!+\\!"}(\ell)$
$\displaystyle=\left[1,1+\ell_{1},1+\ell_{1}+\ell_{2},\ldots,1+\sum_{i=1}^{t}\ell_{i}\right],$
just as in (35).
The $Z$-shaped propagator
$``Fib"=\mathcal{P}(\psi)(``1+\Sigma")\in\mathcal{P}(Z)$ will have the
following behavior: with no inputs, it will output the Fibonacci sequence
$``Fib"()=[1,1,2,3,5,8,13\ldots].$
Again, this is calculated using the formula given in Announcement 3.3.3. We
note first that since ${\tt in}(Z)=\emptyset$ we have ${\tt vset}_{{\tt
in}(Z)}=\\{*\\}$, so $\overline{{\tt in}(Z)}=\textnormal{List}({\tt
vset}_{{\tt in}(Z)})=\textnormal{List}(\\{*\\})$.
As above we provide a table that shows the calculation given the formula in
Announcement 3.3.3.
${{{{\displaystyle\begin{array}[]{| l || l | l | l |}\cr\vrule\lx@intercol\hrule height=0.79999pt}\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\vrule\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\lx@intercol\lx@intercol\lx@intercol\lx@intercol\lx@intercol\vrule\lx@intercol\lx@intercol\hfil\textnormal{Calculating }``Fib"\hfil\lx@intercol\vrule\lx@intercol\lx@intercol\\\ \cr\hrule height=0.79999pt{}}&C_{\psi,``1+\Sigma"}(\ell)&S^{\prime}_{\psi}C_{\psi,``1+\Sigma"}(\ell)&E_{\psi,``1+\Sigma"}S^{\prime}_{\psi}C_{\psi,``1+\Sigma"}(\ell)\\\ \ell\in\overline{\emptyset}&\hskip 28.90755pt\in\overline{\\{c_{Y},d_{\psi}\\}}&\hskip 39.74872pt\in\overline{\\{a_{Y},d_{\psi}\\}}&\hskip 79.49744pt\in\overline{\\{c_{Y},d_{\psi}\\}}\\\ \cr\hrule height=0.99998pt}[\;]&[\;]&[\;]&[1]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1]\\\ \hline\cr[*]&[1]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1]&[1]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1]&[1,2]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1]\\\ \hline\cr[*,*]&[1,2]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1]&[1,1]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,2]&[1,2,3]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1,2]\\\ \hline\cr[*,*,*]&[1,2,3]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1,2]&[1,1,2]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,2,3]&[1,2,3,5]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1,2,3]\\\ \hline\cr[*,*,*,*]&[1,2,3,5]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1,2,3]&[1,1,2,3]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,2,3,5]&[1,2,3,5,8]\\!\;\raisebox{2.0pt}{${}_{\varcurlyvee}$}\;\\![1,1,2,3,5]\\\ \cr\hrule height=0.79999pt}\end{array}$
In the case of a list $\ell\in\textnormal{List}(\\{*\\})$ of length $t$, we
have
$``Fib"(n)=\mathcal{P}(\psi)(``1+\Sigma")(\ell)=\left[1,1,2,3,\ldots,1+\sum_{i=1}^{t-2}``Fib"(i)\right].$
Thus we have achieved our goal. Note that, while unknown to the authors, the
fact that $``Fib"(t)=1+\sum_{i=1}^{t-2}``Fib"(i)$ was known at least as far
back as 1891, [Luc]. For us it appeared not by any investigation, but merely
by cordoning off part of our original wiring diagram for $``Fib"$,
$``Fib"$+
Above in (27) we computed the supplier assignment for the composition WD,
$\omega:=\psi\circ\phi\colon X\rightarrow Z$. In case the above tables were
unclear, we make one more attempt at explaining how propagators work by
showing a sequence of images with values traversing the wires of $\omega$
applied to $``\\!+\\!"$. The wires all start with the basepoint on their
supply sides, at which point it is shuttled to the demand sides. It is then
processed, again giving values on the supply sides that are again shuttled to
the demand sides. This is repeated once more.
$``Fib"$–Supply (iter. 1)$+$$1$$1$$``Fib"$–Demand (iter.
1)$+$$1$$1$$1$$1$$``Fib"$–Supply (iter. 2)$+$$2$$1$$``Fib"$–Demand (iter.
2)$+$$1$$2$$2$$1$$``Fib"$–Supply (iter. 3)$+$$3$$2$$``Fib"$–Demand (iter.
3)$+$$2$$3$$3$$2$
One sees the first three elements of the Fibonacci sequence $[1,1,2]$, as
demanded, emerging from the output wire.
### 3.5. Proof that the algebra requirements are satisfied by $\mathcal{P}$
Below we prove that $\mathcal{P}$, as announced, satisfies the requirements
necessary for it to be a $\mathcal{W}$-algebra. Unfortunately, the proof is
quite technical and not very enlightening. Given a composition
$\omega=\psi\circ\phi$, there is a correspondence between the wires in
$\omega$ with the wires in $\psi$ and $\phi$, as laid out in Announcement
2.2.8. The following proof essentially amounts to checking that, under this
correspondence, the way Announcement 3.3.3 instructs us to shuttle information
along the wires of $\omega$ is in agreement with the way it instructs us to
shuttle information along the wires of $\psi$ and $\phi$.
###### Theorem 3.5.1.
The function
$\mathcal{P}\colon\textnormal{Ob}(\mathcal{W})\rightarrow\textnormal{Ob}({\bf
Sets})$ defined in Announcement 3.3.1 and the function
$\mathcal{P}\colon\mathcal{W}(Y;Z)\rightarrow\textnormal{Hom}_{\bf
Sets}(\mathcal{P}(Y);\mathcal{P}(Z))$ given in Announcement 3.3.3 satisfy the
requirements for $\mathcal{P}$ to be a $\mathcal{W}$-algebra.
###### Proof.
We must show that both the identity law and the composition law hold. This
will require several technical lemmas, which for the sake of flow we have
included within the current proof.
We begin with the identity law. Let $Z=({\tt in}(Z),{\tt out}(Z),{\tt
vset}_{Z})$ be an object. The supplier assignment for
$\textnormal{id}_{Z}\colon Z\rightarrow Z$ is given by the identity function
$s_{\textnormal{id}_{Z}}\colon{\tt out}(Z)\amalg{\tt in}(Z)\xrightarrow{\ \
\textnormal{id}\ \ }{\tt in}(Z)\amalg{\tt out}(Z).$
Let $f\in\mathcal{P}(Z)=\textnormal{Hist}^{1}({\tt vset}_{{\tt in}(Z)},{\tt
vset}_{{\tt out}(Z)})$ be a historical propagator. We need to show that
$\mathcal{P}(\textnormal{id}_{Z})(f)=f$.
Recall the maps
$S_{\textnormal{id}_{Z}}\colon\overline{{Sp_{\textnormal{id}_{Z}}}}\rightarrow\overline{{Dm_{\textnormal{id}_{Z}}}},$ | $S^{\prime}_{\textnormal{id}_{Z}}\colon\overline{{Sp_{\textnormal{id}_{Z}}}}\rightarrow\overline{{in{Dm_{\textnormal{id}_{Z}}}}},$ | $S^{\prime\prime}_{\textnormal{id}_{Z}}\colon\overline{{in{Sp_{\textnormal{id}_{Z}}}}}\rightarrow\overline{{\tt out}(Z)},$
---|---|---
$E_{\textnormal{id}_{Z},f}\colon\overline{{\tt in}(Z)}\rightarrow\overline{{in{Sp_{\textnormal{id}_{Z}}}}},$ | $C_{\textnormal{id}_{Z},f}\colon\overline{{\tt in}(Z)}\rightarrow\overline{{Sp_{\textnormal{id}_{Z}}}},$
from Announcement 3.3.3, where ${in{Sp_{\textnormal{id}_{Z}}}}={\tt out}(Z)$.
###### Lemma 3.5.2.
Suppose given a list $\ell\in\overline{{\tt in}(Z)}$. We have
$C_{\textnormal{id}_{Z},f}(\ell)=\begin{cases}[\;]&\textnormal{ if
}|\ell|=0,\\\ \big{(}\ell,f(\partial\ell)\big{)}&\textnormal{ if }|\ell|\geq
1.\end{cases}$
###### Proof.
We work by induction. The result holds trivially for the empty list. Thus we
may assume that the result holds for $\partial\ell$ (i.e. that
$C_{\textnormal{id}_{Z},f}(\partial\ell)=(\partial\ell,f(\partial\partial\ell)$
holds) and deduce that it holds for $\ell$. Note that
$S^{\prime}_{\textnormal{id}_{Z}}([\;])=[\;]$ and
$E_{\textnormal{id}_{Z},f}([\;])=f([\;])$. By the formulas (33) we have
$\displaystyle C_{\textnormal{id}_{Z},f}(\ell)$
$\displaystyle=\big{(}\textnormal{id}_{\overline{{\tt
in}(Z)}}\times(E_{\textnormal{id}_{Z},f}\circ
S^{\prime}_{\textnormal{id}_{Z}}\circ
C_{\textnormal{id}_{Z},f}\circ\partial)\big{)}\circ\Delta(\ell)$
$\displaystyle=\big{(}\textnormal{id}_{\overline{{\tt
in}(Z)}}\times(E_{\textnormal{id}_{Z},f}\circ
S^{\prime}_{\textnormal{id}_{Z}}\circ
C_{\textnormal{id}_{Z},f}\circ\partial)\big{)}(\ell,\ell)$
$\displaystyle=\big{(}\textnormal{id}_{\overline{{\tt
in}(Z)}}(\ell),E_{\textnormal{id}_{Z},f}\circ
S^{\prime}_{\textnormal{id}_{Z}}\circ
C_{\textnormal{id}_{Z},f}\circ\partial(\ell)\big{)}$
$\displaystyle=\big{(}\ell,E_{\textnormal{id}_{Z},f}\circ
S^{\prime}_{\textnormal{id}_{Z}}(\partial\ell,f(\partial\partial\ell))\big{)}$
$\displaystyle=\big{(}\ell,E_{\textnormal{id}_{Z},f}\circ\pi_{\overline{{in{Dm_{\textnormal{id}_{Z}}}}}}\circ
S_{\textnormal{id}_{Z}}(\partial\ell,f(\partial\partial\ell))\big{)}$
$\displaystyle=\big{(}\ell,E_{\textnormal{id}_{Z},f}\circ\pi_{\overline{{in{Dm_{\textnormal{id}_{Z}}}}}}(\partial\ell,f(\partial\partial\ell))\big{)}$
$\displaystyle=\big{(}\ell,E_{\textnormal{id}_{Z},f}(\partial\ell)\big{)}$
$\displaystyle=\big{(}\ell,f(\partial\ell)\big{)}.$
∎
Expanding the definition of $\mathcal{P}(\textnormal{id}_{Z})(f)(\ell)$ we now
complete the proof that the identity law holds for $\mathcal{P}$:
$\displaystyle\mathcal{P}(\textnormal{id}_{Z})(f)(\ell)$
$\displaystyle=S^{\prime\prime}_{\textnormal{id}_{Z}}\circ
E_{\textnormal{id}_{Z},f}\circ S^{\prime}_{\textnormal{id}_{Z}}\circ
C_{\textnormal{id}_{Z},f}(\ell)$
$\displaystyle=S^{\prime\prime}_{\textnormal{id}_{Z}}\circ
E_{\textnormal{id}_{Z},f}\circ
S^{\prime}_{\textnormal{id}_{Z}}(\ell,f(\partial\ell))$
$\displaystyle=S^{\prime\prime}_{\textnormal{id}_{Z}}\circ
E_{\textnormal{id}_{Z},f}\circ\pi_{\overline{{in{Dm_{\textnormal{id}_{Z}}}}}}\circ
S_{\textnormal{id}_{Z}}(\ell,f(\partial\ell))$
$\displaystyle=S^{\prime\prime}_{\textnormal{id}_{Z}}\circ
E_{\textnormal{id}_{Z},f}\circ\pi_{\overline{{in{Dm_{\textnormal{id}_{Z}}}}}}(\ell,f(\partial\ell))$
$\displaystyle=S^{\prime\prime}_{\textnormal{id}_{Z}}\circ
E_{\textnormal{id}_{Z},f}(\ell)$
$\displaystyle=S^{\prime\prime}_{\textnormal{id}_{Z}}\big{(}f(\ell)\big{)}$
$\displaystyle=f(\ell).$
We now move on to the composition law. Let $s\colon m\rightarrow n$ be a
morphism in ${\bf Fin}$. Let $Z\in\textnormal{Ob}(\mathcal{W})$ be a black
box, let $Y\colon n\rightarrow\textnormal{Ob}(\mathcal{W})$ be an $n$-indexed
set of black boxes, and let $x\colon m\rightarrow\textnormal{Ob}(\mathcal{W})$
be an $m$-indexed set of black boxes. We must show that the following diagram
of sets commutes:
$\textstyle{\mathcal{W}_{n}(Y;Z)\times\prod_{i\in
n}\mathcal{W}_{m_{i}}(X_{i};Y(i))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{P}}$$\scriptstyle{\circ_{\mathcal{W}}}$$\textstyle{\mathcal{W}_{m}(X;Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{P}}$$\textstyle{{\bf
Sets}_{n}(\mathcal{P}(Y);\mathcal{P}(Z))\times\prod_{i\in n}{\bf
Sets}_{m_{i}}(\mathcal{P}(X_{i});\mathcal{P}(Y(i)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\circ_{\bf
Sets}}$$\textstyle{{\bf Sets}_{m}(\mathcal{P}(X);\mathcal{P}(Z))}$
Suppose given $\psi\colon Y\rightarrow Z$ and $\phi_{i}\colon X_{i}\rightarrow
Y(i)$ for each $i$, and let $\phi=\bigotimes_{i}\phi_{i}\colon X\rightarrow
Y$. We can trace through the diagram to obtain $\mathcal{P}(\psi)\circ_{\bf
Sets}\mathcal{P}(\phi)$ and $\mathcal{P}(\psi\circ_{\mathcal{W}}\phi)$, both
in ${\bf Sets}_{m}(\mathcal{P}(X);\mathcal{P}(Z)))$ and we want to show they
are equal as functions. From here on, we drop the subscripts on $\circ_{-}$,
i.e. we want to show
$\mathcal{P}(\psi)\circ\mathcal{P}(\phi)=\mathcal{P}(\psi\circ\phi).$
Let $\omega=\psi\circ\phi$. An element
$f\in\mathcal{P}(X)=\textnormal{Hist}^{1}({\tt vset}_{{\tt in}(X)},{\tt
vset}_{{\tt out}(X)})$ is a 1-historical propagator, $f\colon\overline{{\tt
in}(X)}\rightarrow\overline{{\tt out}(X)}$. We are required to show that the
following equation holds in $\mathcal{P}(Z)$:
(36)
$\mathcal{P}(\psi)\circ\mathcal{P}(\phi)(f)\stackrel{{\scriptstyle?}}{{=}}\mathcal{P}(\omega)(f).$
Expanding using the definition (34) of
$\mathcal{P}(\psi)\circ\mathcal{P}(\phi)(f)$ and $\mathcal{P}(\omega)(f)$ we
see that this translates into proving the commutativity of the following
diagram:
$\textstyle{\overline{{in{Sp_{\psi}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{?}$$\scriptstyle{S^{\prime\prime}_{\psi}}$$\textstyle{\overline{{\tt
out}(Z)}}$$\textstyle{\overline{{in{Sp_{\omega}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime\prime}_{\omega}}$$\textstyle{\overline{{in{Dm_{\psi}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{\psi,g}}$$\textstyle{\overline{{in{Dm_{\omega}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{\omega,f}}$$\textstyle{\overline{{Sp_{\psi}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime}_{\psi}}$$\textstyle{\overline{{\tt
in}(Z)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{\psi,g}}$$\scriptstyle{C_{\omega,f}}$$\textstyle{\overline{{Sp_{\omega}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime}_{\omega}}$
where we abbreviated $g=\mathcal{P}(\phi)(f)$. To do so, we must prove some
technical results (Lemmas 3.5.3, 3.5.4, and 3.5.5) which assert the equality
of various demand and supply streams flowing on the composed wiring diagram
$\omega=\psi\circ\phi$.
The ultimate proof of (36) will be inductive in nature. That is, to prove that
the result holds for a nonempty list $\ell$ of length $t\geq 1$, we will
assume that it holds for the list $\partial\ell$ of length $t-1$. More
precisely, to prove (36) we will need to know the following equality of
functions $\overline{{\tt in}(Z)}\rightarrow\overline{{in{Dm_{\omega}}}}$
(37) $\displaystyle S^{\prime}_{\omega}\circ
C_{\omega,f}=(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}$
and this is proven by induction on the length of $\ell\in\overline{{\tt
in}(Z)}$. The base of the induction is clear after recalling that definition
(33) gives $C_{\omega,f}([\;])=[\;]$, $C_{\phi,f}([\;])=[\;]$ and
$C_{\psi,g}([\;])=[\;]$, and that $S^{\prime}_{\psi}$ and
$s^{\prime}_{\omega}$ are 0-historical.
The next three lemmas carry out the induction step and assume the following
induction hypothesis regarding the equality of functions $\overline{{\tt
in}(Z)}\rightarrow\overline{{in{Dm_{\omega}}}}$
(38) $\displaystyle S^{\prime}_{\omega}\circ
C_{\omega,f}\circ\partial=(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}\circ\partial.$
###### Lemma 3.5.3.
If we assume that equation (38) holds then the following diagram commutes:
$\textstyle{\overline{{\tt
in}(Z)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{\omega,f}}$$\scriptstyle{C_{\psi,\mathcal{P}(\phi)(f)}}$$\textstyle{\overline{{Sp_{\omega}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{in{Sp_{\phi}}}}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id}}$$\textstyle{\overline{{Sp_{\psi}}}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{\tt
out}(Y)}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
in other words, we have the following equality between functions
$\overline{{\tt in}(Z)}\rightarrow\overline{{Sp_{\psi}}}$:
(39) $C_{\psi,\mathcal{P}(\phi)(f)}=(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}.$
###### Proof.
For convenience we will abbreviate $g=\mathcal{P}(\phi)(f)$. It follows from
our induction hypothesis (38), the internal square in the following diagram
(when composed with
$(\textnormal{id}\times\partial)\circ\Delta\colon\overline{{\tt
in}(Z)}\rightarrow\overline{{\tt in}(Z)}\times\overline{{\tt in}(Z)}$)
commutes:
$\textstyle{\overline{{\tt
in}(Z)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\textnormal{id}\times\partial)\circ\Delta}$$\scriptstyle{C_{\omega,f}}$$\scriptstyle{C_{\psi,g}}$$\textstyle{\overline{{Sp_{\omega}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{\tt
in}(Z)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
C_{\omega,f}}$$\scriptstyle{\textnormal{id}\times
C_{\psi,g}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{Sp_{\omega}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
S^{\prime}_{\omega}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{in{Dm_{\omega}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
E_{\omega,f}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{in{Sp_{\omega}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{Sp_{\psi}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
S^{\prime}_{\psi}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{in{Dm_{\phi}}}}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
E_{\phi,f}\times\delta^{1}_{\psi}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{in{Sp_{\phi}}}}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{in{Dm_{\psi}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
E_{\psi,g}}$$\textstyle{\overline{{\tt in}(Z)}\times\overline{{\tt
in}(Y)}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
C_{\phi,f}\times\textnormal{id}}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{Sp_{\phi}}}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textnormal{id}\times
S^{\prime}_{\phi}\times\textnormal{id}}$$\textstyle{\overline{{Sp_{\psi}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{in{Sp_{\psi}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\tt
in}(Z)}\times\overline{{\tt out}(Y)}\times\overline{DN_{\psi}}}$
The top square and left square commute by definition of $C_{\omega,f}$ and
$C_{\psi,f}$ respectively, see (33). The square
$E_{\omega,f}=E_{\phi,f}\times\delta^{1}_{\psi}$ commutes also by definition
(33). The commutativity of the bottom-right corner of the diagram translates
into the following identity between functions
$\overline{{in{Dm_{\psi}}}}\rightarrow\overline{{\tt
out}(Y)}\times\overline{DN_{\psi}}$:
$E_{\psi,\mathcal{P}(\phi)(f)}=(S^{\prime\prime}_{\phi}\times\textnormal{id})\circ(E_{\phi,f}\times\delta^{1}_{\psi})\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id}).$
But this is a direct consequence of the definitions
$E_{\psi,\mathcal{P}(\phi)(f)}=\mathcal{P}(\phi)(f)\times\delta^{1}_{\psi}$
and $\mathcal{P}(\phi)(f)=S^{\prime\prime}_{\phi}\circ E_{\phi,f}\circ
S^{\prime}_{\phi}\circ C_{\phi,f}$. It follows that the outer square commutes.
∎
###### Lemma 3.5.4.
If we assume that equation (38) holds, then so does the following equality of
functions $\overline{{\tt in}(Z)}\rightarrow\overline{{in{Sp_{\phi}}}}$: 555It
is possible for one to draw a diagram representing this equation as we did in
the preceding lemma, however we did not find such a diagram enlightening in
this case.
(40) $\displaystyle\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\omega,f}=\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\phi,f}\circ\pi_{\overline{{\tt in}(Y)}}\circ S^{\prime}_{\psi}\circ
C_{\psi,g}.$
###### Proof.
We will use the following three “forgetful” equations,
(41) $\displaystyle E_{\phi,f}\circ\pi_{\overline{{in{Dm_{\phi}}}}}\circ
S^{\prime}_{\omega}$
$\displaystyle=\pi_{\overline{{in{Sp_{\phi}}}}}\circ(E_{\phi,f}\times\delta_{\psi}^{1})\circ
S^{\prime}_{\omega},$ (42) $\displaystyle\pi_{\overline{{in{Sp_{\phi}}}}}\circ
E_{\omega,f}\circ S^{\prime}_{\omega}\circ C_{\omega,f}\circ\partial$
$\displaystyle=\pi_{\overline{{in{Sp_{\phi}}}}}\circ\big{(}\textnormal{id}_{\overline{{\tt
in}(Z)}}\times(E_{\omega,f}\circ S^{\prime}_{\omega}\circ
C_{\omega,f}\circ\partial)\big{)}\circ\Delta,$ (43) $\displaystyle
E_{\phi,f}\circ S^{\prime}_{\phi}\circ C_{\phi,f}\circ\partial$
$\displaystyle=\pi_{\overline{{in{Sp_{\phi}}}}}\circ\big{(}\textnormal{id}_{\overline{{\tt
in}(Y)}}\times(E_{\phi,f}\circ S^{\prime}_{\phi}\circ
C_{\phi,f}\circ\partial)\big{)}\circ\Delta,$ (44) $\displaystyle
S^{\prime}_{\phi}\circ C_{\phi,f}\circ\pi_{{\tt in}(Y)}$
$\displaystyle=\pi_{\overline{{in{Dm_{\phi}}}}}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id}).$
which are “obvious” in the sense that they are simply a matter of tracking
coordinate projections. The proof will go as follows. We apply
$E_{\phi,f}\circ\pi_{\overline{{in{Dm_{\psi}}}}}$ to both sides of the assumed
equality (38) and simplify. On the left-hand side we use (41) then the fact
that by definition we have
(45) $\displaystyle E_{\omega,f}=E_{\phi,f}\times\delta^{1}_{\psi},$
then (42), then the definition of $C_{\omega,f}$ which we reproduce here:
(46) $\displaystyle C_{\omega,f}=\big{(}\textnormal{id}_{\overline{{\tt
in}(Z)}}\times(E_{\omega,f}\circ S^{\prime}_{\omega}\circ
C_{\omega,f}\circ\partial)\big{)}\circ\Delta$
to obtain the following equality of functions $\overline{{\tt
in}(Z)}\rightarrow\overline{{in{Sp_{\phi}}}}$:
$\displaystyle E_{\phi,f}\circ\pi_{\overline{{in{Dm_{\phi}}}}}\circ
S^{\prime}_{\omega}\circ C_{\omega,f}\circ\partial$
$\displaystyle\quad=^{(\ref{dia:forgetful1})}\pi_{\overline{{in{Sp_{\phi}}}}}\circ(E_{\phi,f}\times\delta_{\psi}^{1})\circ
S^{\prime}_{\omega}\circ C_{\omega,f}\circ\partial$
$\displaystyle\quad=^{(\ref{dia:E fact})}\pi_{\overline{{in{Sp_{\phi}}}}}\circ
E_{\omega,f}\circ S^{\prime}_{\omega}\circ C_{\omega,f}\circ\partial$
$\displaystyle\quad=^{(\ref{dia:forgetful2})}\pi_{\overline{{in{Sp_{\phi}}}}}\circ\big{(}\textnormal{id}_{\overline{{\tt
in}(Z)}}\times(E_{\omega,f}\circ S^{\prime}_{\omega}\circ
C_{\omega,f}\circ\partial)\big{)}\circ\Delta$
$\displaystyle\quad=^{(\ref{dia:P fact})}\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\omega,f}.$
On the right hand side we use (44), then commute the $\partial$, then apply
(43), and then the definition of $C_{\phi,f}$ which we reproduce here:
(47) $\displaystyle C_{\phi,f}=\big{(}\textnormal{id}_{\overline{{\tt
in}(Y)}}\times(E_{\phi,f}\circ S^{\prime}_{\phi}\circ
C_{\phi,f}\circ\partial)\big{)}\circ\Delta$
to obtain the following equality of functions $\overline{{\tt
in}(Z)}\rightarrow\overline{{in{Sp_{\phi}}}}$:
$\displaystyle
E_{\phi,f}\circ\pi_{\overline{{in{Dm_{\phi}}}}}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}\circ\partial$
$\displaystyle\quad=^{(\ref{dia:forgetful4})}E_{\phi,f}\circ
S^{\prime}_{\phi}\circ C_{\phi,f}\circ\pi_{\overline{{\tt in}(Y)}}\circ
S^{\prime}_{\psi}\circ C_{\psi,g}\circ\partial$
$\displaystyle\quad=E_{\phi,f}\circ S^{\prime}_{\phi}\circ
C_{\phi,f}\circ\partial\circ\pi_{\overline{{\tt in}(Y)}}\circ
S^{\prime}_{\psi}\circ C_{\psi,g}$
$\displaystyle\quad=^{(\ref{dia:forgetful3})}\pi_{\overline{{in{Sp_{\phi}}}}}\circ\big{(}\textnormal{id}_{\overline{{\tt
in}(Y)}}\times(E_{\phi,f}\circ S^{\prime}_{\phi}\circ
C_{\phi,f}\circ\partial)\big{)}\circ\Delta\circ\pi_{\overline{{\tt
in}(Y)}}\circ S^{\prime}_{\psi}\circ C_{\psi,g}$
$\displaystyle\quad=^{(\ref{dia:P fact
2})}\pi_{\overline{{in{Sp_{\phi}}}}}\circ C_{\phi,f}\circ\pi_{\overline{{\tt
in}(Y)}}\circ S^{\prime}_{\psi}\circ C_{\psi,g}.$
Combining these computations with the induction hypothesis (38) gives the
result:
$\displaystyle\pi_{\overline{{in{Sp_{\phi}}}}}\circ C_{\omega,f}$
$\displaystyle=E_{\phi,f}\circ\pi_{\overline{{in{Dm_{\phi}}}}}\circ
S^{\prime}_{\omega}\circ C_{\omega,f}\circ\partial$
$\displaystyle=E_{\phi,f}\circ\pi_{\overline{{in{Dm_{\phi}}}}}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}\circ\partial$
$\displaystyle=\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\phi,f}\circ\pi_{\overline{{\tt in}(Y)}}\circ S^{\prime}_{\psi}\circ
C_{\psi,g}.$
∎
###### Lemma 3.5.5 (Main Induction Step).
If we assume that equation (38), reproduced here
(38) $\displaystyle S^{\prime}_{\omega}\circ
C_{\omega,f}\circ\partial=(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}\circ\partial,$
holds, then equation (38) holds without the precomposed $\partial$, i.e. we
have the following equality of functions $\overline{{\tt
in}(Z)}\rightarrow\overline{{in{Dm_{\omega}}}}$:
$\displaystyle S^{\prime}_{\omega}\circ
C_{\omega,f}=(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}.$
###### Proof.
To keep the notation from becoming too cluttered we adopt the following
convention: an identity map written as the right hand term of a product will
always mean $\textnormal{id}_{\overline{DN_{\psi}}}$, while an identity map
written as the left hand term of a product will mean one of
$\textnormal{id}_{\overline{{\tt in}(Y)}}$, $\textnormal{id}_{\overline{{\tt
out}(Y)}}$, $\textnormal{id}_{\overline{{\tt in}(Z)}}$, or
$\textnormal{id}_{\overline{{\tt out}(Z)}}$, which one should be clear from
the context.
The proof will be by cases, we show for each $j\in{in{Dm_{\omega}}}$ that
$\pi_{j}\circ(S^{\prime}_{\phi}\times\textnormal{id}\big{)}\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}=\pi_{j}\circ S^{\prime}_{\omega}\circ
C_{\omega,f},$
i.e. we show that the two ways of producing internal demand streams agree by
checking wire by wire. Since ${in{Dm_{\omega}}}={in{Dm_{\phi}}}\amalg
DN_{\psi}$, there are three main cases to consider: $j\in DN_{\psi}$,
$j\in{in{Dm_{\phi}}}$ with $s_{\phi}(j)\in{\tt in}(Y)$, and
$j\in{in{Dm_{\phi}}}$ with $s_{\phi}(j)\in{in{Sp_{\phi}}}$. We go through
these in turn below. Most of the necessary equalities will use that shuttling
streams between outputs and inputs does not change the value stream.
1. (1)
Suppose $j\in DN_{\psi}$. We use Lemma 3.5.3 and the fact that the right hand
identity maps are $\textnormal{id}_{\overline{DN_{\psi}}}$ to see
$\displaystyle\pi_{j}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}$ $\displaystyle\quad=^{(\ref{eq:cascade
equality})}\pi_{j}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}$
$\displaystyle\quad=\pi_{j}\circ S^{\prime}_{\psi}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}$ ($*$)
$\displaystyle\quad=\pi_{s_{\psi}(j)}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}.$
Now there are two cases depending on what has supplied wire $j$.
* •
Suppose $s_{\psi}(j)\in{\tt in}(Z)\amalg DN_{\psi}$. Notice that in this case
(24) gives $s_{\psi}(j)=s_{\omega}(j)$. Then ($*$) above becomes
$\displaystyle\pi_{s_{\psi}(j)}\circ(\textnormal{id}_{\overline{{\tt
in}(Z)}}\times
S^{\prime\prime}_{\phi}\times\textnormal{id}_{\overline{DN_{\psi}}})\circ
C_{\omega,f}$ $\displaystyle=\pi_{s_{\psi}(j)}\circ
C_{\omega,f}=\pi_{s_{\omega}(j)}\circ C_{\omega,f}$
$\displaystyle=\pi_{j}\circ S^{\prime}_{\omega}\circ C_{\omega,f}.$
* •
Suppose $s_{\psi}(j)\in{\tt out}(Y)$. In this case (24) gives $s_{\phi}\circ
s_{\psi}(j)=s_{\omega}(j)$. Because
$S^{\prime\prime}_{\phi}=\pi_{s_{\phi}\big{|}_{{\tt
out}(Y)}}\colon\overline{{in{Sp_{\phi}}}}\rightarrow\overline{{\tt out}(Y)}$,
we see that ($*$) simplifies as
$\displaystyle\pi_{s_{\psi}(j)}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}$
$\displaystyle=\pi_{s_{\psi}(j)}\circ
S^{\prime\prime}_{\phi}\circ\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\omega,f}$ $\displaystyle=\pi_{s_{\phi}\circ s_{\psi}(j)}\circ
C_{\omega,f}=\pi_{s_{\omega}(j)}\circ C_{\omega,f}$
$\displaystyle=\pi_{j}\circ S^{\prime}_{\omega}\circ C_{\omega,f}.$
2. (2)
Suppose $j\in{in{Dm_{\phi}}}$ and $s_{\phi}(j)\in{\tt in}(Y)$. We will use
Lemma 3.5.3 and the equation
$\pi_{j}\circ(S^{\prime}_{\phi}\times\textnormal{id})=\pi_{s_{\phi}(j)}.$
We will also use the fact that
$\pi_{s_{\phi}(j)}\circ(C_{\phi,f}\times\textnormal{id})=\pi_{s_{\phi}(j)}$,
which holds because $s_{\phi}(j)\in{\tt in}(Y)$ and $C_{\phi,f}$ is the
identity on $\overline{{\tt in}(Y)}$. With these in hand we compute:
$\displaystyle\pi_{j}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}$ $\displaystyle\quad=^{(\ref{eq:cascade
equality})}\pi_{j}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}$
$\displaystyle\quad=\pi_{s_{\phi}(j)}\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}$
$\displaystyle\quad=\pi_{s_{\phi}(j)}\circ
S^{\prime}_{\psi}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f},$ ($**$)
$\displaystyle\quad=\pi_{s_{\psi}\circ s_{\phi}(j)}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f},$
There are again two cases to consider depending on what has supplied wire $j$:
* •
Suppose $s_{\psi}\circ s_{\phi}(j)\in{\tt in}(Z)\amalg DN_{\psi}$. Then we get
$\pi_{s_{\psi}\circ s_{\phi}(j)}\circ(\textnormal{id}_{\overline{{\tt
in}(Z)}}\times
S^{\prime\prime}_{\phi}\times\textnormal{id}_{\overline{DN_{\psi}}})=\pi_{s_{\psi}\circ
s_{\phi}(j)}.$
Now (24) implies the identity $s_{\psi}\circ s_{\phi}(j)=s_{\omega}(j)$ and
thus ($**$) becomes
$\displaystyle\pi_{s_{\psi}\circ s_{\phi}(j)}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}$
$\displaystyle=\pi_{s_{\psi}\circ s_{\phi}(j)}\circ
C_{\omega,f}=\pi_{s_{\omega}(j)}\circ C_{\omega,f}$
$\displaystyle=\pi_{j}\circ S^{\prime}_{\omega}\circ C_{\omega,f}.$
* •
Suppose $s_{\psi}\circ s_{\phi}(j)\in{\tt out}(Y)$. Then notice that by (24)
we have $s_{\omega}(j)=s_{\phi}\circ s_{\psi}\circ s_{\phi}(j)$ and ($**$)
simplifies as
$\displaystyle\pi_{s_{\psi}\circ s_{\phi}(j)}\circ(\textnormal{id}\times
S^{\prime\prime}_{\phi}\times\textnormal{id})\circ C_{\omega,f}$
$\displaystyle=\pi_{s_{\psi}\circ s_{\phi}(j)}\circ
S^{\prime\prime}_{\phi}\circ\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\omega,f}$ $\displaystyle=\pi_{s_{\phi}\circ s_{\psi}\circ
s_{\phi}(j)}\circ C_{\omega,f}=\pi_{s_{\omega}(j)}\circ C_{\omega,f}$
$\displaystyle=\pi_{j}\circ S^{\prime}_{\omega}\circ C_{\omega,f}.$
3. (3)
Suppose $j\in{in{Dm_{\phi}}}$ and $s_{\phi}(j)\in{in{Sp_{\phi}}}$. As usual we
have $\pi_{j}\circ S^{\prime}_{\phi}=\pi_{s_{\phi}(j)}$, but noting that ${\tt
vset}_{j}={\tt vset}_{s_{\phi}(j)}$, the assumptions on $j$ imply that we have
$\pi_{j}\circ
S^{\prime}_{\phi}=\pi_{s_{\phi}(j)}\circ\pi_{\overline{{in{Sp_{\phi}}}}}.$
In this case (24) gives $s_{\omega}(j)=s_{\phi}(j)$ and thus by Lemma 3.5.4,
$\displaystyle\pi_{j}\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}$ $\displaystyle\quad=\pi_{j}\circ
S^{\prime}_{\phi}\circ C_{\phi,f}\circ\pi_{\overline{{\tt in}(Y)}}\circ
S^{\prime}_{\psi}\circ C_{\psi,g}$
$\displaystyle\quad=\pi_{s_{\phi}(j)}\circ\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\phi,f}\circ\pi_{\overline{{\tt in}(Y)}}\circ S^{\prime}_{\psi}\circ
C_{\psi,g}$ $\displaystyle\quad=^{(\ref{dia:what we
want})}\pi_{s_{\phi}(j)}\circ\pi_{\overline{{in{Sp_{\phi}}}}}\circ
C_{\omega,f}$ $\displaystyle\quad=\pi_{s_{\omega}(j)}\circ C_{\omega,f}$
$\displaystyle\quad=\pi_{j}\circ S^{\prime}_{\omega}\circ C_{\omega,f}.$
∎
To complete the proof of Theorem 3.5.1 recall that we have been given
morphisms $\phi\colon X\rightarrow Y$ and $\psi\colon Y\rightarrow Z$ and
$\omega=\psi\circ\phi$ in $\mathcal{W}$ with notation as in Announcement
2.2.8. These have corresponding supplier assignments $s_{\phi},s_{\psi}$, and
$s_{\omega}$. Abbreviate $g=\mathcal{P}(\phi)(f)\colon\overline{{\tt
in}(Y)}\rightarrow\overline{{\tt out}(Y)}$. Consider the following diagram of
sets:
---
$\textstyle{\overline{{in{Sp_{\psi}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime\prime}_{\psi}}$$\textstyle{\overline{{\tt
out}(Z)}}$$\textstyle{\overline{{in{Sp_{\omega}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime\prime}_{\omega}}$$\textstyle{\overline{{\tt
out}(Y)}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{in{Sp_{\phi}}}}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime\prime}_{\phi}\times\textnormal{id}}$$\textstyle{\overline{{in{Dm_{\psi}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{\psi,g}}$$\textstyle{\overline{{in{Dm_{\omega}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{\omega,f}}$$\textstyle{\overline{{\tt
in}(Y)}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\times\delta^{1}_{\psi}}$$\scriptstyle{C_{\phi,f}\times\textnormal{id}}$$\textstyle{\overline{{Sp_{\phi}}}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime}_{\phi}\times\textnormal{id}}$$\textstyle{\overline{{in{Dm_{\phi}}}}\times\overline{DN_{\psi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E_{\phi,f}\times\delta^{1}_{\psi}}$$\textstyle{\overline{{Sp_{\psi}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime}_{\psi}}$$\textstyle{\overline{{\tt
in}(Z)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{\psi,g}}$$\scriptstyle{C_{\omega,f}}$$\textstyle{\overline{{Sp_{\omega}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{\prime}_{\omega}}$
Recall that our goal was to show that the outermost square commutes. We will
see that each inner square is commutative in the sense that the following
equations hold:
$\displaystyle
S^{\prime\prime}_{\omega}=S^{\prime\prime}_{\psi}\circ(S^{\prime\prime}_{\phi}\times\textnormal{id})$
$\displaystyle\colon\overline{{in{Sp_{\phi}}}}\times\overline{DN_{\psi}}\longrightarrow\overline{{\tt
out}(Z)}$ $\displaystyle E_{\psi,g}=g\times\delta^{1}_{\psi}$
$\displaystyle\colon\overline{{in{Dm_{\psi}}}}\longrightarrow\overline{{\tt
out}(Y)}\times\overline{DN_{\psi}}$ $\displaystyle
g\times\delta^{1}_{\psi}=(S^{\prime\prime}_{\phi}\times\textnormal{id})\circ(E_{\phi,f}\times\delta^{1}_{\psi})\circ(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})$
$\displaystyle\colon\overline{{\tt
in}(Y)}\times\overline{DN_{\psi}}\rightarrow\overline{{\tt
out}(Y)}\times\overline{DN_{\psi}}$ $\displaystyle
E_{\phi,f}\times\delta^{1}_{\psi}=E_{\omega,f}$
$\displaystyle\colon\overline{{in{Dm_{\phi}}}}\times\overline{DN_{\psi}}\longrightarrow\overline{{in{Sp_{\omega}}}}$
$\displaystyle S^{\prime}_{\omega}\circ
C_{\omega,f}=(S^{\prime}_{\phi}\times\textnormal{id})\circ(C_{\phi,f}\times\textnormal{id})\circ
S^{\prime}_{\psi}\circ C_{\psi,g}$ $\displaystyle\colon\overline{{\tt
in}(Z)}\longrightarrow\overline{{in{Dm_{\omega}}}}$
The first follows from Lemma 2.2.7, especially (16), and Announcement 2.2.8,
especially (24). The next three follow directly from definitions (33). The
last equality has been proven in Lemma 3.5.5.
It follows that the equation below holds for functions $\overline{{\tt
in}(Z)}\longrightarrow\overline{{\tt out}(Z)}$:
$\mathcal{P}(\omega)(f)=S^{\prime\prime}_{\omega}\circ E_{\omega,f}\circ
S^{\prime}_{\omega}\circ C_{\omega,f}=S^{\prime\prime}_{\psi}\circ
E_{\psi,g}\circ S^{\prime}_{\psi}\circ
C_{\psi,g}=\mathcal{P}(\psi)(g)=\mathcal{P}(\psi)\circ\mathcal{P}(\phi)(f)$
Indeed, the left-hand equality and the second-to-last equality are by
definition of $\mathcal{P}$ on morphisms, as given in (34). The second
equality is found by a diagram chase using the six equations above. ∎
## 4\. Future work
The authors hope that this work can be put to use rather directly in modeling
and design applications. The relationship between the operad $\mathcal{W}$ and
its algebra $\mathcal{P}$ is quite explicitly a relationship between form and
function. The ability to zoom in and out, i.e. to change levels of abstraction
with ease is a facility which we believe is essential to any good theory of
the brain, computer programs, cyber-physical systems, etc.
Below we will discuss some possibilities for future work. We see three major
directions in which to go. The first is to connect this work to other work on
wiring diagrams. The second is to consider applications, e.g. to computer
science and cognitive neuroscience. The third is to investigate the notion of
dependency, or cause and effect, in our formalism. We discuss these in turn
below.
### 4.1. Connecting to other work on wiring diagrams
While wiring diagrams have been useful in engineering for many years, there
are a few mathematical approaches that should connect to our own, including
[AADF], [BB], [DL], and [Sp2].
The work by [AADF] studies dynamics inside of strongly connected (transitive)
networks of identical units. Their main aim is to relate the dynamics on the
network to properties of the underlying network architecture. The underlying
network should be viewed as analogous to a morphism $\psi$ in $\mathcal{W}$,
while the dynamics lying over the network should be viewed as analogous to the
morphism $\mathcal{P}(\psi)$. The cells in their networks are considered to
have internal states which collude with the inputs to produce the output of a
cell. There exists an algebra over $\mathcal{W}$ of “propagators with internal
states” and a retract from this algebra to $\mathcal{P}$, which should allow
the transfer of results of [AADF] to our framework. Arguably one of the main
aims of [AADF] is to introduce a notion of inflation for these networks. A
careful comparison, see for example [AADF, Figure 15] and [AADF, Figure 29],
reveals that their inflation procedure is a special case of the composition of
morphisms in $\mathcal{W}$ where the black boxes being inserted into a wiring
diagram come from a special class called inflations.
In [BB], the authors investigate reaction networks and in particular
stochastic Petri nets. There, various species (e.g. chemicals or populations)
interact in prescribed ways, and the dynamics of their changing populations
are studied. A similar but more complex situation is studied in [DL]. Both of
these papers work with continuous time processes, whereas we work with
discrete time processes. Still, we plan to investigate the relationship
between these ideas in the future.
The only other place, other than the present paper, where operads are
explicitly mentioned in the context of wiring diagrams seems to be [Sp2],
where the author studies systems of interacting relations using an operad
$\mathcal{T}$. One might think that an operad functor would appropriately
relate it to the present operad $\mathcal{W}$, but that does not appear to be
the case because of the delay nodes that exist in $\mathcal{W}$ but not
$\mathcal{T}$. Instead, these two operads need to be compared via a third, in
which delay nodes do not occur, but wires are still directed. We hope to make
this precise in the future.
### 4.2. Applications, e.g. to computer science and cognitive neuroscience
The authors’ primary purposes in the above work was to formalize what we
considered fundamental principles in the relation of form and function in both
computers and brains. On the operad/form level we are speaking of hierarchical
chunking; on the algebra/function level we are speaking of historical
propagators.
One can ask several interesting questions at this point. For example, can we
create from $\mathcal{W}$ and $\mathcal{P}$ a viable computer programming
language? We would hope that the propagators given by computable functions are
closed in, i.e. form a subalgebra of, $\mathcal{P}$. But perhaps one could ask
for more as well. For example, if each transistor in a computer acts like a
NOR gate, one could ask whether or not the subalgebra generated by NOR gates
is Turing complete. We conjecture that something like this is true. If so, we
believe our language will provide a simple, grounded, and useful perspective
on the actual operation of computers.
There are also many interesting questions on the neuroscience side that
motivated this work. These essentially amount to a question of “what”. What is
a neuron? What is a brain? What is the relationship between the actions of
individual neurons and the brain as a whole? It is easy to imagine that a
neuron is simply a black box where we assign certain multisets of
neurotransmitters to each input and output, the historical propagators would
then record activity patterns of discretized neurons. If this turns out to be
the case then the distinction between neuron and brain becomes blurred, each
is simply a black box with some specified inputs and outputs. From this
perspective the questions of how the activity of individual neurons relates to
the activity of a functional brain region or of the entire brain becomes
subsumed by the operad formalism where we can think of each as a different
choice of chunking within a single (massively complex) wiring diagram
representing the connections occurring within an entire brain. Deep questions
regarding precisely how the actions of neurons in one part of the brain
influence the activity in other areas will rely on the work of
neuroscientists’ understanding of the precise wiring pattern of the brain and
remain to be understood. We will speak more on these questions of dependency
within our formalism in the next section.
### 4.3. Investigating the notion of dependency
Given a propagator with $m$-inputs and $n$-outputs, one may ask about the
relation of dependency between them. When one says that the outcome of a
process is dependent on the inputs, this should mean that changing the inputs
will cause a change in the outputs.
In one form or another, the ability to track changes as they propagate through
a network of processes is one of the basic questions in almost any field of
research. Indeed, concern with notions of cause and effect is an essential
characteristic of human thought. Making mathematical sense of this notion
would presumably be immensely valuable. In particular, it should have direct
applications to neuroscience and computer programming disciplines.
It is not clear that there exists a reasonable notion of causality that is
algebraic in nature, i.e. one that can be formulated as a
$\mathcal{W}$-algebra receiving a morphism from $\mathcal{P}$. In that case we
may look to other approaches, e.g. that of Bayesian networks as in [Pea] and
[Fon]. Whether Bayesian networks also form an algebra on $\mathcal{W}$ or a
related operad, and how such an algebra compares with $\mathcal{P}$ should
certainly be investigated.
## References
* [Awo] S. Awodey. (2010) Category theory. Second edition. Oxford Logic Guides, 52. Oxford University Press, Oxford.
* [AADF] Aguiar, M., Ashwin, P., Dias, A., Field, M. (2010) “Dynamics of coupled cell networks: synchrony, heteroclinic cycles, and inflation”. Journal of nonlinear science, Springer.
* [Bou] Bourbaki, N. (1972) “Univers”. In M. Artin et al. eds. SGA 4 - vol 1, Lecture Notes in Mathematics 269 (in French). Springer-Verlag pp. 185–217.
* [BB] Baez, J.C., Biamonte, J. (2012). “A Course on Quantum Techniques for Stochastic Mechanics”. Available online, http://arxiv.org/abs/1209.3632.
* [BV] Boardman, M.; Vogt, R. (1973) “Homotopy invariant algebraic structures on topological spaces.” Lecture notes in mathematics 347\. Springer-Verlag.
* [BW] Barr M., Wells, C. (1990) Category theory for computing science. Prentice Hall International Series in Computer Science. Prentice Hall International, New York.
* [DL] Deville, L., Lerman, E. (2013) “Dynamics on networks of manifolds”. Available online: http://arxiv.org/pdf/1208.1513v2.pdf.
* [Fon] Fong, B. (2013) “Causal Theories: A Categorical Perspective on Bayesian Networks”. Available online http://arxiv.org/abs/1301.6201
* [Lei] Leinster, T. (2004) Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series 298, Cambridge University Press.
* [Luc] Lucas, É. (1891), Théorie des nombres (in French) 1, Gauthier-Villars.
* [Lur] Lurie, J. (2012) “Higher algebra”. http://www.math.harvard.edu/~lurie/papers/HigherAlgebra.pdf.
* [Mac] (1998) Mac Lane, S. Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York.
* [Man] Manzyuk, O. (2009) “Closed categories vs. closed multicategories”. http://arxiv.org/abs/0904.3137
* [May] May, P. (1972). The geometry of iterated loop spaces. Springer-Verlag.
* [NIST] National Institute of Standards and Technology (1993). IDEF0: functional modeling method.
* [Pea] Pearl, J. (2009) Causality: Models, reasoning, and inference. Cambridge University Press.
* [Pen] Penrose, R. (2011). Cycles of time: An extraordinary new view of the universe. Random House.
* [RS] Radul, A.; Sussman, G.J. (2009). “The art of the propagator”. MIT Computer science and artificial intelligence laboratory technical report.
* [Sp1] Spivak, D.I. (2013) Category theory for scientists. http://arxiv.org/abs/1302.6946
* [Sp2] Spivak, D.I. (2013) “The operad of wiring diagrams: Formalizing a graphical language for databases, recursion, and plug-and-play circuits.” ePrint available: http://arxiv.org/abs/1305.0297
|
arxiv-papers
| 2013-07-25T23:33:24 |
2024-09-04T02:49:48.480218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dylan Rupel and David I. Spivak",
"submitter": "David Spivak",
"url": "https://arxiv.org/abs/1307.6894"
}
|
1307.6905
|
# Matrix elements of one-body and two-body operators between arbitrary HFB
multi-quasiparticle states
Qing-Li Hu Zao-Chun Gao [email protected] Y. S. Chen China Institute of
Atomic Energy, P.O. Box 275 (10), Beijing 102413, PR China
###### Abstract
We present new formulae for the matrix elements of one-body and two-body
physical operators, which are applicable to arbitrary Hartree-Fock-Bogoliubov
wave functions, including those for multi-quasiparticle excitations. The
testing calculations show that our formulae may substantially reduce the
computational time by several orders of magnitude when applied to many-body
quantum system in a large Fock space.
###### keywords:
Hartree-Fock-Bogoliubov method, beyond mean-field, Pfaffian, two-body operator
††journal: Physics Letter B
## 1 Introduction
Although the Schrödinger equation was proposed as early as in 1926, its exact
solution (by means of the full configuration interaction, FCI) for the quantum
mechanical many-body system is still hopeless except for the smallest system
due to the combinatorial computational cost. The mean-field theory has been a
great success in describing the microscopic systems, such as the nuclei, the
atoms, and the molecules. The Hartree-Fock-Bogoliubov (HFB) approximation, as
the best mean-field method, has played a central role in understanding
interacting many-body quantum systems in all fields of physics. However, the
HFB wave functions are far from the eigenstates of the Hamiltonian, and the
effects that go beyond mean-field are missing. Post-HFB treatments (beyond-
mean field methods), such as the configuration interaction(CI), the generator
coordinate method(GCM), and the symmetry restoration, are expected to improve
the wave functions and present better description of the quantum mechanical
many-body systems. For instance, symmetry restoration of the HFB states has
been performed not only in the nuclei (e.g.[1]), but also in the
molecules(e.g.[2]). Moreover, symmetry restoration also improves the
descriptions of quantum dots and ultra-cold Bose systems in the condense
matter world[28].
The overlaps and the matrix elements of the Hamiltonian between the HFB states
are basic blocks to establish such post-HFB calculations. Efficient evaluating
of those quantities is of extreme importance to implement the post-HFB
calculations. Efforts have been devoted to finding convenient formulae for
such matrix elements and overlaps for decades. The Onishi formula [3, 4] is
the first expression of the overlap between two different HFB vacua, but the
sign of the overlap is not determined. Many works have been done to overcome
this sign problem [5, 6, 7, 8, 29, 9, 10, 11, 12]. In Ref.[12], Robledo made
the final solution and proposed a new formula using the Pfaffian rather than
the determinant. After that, overlaps between quasi-particle states have been
intensively studied, which are also based on the Pfaffian [30, 13, 14, 15, 16,
17]. It is realized that overlaps between multi-quasiparticle HFB states,
originally evaluated with the generalized Wick’s theorem(GWT)[18], can be
equivalently calculated by compact formulae with Pfaffian[13, 14, 15, 16, 17].
Thanks to the same mathematical structure of the Pfaffian and the GWT, the
combinatorial explosion is avoided. We also should mention that, before
Robledo’s work [12], there is another compact formula for the GWT [19]. It is
obtained by using Gaudin’s theorem in the finite-temperature formalism, but
not expressed with the Pfaffian.
Although the overlap between HFB states can be quickly calculated using the
proposed Pfaffian formulae or the method in [19] to avoid the combinatorial
explosion, one may certainly encounter another difficulty in evaluating the
matrix elements of many-body operators, which has never been treated. We
address this problem as follows. In the representation of second quantization,
one can write the one-body operator $\hat{T}$ and two-body operator $\hat{V}$
as
$\displaystyle\hat{T}$ $\displaystyle=$
$\displaystyle\sum_{\mu\nu}T_{\mu\nu}\hat{c}^{\dagger}_{\mu}\hat{c}_{\nu},$
(1) $\displaystyle\hat{V}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{\mu\nu\delta\gamma}V_{\mu\nu\gamma\delta}\hat{c}^{\dagger}_{\mu}\hat{c}^{\dagger}_{\nu}\hat{c}_{\delta}\hat{c}_{\gamma},$
(2)
where $(\hat{c}^{\dagger},\hat{c})$ are the creation and annihilation
operators of the spherical harmonic oscillator, i.e.
$\hat{c}^{\dagger}_{\mu}|-\rangle=|Nljm\rangle$ and
$\hat{c}_{\mu}|-\rangle=0$. $|-\rangle$ stands for the true vacuum. Here, we
assume all operators are defined in the same $M-$dimensional Fock space.
The matrix element of an operator $\hat{O}$($=\hat{T}$ or $\hat{V}$) with
multi-quasiparticle excitations is generally given as
$\displaystyle\langle\Phi|\hat{\beta}_{i_{1}}\cdots\hat{\beta}_{i_{L}}\hat{O}\hat{\mathbb{R}}\hat{\beta}^{\prime\dagger}_{j_{L+1}}\cdots\hat{\beta}^{\prime\dagger}_{j_{2n}}|\Phi^{\prime}\rangle,$
(3)
where $\hat{\mathbb{R}}$ stands for a unitary transformation. $|\Phi\rangle$
and $|\Phi^{\prime}\rangle$ are different normalized HFB vacua.
$(\hat{\beta},\hat{\beta}^{\dagger})$ and
$(\hat{\beta}^{\prime},\hat{\beta}^{\prime\dagger})$ are corresponding
quasiparticle operators with
$\hat{\beta}_{i}|\Phi\rangle=\hat{\beta}^{\prime}_{i}|\Phi^{\prime}\rangle=0$
for any $i$.
Conventionally, the matrix element in Eq.(3) can be obtained in two steps. The
first step is evaluating the matrix element of each $c^{\dagger}_{\mu}c_{\nu}$
(or
$\hat{c}^{\dagger}_{\mu}\hat{c}^{\dagger}_{\nu}\hat{c}_{\delta}\hat{c}_{\gamma}$)
in Eq.(1) [or Eq.(2)] through Pfaffian or the method in ref [19] to avoid the
combinatorial explosion. The second step is collecting all the
$c^{\dagger}_{\mu}c_{\nu}$ (or
$\hat{c}^{\dagger}_{\mu}\hat{c}^{\dagger}_{\nu}\hat{c}_{\delta}\hat{c}_{\gamma}$)
matrix elements to get the final value of Eq.(3). Unlike the overlap between
HFB states, each matrix element of Eq.(3)(with $\hat{O}=\hat{V}$) requires the
summation over $\mu,\nu,\delta,\gamma$. This is too much time consuming for a
symmetry restoration in a relatively large configuration space, where
thousands or millions of the matrix elements need to be calculated at each
mesh point in the integral of the projection. Such calculations in a large
Fock space will be even too expensive to be tractable.
In this Letter, we present new formulae for evaluating the matrix elements of
Eq. (3) between arbitrary HFB states, which are in compact forms and may
greatly reduce the computational cost of the post-HFB calculations.
## 2 Overlaps
Let’s start with a useful equation that the expectation value of a product of
arbitrary single-fermion operators, $\hat{z}_{i}$, is given by the Pfaffian of
all possible contractions [16, 20, 21],
$\displaystyle\langle-|\hat{z}_{1}\cdots\hat{z}_{2k}|-\rangle=\mathrm{pf}(S),$
(4)
where $S$ is a $2k\times 2k$ skew-symmetric matrix with the matrix element
$S_{ij}=\langle-|\hat{z}_{i}\hat{z}_{j}|-\rangle,\,S_{ji}=-S_{ij},\,(i<j)$.
One can extend Eq. (4) to a more general form (details of proof are given in
the Supplemental material to this article),
$\displaystyle\langle\Phi^{a}|\hat{z}_{1}\cdots\hat{z}_{2n}|\Phi^{b}\rangle=\mathrm{pf}(\mathbb{S})\langle\Phi^{a}|\Phi^{b}\rangle,$
(5)
where $|\Phi^{a}\rangle$ (or $|\Phi^{b}\rangle$) can be regarded as the true
vacuum or arbitrary HFB vacuum. $\mathbb{S}$ is a $2n\times 2n$ skew-symmetric
matrix, but the matrix element in the upper triangular is
$\displaystyle\mathbb{S}_{ij}=\frac{\langle\Phi^{a}|\hat{z}_{i}\hat{z}_{j}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle}\quad(i<j).$
(6)
For the lower triangular of $\mathbb{S}$,
$\mathbb{S}_{ji}=-\mathbb{S}_{ij}(i<j)$. Attention must be payed to the
useless contraction
$\frac{\langle\Phi^{a}|\hat{z}_{j}\hat{z}_{i}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle}\,(i<j)$,
which never appears in the GWT and should not be taken as
$\mathbb{S}_{ji}(i<j)$. Here, we assume that $\langle\Phi^{a}|\Phi^{b}\rangle$
is nonzero, and can be evaluated by the available formulae proposed by several
authors [12, 13, 14, 15, 16, 22].
Here, we define the HFB vacuum $|\Phi^{\sigma}\rangle$ ($\sigma=a,b$) as
$\displaystyle|\Phi^{\sigma}\rangle$ $\displaystyle=$
$\displaystyle\mathcal{N}_{\sigma}\hat{\beta}^{\sigma}_{1}\cdots\hat{\beta}^{\sigma}_{N_{\sigma}}|-\rangle,$
(7)
where $\mathcal{N}_{\sigma}$is the normalization factor of
$|\Phi^{\sigma}\rangle$. $N_{\sigma}$ is the number of $\hat{\beta}^{\sigma}$
operators acting on $|-\rangle$ to form the HFB vacuum
$|\Phi^{\sigma}\rangle$. The operator $\hat{z}$ can be expressed in terms of
either $(\hat{\beta}^{a},\hat{\beta}^{a\dagger})$ or
$(\hat{\beta}^{b},\hat{\beta}^{b\dagger})$,
$\displaystyle\hat{z}_{i}=\sum_{j}\left(A_{ij}^{a}\hat{\beta}^{a}_{j}+B_{ij}^{a}\hat{\beta}^{a\dagger}_{j}\right)=\sum_{j}\left(A_{ij}^{b}\hat{\beta}^{b}_{j}+B_{ij}^{b}\hat{\beta}^{b\dagger}_{j}\right).$
(8)
We should stress that the coefficients $A^{a}_{ij}$ and $B^{a}_{ij}$ (or
$A^{b}_{ij}$ and $B^{b}_{ij}$) are arbitrary, which means $\hat{z}_{i}$ can
stand for any single-fermion operator, such as $\hat{c}_{i}$,
$\hat{c}^{\dagger}_{i}$, $\hat{\beta}^{a}_{i}$, $\hat{\beta}^{a\dagger}_{i}$,
$\hat{\beta}^{b}_{i}$, $\hat{\beta}^{b\dagger}_{i}$, or even
$\hat{\mathbb{R}}\hat{c}_{i}\hat{\mathbb{R}}^{-1}$,
$\hat{\mathbb{R}}\hat{\beta}^{b\dagger}_{i}\hat{\mathbb{R}}^{-1}$, etc. For
instance, if $\hat{z}_{i}=\hat{\beta}^{a}_{i}$, then $A_{ij}^{a}=\delta_{ij}$
and $B_{ij}^{a}=0$. The operators ($\hat{c}_{i}$, $\hat{c}^{\dagger}_{i}$),
($\hat{\beta}^{a}_{i}$, $\hat{\beta}^{a\dagger}_{i}$) and
($\hat{\beta}^{b}_{i}$, $\hat{\beta}^{b\dagger}_{i}$) do obey the fermion-
commutation relations, but the general operator $\hat{z}_{i}$ does not have
any constraint. Hence, we do not impose
$\hat{z}_{i}\hat{z}_{j}=-\hat{z}_{j}\hat{z}_{i}$. By assuming the unitary
transformation between $(\hat{\beta}^{a},\hat{\beta}^{a\dagger})$ and
$(\hat{\beta}^{b},\hat{\beta}^{b\dagger})$ being
$\displaystyle\left(\begin{array}[]{c}\hat{\beta}^{b}\\\
\hat{\beta}^{b\dagger}\end{array}\right)=\left(\begin{array}[]{cc}\mathbb{X}&\mathbb{Y}\\\
\mathbb{Y}^{*}&\mathbb{X}^{*}\end{array}\right)\left(\begin{array}[]{c}\hat{\beta}^{a}\\\
\hat{\beta}^{a\dagger}\end{array}\right),$ (15)
one can obtain the explicit expressions of $\mathbb{S}_{ij}$ in the following
three equivalent forms (see details in Supplemental material),
$\displaystyle\mathbb{S}_{ij}$ $\displaystyle=$
$\displaystyle[A^{a}B^{aT}+A^{a}\mathbb{X}^{-1}\mathbb{Y}A^{aT}]_{ij},$ (16)
$\displaystyle\mathbb{S}_{ij}$ $\displaystyle=$
$\displaystyle[A^{a}\mathbb{X}^{-1}B^{bT}]_{ij},$ (17)
$\displaystyle\mathbb{S}_{ij}$ $\displaystyle=$
$\displaystyle[A^{b}B^{bT}+B^{b}\mathbb{Y}^{*}\mathbb{X}^{-1}B^{bT}]_{ij},$
(18)
where the existence of the matrix $\mathbb{X}^{-1}$ is guaranteed by the
assumption $\langle\Phi^{a}|\Phi^{b}\rangle\neq 0$, according to the Onishi
formula [3, 4], in which $\mathrm{det}\mathbb{X}\neq 0$ .
Note that Eq.(5) can be regarded as a generalization of the conclusion
proposed recently in Ref.[17].
## 3 Matrix elements of operators
The matrix elements of Eq.(3) can be rewritten in a general form
$\displaystyle I$ $\displaystyle=$
$\displaystyle\langle\Phi^{a}|\hat{z}_{1}\cdots\hat{z}_{L}\hat{O}\hat{z}_{L+1}\cdots\hat{z}_{2n}|\Phi^{b}\rangle,$
(19)
where
$\displaystyle\hat{z}_{k}$ $\displaystyle=$
$\displaystyle\bigg{\\{}\begin{array}[]{cc}{\hat{\beta}}_{i_{k}},&1\leq k\leq
L\\\
\hat{\mathbb{R}}{\hat{\beta}}^{\prime\dagger}_{j_{k}}\hat{\mathbb{R}}^{-1},&L+1\leq
k\leq 2n\end{array}$ (22) $\displaystyle|\Phi^{a}\rangle$ $\displaystyle=$
$\displaystyle|\Phi\rangle,\quad|\Phi^{b}\rangle=\hat{\mathbb{R}}|\Phi^{\prime}\rangle.$
(23)
For fast calculation, we derive new formulae of $I$ instead of directly using
Eq.(19). Here, we denote $I$ as $I_{1}$ for $\hat{O}=\hat{T}$, and $I_{2}$ for
$\hat{O}=\hat{V}$.
To establish the notation, we define the following matrix elements of
$\mathbb{S}^{(\pm)}$ and $\mathbb{C}^{(\pm,0)}$,
$\displaystyle\mathbb{S}^{(+)}_{\mu k}$ $\displaystyle=$
$\displaystyle\bigg{\\{}\begin{array}[]{cc}-\frac{\langle\Phi^{a}|\hat{z}_{k}\hat{c}^{\dagger}_{\mu}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle},&1\leq
k\leq L\\\
\frac{\langle\Phi^{a}|\hat{c}^{\dagger}_{\mu}\hat{z}_{k}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle},&L+1\leq
k\leq 2n\end{array},$ (26) $\displaystyle\mathbb{S}^{(-)}_{\mu k}$
$\displaystyle=$
$\displaystyle\bigg{\\{}\begin{array}[]{cc}-\frac{\langle\Phi^{a}|\hat{z}_{k}\hat{c}_{\mu}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle},&1\leq
k\leq L\\\
\frac{\langle\Phi^{a}|\hat{c}_{\mu}\hat{z}_{k}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle},&L+1\leq
k\leq 2n\end{array},$ (29) $\displaystyle\mathbb{C}^{(+)}_{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{\langle\Phi^{a}|\hat{c}^{\dagger}_{\mu}\hat{c}^{\dagger}_{\nu}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle},\quad\mathbb{C}^{(-)}_{\mu\nu}=\frac{\langle\Phi^{a}|\hat{c}_{\mu}\hat{c}_{\nu}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle},$
$\displaystyle\mathbb{C}^{(0)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\langle\Phi^{a}|\hat{c}^{\dagger}_{\mu}\hat{c}_{\nu}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle},$
(31)
where the shapes of $\mathbb{S}^{(\pm)}$ and $\mathbb{C}^{(\pm,0)}$ are
$M\times 2n$ and $M\times M$, respectively.
For the one-body operator $\hat{T}$, we denote the quantity $T_{0}$ and the
matrix $\mathbb{T}$ using above notations,
$\displaystyle T_{0}$ $\displaystyle=$
$\displaystyle\sum_{\mu\nu}T_{\mu\nu}\mathbb{C}^{(0)}_{\mu\nu},\quad\mathbb{T}_{ij}=\sum_{\mu\nu}T_{\mu\nu}\mathbb{S}^{(+)}_{\mu
i}\mathbb{S}^{(-)}_{\nu j}.$ (32)
Similar to the Laplace expansion for determinant, there is also a general
expansion formula for Pfaffian (Lemma 4.2 in Ref [23], or Lemma 2.3 in
Ref.[24]). Due to the same mathematical structure of the GWT and Pfaffian,
this Pfaffian expansion is essentially equivalent to the contraction role of
the GWT. We present several explicit expansions of Pfaffian in the
Supplemental material, and using the one with respect to two rows (Eq.(S40) in
Supplemental material) to get
$\displaystyle\frac{I_{1}}{\langle\Phi^{a}|\Phi^{b}\rangle}=\frac{\langle\Phi^{a}|\hat{z}_{1}\cdots\hat{z}_{L}\hat{T}\hat{z}_{L+1}\cdots\hat{z}_{2n}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle}$
(33) $\displaystyle=$ $\displaystyle
T_{0}\mathrm{pf}(\mathbb{S})-\sum_{i,j=1}^{2n}(-1)^{i+j+1}\alpha_{ij}\mathbb{T}_{ij}\mathrm{pf}(\mathbb{S}\\{i,j\\}),$
where $\alpha_{ij}=1$ for $i<j$ and $-1$ for $i>j$. Here and below, we denote
$\mathbb{S}\\{i,j,...\\}$ as a sub-matrix of $\mathbb{S}$ obtained by removing
the rows and columns of $i$,$j$,$\cdots$. The indexes $i,j,\cdots$ are
different from each other by definition. Thus we may set $\alpha_{ii}=0$, and
hope this does not confuse the readers.
If $\mathrm{pf}(\mathbb{S})\neq 0$, then $\mathbb{S}^{-1}$ exists.
pf$(\mathbb{S}\\{i,j,...\\})$ can be expressed with pf$(\mathbb{S})$ and some
matrix elements of $\mathbb{S}^{-1}$ through the Pfaffian version of Lewis
Carroll formula[25]. An alternative form of this formula has been given by
Mizusaki and Oi[14] in the study of HFB matrix elements. Some explicit
expressions for this formula are given in the Supplemental material. Here, we
use the one for $\mathrm{pf}(\mathbb{S}\\{i,j\\})$ (see Eq.(S54) in
Supplemental material) to get
$\displaystyle{I_{1}}=\left[T_{0}-\mathrm{Tr}(\mathbb{T}\mathbb{S}^{-1})\right]\mathrm{pf}(\mathbb{S}){\langle\Phi^{a}|\Phi^{b}\rangle},$
(34)
where Tr is the trace of a matrix.
If $\mathbb{S}^{-1}$ does not exist, Eq.(34) is invalid, but one can compact
Eq.(33) to
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\left\\{T_{0}\mathrm{pf}(\mathbb{S})-\sum_{i=1}^{2n}\mathrm{pf}(\bar{\mathbb{S}}^{i})\right\\}{\langle\Phi^{a}|\Phi^{b}\rangle},$
(35)
where the skew-symmetric matrices $\bar{\mathbb{S}}^{i}$ are the same as
$\mathbb{S}$ but the matrix elements in the $i$-th row and column
$\bar{\mathbb{S}}^{i}_{ij}=-\bar{\mathbb{S}}^{i}_{ji}=\mathbb{T}_{ij}$. [We
set $\mathbb{T}_{ii}=0$ due to $i\neq j$ in Eq.(33)].
Calculation of the matrix element involving two-body operator is more
complicated. Like the one-body operator $\hat{T}$, we define the following
notations associated with the two-body operator $\hat{V}$,
$\displaystyle V_{0}$ $\displaystyle=$
$\displaystyle\frac{\langle\Phi^{a}|\hat{V}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle}=\frac{1}{4}\sum_{\mu\nu\delta\gamma}V_{\mu\nu\gamma\delta}{\mathbb{C}_{\mu\nu\delta\gamma}},$
(36) $\displaystyle\mathbb{V}^{(1)}_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{\mu\nu\delta\gamma}V_{\mu\nu\gamma\delta}{\mathbb{D}^{ij}_{\mu\nu\delta\gamma}},$
(37) $\displaystyle\mathbb{V}^{(2)}_{ijkl}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{\mu\nu\delta\gamma}V_{\mu\nu\gamma\delta}{\mathbb{E}^{ijkl}_{\mu\nu\delta\gamma}},$
(38)
where
$\displaystyle\mathbb{C}_{\mu\nu\delta\gamma}$ $\displaystyle=$
$\displaystyle\mathbb{C}^{(+)}_{\mu\nu}\mathbb{C}^{(-)}_{\delta\gamma}-\mathbb{C}^{(0)}_{\mu\delta}\mathbb{C}^{(0)}_{\nu\gamma}+\mathbb{C}^{(0)}_{\mu\gamma}\mathbb{C}^{(0)}_{\nu\delta},$
(39) $\displaystyle\mathbb{D}^{ij}_{\mu\nu\delta\gamma}$ $\displaystyle=$
$\displaystyle\mathbb{C}^{(+)}_{\mu\nu}\mathbb{S}^{(-)}_{\delta
i}\mathbb{S}^{(-)}_{\gamma
j}-\mathbb{C}^{(0)}_{\mu\delta}\mathbb{S}^{(+)}_{\nu
i}\mathbb{S}^{(-)}_{\gamma j}$ (40) $\displaystyle+$
$\displaystyle\mathbb{C}^{(0)}_{\mu\gamma}\mathbb{S}^{(+)}_{\nu
i}\mathbb{S}^{(-)}_{\delta
j}+\mathbb{C}^{(0)}_{\nu\delta}\mathbb{S}^{(+)}_{\mu
i}\mathbb{S}^{(-)}_{\gamma j}$ $\displaystyle-$
$\displaystyle\mathbb{C}^{(0)}_{\nu\gamma}\mathbb{S}^{(+)}_{\mu
i}\mathbb{S}^{(-)}_{\delta
j}+\mathbb{C}^{(-)}_{\delta\gamma}\mathbb{S}^{(+)}_{\mu
i}\mathbb{S}^{(+)}_{\nu j},$
$\displaystyle\mathbb{E}^{ijkl}_{\mu\nu\delta\gamma}$ $\displaystyle=$
$\displaystyle\mathbb{S}^{(+)}_{\mu i}\mathbb{S}^{(+)}_{\nu
j}\mathbb{S}^{(-)}_{\delta k}\mathbb{S}^{(-)}_{\gamma l}.$ (41)
Similar to Eq.(33), one can use Pfaffian expansions (Eq.(S40) and Eq.(S52) in
Supplemental material) to obtain the following $I_{2}$ expression,
$\displaystyle\frac{I_{2}}{\langle\Phi^{a}|\Phi^{b}\rangle}=\frac{\langle\Phi^{a}|\hat{z}_{1}\cdots\hat{z}_{L}\hat{V}\hat{z}_{L+1}\cdots\hat{z}_{2n}|\Phi^{b}\rangle}{\langle\Phi^{a}|\Phi^{b}\rangle}$
(42) $\displaystyle=$ $\displaystyle
V_{0}\mathrm{pf}(\mathbb{S})+\sum_{i,j=1}^{2n}(-1)^{i+j}\alpha_{ij}\mathbb{V}^{(1)}_{ij}\mathrm{pf}(\mathbb{S}\\{i,j\\})$
$\displaystyle+$
$\displaystyle\sum_{i,j,k,l=1}^{2n}(-1)^{i+j+k+l}\alpha_{ijkl}\mathbb{V}^{(2)}_{ijkl}\mathrm{pf}(\mathbb{S}\\{i,j,k,l\\}),$
where
$\alpha_{ijkl}=\alpha_{ij}\alpha_{ik}\alpha_{il}\alpha_{jk}\alpha_{jl}\alpha_{kl}$.
Eq.(42) clearly shows the contraction role of the GWT.
In analogy to Eq.(34), if $\mathrm{pf}(\mathbb{S})\neq 0$, by replacing
$\mathrm{pf}(\mathbb{S}\\{i,j\\})$ and $\mathrm{pf}(\mathbb{S}\\{i,j,k,l\\})$
using the Pfaffian version of Lewis Carroll formula (Eq.(S54) and Eq.(S55) in
Supplemental material), one can simplify Eq.(42) as
$\displaystyle{I_{2}}={\langle\Phi^{a}|\Phi^{b}\rangle}\mathrm{pf}(\mathbb{S})[V_{0}-\mathrm{Tr}(\mathbb{V}^{(1)}\mathbb{S}^{-1})$
$\displaystyle+\sum_{i,j,k,l=1}^{2n}\mathbb{V}^{(2)}_{ijkl}(\mathbb{S}^{-1}_{ij}\mathbb{S}^{-1}_{kl}-\mathbb{S}^{-1}_{ik}\mathbb{S}^{-1}_{jl}+\mathbb{S}^{-1}_{il}\mathbb{S}^{-1}_{jk})].$
(43)
However, if $\mathrm{pf}(\mathbb{S})=0$, like Eq.(35), Eq.(42) can be
compacted to
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\langle\Phi^{a}|\Phi^{b}\rangle\left\\{V_{0}\mathrm{pf}(\mathbb{S})-\sum_{i=1}^{2n}\right.\mathrm{pf}(\tilde{\mathbb{S}}^{i})$
$\displaystyle\left.+\sum_{i,j=1}^{2n}(-1)^{i+j+1}\alpha_{ij}\sum_{k=1}^{2n}\mathrm{pf}(\tilde{\mathbb{S}}^{ijk}\\{i,j\\})\right\\},$
where $\tilde{\mathbb{S}}^{i}$ is the same as $\bar{\mathbb{S}}^{i}$ but
$\mathbb{T}$ is replaced by $\mathbb{V}^{(1)}$. $\tilde{\mathbb{S}}^{ijk}$ is
the same as $\mathbb{S}$ but the matrix elements in the $k$-th row and $k$-th
column
$\tilde{\mathbb{S}}^{ijk}_{kl}=-\tilde{\mathbb{S}}^{ijk}_{lk}=\mathbb{V}^{(2)}_{ijkl}$.
All the above formulae are based on the assumption
$\langle\Phi^{a}|\Phi^{b}\rangle\neq 0$. However, the case of
$\langle\Phi^{a}|\Phi^{b}\rangle=0$ that leads to the well known Egido pole
[26] should be carefully studied. In this situation, Eq.(5) is invalid and
Eq.(4) should be used. By inserting Eq.(7) into Eq.(19), and regarding all
$\hat{\beta}^{b}$ and $\hat{\beta}^{a\dagger}$ as $\hat{z}$, one can rewrite
$I$ as
$\displaystyle
I={\mathcal{N}_{a}\mathcal{N}_{b}}\langle-|\hat{z}_{1}\cdots\hat{z}_{L^{\prime}}\hat{O}\hat{z}_{L^{\prime}+1}\cdots\hat{z}_{2n^{\prime}}|-\rangle,$
(45)
which is similar to Eq.(19), but $L^{\prime}=L+N_{a}$ and
$2n^{\prime}=2n+N_{a}+N_{b}$. Although $I$ can be directly calculated with
Eq.(4) or the formulae in Ref.[16]. However, one can also derive corresponding
compact forms in this situation. Replacing $|\Phi^{a}\rangle$ and
$|\Phi^{b}\rangle$ with $|-\rangle$, it is seen all the above derived formulae
from Eq.(26) to Eq.(3) are valid because $\langle-|-\rangle=1$. But, the
matrix $\mathbb{S}$ becomes $S$, whose shape is
$(2n+N_{a}+N_{b})\times(2n+N_{a}+N_{b})$, and much larger than the $(2n\times
2n)$ dimension of $\mathbb{S}$. Thus more computing time is required in this
case.
## 4 Discussions
Numerical calculations have been performed to test the validity of new
formulae. The matrix elements of $\mathbb{S}$, $\mathbb{S}^{(\pm)}$ and
$\mathbb{C}^{(\pm,0)}$ are required and should be evaluated with one of Eqs.
(16-18). Here, these matrix elements, together with $T_{\mu\nu}$ and
$V_{\mu\nu\gamma\delta}$, are chosen as complex random numbers. The results
show that the values of $I_{1}$ with Eqs. (34), and (35) are indeed identical
to that with the conventional method. Similarly, the same values of $I_{2}$
with (3), (3) and the conventional method are also confirmed (we present the
testing FORTRAN code for $I_{2}$ in the Supplemental material).
Figure 1: (color online) (a), CPU time, $t_{1}$, for the conventional method,
as a function of $M$ and $2n$; (b), CPU time, $t_{2}$, for Eq.(3), as a
function of $M$ and $2n$, (c), Ratio of $t_{1}$ to $t_{2}$; (d) Total CPU
time, $t_{V}$, for $V_{0}$, $\mathbb{V}^{(1)}$ and $\mathbb{V}^{(2)}$, $N$ is
the dimension of $\mathbb{V}^{(1)}$ and $\mathbb{V}^{(2)}$ with $1\leq
i,j,k,l\leq N$.
The efficiency of the most important Eq. (3) is studied and the results are
shown in Fig.1. Assuming $V_{0}$, $\mathbb{V}^{(1)}$ and $\mathbb{V}^{(2)}$
are available, the computational cost of Eq.(3) is $O((2n)^{4})$, which is
independent of $M$. This implies Eq.(3) can be very conveniently extended to
large model spaces. In contrast, the conventional method requires a time
$O(M^{4}(2n)^{3})$ which highly depends on the model space due to the four-
fold summation in Eq.(2). Testing calculations have been carried out on a
Intel CPU with 2.4GHz. The elapsed time (in second), $t_{1}$ for the
conventional method and $t_{2}$ for Eq. (3), are shown in Fig.1(a) and (b),
respectively. To obtain the reliable $t_{1}(t_{2})$ value, identical
calculations are repeated for many times (denoted by $m$, ranging from 10 to
$10^{6}$) until the total elapsed time, $T$, is long enough, then
$t_{1}(t_{2})=T/m$. From Fig.1(c), the ratio $t_{1}/t_{2}$ can be easily above
the order of $10^{6}$ for $M=80$. Here, we chose $2n$ up to 12 because in the
practical calculations, it seems enough to include up to 6-quasiparticle
states.
However, the elapsed time, $t_{V}$, for $V_{0}$, $\mathbb{V}^{(1)}$ and
$\mathbb{V}^{(2)}$ strongly depends on $M$. Moreover, $t_{V}$ is not included
in $t_{2}$ and should be separately considered. Fortunately, all the $I_{2}$
matrix elements on top of the same ($\langle\Phi^{a}|$, $|\Phi^{b}\rangle$)
pair share the common $V_{0}$, $\mathbb{V}^{(1)}$ and $\mathbb{V}^{(2)}$. Thus
they are evaluated just one time for given HFB vacua, $|\Phi^{a}\rangle$ and
$|\Phi^{b}\rangle$. Notice that the computational cost of $\mathbb{V}^{(1)}$
and $\mathbb{V}^{(2)}$ also depends on their dimension, $N$, with $1\leq
i,j,k,l\leq N$. To cover all the $I_{2}$ matrix elements, $N$ should be
properly chosen in the range of $2n\leq N\leq 2M$. Most of $t_{V}$ is taken by
$\mathbb{V}^{(2)}$, whose computational cost is $O(M^{4}N)$. The $t_{V}$
values for various $M,N$ are shown in Fig.1(d). Comparing with $t_{1}$, it
looks that $t_{V}\approx 0.1t_{1}$ at large $M$. Let us denote by $M_{I}$ the
dimension of the $I_{2}$ matrix, and the global efficiency of Eq.(3) relative
to the conventional method can be evaluated through
$r=\frac{M_{I}^{2}t_{1}}{t_{V}+M_{I}^{2}t_{2}}$. Suppose $M_{I}=100,M=80$, $r$
can be easily in the order of $10^{5}$.
In Fig.1(d), the CPU time, $t_{V}$ is within several seconds for $M\leq 80$,
calculations may be implemented when one directly uses Eq.(2), as is also
taken in the standard $M-$scheme shell model methods. However, $t_{V}$ can
drastically increase with $M$ bigger and bigger. Therefore, for heavy nuclei,
one has to seek a more concise form of two-body interaction, such as separable
interactions [31, 32], instead of directly using Eq.(2). For instance, the
Projected Shell Model (PSM) uses the quadruple plus pairing interaction. The
present method may be conveniently applied to develop the PSM, so that it may
includes the states with more quasiparticles(e.g., 6-q.p., 8-q.p., etc).
## 5 Summary
In this letter, we focused on the matrix elements of one-body and two-body
physical operators between arbitrary HFB states. The formula of Eq.(4), used
by Bertsch and Robledo [16], has been extended to evaluate the matrix element
of a product of single-fermion operators between two arbitrary HFB vacua [see
Eq.(5)]. Start from Eq.(5), the matrix elements of physical operators have
been successfully transformed into compact forms. Formulae for the
pf$(\mathbb{S})=0$ case have also been given. Besides, the case of the Egido
pole with $\langle\Phi^{a}|\Phi^{b}\rangle=0$ has been discussed. Testing
calculations for the two-body operator matrix elements show that the new
formulae can easily be in several orders faster than the conventional method.
Thus those hopeless beyond mean field calculations for heavy nuclei in a large
Fock space may be implemented by using the present method.
Acknowledgements Z.G. thanks Prof. Y. Sun and Dr. F.Q. Chen for the fruitful
discussions and the manuscript. The work is supported by the National Natural
Science Foundation of China under Contract Nos. 11175258, 11021504 and
11275068.
## Appendix A Supplemental material
Supplementary material for mathematical details and the testing code can be
found online at http://dx.doi.org/10.1016/j.physletb.2014.05.045.
## References
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* [27] K. Hara and Y. Sun, Int. J. Mod. Phys. E 04 (1995) 637\.
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* [29] L. M. Robledo, Phys Rev C 50, (1994)2874.
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|
arxiv-papers
| 2013-07-26T01:45:52 |
2024-09-04T02:49:48.497155
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qing-Li Hu, Zao-Chun Gao and Y. S. Chen",
"submitter": "Zao-Chun Gao",
"url": "https://arxiv.org/abs/1307.6905"
}
|
1307.7018
|
LHCP 2013
11institutetext: Technische Universität Dortmund, Experimentelle Physik V,
44227 Dortmund, Germany
# Measurement of $\boldsymbol{\gamma}$ from $\boldsymbol{B\rightarrow DK}$
decays at LHCb
Maximilian Schlupp on behalf of the LHCb Collaboration 11
[email protected]
###### Abstract
We report results from the first measurements of the CKM angle $\gamma$ using
$B\\!\rightarrow DK$ decays with the LHCb experiment. Three well established
methods are used to extract the $C\\!P$ observables. The updated measurement
of $\gamma$ in the three-body $D^{0}$ Dalitz space results in $\gamma=(57\pm
16)^{\circ}$. When combining the observables from all $B\\!\rightarrow DK$
studies, the best fit value for $\gamma\in[0,180]^{\circ}$ is
$\gamma=67.2^{\circ}$ with $\gamma\in[55.1,79.1]^{\circ}$ at 68%CL and
$\gamma\in[43.9,89.5]^{\circ}$ at 95%CL. This represents the most precise
$\gamma$ values directly measured by a single experiment.
Furthermore, a new time-dependent approach using $B^{0}_{s}\\!\rightarrow
D^{\pm}_{s}K^{\mp}$ decays is used for the first time to measure $C\\!P$
observables and future prospects for $\gamma$ at LHCb are given.
## 1 Introduction
The CKM parameter
$\gamma=\text{arg}(-V_{ud}V_{ub}^{\ast}/V_{cd}V_{cb}^{\ast})$ is the least
well measured angle of the Unitarity Triangle. So far, the best measurements
from single experiments have been performed by the $B$-factories BaBar and
Belle. The latest results from both experiments are
$\gamma=(69^{+17}_{-16})^{\circ}$ Lees:2013zd and
$\gamma=(68^{+15}_{-14})^{\circ}$ Trabelsi:2013uj , respectively.
One of the core physics goals of the LHCb experiment is to precisely measure
the CKM angle $\gamma$. This can be done by exploiting tree-level processes
like $B^{\pm}\\!\rightarrow DK^{\pm}$ or $B^{0}_{s}\\!\rightarrow
D^{\pm}_{s}K^{\mp}$, which are sensitive to Standard Model (SM) interactions
only. In contrast, it is also possible to extract $\gamma$ from loop processes
such as two or three-body charmless $B$ transitions. Potential differences in
these results could indicate new physics contributions. Comparing direct
measurements to indirect SM fits could also indicate tensions within the SM.
Examples of two different approaches to measure $\gamma$ are described in
these proceedings. First the more traditional time-independent measurements
already performed by the $B$-factories in section 2 and then a new, LHCb
exclusive, time-dependent way in section 3.
## 2 Time-Independent measurements using charged $\boldsymbol{B}$ decays
Measuring $\gamma$ with charged $b$-hadron decays one considers the
interference from $b\\!\rightarrow u$ and $b\\!\rightarrow c$ transitions in
$B\\!\rightarrow Dh$. Here, $D$ is either a $D^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $h$ is a $K^{\pm}$ or
$\pi^{\pm}$.
The interference is ensured by reconstructing the $D$ meson in a final state
common to $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$, so
that the two decay paths $B^{+}\\!\rightarrow DK^{+}$ and
$B^{+}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}K^{+}$ are
indistinguishable 111Charge-conjugation is implied throughout the document, if
not stated otherwise.. The sensitivity on $\gamma$ is roughly given by the
ratio of the suppressed over the favoured $B$ decay amplitude, $r_{B}$. The
interference additionally is dependent on the relative strong phase difference
$\delta_{B}$ of the two $B$ amplitudes.
There are three established methods to extract $\gamma$ from these types of
processes, which depend on the $D$ final state: the ADS method Atwood:1996ci
using quasi flavour-specific, doubly Cabbibo suppressed states (e.g.
$D\\!\rightarrow K^{+}\pi^{-}$ or $D\\!\rightarrow
K^{+}\pi^{-}\pi^{+}\pi^{-}$). The $D$ final states are chosen so that the
decay suppressions ($r_{B}$ and the $D$ system equivalent $r_{D}$) are similar
between the two interfering $B$ amplitudes. The $C\\!P$ asymmetries are
therefore expected to be large. However, the interference acquires an
additional dependence on the strong phase difference in the $D$ meson system,
$\delta_{D}$.
The GLW method Gronau:1990ra ; Gronau:1991dp on the other hand, makes use of
the $D$ meson decaying into a $C\\!P$ eigenstate, where one can eliminate the
$D$ system parameters.
In the GGSZ method Giri:2003ty three-body self-conjugate $D$ final states are
studied (e.g. $D\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$
or $D\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$). Performing a
Dalitz plot analysis of the $D$ meson decays leads to a good sensitivity on
$\gamma$.
LHCb results from the three methods are presented in the following sections.
Additionally, a combination of the various observables from the different $B$
decay modes is shown in section 2.3, which increases the sensitivity on
$\gamma$ beyond the single measurements.
### 2.1 ADS/GLW
The LHCb collaboration has performed analyses in $B^{+}\\!\rightarrow DK^{+}$
and $B^{+}\\!\rightarrow D\pi^{+}$, where the $D$ meson is reconstructed in
$K^{\pm}$ $\pi^{\mp}$, $K^{+}$ $K^{-}$, $\pi^{+}$ $\pi^{-}$, $\pi^{\pm}$
$K^{\mp}$, and $\pi^{\pm}$ $K^{\mp}$ $\pi^{+}$ $\pi^{-}$ Aaij:2012kz ;
Aaij:2013mba with a dataset corresponding to an integrated luminosity of
$1\,\mbox{\,fb}^{-1}$ at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The ADS
doubly Cabbibo suppressed modes in $B\rightarrow(\pi K)_{D}K$,
$B\rightarrow(\pi K\pi\pi)_{D}K$ and $B\rightarrow(\pi K\pi\pi)_{D}\pi$ are
observed for the first time with a significance of $>\\!\\!10\sigma$,
$5.1\sigma$ and $>\\!\\!10\sigma$, respectively. Here $(f)_{D}$ is the
abbreviated form for a $D$ meson decaying into the final state $f$,
$D\rightarrow f$. The respective invariant mass distributions are shown in
Figure 1 and 2.
Figure 1: Invariant mass distribution of the two-body ADS suppressed modes in
$B\rightarrow(\pi K)_{D}K$ (top) and $B\rightarrow(\pi K)_{D}\pi$ (bottom).
Figure 2: Invariant mass distribution of the four-body ADS suppressed modes in
$B\rightarrow(\pi K\pi\pi)_{D}K$ (top) and $B\rightarrow(\pi K\pi\pi)_{D}\pi$
(bottom).
Using the ADS and GLW methods the following $C\\!P$ observables sensitive to
$\gamma$, $r_{B}$, $\delta_{B}$, $r_{D}$ and $\delta_{D}$ can be measured: the
charge-averaged ratios of $B\\!\rightarrow DK$ and $B\\!\rightarrow D\pi$
$\displaystyle R^{f}_{K/\pi}=\frac{\Gamma(B^{-}\\!\rightarrow
DK^{-})+\Gamma(B^{+}\\!\rightarrow DK^{+})}{\Gamma(B^{-}\\!\rightarrow
D\pi^{-})+\Gamma(B^{+}\\!\rightarrow D\pi^{+})}\quad,$
where $f$ indicates the $D$ final state, the charge asymmetries
$\displaystyle A^{f}_{h}=\frac{\Gamma(B^{-}\\!\rightarrow
Dh^{-})-\Gamma(B^{+}\\!\rightarrow Dh^{+})}{\Gamma(B^{-}\\!\rightarrow
Dh^{-})+\Gamma(B^{+}\\!\rightarrow Dh^{+})}\quad,$
and the non charge-averaged ratio of suppressed and favoured $D$ final state
$\displaystyle R^{\pm}_{h}=\frac{\Gamma(B^{\pm}\\!\rightarrow
Dh^{\pm})_{\text{sup}}}{\Gamma(B^{\pm}\\!\rightarrow Dh^{\pm})}\quad.$
The resulting values can be found in the refs. Aaij:2012kz ; Aaij:2013mba and
serve as inputs for the combined $\gamma$ measurement in section 2.3.
Furthermore, direct $C\\!P$ violation in $B^{\pm}\\!\rightarrow DK^{\pm}$ is
observed with a total significance of $5.8\sigma$.
### 2.2 GGSZ
The GGSZ method exploits the three-body $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ Dalitz space in
$B^{\pm}\\!\rightarrow DK^{\pm}$ decays to extract the $C\\!P$ observables
$x_{\pm}=r_{B}\cos(\delta_{B}\pm\gamma)$ and
$y_{\pm}=r_{B}\sin(\delta_{B}\pm\gamma)$. Due to the rich resonance structure
of the $D$ decays, this method has proven to be most sensitive one at the
$B$-factories. We report the model-independent measurement using a dataset
corresponding to $2\,\mbox{\,fb}^{-1}$ of integrated luminosity with a centre
of mass energy of $\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb
Collaboration LHCb-CONF-2013-004 , which is the successor of the
$1\,\mbox{\,fb}^{-1}$ publication Aaij:2012hu at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. The variation of the strong phase
difference $\delta_{D}$ in bins of the $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{-}$ Dalitz plot is taken as an external
input from the CLEO collaboration. The resulting numbers for the $C\\!P$
violation parameters $x_{\pm}$ and $y_{\pm}$ are illustrated in Figure 3 for
$2\,\mbox{\,fb}^{-1}$, where the combined $3\,\mbox{\,fb}^{-1}$ values are:
$\displaystyle\langle x_{+}\rangle$ $\displaystyle=(-8.9\pm 3.1)\times
10^{-2}{,}\ \ \langle x_{-}\rangle=(3.5\pm 2.9)\times 10^{-2}$
$\displaystyle\langle y_{+}\rangle$ $\displaystyle=(0.1\pm 3.7)\times
10^{-2}{,}\ \ \ \ \,\langle y_{-}\rangle=(7.9\pm 3.8)\times 10^{-2}.$
Figure 3: Best fit values (stars) and $1\sigma$, $2\sigma$ and $3\sigma$
confidence intervals (contours) in the ($x$,$y$) plane using the statistical
uncertainties and correlations only.
The dominant systematic uncertainties are coming from the assumption of no
interference in the control channel and the external hadronic input
parameters. However, the results are limited statistically. The underlying
physics parameters are extracted using a frequentist approach resulting in
$\gamma=(57\pm 16)^{\circ}$, $r_{B}=(8.8^{+2.3}_{-2.4})\times 10^{-2}$ and
$\delta_{B}=(124^{+15}_{-17})^{\circ}$. This results competes with the
methodically equivalent Belle measurement Aihara:2012aw of
$\gamma=(77.4^{+15.1}_{-14.9}\pm 4.1\pm 4.3)^{\circ}$ for the current world’s
most precise single direct measurement of $\gamma$.
### 2.3 Combination
To reach the best possible sensitivity on $\gamma$ the observables from the
ADS, GLW and GGSZ analyses, the amplitudes and ratios from section 2.1 and the
combined $3\,\mbox{\,fb}^{-1}$ $C\\!P$ observables from section 2.2, are
evaluated at the same time for the $B\\!\rightarrow DK$ transitions.
Additionally, inputs from the CLEO collaboration Lowery:2009id and the Heavy
Flavour Averaging Group (HFAG) Amhis:2012bh have been used to constrain the
hadronic parameters of the $D$ system and the effect of direct $C\\!P$
violation in $D$ decays, respectively. A likelihood is constructed from the
input measurements as follows:
$\displaystyle\mathcal{L}(\vec{\alpha})=\prod_{i}{\xi_{i}(\vec{A}_{i}^{\text{obs}}|\vec{\alpha})}\quad,$
where $i$ denotes the different measurements, $\vec{A}_{i}^{\text{obs}}$ the
observables, $\xi_{i}$ the probability density functions (PDFs) of the
observables $\vec{A}_{i}$ and $\vec{\alpha}$ is the set of parameters
($\gamma$, $r_{B}$, etc.). For most of the PDFs $\xi_{i}$ a multidimensional
Gaussian is assumed taking correlations into account. Whenever highly non-
Gaussian behaviour is present, $\xi_{i}$ is replaced by the experimental
likelihood.
The confidence intervals are calculated using a frequentist method. Its
coverage is not guaranteed from first principles, so the coverage is tested.
It is found that the coverage is almost correct so that the results are scaled
according to the small differences. Additionally, the confidence intervals are
cross-checked and found to be consistent with a method inspired by Berger and
Boos BergerBoos . In this method the values of the nuisance parameters are
sampled from a uniform distribution covering a multidimensional confidence
belt $C_{\beta}$, instead of fixing the nuisance parameters to their best-fit
values. $C_{\beta}$ is chosen such that the corresponding corrections to the
p-value are negligible. For more details on the inputs, the statistical
procedures and the validation of the results, see Aaij:2013zfa ; LHCb-
CONF-2013-006 ; BergerBoos . The best fit values and confidence intervals for
$\gamma$, $r_{B}$ and $\delta_{B}$ are listed in Table 1, all values are
modulo $180^{\circ}$.
Table 1: Best-fit values and confidence intervals for $\gamma$, $r_{B}$ and $\delta_{B}$ from the combination of the $B\\!\rightarrow DK$ measurements. quantity | $D$ $K$ combination
---|---
$\gamma$ | $67.2^{\circ}$
68% CL | $[55.1,79.1]^{\circ}$
95% CL | $[43.9,89.5]^{\circ}$
$r_{B}$ | $114.3^{\circ}$
68% CL | $[101.3,126.3]^{\circ}$
95% CL | $[88.7,136.3]^{\circ}$
$\delta_{B}$ | $0.0923$
68% CL | $[0.0843,0.1001]$
95% CL | $[0.0762,0.1075]$
The $1-\text{CL}$ curve for $\gamma$ and the two-dimensional likelihood
projection for $\gamma$ and $r_{B}$ are shown in Figure 4 and 5, respectively.
The 68% CL interval for $\gamma$ can be translated to $\gamma=(67\pm
12)^{\circ}$.
Figure 4: $1-\text{CL}$ curve for $\gamma$ from the combined ADS/GLW
$1\,\mbox{\,fb}^{-1}$ and GGSZ $3\,\mbox{\,fb}^{-1}$ measurements. The
$1\sigma$ and $2\sigma$ confidence interval can be read off at the
intersections of the blue curve with the dotted lines labelled $68.3\,\%$ and
$95.5\,\%$, respectively. Figure 5: Best-fit values (markers) and contours
where the difference in log-likelihood corresponds to $1\sigma$ and $2\sigma$.
The $3\,\mbox{\,fb}^{-1}$ GGSZ and $1\,\mbox{\,fb}^{-1}$ ADS/GLW analyses are
shown separately in blue and orange.
This preliminary result has a lower uncertainty compared to the latest results
from BaBar Lees:2013zd and Belle Trabelsi:2013uj .
## 3 Time-dependent measurement in
$\boldsymbol{B^{0}_{s}}\\!\rightarrow\boldsymbol{D^{\pm}_{s}K^{\mp}}$ decays
A different approach to extract $\gamma$ is to use neutral $B$ mesons and
perform a time-dependent measurement of the $C\\!P$ parameters. This can be
done using tree-level $B^{0}_{s}\\!\rightarrow D^{\pm}_{s}K^{\mp}$ decays. The
sensitivity to $\gamma$ arises from the interference of both $B$ mesons,
$B^{0}_{s}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$,
decaying into the same final state: $D^{+}_{s}$ $K^{-}$ or $D^{-}_{s}$
$K^{+}$. Note that the $D_{s}$ final states are not of major importance in
this method. Each decay amplitude is roughly of the same order of magnitude,
thus the expected interference is large $r_{B}^{D_{s}K}=0.37$.
In order to resolve the $B^{0}_{s}$ oscillations, a good time resolution is
mandatory. For the analysis of $B^{0}_{s}\\!\rightarrow D^{\pm}_{s}K^{\mp}$
decays at LHCb LHCb-CONF-2012-029 it is determined from Monte Carlo (MC)
simulations. The difference of the reconstructed and the true decay time is
fitted with a resolution model, which is the sum of three Gaussians. To
account for differences in data and simulations we scale the Gaussian’s widths
according to $B^{0}_{s}$ $\rightarrow$ $D_{s}$ $\pi$ MC and a data sample of
"fake" $B^{0}_{s}$ constructed from prompt $D_{s}$ mesons which are combined
with a random $\pi$. We assume that the differences between
$B^{0}_{s}\\!\rightarrow D^{\pm}_{s}K^{\mp}$ and the control channel
$B^{0}_{s}$ $\rightarrow$ $D_{s}$ $\pi$ are negligible for the relevant
quantities. The resulting effective time resolution is estimated as
$\sigma_{t}\approx 50\rm\,fs$. Another crucial part is the determination of
the time acceptance, which is also obtained from MC. The invariant mass
distribution of the $B^{0}_{s}$ candidates is fitted using an unbinned maximum
likelihood method in order to get weights, which separate signal from
background components. The full mass-fit is shown in Figure 6.
Figure 6: Invariant mass distribution of $B^{0}_{s}$ candidates together with
the signal and background components and the full fit. Below the corresponding
pulls are shown.
The weighted decay time distribution is then fitted using the _sFit_ sFit
technique, where the fit determines the corresponding $C\\!P$ observables. The
resulting values can be found in LHCb-CONF-2012-029 and the decay time fit is
shown in Figure 7.
Figure 7: Fit to the weighted decay time distribution, showing all fit
components separately.
The weighing procedure is cross-checked with a conventional 2-dimensional fit
in the invariant mass and decay time.
It is found that correlations within the systematics have a non-negligible
effect on extracting the actual $C\\!P$ parameters $\gamma+\beta_{s}$, where
$\beta_{s}$ is the $B^{0}_{s}$ mixing phase. Measuring the $C\\!P$ parameters
marks the first important step towards a time-dependent estimation of $\gamma$
from $B^{0}_{s}\\!\rightarrow D^{\pm}_{s}K^{\mp}$ decays.
## 4 Conclusions and prospects
We reported several measurements of $\gamma$ with the LHCb experiment. Up to
now, the GGSZ analysis is the most sensitive single measurement of
$\gamma=(57\pm 16)^{\circ}$ using the full combined $3\,\mbox{\,fb}^{-1}$ LHCb
dataset. Exploiting the ADS/GLW method on $1\,\mbox{\,fb}^{-1}$ of LHCb data
in $B\\!\rightarrow Dh$ with two- and four-body $D$ decays leads to the
observations of the corresponding suppressed ADS modes with significances
greater than $5\sigma$. Furthermore, $C\\!P$ observables are provided by the
analyses from which $\gamma$ can be extracted.
Combining all $C\\!P$ observables from the $B\\!\rightarrow DK$ measurements
the resulting LHCb result is $\gamma=(67\pm 12)^{\circ}$, which is more
precise than recent BaBar Lees:2013zd and Belle Trabelsi:2013uj results.
Further improvements are expected with the analyses updated to the full
available dataset. When more channels, which were not discussed throughout
these proceedings are analysed with the current or with a future dataset, the
sensitivity on $\gamma$ will increase by including these to the combined
measurement. Then LHCb will be able to compare $\gamma$ estimations from tree-
level and loop-level processes.
In the future we expect to decrease the uncertainty on $\gamma$ to
$\delta\gamma\sim\mathcal{O}(1^{\circ})$ Bediaga:2012py using a dataset of
$50\,\mbox{\,fb}^{-1}$ and combining different decay channels. This dataset is
planed to be recorded within the coming decade. I would like to thank the
organisers of the LHCP 2013 for the possibility to participate in this
excellent conference.
This work is financed by the german Federal Ministry of Education and Research
(BMBF).
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|
arxiv-papers
| 2013-07-26T12:44:46 |
2024-09-04T02:49:48.513818
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maximilian Schlupp",
"submitter": "Maximilian Schlupp",
"url": "https://arxiv.org/abs/1307.7018"
}
|
1307.7025
|
The -calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the -calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the -calculus is complete for pure qubit stabilizer quantum mechanics, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations.
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|
arxiv-papers
| 2013-07-26T13:11:11 |
2024-09-04T02:49:48.520651
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miriam Backens",
"submitter": "Miriam Backens",
"url": "https://arxiv.org/abs/1307.7025"
}
|
1307.7176
|
# Phase retrieval from very few measurements
Matthew Fickus Dustin G. Mixon [email protected] Aaron A. Nelson Yang
Wang Department of Mathematics and Statistics, Air Force Institute of
Technology, Wright-Patterson AFB, OH 45433, USA Department of Mathematics,
Michigan State University, East Lansing, MI 48824, USA
###### Abstract
In many applications, signals are measured according to a linear process, but
the phases of these measurements are often unreliable or not available. To
reconstruct the signal, one must perform a process known as phase retrieval.
This paper focuses on completely determining signals with as few intensity
measurements as possible, and on efficient phase retrieval algorithms from
such measurements. For the case of complex $M$-dimensional signals, we
construct a measurement ensemble of size $4M-4$ which yields injective
intensity measurements; this is conjectured to be the smallest such ensemble.
For the case of real signals, we devise a theory of “almost” injective
intensity measurements, and we characterize such ensembles. Later, we show
that phase retrieval from $M+1$ almost injective intensity measurements is
$\NP$-hard, indicating that computationally efficient phase retrieval must
come at the price of measurement redundancy.
###### keywords:
phase retrieval , informationally complete , unit norm tight frames ,
computational complexity
††journal: Linear Algebra and its Applications
## 1 Introduction
Given an ensemble $\Phi=\\{\varphi_{n}\\}_{n=1}^{N}$ of $M$-dimensional
vectors (real or complex), the phase retrieval problem is to recover a signal
$x$ from intensity measurements $\mathcal{A}(x):=\\{|\langle
x,\varphi_{n}\rangle|^{2}\\}_{n=1}^{N}$. Note that for any scalar $\omega$ of
unit modulus, $\mathcal{A}(\omega x)=\mathcal{A}(x)$, and so the best one can
hope to do is recover $x$ up to a global phase factor $\\{\omega
x:|\omega|=1\\}$. Intensity measurements arise in a number of applications in
which phase is either unreliable or not available [9, 19, 27, 31, 32, 38], and
in most of these applications, it is desirable to perform phase retrieval from
as few measurements as possible; indeed, increasing $N$ invariably makes the
measurement process more expensive or time consuming.
Recently, there has been a lot of work on algorithmic phase retrieval. For
example, phase retrieval can be formulated as a low-rank (actually, rank-1)
matrix recovery problem [11, 12, 13, 17, 21, 36], and with this formulation,
phase retrieval is possible from $N=O(M)$ intensity measurements [12]. Another
approach is to exploit the polarization identity along with expander graphs to
design a measurement ensemble and apply spectral methods to perform phase
retrieval [1, 5]. One can also formulate phase retrieval in terms of MaxCut,
and solvers for this formulation are equivalent to a popular solver
(PhaseLift) for the matrix recovery formulation [35, 37]. While this recent
work has focused on stable and efficient phase retrieval from asymptotically
few measurements (namely, $N=O(M)$), the present paper focuses on injectivity
and algorithmic efficiency with the absolute minimum number of measurements.
In the next section, we construct an ensemble of $N=4M-4$ measurement vectors
in $\mathbb{C}^{M}$ which yield injective intensity measurements. This is the
second known injective ensemble of this size (the first is due to Bodmann and
Hammen [8]), and it is conjectured to be the smallest-possible injective
ensemble [4]. The same conjecture suggests that $4M-4$ generic measurement
vectors yield injectivity (that is, there exists a measure-zero set of
ensembles of $4M-4$ vectors such that every ensemble of $4M-4$ vectors outside
of this set yields injectivity). The following summarizes what is currently
known about the so-called “$4M-4$ conjecture”:
* 1.
The conjecture holds for $M=2,3$ [4].
* 2.
If $N<4M-2\alpha(M-1)-3$, then $\mathcal{A}$ is not injective [28]; here,
$\alpha(M-1)\leq\log_{2}M$ denotes the number of $1$’s in the binary expansion
of $M-1$.
* 3.
For each $M\geq 2$, there exists an ensemble $\Phi$ of $N=4M-4$ measurement
vectors such that $\mathcal{A}$ is injective [8] (see also Section 2 of this
paper).
* 4.
If $N\geq 4M-2$, then $\mathcal{A}$ is injective for generic $\Phi$ [3].
Bodmann and Hammen [8] leverage the Dirichlet kernel and the Cayley map to
prove injectivity of their ensemble, but it is unclear whether phase retrieval
is algorithmically feasible from their ensemble. By contrast, for the ensemble
in this paper, we use basic ideas from harmonic analysis over cyclic groups to
devise a corresponding phase retrieval algorithm, and we demonstrate
injectivity by proving that the algorithm succeeds.
In Section 3, we devise a theory of ensembles for which the corresponding
intensity measurements are “almost” injective, that is,
$\mathcal{A}^{-1}(\mathcal{A}(x))=\\{\omega x:|\omega|=1\\}$ for almost every
$x$. In this section, we focus on the real case, meaning phase retrieval is up
to a global sign factor $\omega=\pm 1$, and our approach is inspired by the
characterization of injectivity in the real case by Balan, Casazza and Edidin
[3]. After characterizing almost injectivity in the real case, we find a
particularly satisfying sufficient condition for almost injectivity: that
$\Phi$ forms a unit norm tight frame with $M$ and $N$ relatively prime.
Characterizing almost injectivity in the complex case remains an open problem.
We conclude with Section 4, in which we consider algorithmic phase retrieval
in the real case from $N=M+1$ almost injective intensity measurements.
Specifically, we show that phase retrieval in this case is $\NP$-hard by
reduction from the subset sum problem. The hardness of phase retrieval in this
minimal case suggests a new problem for phase retrieval: What is the smallest
$C$ for which there exists a family of ensembles of size $N=CM+o(M)$ such that
phase retrieval can be performed in polynomial time?
## 2 $4M-4$ injective intensity measurements
In this section, we provide an ensemble of $4M-4$ measurement vectors which
yield injective intensity measurements for $\mathbb{C}^{M}$. The vectors in
our ensemble are modulated discrete cosine functions, and they are explicitly
constructed at the end of this section. We start here by motivating our
construction, specifically by identifying the significance of circular
autocorrelation.
Consider the $P$-dimensional complex vector space
$\ell(\mathbb{Z}_{P}):=\\{u\colon\mathbb{Z}\rightarrow\mathbb{C}:u[p+P]=u[p],~{}\forall
p\in\mathbb{Z}\\}$. The discrete Fourier basis in $\ell(\mathbb{Z}_{P})$ is
the sequence of $P$ vectors $\\{f_{q}\\}_{q\in\mathbb{Z}_{P}}$ defined by
$f_{q}[p]:=e^{2\pi ipq/P}$ (the notation “$q\in\mathbb{Z}_{P}$” is taken to
mean a set of coset representatives of $\mathbb{Z}$ with respect to the
subgroup $P\mathbb{Z}$). The discrete Fourier transform (DFT) on
$\mathbb{Z}_{P}$ is the analysis operator
$F^{*}\colon\ell(\mathbb{Z}_{P})\rightarrow\ell(\mathbb{Z}_{P})$ of this
basis, with corresponding inverse DFT $(F^{*})^{-1}=\frac{1}{P}F$, where
$(F^{*}u)[q]=\langle u,f_{q}\rangle=\sum_{p\in\mathbb{Z}_{P}}u[p]e^{-2\pi
ipq/P},\qquad(Fv)[p]=\sum_{q\in\mathbb{Z}_{P}}v[q]f_{q}[p]=\sum_{q\in\mathbb{Z}_{P}}v[q]e^{2\pi
ipq/P}.$
Now let $T^{p}\colon\ell(\mathbb{Z}_{P})\rightarrow\ell(\mathbb{Z}_{P})$ be
the translation operator defined by $(T^{p}u)[p^{\prime}]:=u[p^{\prime}-p]$.
The circular autocorrelation of $u$ is then
$\operatorname{CirAut}(u)\in\ell(\mathbb{Z}_{P})$, defined entrywise by
$\operatorname{CirAut}(u)[p]:=\langle
u,T^{p}u\rangle=\sum_{p^{\prime}\in\mathbb{Z}_{P}}u[p^{\prime}]\overline{u[p^{\prime}-p]}.$
(1)
Consider the DFT of a circular autocorrelation:
$\displaystyle(F^{*}\operatorname{CirAut}(u))[q]$
$\displaystyle=\sum_{p\in\mathbb{Z}_{P}}\sum_{p^{\prime}\in\mathbb{Z}_{P}}u[p^{\prime}]\overline{u[p^{\prime}-p]}e^{-2\pi
ipq/P}$ $\displaystyle=\sum_{p^{\prime}\in\mathbb{Z}_{P}}u[p^{\prime}]e^{-2\pi
ip^{\prime}q/P}\overline{\bigg{(}\sum_{p\in\mathbb{Z}_{P}}u[p^{\prime}-p]e^{-2\pi
i(p^{\prime}-p)q/P}\bigg{)}}$
$\displaystyle=\sum_{p^{\prime}\in\mathbb{Z}_{P}}u[p^{\prime}]e^{-2\pi
ip^{\prime}q/P}\overline{\bigg{(}\sum_{p^{\prime\prime}\in\mathbb{Z}_{P}}u[p^{\prime\prime}]e^{-2\pi
ip^{\prime\prime}q/P}\bigg{)}}=|\langle u,f_{q}\rangle|^{2}.$
As such, if one has the intensity measurements $\\{|\langle
u,f_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{P}}$, then one may compute the
circular autocorrelation $\operatorname{CirAut}(u)$ by applying the inverse
DFT. In order to perform phase retrieval from $\\{|\langle
u,f_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{P}}$, it therefore suffices to
determine $u$ from $\operatorname{CirAut}(u)$. This is the motivation for our
approach in this section.
To see how to “invert” $\operatorname{CirAut}$, let’s consider an example.
Take $x=(a,b,c)\in\mathbb{C}^{3}$ and consider the circular autocorrelation of
$x$ as a signal in $\ell(\mathbb{Z}_{3})$:
$\displaystyle\operatorname{CirAut}(x)=(|a|^{2}+|b|^{2}+|c|^{2},a\overline{c}+b\overline{a}+c\overline{b},a\overline{b}+b\overline{c}+c\overline{a}).$
Notice that every entry of $\operatorname{CirAut}(x)$ is a nonlinear
combination of the entries of $x$, from which it is unclear how to compute the
entries of $x$. To simplify the structure, we pad $x$ with zeros and enforce
even symmetry; then the circular autocorrelation of
$u:=(2a,b,c,0,0,0,0,c,b)\in\ell(\mathbb{Z}_{9})$ is
$\displaystyle\operatorname{CirAut}(u)=(4|a|^{2}$
$\displaystyle+|b|^{2}+|c|^{2},2\operatorname{Re}(2a\overline{b}+b\overline{c}),|b|^{2}+4\operatorname{Re}(a\overline{c}),2\operatorname{Re}(b\overline{c}),|c|^{2},$
$\displaystyle|c|^{2},2\operatorname{Re}(b\overline{c}),|b|^{2}+4\operatorname{Re}(a\overline{c}),2\operatorname{Re}(2a\overline{b}+b\overline{c})).$
(2)
Although it still appears rather complicated, this circular autocorrelation
actually lends itself well to recovering the entries of $x$.
Before explaining this further, first note that $9=4(3)-3$, and we can
generalize our mapping $x\mapsto u$ by sending vectors in $\mathbb{C}^{M}$ to
members of $\ell(\mathbb{Z}_{4M-3})$. To make this clear, consider the
reversal operator $R\colon\ell(\mathbb{Z}_{P})\rightarrow\ell(\mathbb{Z}_{P})$
defined by $(Ru)[p]=u[-p]$. Then given a vector $x\in\mathbb{C}^{M}$, padding
with zeros and enforcing even symmetry is equivalent to embedding $x$ in
$\ell(\mathbb{Z}_{4M-3})$ by appending $3M-3$ zeros to $x$ and then taking
$u=x+Rx\in\ell(\mathbb{Z}_{4M-3})$. (From this point forward we use $x$ to
represent both the original signal in $\mathbb{C}^{M}$ and the version of $x$
embedded in $\ell(\mathbb{Z}_{4M-3})$ via zero-padding; the distinction will
be clear from context.) Computing $x\in\mathbb{C}^{M}$ then reduces to
determining the first $M$ entries of $x\in\ell(\mathbb{Z}_{4M-3})$ from
$\operatorname{CirAut}(x+Rx)$. If $x$ is completely real-valued, then this is
indeed possible. For instance, consider the circular autocorrelation (2). If
the entries of $x$ are all real, then this becomes
$\displaystyle\operatorname{CirAut}(x+Rx)=(4a^{2}+b^{2}+c^{2},4ab+2bc,b^{2}+4ac,2bc,c^{2},c^{2},2bc,b^{2}+4ac,4ab+2bc).$
Since $\operatorname{CirAut}(x+Rx)[4]=c^{2}$, we simply take a square root to
obtain $c$ up to a sign. Assuming $c$ is nonzero, we then divide
$\operatorname{CirAut}(x+Rx)[3]$ by 2c to determine $b$ up to the same sign.
Then subtracting $b^{2}$ from $\operatorname{CirAut}(x+Rx)[2]$ and dividing by
$4c$ gives $a$ up to the same sign.
From this example, we see that the process of recovering the entries of $x$
from $\operatorname{CirAut}(x+Rx)$ is iterative, working backward through its
first $2M-2$ entries. But what happens if $c$ is zero? Fortunately, our
process doesn’t break: In this case, we have
$\displaystyle\operatorname{CirAut}(x+Rx)=(4a^{2}+b^{2},4ab,b^{2},0,0,0,0,b^{2},4ab).$
Thus, we need only start with $\operatorname{CirAut}(x+Rx)[2]$ to determine
the remaining entries of $x$ up to a sign. This observation brings to light
the important role of the last nonzero entry of $x$ in our iteration. The
relationship between this coordinate and the entries of
$\operatorname{CirAut}(x+Rx)$ will become more rigorous later.
The above example illustrated how a real signal $x$ is determined by
$\operatorname{CirAut}(x+Rx)$. A complex-valued signal, on the other hand, is
not completely determined from $\operatorname{CirAut}(x+Rx)$. Luckily, this
can be fixed by introducing a second vector in $\ell(\mathbb{Z}_{4M-3})$
obtained from $x$, and we will demonstrate this later, but for now we focus on
$x+Rx$. To this end, let’s first take a closer look at the entries of
$\operatorname{CirAut}(x+Rx)$. Since this circular autocorrelation has even
symmetry by construction, we need only consider all entries of
$\operatorname{CirAut}(x+Rx)$ up to index $2M-2$. This leads to the following
lemma:
###### Lemma 1.
Let $x$ denote an $M$-dimensional complex signal embedded in
$\ell(\mathbb{Z}_{4M-3})$ such that $x[p]=0$ for all $p=M,\ldots,4M-4$. Then
$\operatorname{CirAut}(x+Rx)[p]=2\operatorname{Re}\langle
x,T^{p}x\rangle+\langle x,RT^{-p}x\rangle$ for all $p=1,\ldots,2M-2$.
###### Proof.
First note that by the definition of the circular autocorrelation in (1) we
have
$\operatorname{CirAut}(x+Rx)[p]=\langle
x+Rx,T^{p}(x+Rx)\rangle=2\operatorname{Re}\langle x,T^{p}x\rangle+\langle
x,RT^{-p}x\rangle+\langle x,RT^{p}x\rangle.$
Thus, to complete the proof it suffices to show that $\langle
x,RT^{p}x\rangle=0$ for all $p=1,\ldots,2M-2$. Since $x$ is only nonzero in
its first $M$ entries, we have
$\langle
x,RT^{p}x\rangle=\sum_{p^{\prime}=0}^{M-1}x[p^{\prime}]\overline{(RT^{p}x)[p^{\prime}]}=\sum_{p^{\prime}=0}^{M-1}x[p^{\prime}]\overline{(T^{p}x)[-p^{\prime}]}=\sum_{p^{\prime}=0}^{M-1}x[p^{\prime}]\overline{x[-p^{\prime}-p]},$
where the summand is zero whenever $-p^{\prime}-p\notin[0,M-1]$ modulo $4M-3$.
This is equivalent to having $-p$ not lie in the Minkowski sum
$p^{\prime}+[0,M-1]$, and since $p^{\prime}\in[0,M-1]$ we see that $\langle
x,RT^{p}x\rangle=0$ for all $p=1,\ldots,2M-2$. ∎
As a consequence of Lemma 1, the following theorem expresses the entries of
$\operatorname{CirAut}(x+Rx)$ in terms of the entries of $x$:
###### Theorem 2.
Let $x$ denote an $M$-dimensional complex signal embedded in
$\ell(\mathbb{Z}_{4M-3})$ such that $x[p]=0$ for all $p=M,\ldots,4M-4$. Then
we have
$\displaystyle\operatorname{CirAut}(x+Rx)[p]=\begin{cases}\displaystyle{2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p+1}{2}}^{M-1}x[p^{\prime}](\overline{x[p^{\prime}-p]}+\overline{x[p-p^{\prime}]})\bigg{)}}&\;\text{if
$p$ is odd}\\\
\displaystyle{2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p}{2}+1}^{M-1}x[p^{\prime}](\overline{x[p^{\prime}-p]}+\overline{x[p-p^{\prime}]})\bigg{)}+\left|x[\tfrac{p}{2}]\right|^{2}}&\;\text{if
$p$ is even}\end{cases}$ (3)
for all $p=1,\ldots,2M-2$.
###### Proof.
We first use Lemma 1 to get
$\displaystyle\operatorname{CirAut}(x+Rx)[p]$
$\displaystyle=2\operatorname{Re}\langle x,T^{p}x\rangle+\langle
x,RT^{-p}x\rangle$
$\displaystyle=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=0}^{M-1}x[p^{\prime}]\overline{x[p^{\prime}-p]}\bigg{)}+\sum_{p^{\prime}=0}^{M-1}x[p^{\prime}]\overline{x[p-p^{\prime}]}$
$\displaystyle=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=p}^{M-1}x[p^{\prime}]\overline{x[p^{\prime}-p]}\bigg{)}+\sum_{p^{\prime}=\max\\{p-(M-1),0\\}}^{\min\\{p,M-1\\}}x[p^{\prime}]\overline{x[p-p^{\prime}]},$
(4)
where the last equality takes into account that the first summand is nonzero
only when $p^{\prime}-p\in[0,M-1]$ and the second summand is nonzero only when
$p-p^{\prime}\in[0,M-1]$, i.e., when $p^{\prime}\in[p,p+(M-1)]$ and
$p^{\prime}\in[p-(M-1),p]$, respectively. To continue, we divide our analysis
into cases.
For $p=1,\ldots,M-1$, (4) gives
$\operatorname{CirAut}(x+Rx)[p]=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=p}^{M-1}x[p^{\prime}]\overline{x[p^{\prime}-p]}\bigg{)}+\sum_{p^{\prime}=0}^{p}x[p^{\prime}]\overline{x[p-p^{\prime}]}.$
(5)
If $p$ is odd we can then write
$\displaystyle\sum_{p^{\prime}=0}^{p}x[p^{\prime}]\overline{x[p-p^{\prime}]}$
$\displaystyle=\sum_{p^{\prime}=0}^{\frac{p-1}{2}}x[p^{\prime}]\overline{x[p-p^{\prime}]}+\sum_{p^{\prime}=\frac{p+1}{2}}^{p}x[p^{\prime}]\overline{x[p-p^{\prime}]}$
$\displaystyle=\sum_{p^{\prime\prime}=\frac{p+1}{2}}^{p}x[p-p^{\prime\prime}]\overline{x[p^{\prime\prime}]}+\sum_{p^{\prime}=\frac{p+1}{2}}^{p}x[p^{\prime}]\overline{x[p-p^{\prime}]}=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p+1}{2}}^{p}x[p^{\prime}]\overline{x[p-p^{\prime}]}\bigg{)},$
(6)
while if $p$ is even we similarly write
$\sum_{p^{\prime}=0}^{p}x[p^{\prime}]\overline{x[p-p^{\prime}]}=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p}{2}+1}^{p}x[p^{\prime}]\overline{x[p-p^{\prime}]}\bigg{)}+\left|x\big{[}\tfrac{p}{2}\big{]}\right|^{2}.$
(7)
Substituting (2) and (7) into (5) then gives (3).
For the remaining case, $p=M,\ldots,2M-2$ and (4) gives
$\operatorname{CirAut}(x+Rx)[p]=\sum_{p^{\prime}=p-(M-1)}^{M-1}x[p^{\prime}]\overline{x[p-p^{\prime}]}.$
(8)
Similar to the previous case, taking $p$ to be odd yields
$\sum_{p^{\prime}=p-(M-1)}^{M-1}x[p^{\prime}]\overline{x[p-p^{\prime}]}=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p+1}{2}}^{M-1}x[p^{\prime}]\overline{x[p-p^{\prime}]}\bigg{)},$
(9)
while taking $p$ to be even yields
$\sum_{p^{\prime}=p-(M-1)}^{M-1}x[p^{\prime}]\overline{x[p-p^{\prime}]}=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p}{2}+1}^{M-1}x[p^{\prime}]\overline{x[p-p^{\prime}]}\bigg{)}+\left|x\big{[}\tfrac{p}{2}\big{]}\right|^{2},$
(10)
and substituting (9) and (10) into (8) also gives (3). ∎
Notice (3) shows that each member of
$\\{\operatorname{CirAut}(x+Rx)[p]\\}_{p=1}^{2M-2}$ can be written as a
combination of the first $M$ entries of $x$, but only those at or beyond the
$\lceil\frac{p}{2}\rceil$th index. As such, the index of the last nonzero
entry of $x$ is closely related to that of the last nonzero entry of
$\\{\operatorname{CirAut}(x+Rx)[p]\\}_{p=1}^{2M-2}$. This corresponds to our
observation earlier in the case of $x\in\mathbb{R}^{3}$ where the third
coordinate was assumed to be zero. We identify the relationship between the
locations of these nonzero entries in the following lemma:
###### Lemma 3.
Let $x$ denote an $M$-dimensional complex signal embedded in
$\ell(\mathbb{Z}_{4M-3})$ such that $x[p]=0$ for all $p=M,\ldots,4M-4$. Then
the last nonzero entry of $\\{\operatorname{CirAut}(x+Rx)[p]\\}_{p=0}^{2M-2}$
has index $p=2q$, where $q$ is the index of the last nonzero entry of $x$.
###### Proof.
If $q\geq 1$, then (3) gives that
$\operatorname{CirAut}(x+Rx)[2q]=|x[q]|^{2}\neq 0$. Note that since
$x[p^{\prime}]=0$ for every $p^{\prime}>q$, (3) also gives that
$\operatorname{CirAut}(x+Rx)[p]=0$ for every $p>2q$. For the remaining case
where $q=0$, (3) immediately gives that $\operatorname{CirAut}(x+Rx)[p]=0$ for
every $p\geq 1$. To show that $\operatorname{CirAut}(x+Rx)[0]\neq 0$ in this
case, we apply the definition of circular autocorrelation (1):
$\operatorname{CirAut}(x+Rx)[0]=\langle
x+Rx,x+Rx\rangle=\|x+Rx\|^{2}=|2x[0]|^{2}\neq 0,$
where the last equality uses the fact that $x$ is only supported at $0$ since
$q=0$. ∎
As previously mentioned, we are unable to recover the entries of a complex
signal $x$ solely from $\operatorname{CirAut}(x+Rx)$. One way to address this
is to rotate the entries of $x$ in the complex plane and also take the
circular autocorrelation of this modified signal. If we rotate by an angle
which is not an integer multiple of $\pi$, this will produce new entries which
are linearly independent from the corresponding entries of $x$ when viewed as
vectors in the complex plane. As we will see, the problem of recovering the
entries of $x$ then reduces to solving a linear system.
Take any $(4M-3)\times(4M-3)$ diagonal modulation operator $E$ whose diagonal
entries $\\{\omega_{k}\\}_{k=0}^{4M-4}$ are of unit modulus satisfying
$\omega_{j}\overline{\omega_{k}}\notin\mathbb{R}$ for all $j\neq k$ and
consider the new vector $Ex\in\ell(\mathbb{Z}_{4M-3})$. Then Theorem 2 gives
$\displaystyle\operatorname{CirAut}(Ex+REx)[p]$
$\displaystyle\qquad\qquad=\begin{cases}\displaystyle{2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p+1}{2}}^{M-1}\omega_{p^{\prime}}x[p^{\prime}](\overline{\omega_{p^{\prime}-p}}\overline{x[p^{\prime}-p]}+\overline{\omega_{p-p^{\prime}}}\overline{x[p-p^{\prime}]})\bigg{)}}&\;\text{if
$p$ is odd}\\\
\displaystyle{2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=\frac{p}{2}+1}^{M-1}\omega_{p^{\prime}}x[p^{\prime}](\overline{\omega_{p^{\prime}-p}}\overline{x[p^{\prime}-p]}+\overline{\omega_{p-p^{\prime}}}\overline{x[p-p^{\prime}]})\bigg{)}+\left|x[\tfrac{p}{2}]\right|^{2}}&\;\text{if
$p$ is even}\end{cases}$ (11)
for all $p=1,\ldots,2M-2$. We will see that (3) and (2) together allow us to
solve for the entries of $x$ (up to a global phase factor) by working
iteratively backward through the entries of $\operatorname{CirAut}(x+Rx)$ and
$\operatorname{CirAut}(Ex+REx)$. As alluded to earlier, each entry index forms
a linear system which can be solved using the following lemma:
###### Lemma 4.
Let $a,b\in\mathbb{C}\setminus\\{0\\}$ and
$\omega\in\mathbb{C}\setminus\mathbb{R}$ with $|\omega|=1$. Then
$b=\frac{i}{\overline{a}\operatorname{Im}(\omega)}\big{(}\operatorname{Re}(\omega
a\overline{b})-\omega\operatorname{Re}(a\overline{b})\big{)}.$ (12)
###### Proof.
Define $\theta:=\operatorname{arg}(\omega)$ and
$\phi:=\operatorname{arg}(a\overline{b})$. Then
$\theta+\phi\equiv\operatorname{arg}(\omega ab)\bmod 2\pi$ and
$\cos(\phi)=\frac{\operatorname{Re}(a\overline{b})}{|a\overline{b}|},\qquad\sin(\phi)=\frac{\operatorname{Im}(a\overline{b})}{|a\overline{b}|},\qquad\cos(\theta+\phi)=\frac{\operatorname{Re}(\omega
a\overline{b})}{|\omega a\overline{b}|}.$
With this, we apply a trigonometric identity to obtain
$\operatorname{Re}(\omega a\overline{b})=|\omega
a\overline{b}|\cos(\theta+\phi)=|a\overline{b}|\left(\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)\right)=\cos(\theta)\operatorname{Re}(a\overline{b})-\sin(\theta)\operatorname{Im}(a\overline{b}).$
Since $\omega\in\mathbb{C}\setminus\mathbb{R}$, then $\sin(\theta)$ is
necessarily nonzero, and so we can isolate $\operatorname{Im}(a\overline{b})$
in the above equation. We then use this expression for
$\operatorname{Im}(a\overline{b})$ to solve for $b$:
$b=\frac{\overline{a}b}{~{}\overline{a}~{}}=\frac{1}{~{}\overline{a}~{}}\big{(}\operatorname{Re}(a\overline{b})-i\operatorname{Im}(a\overline{b})\big{)}=\frac{i}{\overline{a}\sin(\theta)}\big{(}\operatorname{Re}(\omega
a\overline{b})-e^{i\theta}\operatorname{Re}(a\overline{b})\big{)}.\qed$
We now use this lemma to describe how to recover $x$ up to global phase. By
Lemma 3, the last nonzero entry of
$\\{\operatorname{CirAut}(x+Rx)[p]\\}_{p=0}^{2M-2}$ has index $p=2q$, where
$q$ indexes the last nonzero entry of $x$. As such, we know that $x[k]=0$ for
every $k>q$, and $x[q]$ can be estimated up to a phase factor
($\hat{x}[q]=e^{i\psi}x[q]$) by taking the square root of
$\operatorname{CirAut}(x+Rx)[2q]=|x[q]|^{2}$ (we will verify this soon, but
this corresponds to the examples we have seen so far). Next, if we know
$\operatorname{Re}(x[q]\overline{x[k]})$ and
$\operatorname{Re}(\omega_{q}\overline{\omega_{k}}x[q]\overline{x[k]})$ for
some $k<q$, then we can use these to estimate $x[k]$:
$\hat{x}[k]:=\frac{i}{\overline{\hat{x}[q]}\operatorname{Im}(\omega_{q}\overline{\omega_{k}})}\left(\operatorname{Re}(\omega_{q}\overline{\omega_{k}}x[q]\overline{x[k]})-\omega_{q}\overline{\omega_{k}}\operatorname{Re}(x[q]\overline{x[k]})\right)=e^{i\psi}x[k],$
(13)
where the last equality follows from substituting $a=x[q]$, $b=x[k]$ and
$\omega=\omega_{q}\overline{\omega_{k}}$ into (12). Overall, once we know
$x[q]$ up to phase, then we can find $x[k]$ relative to this same phase for
each $k=0,\ldots,q-1$, provided we know
$\operatorname{Re}(x[q]\overline{x[k]})$ and
$\operatorname{Re}(\omega_{q}\overline{\omega_{k}}x[q]\overline{x[k]})$ for
these $k$’s. Thankfully, these values can be determined from the entries of
$\operatorname{CirAut}(x+Rx)$ and $\operatorname{CirAut}(Ex+REx)$:
###### Theorem 5.
Let $x$ denote an $M$-dimensional complex signal embedded in
$\ell(\mathbb{Z}_{4M-3})$ such that $x[p]=0$ for all $p=M,\ldots,4M-4$ and $E$
be a $(4M-3)\times(4M-3)$ diagonal modulation operator with diagonal entries
$\\{\omega_{k}\\}_{k=0}^{4M-4}$ satisfying $|\omega_{k}|=1$ for all
$k=0,\ldots,4M-4$ and $\omega_{j}\overline{\omega_{k}}\notin\mathbb{R}$ for
all $j\neq k$. Then $x$ can be recovered up to a global phase factor from
$\operatorname{CirAut}(x+Rx)$ and $\operatorname{CirAut}(Ex+REx)$.
###### Proof.
Letting $q$ denote the last nonzero entry of $x$, it suffices to estimate
$\\{x[k]\\}_{k=0}^{q}$ up to a global phase factor. To this end, recall from
Lemma 3 that the last nonzero entry of
$\\{\operatorname{CirAut}(x+Rx)[p]\\}_{p=0}^{2M-2}$ has index $p=2q$. If
$q=0$, then we have already seen that
$\operatorname{CirAut}(x+Rx)[0]=4|x[0]|^{2}$. Since there exists
$\psi\in[0,2\pi)$ such that $x[0]=e^{-i\psi}|x[0]|$, we may take
$\hat{x}[0]:=\frac{1}{2}\sqrt{\operatorname{CirAut}(x+Rx)[0]}=|x[0]|=e^{i\psi}x[0]$.
Otherwise $q\in[1,M-1]$, and (3) gives
$\displaystyle\operatorname{CirAut}(x+Rx)[2q]$
$\displaystyle=\left|x[q]\right|^{2}+2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=q+1}^{M-1}x[p^{\prime}](\overline{x[p^{\prime}-2q]}+\overline{x[2q-p^{\prime}]})\bigg{)}=\left|x[q]\right|^{2}.$
Thus, taking $\hat{x}[q]:=\sqrt{\operatorname{CirAut}(x+Rx)[2q]}=|x[q]|$ gives
us $\hat{x}[q]=e^{i\psi}x[q]$ for some $\psi\in[0,2\pi)$.
In the case where $q=1$, all that remains to determine is $\hat{x}[0]$, a
calculation which we save for the end of the proof. For now, suppose $q\geq
2$. Since we already know $\hat{x}[q]=e^{i\psi}x[q]$, we would like to
determine $\hat{x}[k]$ for $k=1,\ldots,q-1$. To this end, take $r\in[0,q-2]$
and suppose we have $\hat{x}[k]=e^{i\psi}x[k]$ for all $k=q-r,\ldots,q$. If we
can obtain $\hat{x}[q-(r+1)]$ up to the same phase from this information, then
working iteratively from $r=0$ to $r=q-2$ will give us $\hat{x}[k]$ up to
global phase for all but the zeroth entry (which we address later). Note when
$r$ is even, (3) gives
$\displaystyle\operatorname{CirAut}(x+Rx)[2q-(r+1)]$
$\displaystyle=2\operatorname{Re}\bigg{(}\sum_{p^{\prime}=q-\frac{r}{2}}^{q}x[p^{\prime}](\overline{x[p^{\prime}-(2q-(r+1))]}+\overline{x[(2q-(r+1))-p^{\prime}]})\bigg{)}$
$\displaystyle=2\operatorname{Re}\left(x[q]\overline{x[q-(r+1)]}\right)+2\sum_{p^{\prime}=q-\frac{r}{2}}^{q-1}\operatorname{Re}\left(x[p^{\prime}]\overline{x[(2q-(r+1))-p^{\prime}]}\right),$
where the last equality follows from the observation that
$p^{\prime}-(2q-(r+1))\leq-q+(r+1)\leq-1$ over the range of the sum, meaning
$x[p^{\prime}-(2q-(r+1))]=0$ throughout the sum. Similarly when $r$ is odd,
(3) gives
$\displaystyle\operatorname{CirAut}(x+Rx)[2q-(r+1)]$
$\displaystyle\qquad\qquad=2\operatorname{Re}\left(x[q]\overline{x[q-(r+1)]}\right)+2\sum_{p^{\prime}=q-\frac{r-1}{2}}^{q-1}\operatorname{Re}\left(x[p^{\prime}]\overline{x[(2q-(r+1))-p^{\prime}]}\right)+\left|x\big{[}q-\tfrac{r+1}{2}\big{]}\right|^{2}.$
In either case, we can isolate $\operatorname{Re}(x[q]\overline{x[q-(r+1)]})$
to get an expression in terms of $\operatorname{CirAut}(x+Rx)[2q-(r+1)]$ and
other terms of the form $\operatorname{Re}(x[k]\overline{x[k^{\prime}]})$ or
$|x[k]|^{2}$ for $k,k^{\prime}\in[q-r,q-1]$. By the induction hypothesis, we
have $\hat{x}[k]=e^{i\psi}x[k]$ for $k=q-r,\ldots,q-1$, and so we can use
these estimates to determine these other terms:
$\operatorname{Re}(\hat{x}[k]\overline{\hat{x}[k^{\prime}]})=\operatorname{Re}(e^{i\psi}x[k]\overline{e^{i\psi}x[k^{\prime}]})=\operatorname{Re}(x[k]\overline{x[k^{\prime}]}),\qquad|\hat{x}[k]|^{2}=|e^{i\psi}x[k]|^{2}=|x[k]|^{2}.$
As such, we can use $\operatorname{CirAut}(x+Rx)[2q-(r+1)]$ along with the
higher-indexed estimates $\hat{x}[k]$ to determine
$\operatorname{Re}(x[q]\overline{x[q-(r+1)]})$. Similarly, we can use
$\operatorname{CirAut}(Ex+REx)[2q-(r+1)]$ along with the higher-indexed
estimates $\hat{x}[k]$ to determine
$\operatorname{Re}(\omega_{q}\overline{\omega_{(q-(r+1))}}x[q]\overline{x[q-(r+1)]})$.
We then plug these into (13), along with the estimate
$\hat{x}[q]=e^{i\psi}x[q]$ (which is also available by the induction
hypothesis), to get $\hat{x}[2q-(r+1)]=e^{i\psi}x[2q-(r+1)]$.
At this point, we have determined $\\{x[k]\\}_{k=1}^{q}$ up to a global phase
factor whenever $q\geq 1$, and so it remains to find $\hat{x}[0]$. For this,
note that when $q$ is odd, (3) gives
$\operatorname{CirAut}(x+Rx)[q]=4\operatorname{Re}(x[q]\overline{x[0]})+2\sum_{p^{\prime}=\frac{q+1}{2}}^{q-1}\operatorname{Re}\left(x[p^{\prime}]\overline{x[q-p^{\prime}]}\right),$
while for even $q$, we have
$\displaystyle\operatorname{CirAut}(x+Rx)[q]=4\operatorname{Re}(x[q]\overline{x[0]})+2\sum_{p^{\prime}=\frac{q}{2}+1}^{q-1}\operatorname{Re}\left(x[p^{\prime}]\overline{x[q-p^{\prime}]}\right)+\left|x\big{[}\tfrac{q}{2}\big{]}\right|^{2}.$
As before, isolating $\operatorname{Re}(x[q]\overline{x[0]})$ in either case
produces an expression in terms of $\operatorname{CirAut}(x+Rx)[q]$ and other
terms of the form $\operatorname{Re}(x[k]\overline{x[k^{\prime}]})$ or
$|x[k]|^{2}$ for $k,k^{\prime}\in[1,q-1]$. These other terms can be calculated
using the estimates $\\{\hat{x}[k]\\}_{k=1}^{q-1}$, and so we can also
calculate $\operatorname{Re}(x[q]\overline{x[0]})$ from
$\operatorname{CirAut}(x+Rx)[q]$. Similarly, we can calculate
$\operatorname{Re}(\omega_{q}\overline{\omega_{0}}x[q]\overline{x[0]})$ from
$\\{\hat{x}[k]\\}_{k=1}^{q-1}$ and $\operatorname{CirAut}(Ex+REx)[q]$, and
plugging these into (13) along with $\hat{x}[q]$ produces the estimate
$\hat{x}[0]=e^{i\psi}x[0]$. ∎
Theorem 5 establishes that it is possible to recover a signal
$x\in\mathbb{C}^{M}$ up to a global phase factor from
$\\{\operatorname{CirAut}(x+Rx)\\}_{q=0}^{2M-2}$ and
$\\{\operatorname{CirAut}(Ex+REx)\\}_{q=0}^{2M-2}$. We now return to how these
circular autocorrelations relate to intensity measurements. Recall that the
DFT of the circular autocorrelation is the modulus squared of the DFT of the
original signal: $(F^{*}\operatorname{CirAut}(u))[q]=|(F^{*}u)[q]|^{2}$. Also
note that the DFT commutes with the reversal operator:
$(F^{*}Ru)[q]=\sum_{p\in\mathbb{Z}_{P}}u[-p]e^{-2\pi
ipq/P}=\sum_{p^{\prime}\in\mathbb{Z}_{P}}u[p^{\prime}]e^{-2\pi
ip^{\prime}(-q)/P}=(F^{*}u)[-q]=(RF^{*}u)[q].$
With this, we can express $\operatorname{CirAut}(x+Rx)$ in terms of intensity
measurements with a particular ensemble:
$\displaystyle(F^{*}\operatorname{CirAut}(x+Rx))[q]$
$\displaystyle=|(F^{*}(x+Rx))[q]|^{2}$
$\displaystyle=|(F^{*}x)[q]+(F^{*}Rx)[q]|^{2}=|(F^{*}x)[q]+(F^{*}x)[-q]|^{2}=|\langle
x,f_{q}+f_{-q}\rangle|^{2}.$
Defining the $q$th discrete cosine function $c_{q}\in\ell(\mathbb{Z}_{4M-3})$
by
$c_{q}[p]:=2\cos\left(\tfrac{2\pi pq}{4M-3}\right)=e^{2\pi
ipq/(4M-3)}+e^{-2\pi ipq/(4M-3)}=(f_{q}+f_{-q})[p],$
this means that $(F^{*}\operatorname{CirAut}(x+Rx))[q]=|\langle
x,c_{q}\rangle|^{2}$ for all $q\in\mathbb{Z}_{4M-3}$. Similarly, if we take
the modulation matrix $E$ to have diagonal entries $\omega_{k}=e^{2\pi
ik/(2M-1)}$ for all $k=0,\ldots,4M-4$, we find
$(F^{*}\operatorname{CirAut}(Ex+REx))[q]=|\langle
Ex,c_{q}\rangle|^{2}=|\langle x,E^{*}c_{q}\rangle|^{2}.$
Thus, coupling the DFT with Theorem 5 allows us to recover the signal $x$ from
$4M-2$ intensity measurements, namely with the ensemble
$\\{c_{q}\\}_{q=0}^{2M-2}\cup\\{E^{*}c_{q}\\}_{q=0}^{2M-2}$. Note that since
$x\in\ell(\mathbb{Z}_{4M-3})$ is actually a zero-padded version of
$x\in\mathbb{C}^{M}$, we may view $c_{q}$ and $E^{*}c_{q}$ as members of
$\mathbb{C}^{M}$ by discarding the entries indexed by $p=M,\ldots,4M-4$.
Considering this section promised phase retrieval from only $4M-4$ intensity
measurements, we must somehow find a way to discard two of these $4M-2$
measurement vectors. To do this, first note that
$\displaystyle\operatorname{CirAut}(Ex+REx)[0]$ $\displaystyle=\|Ex+REx\|^{2}$
$\displaystyle=\sum_{k\in\mathbb{Z}_{4M-3}}\left|e^{2\pi
ik/(2M-1)}x[k]+e^{2\pi i(-k)/(2M-1)}x[-k]\right|^{2}$
$\displaystyle=\sum_{k=-(2M-2)}^{-1}\left|e^{2\pi
i(-k)/(2M-1)}x[-k]\right|^{2}+|2x[0]|^{2}+\sum_{k=1}^{2M-2}\left|e^{2\pi
ik/(2M-1)}x[k]\right|^{2}$ $\displaystyle=\|x+Rx\|^{2}$
$\displaystyle=\operatorname{CirAut}(x+Rx)[0].$
Moreover, we have
$\displaystyle\operatorname{CirAut}(Ex+REx)[2M-2]$
$\displaystyle=\sum_{k\in\mathbb{Z}_{4M-3}}(Ex+REx)[k]\overline{(Ex+REx)[k-(2M-2)]}$
$\displaystyle=(Ex+REx)[M-1]\overline{(Ex+REx)[-(M-1)]}$
$\displaystyle=(Ex+REx)[M-1]\overline{(Ex+REx)[M-1]},$
where the last equality is by even symmetry. Since $x$ is only supported on
$k=0,\ldots,M-1$, we then have
$\displaystyle\operatorname{CirAut}(Ex+REx)[2M-2]$
$\displaystyle=|(Ex+REx)[M-1]|^{2}$ $\displaystyle=\left|e^{2\pi
i(M-1)/(2M-1)}x[M-1]+e^{-2\pi i(M-1)/(2M-1)}x[-(M-1)]\right|^{2}$
$\displaystyle=\left|e^{2\pi
i(M-1)/(2M-1)}x[M-1]\right|^{2}=|x[M-1]|^{2}=\operatorname{CirAut}(x+Rx)[2M-2].$
Furthermore, the even symmetry of the circular autocorrelation also gives
$\displaystyle\operatorname{CirAut}(Ex+REx)[-(2M-2)]$
$\displaystyle=\operatorname{CirAut}(Ex+REx)[2M-2]$
$\displaystyle=\operatorname{CirAut}(x+Rx)[2M-2]=\operatorname{CirAut}(x+Rx)[-(2M-2)].$
These redundancies between $\operatorname{CirAut}(x+Rx)$ and
$\operatorname{CirAut}(Ex+REx)$ indicate that we might be able to remove
measurement vectors from our ensemble while maintaining our ability to perform
phase retrieval. The following theorem confirms this suspicion:
###### Theorem 6.
Let $c_{q}\in\mathbb{C}^{M}$ be the truncated discrete cosine function defined
by $c_{q}[p]:=2\cos(\frac{2\pi pq}{4M-3})$ for all $p=0,\ldots,M-1$, and let
$E$ be the $M\times M$ diagonal modulation operator with diagonal entries
$\omega_{k}=e^{2\pi ik/(2M-1)}$ for all $k=0,\ldots,M-1$. Then the intensity
measurement mapping
$\mathcal{A}\colon\mathbb{C}^{M}/\mathbb{T}\rightarrow\mathbb{R}^{4M-4}$
defined by $\mathcal{A}(x):=\\{|\langle
x,c_{q}\rangle|^{2}\\}_{q=0}^{2M-2}\cup\\{|\langle
x,E^{*}c_{q}\rangle|^{2}\\}_{q=1}^{2M-3}$ is injective.
###### Proof.
Since Theorem 5 allows us to reconstruct any $x\in\mathbb{C}^{M}$ up to a
global phase factor from the entries of $\operatorname{CirAut}(x+Rx)$ and
$\operatorname{CirAut}(Ex+REx)$, it suffices to show that the intensity
measurements $\\{|\langle x,c_{q}\rangle|^{2}\\}_{q=0}^{2M-2}\cup\\{|\langle
x,E^{*}c_{q}\rangle|^{2}\\}_{q=1}^{2M-3}$ allow us to recover the entries of
these circular autocorrelations. To this end, recall that
$\displaystyle\operatorname{CirAut}(x+Rx)=(F^{*})^{-1}\\{|\langle
x,c_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{4M-3}},\quad\operatorname{CirAut}(Ex+REx)=(F^{*})^{-1}\\{|\langle
x,E^{*}c_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{4M-3}}.$
Since we have $\\{|\langle x,c_{q}\rangle|^{2}\\}_{q=0}^{2M-2}$, we can
exploit even symmetry to determine the rest of $\\{|\langle
x,c_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{4M-3}}$, and then apply the inverse
DFT to get $\operatorname{CirAut}(x+Rx)$. Moreover, by the previous
discussion, we also obtain the $0$, $2M-2$, and $-(2M-2)$ entries of
$\operatorname{CirAut}(Ex+REx)$ from the corresponding entries of
$\operatorname{CirAut}(x+Rx)$. Organize this information about
$\operatorname{CirAut}(Ex+REx)$ into a vector $w\in\ell(\mathbb{Z}_{4M-3})$
whose $0$, $2M-2$, and $-(2M-2)$ entries come from
$\operatorname{CirAut}(Ex+REx)$ and whose remaining entries are populated by
even symmetry from $\\{|\langle x,E^{*}c_{q}\rangle|^{2}\\}_{q=1}^{2M-3}$. We
can express $w$ as a matrix-vector product $w=A\\{|\langle
x,E^{*}c_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{4M-3}}$, where $A$ is the
identity matrix with the $0$, $2M-2$, and $-(2M-2)$ rows replaced by the
corresponding rows of the inverse DFT matrix. To complete the proof, it
suffices to show that the matrix $A$ is invertible, since this would imply
$\operatorname{CirAut}(Ex+REx)=(F^{*})^{-1}A^{-1}w$.
Using the cofactor expansion, note that $\det(A)$ reduces to a determinant of
a 3$\times$3 submatrix of $(F^{*})^{-1}$. Specifically, letting
$\theta:=2\pi(2M-2)^{2}/(4M-3)$ we have
$\displaystyle\det(A)=\det\left(\left[\begin{array}[]{ccc}1&1&1\\\
1&e^{i\theta}&e^{-i\theta}\\\
1&e^{-i\theta}&e^{i\theta}\end{array}\right]\right)$
$\displaystyle=(e^{2i\theta}-e^{-2i\theta})-(e^{i\theta}-e^{-i\theta})+(e^{-i\theta}-e^{i\theta})$
$\displaystyle=(e^{i\theta}+e^{-i\theta}-2)(e^{i\theta}-e^{-i\theta})=4i(\cos(\theta)-1)\sin(\theta),$
and so $A$ is invertible if and only if $\cos(\theta)-1\neq 0$ and
$\sin(\theta)\neq 0$. This equivalent to having $\pi$ not divide $\theta$, and
indeed, the ratio
$\displaystyle\frac{\theta}{\pi}=\frac{2(2M-2)^{2}}{4M-3}=2M-\frac{5}{2}+\frac{1}{2(4M-3)}$
is not an integer because $M\geq 2$. As such, $A$ is invertible. ∎
We conclude this section by summarizing our measurement design and phase
retrieval procedure:
Measurement design
* 1.
Define the $q$th truncated discrete cosine function
$c_{q}:=\\{2\cos(\frac{2\pi pq}{4M-3})\\}_{p=0}^{M-1}$
* 2.
Define the $M\times M$ diagonal matrix $E$ with entries $\omega_{k}:=e^{2\pi
ik/(2M-1)}$ for all $k=0,\ldots,M-1$
* 3.
Take $\Phi:=\\{c_{q}\\}_{q=0}^{2M-2}\cup\\{E^{*}c_{q}\\}_{q=1}^{2M-3}$
Phase retrieval procedure
* 1.
Calculate $\\{|\langle x,c_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{4M-3}}$ from
$\\{|\langle x,c_{q}\rangle|^{2}\\}_{q=0}^{2M-2}$ by even extension
* 2.
Calculate $\operatorname{CirAut}(x+Rx)=(F^{*})^{-1}\\{|\langle
x,c_{q}\rangle|^{2}\\}_{q\in\mathbb{Z}_{4M-3}}$
* 3.
Define $w\in\ell(\mathbb{Z}_{4M-3})$ so that its $0$, $2M-2$, and $-(2M-2)$
entries are the corresponding entries in $\operatorname{CirAut}(x+Rx)$ and its
remaining entries are populated by even symmetry from $\\{|\langle
x,E^{*}c_{q}\rangle|^{2}\\}_{q=1}^{2M-3}$
* 4.
Define $A$ to be the identity matrix with the $0$, $2M-2$, and $-(2M-2)$ rows
replaced by the corresponding rows of the inverse DFT matrix $(F^{*})^{-1}$
* 5.
Calculate $\operatorname{CirAut}(Ex+REx)=(F^{*})^{-1}A^{-1}w$
* 6.
Recover $x$ up to global phase from $\operatorname{CirAut}(x+Rx)$ and
$\operatorname{CirAut}(Ex+REx)$ using the process described in the proof of
Theorem 5
## 3 Almost injectivity
While $4M+o(M)$ measurements are necessary and generically sufficient for
injectivity in the complex case, you can save a factor of $2$ in the number of
measurements if you are willing to slightly weaken the desired notion of
injectivity [3, 25]. To be explicit, we start with the following definition:
###### Definition 7.
Consider $\Phi=\\{\varphi_{n}\\}_{n=1}^{N}\subseteq\mathbb{R}^{M}$. The
intensity measurement mapping $\mathcal{A}\colon\mathbb{R}^{M}/\\{\pm
1\\}\rightarrow\mathbb{R}^{N}$ defined by $(\mathcal{A}(x))(n):=|\langle
x,\varphi_{n}\rangle|^{2}$ is said to be almost injective if
$\mathcal{A}^{-1}(\mathcal{A}(x))=\\{\pm x\\}$ for almost every
$x\in\mathbb{R}^{M}$.
The above definition specifically treats the real case, but it can be
similarly defined for the complex case in the obvious way. For the complex
case, it is known that $2M$ measurements are necessary for almost injectivity
[25], and that $2M$ generic measurements suffice [3]; this is the factor-
of-$2$ savings mentioned above. For the real case, it is also known how many
measurements are necessary and generically sufficient for almost injectivity:
$M+1$ [3]. Like the complex case, this is also a factor-of-$2$ savings from
the injectivity requirement: $2M-1$. This requirement for injectivity in the
real case follows from the following result from [3], which we prove here
because the proof is short and inspires the remainder of this section:
###### Theorem 8.
Consider $\Phi=\\{\varphi_{n}\\}_{n=1}^{N}\subseteq\mathbb{R}^{M}$ and the
intensity measurement mapping $\mathcal{A}\colon\mathbb{R}^{M}/\\{\pm
1\\}\rightarrow\mathbb{R}^{N}$ defined by $(\mathcal{A}(x))(n):=|\langle
x,\varphi_{n}\rangle|^{2}$. Then $\mathcal{A}$ is injective if and only if for
every $S\subseteq\\{1,\ldots,N\\}$, either $\\{\varphi_{n}\\}_{n\in S}$ or
$\\{\varphi_{n}\\}_{n\in S^{\mathrm{c}}}$ spans $\mathbb{R}^{M}$.
###### Proof.
We will prove both directions by obtaining the contrapositives.
($\Rightarrow$) Assume there exists $S\subseteq\\{1,\ldots,N\\}$ such that
neither $\\{\varphi_{n}\\}_{n\in S}$ nor $\\{\varphi_{n}\\}_{n\in
S^{\mathrm{c}}}$ spans $\mathbb{R}^{M}$. This implies that there are nonzero
vectors $u,v\in\mathbb{R}^{M}$ such that $\langle u,\varphi_{n}\rangle=0$ for
all $n\in S$ and $\langle v,\varphi_{n}\rangle=0$ for all $n\in
S^{\mathrm{c}}$. For each $n$, we then have
$|\langle u\pm v,\varphi_{n}\rangle|^{2}=|\langle u,\varphi_{n}\rangle|^{2}\pm
2\operatorname{Re}\langle u,\varphi_{n}\rangle\overline{\langle
v,\varphi_{n}\rangle}+|\langle v,\varphi_{n}\rangle|^{2}=|\langle
u,\varphi_{n}\rangle|^{2}+|\langle v,\varphi_{n}\rangle|^{2}.$
Since $|\langle u+v,\varphi_{n}\rangle|^{2}=|\langle
u-v,\varphi_{n}\rangle|^{2}$ for every $n$, we have
$\mathcal{A}(u+v)=\mathcal{A}(u-v)$. Moreover, $u$ and $v$ are nonzero by
assumption, and so $u+v\neq\pm(u-v)$.
($\Leftarrow$) Assume that $\mathcal{A}$ is not injective. Then there exist
vectors $x,y\in\mathbb{R}^{M}$ such that $x\neq\pm y$ and
$\mathcal{A}(x)=\mathcal{A}(y)$. Taking $S:=\\{n:\langle
x,\varphi_{n}\rangle=-\langle y,\varphi_{n}\rangle\\}$, we have $\langle
x+y,\varphi_{n}\rangle=0$ for every $n\in S$. Otherwise when $n\in
S^{\mathrm{c}}$, we have $\langle x,\varphi_{n}\rangle=\langle
y,\varphi_{n}\rangle$ and so $\langle x-y,\varphi_{n}\rangle=0$. Furthermore,
both $x+y$ and $x-y$ are nontrivial since $x\neq\pm y$, and so neither
$\\{\varphi_{n}\\}_{n\in S}$ nor $\\{\varphi_{n}\\}_{n\in S^{\mathrm{c}}}$
spans $\mathbb{R}^{M}$. ∎
Similar to the above result, in this section, we characterize ensembles of
measurement vectors which yield almost injective intensity measurements, and
similar to the above proof, the basic idea behind our analysis is to consider
sums and differences of signals with identical intensity measurements. Our
characterization starts with the following lemma:
###### Lemma 9.
Consider $\Phi=\\{\varphi_{n}\\}_{n=1}^{N}\subseteq\mathbb{R}^{M}$ and the
intensity measurement mapping $\mathcal{A}\colon\mathbb{R}^{M}/\\{\pm
1\\}\rightarrow\mathbb{R}^{N}$ defined by $(\mathcal{A}(x))(n):=|\langle
x,\varphi_{n}\rangle|^{2}$. Then $\mathcal{A}$ is almost injective if and only
if almost every $x\in\mathbb{R}^{M}$ is not in the Minkowski sum
$\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$
for all $S\subseteq\\{1,\ldots,N\\}$. More precisely,
$\mathcal{A}^{-1}(\mathcal{A}(x))=\\{\pm x\\}$ if and only if
$x\notin\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$
for any $S\subseteq\\{1,\ldots,N\\}$.
###### Proof.
By the definition of the mapping $\mathcal{A}$, for $x,y\in\mathbb{R}^{M}$ we
have $\mathcal{A}(x)=\mathcal{A}(y)$ if and only if $|\langle
x,\varphi_{n}\rangle|=|\langle y,\varphi_{n}\rangle|$ for all
$n\in\\{1,\ldots,N\\}$. This occurs precisely when there is a subset
$S\subseteq\\{1,\ldots,N\\}$ such that $\langle x,\varphi_{n}\rangle=-\langle
y,\varphi_{n}\rangle$ for every $n\in S$ and $\langle
x,\varphi_{n}\rangle=\langle y,\varphi_{n}\rangle$ for every $n\in
S^{\mathrm{c}}$. Thus, $\mathcal{A}^{-1}(\mathcal{A}(x))=\\{\pm x\\}$ if and
only if for every $y\neq\pm x$ and for every $S\subseteq\\{1,\ldots,N\\}$,
either there exists an $n\in S$ such that $\langle x+y,\varphi_{n}\rangle\neq
0$ or an $n\in S^{\mathrm{c}}$ such that $\langle x-y,\varphi_{n}\rangle\neq
0$. We claim that this occurs if and only if $x$ is not in the Minkowski sum
$\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$
for all $S\subseteq\\{1,\ldots,N\\}$, which would complete the proof. We
verify the claim by seeking the contrapositive in each direction.
$(\Rightarrow)$ Suppose
$x\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$.
Then there exists $u\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}$
and $v\in\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$
such that $x=u+v$. Taking $y:=u-v$, we see that
$x+y=2u\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}$ and
$x-y=2v\in\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$,
which means that for every $S\subseteq\\{1,\ldots,N\\}$ there is no $n\in S$
such that $\langle x+y,\varphi_{n}\rangle\neq 0$ nor $n\in S^{\mathrm{c}}$
such that $\langle x-y,\varphi_{n}\rangle\neq 0$. Furthermore, $u$ and $v$ are
nonzero, and so $y\neq\pm x$.
$(\Leftarrow)$ Suppose $y\neq\pm x$ and for every $S\subseteq\\{1,\ldots,N\\}$
there is no $n\in S$ such that $\langle x+y,\varphi_{n}\rangle\neq 0$ nor
$n\in S^{\mathrm{c}}$ such that $\langle x-y,\varphi_{n}\rangle\neq 0$. Then
$x+y\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}$ and
$x-y\in\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$.
Since $x=\frac{1}{2}(x+y)+\frac{1}{2}(x-y)$, we have that
$x\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$.
∎
###### Theorem 10.
Consider $\Phi=\\{\varphi_{n}\\}_{n=1}^{N}\subseteq\mathbb{R}^{M}$ and the
intensity measurement mapping $\mathcal{A}\colon\mathbb{R}^{M}/\\{\pm
1\\}\rightarrow\mathbb{R}^{N}$ defined by $(\mathcal{A}(x))(n):=|\langle
x,\varphi_{n}\rangle|^{2}$. Suppose $\Phi$ spans $\mathbb{R}^{M}$ and each
$\varphi_{n}$ is nonzero. Then $\mathcal{A}$ is almost injective if and only
if the Minkowski sum
$\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$
is a proper subspace of $\mathbb{R}^{M}$ for each nonempty proper subset
$S\subseteq\\{1,\ldots,N\\}$.
Note that the above result is not terribly surprising considering Lemma 9, as
the new condition involves a simpler Minkowski sum in exchange for additional
(reasonable and testable) assumptions on $\Phi$. The proof of this theorem
amounts to measuring the difference between the two Minkowski sums:
###### Proof of Theorem 10.
We start with the following claim:
$\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}=\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)\setminus\left(\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right).$
(14)
Before verifying this claim, let’s first use it to prove the theorem. From
Lemma 9 we know that $\mathcal{A}$ is almost injective if and only if almost
every $x\in\mathbb{R}^{M}$ is not in the Minkowski sum
$\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$
for any $S\subseteq\\{1,\ldots,N\\}$. In other words, the Lebesgue measure of
this Minkowski sum is zero for each $S\subseteq\\{1,\ldots,N\\}$. By (14),
this equivalently means that the Lebesgue measure of
$\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)\setminus\left(\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)$
is zero for each $S\subseteq\\{1,\ldots,N\\}$. Since $\Phi$ spans
$\mathbb{R}^{M}$, this set is empty (and therefore has Lebesgue measure zero)
when $S=\emptyset$ or $S=\\{1,\ldots,N\\}$. Also, since each $\varphi_{n}$ is
nonzero, we know that $\operatorname{span}(\Phi_{S})^{\perp}$ and
$\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$ are proper subspaces of
$\mathbb{R}^{M}$ whenever $S$ is a nonempty proper subset of
$\\{1,\ldots,N\\}$, and so in these cases both subspaces must have Lebesgue
measure zero. As such, we have that for every nonempty proper subset
$S\subseteq\\{1,\ldots,N\\}$,
$\displaystyle\operatorname{Leb}\left[\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)\setminus\left(\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)\right]$
$\displaystyle\qquad\qquad\geq\operatorname{Leb}\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)-\operatorname{Leb}\left(\operatorname{span}(\Phi_{S})^{\perp}\right)-\operatorname{Leb}\left(\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)$
$\displaystyle\qquad\qquad=\operatorname{Leb}\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)$
$\displaystyle\qquad\qquad\geq\operatorname{Leb}\left[\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)\setminus\left(\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)\right].$
In summary,
$\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)\setminus\left(\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)$
having Lebesgue measure zero for each $S\subseteq\\{1,\ldots,N\\}$ is
equivalent to
$\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$
having Lebesgue measure zero for each nonempty proper subset
$S\subseteq\\{1,\ldots,N\\}$, which in turn is equivalent to the Minkowski sum
$\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$
being a proper subspace of $\mathbb{R}^{M}$ for each nonempty proper subset
$S\subseteq\\{1,\ldots,N\\}$, as desired.
Thus, to complete the proof we must verify the claim (14). We will do so by
verifying both inclusions. Clearly
$\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$
is a subset of
$\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$,
so to prove $\subseteq$ in (14), it suffices to show that
$\left(\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}\right)\cap\left(\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)=\emptyset.$
(15)
Assuming to the contrary, then without loss of generality there exist elements
$a\in\operatorname{span}(\Phi_{S})^{\perp}$,
$b\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}$, and
$c\in\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$ such
that $a=b+c$. But this means that $a-b=c\neq 0$ is in both
$\operatorname{span}(\Phi_{S})^{\perp}$ and
$\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$, contradicting the
assumption that the vectors $\Phi=\\{\varphi_{n}\\}_{n=1}^{N}$ span
$\mathbb{R}^{M}$. To prove $\supseteq$ in (14), note that (15) tells us it is
equivalent to show the containment
$\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\subseteq\left(\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}\right)\cup\left(\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right).$
To this end, let $a\in\operatorname{span}(\Phi_{S})^{\perp}$ and
$b\in\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$ so that
$a+b\in\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$.
Then the inclusion follows from observing the following cases:
* (I)
Suppose $a$ and $b$ are nonzero. Then
$a\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}$ and
$b\in\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$,
implying that
$a+b\in\operatorname{span}(\Phi_{S})^{\perp}\setminus\\{0\\}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\setminus\\{0\\}$.
* (II)
Suppose exactly one of $a$ and $b$ are nonzero (without loss of generality
that $a\neq 0$ and $b=0$). Then
$a+b=a\in\operatorname{span}(\Phi_{S})^{\perp}$, implying that
$a+b\in\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$.
* (III)
Suppose $a$ and $b$ are both zero. Then
$a+b\in\operatorname{span}(\Phi_{S})^{\perp}\cup\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$.
Having confirmed both inclusions of our initial claim (14), the proof is
complete. ∎
At this point, consider the following stronger restatement of Theorem 10:
“Suppose each $\varphi_{n}$ is nonzero. Then $\mathcal{A}$ is almost injective
if and only if $\Phi$ spans $\mathbb{R}^{M}$ and the Minkowski sum
$\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}$
is a proper subspace of $\mathbb{R}^{M}$ for each nonempty proper subset
$S\subseteq\\{1,\ldots,N\\}$.” Note that we can move the spanning assumption
into the condition because if $\Phi$ does not span, then we can decompose
almost every $x\in\mathbb{R}^{M}$ as $x=u+v$ such that
$u\in\operatorname{span}(\Phi)$ and $v\in\operatorname{span}(\Phi)^{\perp}$
with $v\neq 0$, and defining $y:=u-v$ then gives
$\mathcal{A}(y)=\mathcal{A}(x)$ despite the fact that $y\neq\pm x$. As for the
assumption that the $\varphi_{n}$’s are nonzero, we note that having
$\varphi_{n}=0$ amounts to having the $n$th entry of $\mathcal{A}(x)$ be zero
for all $x$. As such, $\Phi$ yields almost injectivity precisely when the
nonzero members of $\Phi$ together yield almost injectivity. With this
identification, the stronger restatement of Theorem 10 above can be viewed as
a complete characterization of almost injectivity. Next, we will replace the
Minkowski sum condition with a rather elegant condition involving the ranks of
$\Phi_{S}$ and $\Phi_{S^{\mathrm{c}}}$ by applying the following lemma:
###### Lemma 11 (Inclusion-exclusion principle for subspaces).
Let $U$ and $V$ be subspaces of a common vector space. Then $\dim(U+V)=\dim
U+\dim V-\dim(U\cap V)$.
###### Proof.
Let $A$ be a basis for $U\cap V$ and let $B$ and $C$ be bases for $U$ and $V$,
respectively, such that $A\subseteq B$ and $A\subseteq C$. It can be shown
that $A\cup B\cup C$ forms a basis for $U+V$, which implies that
$\dim(U+V)=|A|+|B\setminus A|+|C\setminus A|=|B|+|C|-|A|=\dim U+\dim
V-\dim(U\cap V).\qed$
###### Theorem 12.
Consider $\Phi=\\{\varphi_{n}\\}_{n=1}^{N}\subseteq\mathbb{R}^{M}$ and the
intensity measurement mapping $\mathcal{A}\colon\mathbb{R}^{M}/\\{\pm
1\\}\rightarrow\mathbb{R}^{N}$ defined by $(\mathcal{A}(x))(n):=|\langle
x,\varphi_{n}\rangle|^{2}$. Suppose each $\varphi_{n}$ is nonzero. Then
$\mathcal{A}$ is almost injective if and only if $\Phi$ spans $\mathbb{R}^{M}$
and $\operatorname{rank}\Phi_{S}+\operatorname{rank}\Phi_{S^{\mathrm{c}}}>M$
for each nonempty proper subset $S\subseteq\\{1,\ldots,N\\}$.
###### Proof.
Considering the discussion after the proof of Theorem 10, it suffices to
assume that $\Phi$ spans $\mathbb{R}^{M}$. Furthermore, considering Theorem
10, it suffices to characterize when
$\dim\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)<M$.
By Lemma 11, we have
$\displaystyle\dim\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)$
$\displaystyle\qquad\qquad=\dim\left(\operatorname{span}(\Phi_{S})^{\perp}\right)+\dim\left(\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)-\dim\left(\operatorname{span}(\Phi_{S})^{\perp}\cap\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right).$
Since $\Phi$ is assumed to span $\mathbb{R}^{M}$, we also have that
$\operatorname{span}(\Phi_{S})^{\perp}\cap\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}=\\{0\\}$,
and so
$\displaystyle\dim\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)$
$\displaystyle=\Big{(}M-\dim\left(\operatorname{span}(\Phi_{S})\right)\Big{)}+\Big{(}M-\dim\left(\operatorname{span}(\Phi_{S^{\mathrm{c}}})\right)\Big{)}-0$
$\displaystyle=2M-\operatorname{rank}\Phi_{S}-\operatorname{rank}\Phi_{S^{\mathrm{c}}}.$
As such,
$\dim\left(\operatorname{span}(\Phi_{S})^{\perp}+\operatorname{span}(\Phi_{S^{\mathrm{c}}})^{\perp}\right)<M$
precisely when
$\operatorname{rank}\Phi_{S}+\operatorname{rank}\Phi_{S^{\mathrm{c}}}>M$. ∎
At this point, we point out some interesting consequences of Theorem 12. First
of all, $\Phi$ cannot be almost injective if $N<M+1$ since
$\operatorname{rank}\Phi_{S}+\operatorname{rank}\Phi_{S^{\mathrm{c}}}\leq|S|+|S^{\mathrm{c}}|=N$.
Also, in the case where $N=M+1$, we note that $\Phi$ is almost injective
precisely when $\Phi$ is full spark, that is, every size-$M$ subcollection is
a spanning set (note this implies that all of the $\varphi_{n}$’s are
nonzero). In fact, every full spark $\Phi$ with $N\geq M+1$ yields almost
injective intensity measurements, which in turn implies that a generic $\Phi$
yields almost injectivity when $N\geq M+1$ [3]. This is in direct analogy with
injectivity in the real case; here, injectivity requires $N\geq 2M-1$,
injectivity with $N=2M-1$ is equivalent to being full spark, and being full
spark suffices for injectivity whenever $N\geq 2M-1$ [3]. Another thing to
check is that the condition for injectivity implies the condition for almost
injectivity (it does).
Having established that full spark ensembles of size $N\geq M+1$ yield almost
injective intensity measurements, we note that checking whether a matrix is
full spark is $\NP$-hard in general [30]. Granted, there are a few explicit
constructions of full spark ensembles which can be used [2, 33], but it would
be nice to have a condition which is not computationally difficult to test in
general. We provide one such condition in the following theorem, but first, we
briefly review the requisite frame theory.
A frame is an ensemble
$\Phi=\\{\varphi_{n}\\}_{n=1}^{N}\subseteq\mathbb{R}^{M}$ together with frame
bounds $0<A\leq B<\infty$ with the property that for every
$x\in\mathbb{R}^{M}$,
$A\|x\|^{2}\leq\sum_{n=1}^{N}|\langle x,\varphi_{n}\rangle|^{2}\leq
B\|x\|^{2}.$
When $A=B$, the frame is said to be tight, and such frames come with a
painless reconstruction formula:
$x=\frac{1}{A}\sum_{n=1}^{N}\langle x,\varphi_{n}\rangle\varphi_{n}.$
To be clear, the theory of frames originated in the context of infinite-
dimensional Hilbert spaces [20, 22], and frames have since been studied in
finite-dimensional settings, primarily because this is the setting in which
they are applied computationally. Of particular interest are so-called unit
norm tight frames (UNTFs), which are tight frames whose frame elements have
unit norm: $\|\varphi_{n}\|=1$ for every $n=1,\ldots,N$. Such frames are
useful in applications; for example, if you encode a signal $x$ using frame
coefficients $\langle x,\varphi_{n}\rangle$ and transmit these coefficients
across a channel, then UNTFs are optimally robust to noise [26] and one
erasure [16]. Intuitively, this optimality comes from the fact that frame
elements of a UNTF are particularly well-distributed in the unit sphere [6].
Another pleasant feature of UNTFs is that it is straightforward to test
whether a given frame is a UNTF: Letting $\Phi=[\varphi_{1}\cdots\varphi_{N}]$
denote an $M\times N$ matrix whose columns are the frame elements, then $\Phi$
is a UNTF precisely when each of the following occurs simultaneously:
* (i)
the rows have equal norm
* (ii)
the rows are orthogonal
* (iii)
the columns have unit norm
(This is a direct consequence of the tight frame’s reconstruction formula and
the fact that a UNTF has unit-norm frame elements; furthermore, since the
columns have unit norm, it is not difficult to see that the rows will
necessarily have norm $\sqrt{N/M}$.) In addition to being able to test that an
ensemble is a UNTF, various UNTFs can be constructed using spectral tetris
[15] (though such frames necessarily have $N\geq 2M$), and every UNTF can be
constructed using the recent theory of eigensteps [10, 24]. Now that UNTFs
have been properly introduced, we relate them to almost injectivity for phase
retrieval:
###### Theorem 13.
If $M$ and $N$ are relatively prime, then every unit norm tight frame
$\Phi=\\{\varphi_{n}\\}_{n=1}^{N}\subseteq\mathbb{R}^{M}$ yields almost
injective intensity measurements.
###### Proof.
Pick a nonempty proper subset $S\subseteq\\{1,\ldots,N\\}$. By Theorem 12, it
suffices to show that
$\operatorname{rank}\Phi_{S}+\operatorname{rank}\Phi_{S^{\mathrm{c}}}>M$, or
equivalently,
$\operatorname{rank}\Phi_{S}\Phi_{S}^{*}+\operatorname{rank}\Phi_{S^{\mathrm{c}}}\Phi_{S^{\mathrm{c}}}^{*}>M$.
Note that since $\Phi$ is a unit norm tight frame, we also have
$\Phi_{S}\Phi_{S}^{*}+\Phi_{S^{\mathrm{c}}}\Phi_{S^{\mathrm{c}}}^{*}=\Phi\Phi^{*}=\tfrac{N}{M}I,$
and so $\Phi_{S}\Phi_{S}^{*}$ and
$\Phi_{S^{\mathrm{c}}}\Phi_{S^{\mathrm{c}}}^{*}$ are simultaneously
diagonalizable, i.e., there exists a unitary matrix $U$ and diagonal matrices
$D_{1}$ and $D_{2}$ such that
$\displaystyle
UD_{1}U^{*}+UD_{2}U^{*}=\Phi_{S}\Phi_{S}^{*}+\Phi_{S^{\mathrm{c}}}\Phi_{S^{\mathrm{c}}}^{*}=\tfrac{N}{M}I.$
Conjugating by $U^{*}$, this then implies that $D_{1}+D_{2}=\tfrac{N}{M}I$.
Let $L_{1}\subseteq\\{1,\ldots,M\\}$ denote the diagonal locations of the
nonzero entries in $D_{1}$, and $L_{2}\subseteq\\{1,\ldots,M\\}$ similarly for
$D_{2}$. To complete the proof, we need to show that $|L_{1}|+|L_{2}|>M$
(since
$|L_{1}|+|L_{2}|=\operatorname{rank}\Phi_{S}\Phi_{S}^{*}+\operatorname{rank}\Phi_{S^{\mathrm{c}}}\Phi_{S^{\mathrm{c}}}^{*}$).
Note that $L_{1}\cup L_{2}\neq\\{1,\ldots,M\\}$ would imply that $D_{1}+D_{2}$
has at least one zero in its diagonal, contradicting the fact that
$D_{1}+D_{2}$ is a nonzero multiple of the identity; as such, $L_{1}\cup
L_{2}=\\{1,\ldots,M\\}$ and $|L_{1}|+|L_{2}|\geq M$. We claim that this
inequality is strict due to the assumption that $M$ and $N$ are relatively
prime. To see this, it suffices to show that $L_{1}\cap L_{2}$ is nonempty.
Suppose to the contrary that $L_{1}$ and $L_{2}$ are disjoint. Then since
$D_{1}+D_{2}=\tfrac{N}{M}I$, every nonzero entry in $D_{1}$ must be $N/M$.
Since $S$ is a nonempty proper subset of $\\{1,\ldots,N\\}$, this means that
there exists $K\in(0,M)$ such that $D_{1}$ has $K$ entries which are $N/M$ and
$M-K$ which are $0$. Thus,
$|S|=\operatorname{Tr}[\Phi_{S}^{*}\Phi_{S}]=\operatorname{Tr}[\Phi_{S}\Phi_{S}^{*}]=\operatorname{Tr}[UD_{1}U^{*}]=\operatorname{Tr}[D_{1}]=K(N/M),$
implying that $N/M=|S|/K$ with $K\neq M$ and $|S|\neq N$. Since this
contradicts the assumption that $N/M$ is in lowest form, we have the desired
result. ∎
In general, whether a UNTF $\Phi$ yields almost injective intensity
measurements is determined by whether it is orthogonally partitionable: $\Phi$
is orthogonally partitionable if there exists a partition $S\sqcup
S^{\mathrm{c}}=\\{1,\ldots,N\\}$ such that $\operatorname{span}(\Phi_{S})$ is
orthogonal to $\operatorname{span}(\Phi_{S^{\mathrm{c}}})$. Specifically, a
UNTF yields almost injective intensity measurements precisely when it is not
orthogonally partitionable. Historically, this property of UNTFs has been
pivotal to the understanding of singularities in the algebraic variety of
UNTFs [23], and it has also played a key role in solutions to the Paulsen
problem [7, 14]. However, it is not clear in general how to efficiently test
for this property; this is why Theorem 13 focuses on such a special case.
Figure 1: The simplex in $\mathbb{R}^{3}$. Pointing out of the page is the
vector $\smash{\frac{1}{\sqrt{3}}(1,1,1)}$, while the other vectors are the
three permutations of $\smash{\frac{1}{\sqrt{3}}(1,-1,-1)}$. Together, these
four vectors form a unit norm tight frame, and since $M=3$ and $N=4$ are
relatively prime, these yield almost injective intensity measurements in
accordance with Theorem 13. For this ensemble, the points $x$ such that
$\mathcal{A}^{-1}(\mathcal{A}(x))\neq\\{\pm x\\}$ are contained in the three
coordinate planes. Above, we depict the intersection between these planes and
the unit sphere. According to Theorem 15, performing phase retrieval with
simplices such as this is $\NP$-hard.
## 4 The computational complexity of phase retrieval
The previous section characterized the real ensembles which yield almost
injective intensity measurements. The benefit of seeking almost injectivity
instead of injectivity is that we can get away with much smaller ensembles.
For example, a full spark ensemble in $\mathbb{R}^{M}$ of size $M+1$ suffices
for almost injectivity, while $2M-1$ measurements are required for
injectivity. In this section, we demonstrate that this savings in the number
of measurements can come at a substantial price in computational requirements
for phase retrieval. In particular, we consider the following problem:
###### Problem 14.
Let $\mathcal{F}=\\{\Phi_{M}\\}_{M=2}^{\infty}$ be a family of ensembles
$\Phi_{M}=\\{\varphi_{M;n}\\}_{n=1}^{N(M)}\subseteq\mathbb{R}^{M}$, where
$N(M)=\poly(M)$. Then $\textsc{ConsistentIntensities}[\mathcal{F}]$ is the
following problem: Given $M\geq 2$ and a rational sequence
$\\{b_{n}\\}_{n=1}^{N(M)}$, does there exist $x\in\mathbb{R}^{M}$ such that
$|\langle x,\varphi_{M;n}\rangle|=b_{n}$ for every $n=1,\ldots,N(M)$?
In this section, we will evaluate the computational complexity of
$\textsc{ConsistentIntensities}[\mathcal{F}]$ for a large class of families of
small ensembles $\mathcal{F}$, but first, we briefly review the main concepts
involved. Complexity theory is chiefly concerned with complexity classes,
which are sets of problems that share certain computational requirements, such
as time or space. For example, the complexity class $\P$ is the set of
problems which can be solved in an amount of time that is bounded by some
polynomial of the bit-length of the input. As another example, $\NP$ contains
all problems for which an affirmative answer comes with a certificate that can
be verified in polynomial time; note that $\P\subseteq\NP$ since for every
problem $A\in\P$, one may ignore the certificate and find the affirmative
answer in polynomial time. One key tool that is used to evaluate the
complexity of a problem is called polynomial-time reduction. This is a
polynomial-time algorithm that solves a problem $A$ by exploiting an oracle
which solves another problem $B$, indicating that solving $A$ is no harder
than solving $B$ (up to polynomial factors in time); if such a reduction
exists, we write $A\leq B$. For example, any efficient phase retrieval
procedure for $\mathcal{F}$ can be used as a subroutine to solve
$\textsc{ConsistentIntensities}[\mathcal{F}]$, indicating that phase retrieval
for $\mathcal{F}$ is at least as hard as
$\textsc{ConsistentIntensities}[\mathcal{F}]$. A problem $B$ is called
$\NP$-hard if $B\geq A$ for every problem $A\in\NP$. Note that since $\leq$ is
transitive, it suffices to show that $B\geq C$ for some $\NP$-hard problem
$C$. Finally, a problem $B$ is called $\NP$-complete if $B\in\NP$ is
$\NP$-hard; intuitively, $\NP$-complete problems are the hardest of problems
in $\NP$. It is an open problem whether $\P=\NP$, but inequality is widely
believed [18]; note that under this assumption, $\NP$-hard problems have no
computationally efficient solution. This provides a proper context for the
main result of this section:
###### Theorem 15.
Let $\mathcal{F}=\\{\Phi_{M}\\}_{M=2}^{\infty}$ be a family of full spark
ensembles $\Phi_{M}=\\{\varphi_{M;n}\\}_{n=1}^{M+1}\subseteq\mathbb{R}^{M}$
with rational entries that can be computed in polynomial time. Then
$\textsc{ConsistentIntensities}[\mathcal{F}]$ is $\NP$-complete.
Note that since the ensembles $\Phi_{M}$ are full spark, the existence of a
solution to the phase retrieval problem $|\langle
x,\varphi_{M;n}\rangle|=b_{n}$ for every $n=1,\ldots,M+1$ implies uniqueness
by Theorem 12. Before proving this theorem, we first relate it to a previous
hardness result from [34]. Specifically, this result can be restated using the
terminology in this paper as follows: There exists a family
$\mathcal{F}=\\{\Phi_{M}\\}_{M=2}^{\infty}$ of ensembles
$\Phi_{M}=\\{\varphi_{M;n}\\}_{n=1}^{2M}\subseteq\mathbb{C}^{M}$, each of
which yielding almost injective intensity measurements, such that
$\textsc{ConsistentIntensities}[\mathcal{F}]$ is $\NP$-complete.
Interestingly, these are the smallest possible almost injective ensembles in
the complex case, and we suspect that the result can be strengthened to the
obvious analogy of Theorem 15:
###### Conjecture 16.
Let $\mathcal{F}=\\{\Phi_{M}\\}_{M=2}^{\infty}$ be a family of ensembles
$\Phi_{M}=\\{\varphi_{M;n}\\}_{n=1}^{2M}\subseteq\mathbb{C}^{M}$ which yield
almost injective intensity measurements and have complex rational entries that
can be computed in polynomial time. Then
$\textsc{ConsistentIntensities}[\mathcal{F}]$ is $\NP$-complete.
To prove Theorem 15, we devise a polynomial-time reduction from the following
problem which is well-known to be $\NP$-complete [29]:
###### Problem 17 (SubsetSum).
Given a finite collection of integers $A$ and an integer $z$, does there exist
a subset $S\subseteq A$ such that $\sum_{a\in S}a=z$?
###### Proof of Theorem 15.
We first show that $\textsc{ConsistentIntensities}[\mathcal{F}]$ is in $\NP$.
Note that if there exists an $x\in\mathbb{R}^{M}$ such that $|\langle
x,\varphi_{M;n}\rangle|=b_{n}$ for every $n=1,\ldots,M+1$, then $x$ will have
all rational entries. Indeed, $v:=\Phi_{M}^{*}x$ has all rational entries,
being a signed version of $\\{b_{n}\\}_{n=1}^{M+1}$, and so
$x=(\Phi_{M}\Phi_{M}^{*})^{-1}\Phi_{M}v$ is also rational. Thus, we can view
$x$ as a certificate of finite bit-length, and for each $n=1,\ldots,M+1$, we
know that $|\langle x,\varphi_{M;n}\rangle|=b_{n}$ can be verified in time
which is polynomial in this bit-length, as desired.
Now we show that $\textsc{ConsistentIntensities}[\mathcal{F}]$ is $\NP$-hard
by reduction from SubsetSum. To this end, take a finite collection of integers
$A$ and an integer $z$. Set $M:=|A|$ and label the members of $A$ as
$\\{a_{m}\\}_{m=1}^{M}$. Let $\Psi$ denote the $M\times M$ matrix whose
columns are the first $M$ members of $\Phi_{M}$. Since $\Phi_{M}$ is full
spark, $\Psi$ is invertible and $\Psi^{-1}\Phi_{M}$ has the form $[I~{}w]$,
where $w$ has all nonzero entries; indeed, if the $m$th entry of $w$ were
zero, then $\Phi_{M}\setminus\\{\varphi_{M;m}\\}$ would not span, violating
full spark. Now define
$b_{n}:=\left\\{\begin{array}[]{ll}\displaystyle{\bigg{|}\frac{a_{n}}{w_{n}}\bigg{|}}&\mbox{if
}n=1,\ldots,M\\\
\displaystyle{\bigg{|}2z-\sum_{m=1}^{M}a_{m}\bigg{|}}&\mbox{if
}n=M+1.\end{array}\right.$ (16)
We claim that an oracle for $\textsc{ConsistentIntensities}[\mathcal{F}]$
would return “yes” from the inputs $M$ and $\\{b_{n}\\}_{n=1}^{M+1}$ defined
above if and only if there exists a subset $S\subseteq A$ such that
$\sum_{a\in S}a=z$, which would complete the reduction.
To prove our claim, we start with ($\Rightarrow$): Suppose there exists
$x\in\mathbb{R}^{M}$ such that $|\langle x,\varphi_{M;n}\rangle|=b_{n}$ for
every $n=1,\ldots,M+1$. Then $y:=\Psi^{*}x$ satisfies $|\langle
y,\Psi^{-1}\varphi_{M;n}\rangle|=b_{n}$ for every $n=1,\ldots,M+1$. Since
$\Psi^{-1}\Phi_{M}=[I~{}w]$, then by (16), the entries of $y$ satisfy
$|y_{m}|=\left|\frac{a_{m}}{w_{m}}\right|\quad\forall
m=1,\ldots,M,\qquad\qquad\bigg{|}\sum_{m=1}^{M}y_{m}w_{m}\bigg{|}=\bigg{|}2z-\sum_{m=1}^{M}a_{m}\bigg{|}.$
By the first equation above, there exists a sequence
$\\{\varepsilon_{m}\\}_{m=1}^{M}$ of $\pm 1$’s such that
$y_{m}=\varepsilon_{m}a_{m}/w_{m}$ for every $m=1,\ldots,M$, and so the second
equation above gives
$\bigg{|}2z-\sum_{m=1}^{M}a_{m}\bigg{|}=\bigg{|}\sum_{m=1}^{M}y_{m}w_{m}\bigg{|}=\bigg{|}\sum_{m=1}^{M}\varepsilon_{m}a_{m}\bigg{|}=\bigg{|}\sum_{\begin{subarray}{c}m=1\\\
\varepsilon_{m}=1\end{subarray}}^{M}a_{m}-\sum_{\begin{subarray}{c}m=1\\\
\varepsilon_{m}=-1\end{subarray}}^{M}a_{m}\bigg{|}=\bigg{|}2\sum_{\begin{subarray}{c}m=1\\\
\varepsilon_{m}=1\end{subarray}}^{M}a_{m}-\sum_{m=1}^{M}a_{m}\bigg{|}.$
Removing the absolute values, this means the left-hand side above is equal to
the right-hand side, up to a sign factor. At this point, isolating $z$ reveals
that $z=\sum_{m\in S}a_{m}$, where $S$ is either $\\{m:\varepsilon_{m}=1\\}$
or $\\{m:\varepsilon_{m}=-1\\}$, depending on the sign factor.
For ($\Leftarrow$), suppose there is a subset $S\subseteq\\{1,\ldots,M\\}$
such that $z=\sum_{m\in S}a_{m}$. Define $\varepsilon_{m}:=1$ when $m\in S$
and $\varepsilon_{m}:=-1$ when $m\not\in S$. Then
$\bigg{|}\sum_{m=1}^{M}\varepsilon_{m}a_{m}\bigg{|}=\bigg{|}\sum_{\begin{subarray}{c}m=1\\\
\varepsilon_{m}=1\end{subarray}}^{M}a_{m}-\sum_{\begin{subarray}{c}m=1\\\
\varepsilon_{m}=-1\end{subarray}}^{M}a_{m}\bigg{|}=\bigg{|}2\sum_{\begin{subarray}{c}m=1\\\
\varepsilon_{m}=1\end{subarray}}^{M}a_{m}-\sum_{m=1}^{M}a_{m}\bigg{|}=\bigg{|}2z-\sum_{m=1}^{M}a_{m}\bigg{|}.$
By the analysis from the ($\Rightarrow$) direction, taking
$y_{m}:=\varepsilon_{m}a_{m}/w_{m}$ for each $m=1,\ldots,M$ then ensures that
$|\langle y,\Psi^{-1}\varphi_{M;n}\rangle|=b_{n}$ for every $n=1,\ldots,M+1$,
which in turn ensures that $x:=(\Psi^{*})^{-1}y$ satisfies $|\langle
x,\varphi_{M;n}\rangle|=b_{n}$ for every $n=1,\ldots,M+1$. ∎
Based on Theorem 15, there is no polynomial-time algorithm to perform phase
retrieval for minimal almost injective ensembles, assuming $\P\neq\NP$. On the
other hand, there exist ensembles of size $2M-1$ for which phase retrieval is
particularly efficient. For example, letting $\delta_{M;m}\in\mathbb{R}^{M}$
denote the $m$th identity basis element, consider the ensemble
$\Phi_{M}:=\\{\delta_{M;m}\\}_{m=1}^{M}\cup\\{\delta_{M;1}+\delta_{M;m}\\}_{m=2}^{M}$;
then one can reconstruct (up to global phase) any $x$ whose first entry is
nonzero by first taking $\hat{x}[1]:=|\langle x,\delta_{M;1}\rangle|$, and
then taking
$\hat{x}[m]:=\frac{1}{2\hat{x}[1]}\Big{(}|\langle
x,\delta_{M;1}+\delta_{M;m}\rangle|^{2}-|\langle
x,\delta_{M;1}\rangle|^{2}-|\langle
x,\delta_{M;m}\rangle|^{2}\Big{)}\qquad\forall m=2,\ldots,M.$
Intuitively, we expect a redundancy threshold that determines whether phase
retrieval can be efficient, and this suggests the following open problem: What
is the smallest $C$ for which there exists a family of ensembles of size
$N=CM+o(M)$ such that phase retrieval can be performed in polynomial time?
## Acknowledgments
The authors thank the Norbert Wiener Center for Harmonic Analysis and
Applications at the University of Maryland, College Park for hosting a
workshop on phase retrieval that helped solidify the main ideas in the almost
injectivity portion of this paper. This work was supported by NSF DMS 1042701
and 1321779. The views expressed in this article are those of the authors and
do not reflect the official policy or position of the United States Air Force,
Department of Defense, or the U.S. Government.
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|
arxiv-papers
| 2013-07-26T21:42:15 |
2024-09-04T02:49:48.530103
|
{
"license": "Public Domain",
"authors": "Matthew Fickus, Dustin G. Mixon, Aaron A. Nelson, Yang Wang",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1307.7176"
}
|
1307.7182
|
# Assembly of filamentary void galaxy configurations
Steven Rieder1,2, Rien van de Weygaert3, Marius Cautun3, Burcu Beygu3 and
Simon Portegies Zwart1
1 Sterrewacht Leiden, Leiden University, P.O. Box 9513, 2300 RA Leiden, The
Netherlands
2 Section System and Network Engineering, University of Amsterdam, Amsterdam,
The Netherlands
3 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700
AV, Groningen, The Netherlands E-mail: [email protected]
(12 July 2013)
###### Abstract
We study the formation and evolution of filamentary configurations of dark
matter haloes in voids. Our investigation uses the high-resolution
$\Lambda$CDM simulation CosmoGrid to look for void systems resembling the
VGS_31 elongated system of three interacting galaxies that was recently
discovered by the Void Galaxy Survey (VGS) inside a large void in the SDSS
galaxy redshift survey. HI data revealed these galaxies to be embedded in a
common elongated envelope, possibly embedded in intravoid filament.
In the CosmoGrid simulation we look for systems similar to VGS_31 in mass,
size and environment. We find a total of eight such systems. For these
systems, we study the distribution of neighbour haloes, the assembly and
evolution of the main haloes and the dynamical evolution of the haloes, as
well as the evolution of the large-scale structure in which the systems are
embedded. The spatial distribution of the haloes follows that of the dark
matter environment.
We find that VGS_31-like systems have a large variation in formation time,
having formed between $10~{}\rm{Gyr}$ ago and the present epoch. However, the
environments in which the systems are embedded evolved resemble each other
substantially. Each of the VGS_31-like systems is embedded in an intra-void
wall, that no later than $z=0.5$ became the only prominent feature in its
environment. While part of the void walls retain a rather featureless
character, we find that around half of them are marked by a pronounced and
rapidly evolving substructure. Five haloes find themselves in a tenuous
filament of a few$~{}h^{-1}{\rm Mpc}$ long inside the intra-void wall.
Finally, we compare the results to observed data from VGS_31. Our study
implies that the VGS_31 galaxies formed in the same (proto)filament, and did
not meet just recently. The diversity amongst the simulated halo systems
indicates that VGS_31 may not be typical for groups of galaxies in voids.
###### keywords:
dark matter - large-scale structure of Universe - cosmology: theory \-
galaxies: formation - galaxies: interactions
††pagerange: Assembly of filamentary void galaxy
configurations–References††pubyear: 2013
## 1 Introduction
Voids form the most prominent aspect of the Megaparsec distribution of
galaxies and matter (Chincarini & Rood, 1975; Gregory & Thompson, 1978;
Zeldovich, Einasto & Shandarin, 1982; Kirshner et al., 1981, 1987; de
Lapparent, Geller & Huchra, 1986). They are enormous regions with sizes in the
range of $20-50~{}h^{-1}{\rm Mpc}$ that are practically devoid of any galaxy,
usually roundish in shape and occupying the major share of volume in the
Universe (see van de Weygaert & Platen, 2011, for a recent review). The voids
are surrounded by sheet-like walls, elongated filaments and dense compact
clusters together with which they define the Cosmic Web (Bond, Kofman &
Pogosyan, 1996), i.e. the salient web-like pattern given by the distribution
of galaxies and matter in the Universe. Theoretical models of void formation
and evolution suggest that voids act as the key organizing element for
arranging matter concentrations into an all-pervasive cosmic network (Icke,
1984; Regős & Geller, 1991; van de Weygaert & van Kampen, 1993; Sahni,
Sathyaprakah & Shandarin, 1994; Sheth & van de Weygaert, 2004; Einasto et al.,
2011; Aragon-Calvo & Szalay, 2013).
Voids mark the transition scale at which density perturbations have decoupled
from the Hubble flow and contracted into recognizable structural features. At
any cosmic epoch, the voids that dominate the spatial matter distribution are
a manifestation of the cosmic structure formation process reaching a non-
linear stage of evolution. Voids emerge out of the density troughs in the
primordial Gaussian field of density fluctuations. Idealized models of
isolated spherically symmetric or ellipsoidal voids (Hoffman & Shaham, 1982;
Icke, 1984; Bertschinger, 1985; Blumenthal et al., 1992; Sheth & van de
Weygaert, 2004) illustrate how the weaker gravity in underdense regions
results in an effective repulsive peculiar gravitational influence. As a
result, matter is evacuating from their interior of initially underdense
regions, while they expand faster than the Hubble flow of the background
Universe. As the voids expand, matter gets squeezed in between them, and
sheets and filaments form the void boundaries.
While idealized spherical or ellipsoidal models provide important insights
into the basic dynamics and evolution of voids, computer simulations of the
gravitational evolution of voids in realistic cosmological environments show a
considerably more complex situation. Sheth & van de Weygaert (2004) (also see
Dubinski et al., 1993; Sahni, Sathyaprakah & Shandarin, 1994; Goldberg &
Vogeley, 2004; Furlanetto & Piran, 2006; Aragon-Calvo & Szalay, 2013) treated
the emergence and evolution of voids within the context of hierarchical
gravitational scenarios. It leads to a considerably modified view of the
evolution of voids, in which the interaction with their surroundings forms a
dominant influence. The void population in the Universe evolves
hierarchically, dictated by two complementary processes. Emerging from a
primordial Gaussian field, voids are often embedded within a larger underdense
region. The smaller voids, matured at an early epoch, tend to merge with one
another to form a larger void, in a process leading to ever larger voids.
Some, usually smaller, voids find themselves in collapsing overdense regions
and will get squeezed and demolished as they collapse with their surroundings.
A key aspect of the hierarchical evolution of voids is the substructure within
their interior. N-body simulations show that while void substructure fades, it
does not disappear (van de Weygaert & van Kampen, 1993). Voids do retain a
rich yet increasingly diluted and diminished infrastructure, as remnants of
the earlier phases of the void hierarchy in which the substructure stood out
more prominent. In fact, the slowing of growth of substructure in a void is
quite similar to structure evolution in a low $\Omega$ Universe (Goldberg &
Vogeley, 2004). Structure within voids assumes a range of forms, and includes
filamentary and sheet-like features as well as a population of low mass dark
matter haloes and galaxies (see e.g. van de Weygaert & van Kampen, 1993;
Gottlöber et al., 2003). Although challenging, void substructure has also been
found in the observational reality. For example, the SDSS galaxy survey has
uncovered a substantial level of substructure within the Boötes void (Platen,
2009), confirming tentative indications for a filamentary feature by Szomoru
et al. (1996).
The most interesting denizens of voids are the rare galaxies that populate
these underdense region, the void galaxies (Szomoru et al., 1996; Kuhn, Hopp &
Elsaesser, 1997; Popescu, Hopp & Elsaesser, 1997; Karachentseva, Karachentsev
& Richter, 1999; Grogin & Geller, 1999, 2000; Hoyle & Vogeley, 2002, 2004;
Rojas et al., 2004, 2005; Tikhonov & Karachentsev, 2006; Patiri et al., 2006a,
b; Ceccarelli et al., 2006; Park et al., 2007; von Benda-Beckmann & Müller,
2008; Wegner & Grogin, 2008; Stanonik et al., 2009; Kreckel et al., 2011;
Pustilnik & Tepliakova, 2011; Kreckel et al., 2012; Hoyle, Vogeley & Pan,
2012). The relation between void galaxies and their surroundings forms an
important aspect of the recent interest in environmental influences on galaxy
formation. Void galaxies appear to have significantly different properties
than average field galaxies. They appear to reside in a more youthful state of
star formation and possess larger and less distorted gas reservoirs. Analysis
of void galaxies in the SDSS and 2dFGRS indicate that void galaxies are bluer
and have higher specific star formation rates than galaxies in denser
environments.
### 1.1 The Void Galaxy Survey
A major systematic study of void galaxies is the Void Galaxy Survey (VGS), a
multi-wavelength program to study $\sim$60 void galaxies selected from the
SDSS DR7 redshift survey (Stanonik et al., 2009; Kreckel et al., 2011, 2012).
These galaxies were selected from the deepest inner regions of voids, with no
a priori bias on the basis of the intrinsic properties of the void galaxies.
The voids were identified using of a unique geometric technique, involving the
Watershed Void Finder (Platen, van de Weygaert & Jones, 2007) applied to a
DTFE density field reconstruction (Schaap & van de Weygaert, 2000). An
important part of the program concerns the gas content of the void galaxies,
and thus far the HI structure of 55 VGS galaxies has been mapped. In addition,
it also involves deep B and R imaging of all galaxies, H$\alpha$ and GALEX UV
data for assessing the star formation properties of the void galaxies.
Perhaps the most interesting configuration found by the Void Galaxy Survey is
VGS_31 (Beygu et al., 2013). Embedded in an elongated common HI cloud, at
least three galaxies find themselves in a filamentary arrangement with a size
of a few hundred kpc. One of these objects is a Markarian galaxy, showing
evidence for recent accretion of minor galaxies. Along with the central
galaxy, which shows strong signs of recent interaction, there is also a
starburst galaxy.
We suspect, from assessing the structure of the void, that the gaseous VGS_31
filament is affiliated to a larger filamentary configuration running across
the void and visible at one of the boundaries of the void. This elicits the
impression that VGS_31 represents a rare specimen of a high density spot in a
tenuous dark matter void filament. Given the slower rate of evolution in
voids, it may mean that we find ourselves in the unique situation of
witnessing the recent assembly of a filamentary galaxy group, a characteristic
stage in the galaxy and structure formation process.
### 1.2 Outline
In this study we concentrate on implications of the unique VGS_31
configuration for our understanding of the dynamical evolution of void
filaments and their galaxy population. We are interested in the assembly of
the filament configuration itself, as well as that of the halo population in
its realm. In fact, we use the specific characteristics of the VGS_31
galaxies, roughly translated from galaxy to dark matter halo, to search for
similar dark halo configurations in the CosmoGrid simulation (Portegies Zwart
et al., 2010; Ishiyama et al., 2013, see Figure 1).
Subsequently, we study in detail the formation and evolution of the entire
environment of these haloes. In this way, we address a range of questions.
What has been the assembly and merging history of the configuration? Did the
VGS_31 galaxies recently meet up and assemble into a filament, or have they
always been together? Is the filament an old feature, or did it emerge only
recently? May we suspect the presence of more light mass galaxies in the
immediate surrounding of VGS_31, or should we not expect more than three such
galaxies in the desolate void region?
Figure 1: A snapshot of the full CosmoGrid volume seen from different sides,
with dots indicating the locations of the void halo systems. The images
display the full $(21~{}h^{-1}{\rm Mpc})^{3}$ volume.
Our study uses a pure dark matter N-body simulation. While a full
understanding of the unique properties of VGS_31 evidently should involve the
complexities of its gas dynamical history, along with that of the stellar
populations, here we specifically concentrate on the overall gravitational
aspects of its dynamical evolution. The reason for this is that the overall
evolution of the filamentary structure will be dictated by the gravitational
influence of the mass concentrations in and around the void. For a proper
understanding of the context in which VGS_31 may have formed, it is therefore
better to concentrate solely on the gravitational evolution.
The outline of this paper is as follows. In section 2.1, we discuss the
simulation used in this article, along with the criteria which we used for the
selecting VGS-31 resembling halo configurations. The properties and evolution
of the eight selected halo groups are presented and discussed in section 3.
Section 4 continues the discussion by assessing the large scale environment in
which the VGS_31 resembling configurations are situated, with special
attention to the walls and filaments in which they reside. We also investigate
the evolution of the surrounding filamentary pattern as the haloes emerge and
evolve. Finally, in section 5 we evaluate and discuss the most likely scenario
for the formation of void systems like VGS_31. In section 6 we summarize and
discuss our findings.
## 2 Simulations
### 2.1 Setup
In order to evaluate possible formation scenarios for systems like VGS_31, we
investigate the formation of systems with similar properties in a cosmological
simulation. For this purpose, we use the CosmoGrid $\Lambda$CDM simulation
(Ishiyama et al., 2013). The CosmoGrid simulation contains $2048^{3}$
particles within a volume of $21~{}h^{-1}{\rm Mpc}^{3}$, and has high enough
mass resolution ($8.9\times 10^{4}\mbox{${h^{-1}\rm M}_{\odot}$}$ per
particle) to study both dark matter haloes and the dark environment in which
the haloes form. The CosmoGrid simulation used a gravitational softening
length $\epsilon$ of 175 parsec, and the following cosmological parameters:
$\Omega_{m}=0.3,\Omega_{\Lambda}=0.7,h=0.7,\sigma_{8}=0.8$ and $n=1.0$.
The first reduction step concerns the detection and identification of haloes
and their properties in the CosmoGrid simulation. For this, we use the
Rockstar (Behroozi, Wechsler & Wu, 2013) halo finder. Rockstar uses a six-
dimensional friends-of-friends algorithm to detect haloes in phase-space. It
excels in tracking substructure, even in ongoing major mergers and in halo
centres (e.g. Knebe et al., 2011; Onions et al., 2012).
Since we are interested in the formation history of the haloes, we analyse
multiple snapshots. Merger trees are constructed to identify haloes across the
snapshots, for which we use the gravitationally consistent merger tree code
from Behroozi et al. (2013). For this we use 193 CosmoGrid snapshots, equally
spaced in time at 70 Myr intervals.
Finally, to compare the CosmoGrid haloes to the galaxies observed in void
regions, we need to identify the regions in CosmoGrid that can be classified
as voids. In doing so we compute the density field using the Delaunay
Tessellation Field Estimator (DTFE, Schaap & van de Weygaert, 2000; van de
Weygaert & Schaap, 2009; Cautun & van de Weygaert, 2011). We express the
resulting density in units of the mean background density $<\rho>$ as
$1+\delta=\rho/<\rho>$. The resulting density field is smoothed with
$1~{}h^{-1}{\rm Mpc}$ Gaussian filter to obtain a large scale density field.
We identify voids as the regions with a $1~{}h^{-1}{\rm Mpc}$ smoothed density
contrast of $\delta<-0.5$.
### 2.2 Selection of the simulated haloes
The VGS_31 system consists of three galaxies with spectrophotometric redshift
$z=0.0209$. The principal galaxies VGS_31a and VGS_31b, and the 2 magnitudes
fainter galaxy VGS_31c, are stretched along an elongated configuration of
$\sim\,120~{}{\rm kpc}$ in size (see Figure 2). The properties of the VGS_31
galaxies are listed in Table LABEL:Tab:VGSsystems. The three galaxies are
connected by an HI bridge that forms a filamentary structure in the void
(Beygu et al., 2013). Both VGS_31a and VGS_31b show strong signs of tidal
interactions. VGS_31b has a tidal tail and a ring like structure wrapped
around the disk. This structure can be the result of mutual gravitational
interaction with VGS_31a or may be caused by a fourth object that fell in
VGS_31b.
Figure 2: B band image of the VGS_31 system: VGS_31_b (left), VGS_31_a
(centre) and VGS_31_c (right). The physical scale of the system may be
inferred from the bar in the lower left-hand corner. See Beygu et al. (2013).
Table 1: Some of the properties of VGS_31 member galaxies.
Name | $M_{*}$ | $M_{HI}$ | $M_{\textit{dyn}}$ | $\delta$
---|---|---|---|---
| $10^{8}M_{\odot}$ | $10^{8}M_{\odot}$ | $10^{10}M_{\odot}$ |
(1) | (2) | (3) | (4) | (5)
VGS_31a | 35.1 | $19.89\pm 2.9$ | $<2.31$ | -0.64
VGS_31b | 105.31 | $14.63\pm 1.97$ | |
VGS_31c | 2.92 | $1.66\pm 0.95$ | |
Object name (1). Stellar mass (2). HI mass (3). Dynamic mass (4). Density
contrast after applying a $1~{}h^{-1}{\rm Mpc}$ Gaussian filter (5).
We use the CosmoGrid simulation to select halo configurations that resemble
VGS_31. In doing so we define a set of five criteria that the halo
configuration should fulfil at $z=0$. The first two criteria involve the
properties of individual haloes:
(a) CGV-A (b) CGV-B
(c) CGV-C (d) CGV-D
(e) CGV-E (f) CGV-F
(g) CGV-G (h) CGV-H
Figure 3: CosmoGrid Void systems A - H. The frames show the dark matter
density distribution in regions of $1~{}h^{-1}{\rm Mpc}^{3}$ around the
principal haloes of each CGV system.
* •
We select only haloes with mass $M_{\rm vir}$ in the range $2\times
10^{10}$${h^{-1}\rm M}_{\odot}$ to $10^{11}$${h^{-1}\rm M}_{\odot}$. This
represents a reasonable estimate for the mass of the most massive dark matter
halo in the VGS_31 system.
* •
Out of the haloes found above, we keep only the ones which reside in void-like
region, where the $1~{}h^{-1}{\rm Mpc}$ smoothed density fulfils
$\delta\leq-0.50$.
There are 84 haloes in the CosmoGrid simulation that fulfil the above two
criteria. Subsequently, we further restrict the selection to those haloes that
are located within a system that is similar to VGS_31. To that end, we look at
the properties of all haloes and subhaloes within a distance of
$200~{}h^{-1}{\rm kpc}$ from the main haloes selected above. A system is
selected when the primary halo has, within $200~{}h^{-1}{\rm kpc}$:
* •
a secondary (sub)halo with $M_{\rm vir}>5\times 10^{9}$${h^{-1}\rm
M}_{\odot}$,
* •
a tertiary (sub)halo with $M_{\rm vir}>10^{9}$${h^{-1}\rm M}_{\odot}$,
* •
no more than 5 neighbour (sub)haloes with $M_{\rm vir}>5\times
10^{9}$${h^{-1}\rm M}_{\odot}$.
Following the application of these criteria, we find a total of 8 VGS_31-like
systems in the CosmoGrid simulation. We call these systems the CosmoGrid void
systems, abbreviated to CGV. The individual haloes in the eight void halo
systems are indicated by means of a letter, eg. CGV-A_a and CGV-A_b. In the
subsequent sections we investigate the halo evolution, merger history and
large scale environment of the eight void halo configurations.
Table 2: Properties of the eight CosmoGrid Void systems found to resemble
VGS_31.
Name | $M_{\rm vir}$ | $R_{\rm vir}$ | $V_{\rm max}$ | $r$ | $\theta$ | $\phi$ | $\delta$ | last MM | $\angle_{\rm wall}$ | $\angle_{\rm fil}$
---|---|---|---|---|---|---|---|---|---|---
| $10^{10}\mbox{${h^{-1}\rm M}_{\odot}$}$ | $~{}h^{-1}{\rm kpc}$ | km/s | $~{}h^{-1}{\rm kpc}$ | ∘ | ∘ | | Gyr | ∘ | ∘
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
CGV-A_a | 3.15 | 64.5 | 48.7 | | | | -0.68 | - | 51.45 | -
CGV-A_b | 0.59 | 36.8 | 34.6 | 17 | 94.0 | 111.2 | -0.68 | | 12.48 | -
CGV-A_c | 0.16 | 23.9 | 22.1 | 76 | 61.9 | 122.6 | -0.68 | | 59.24 | -
CGV-B_a | 3.95 | 69.5 | 63.5 | | | | -0.51 | 5.24 | 20.80 | 80.71
CGV-B_b | 0.87 | 42.0 | 47.6 | 23 | 20.1 | 141.5 | -0.51 | | 46.76 | 62.02
CGV-B_c | 0.16 | 23.7 | 29.8 | 20 | 26.9 | 135.8 | -0.51 | | 17.04 | 86.71
CGV-C_a | 2.99 | 63.3 | 54.8 | | | | -0.51 | 1.19 | 11.94 | -
CGV-C_b | 0.54 | 35.8 | 34.0 | 37 | 36.1 | 142.7 | -0.51 | | 34.79 | -
CGV-C_c | 0.14 | 23.0 | 24.2 | 64 | 2.9 | 81.4 | -0.51 | | 5.49 | -
CGV-C_d | 0.13 | 22.3 | 22.6 | 113 | 93.4 | -19.5 | -0.51 | | 82.73 | -
CGV-D_a | 4.60 | 73.1 | 61.3 | | | | -0.63 | 10.9 | 23.18 | 20.41
CGV-D_b | 0.93 | 42.9 | 38.9 | 86 | 114.0 | -119.2 | -0.63 | | 2.36 | 55.11
CGV-D_c | 0.16 | 23.6 | 30.7 | 71 | 65.9 | -4.1 | -0.63 | | 40.89 | 62.16
CGV-D_d | 0.13 | 22.3 | 24.1 | 32 | 52.7 | -25.2 | -0.63 | | 18.54 | 86.89
CGV-E_a | 1.99 | 55.3 | 54.4 | | | | -0.57 | 2.44 | 8.54 | 30.67
CGV-E_b | 1.01 | 44.1 | 44.7 | 82 | 104.9 | 130.1 | -0.57 | | 68.28 | 85.00
CGV-E_c | 0.23 | 26.8 | 33.5 | 120 | 115.7 | 162.7 | -0.57 | | 72.48 | 81.23
CGV-F_a | 2.27 | 57.8 | 55.0 | | | | -0.62 | - | 49.25 | 73.83
CGV-F_b | 0.74 | 39.8 | 40.1 | 14 | 124.3 | -119.7 | -0.62 | | 77.65 | 89.75
CGV-F_c | 0.11 | 21.2 | 28.9 | 6 | 169.6 | -101.4 | -0.62 | | 63.99 | 82.47
CGV-G_a | 2.14 | 56.7 | 57.4 | | | | -0.61 | 5.80 | 4.26 | 12.07
CGV-G_b | 0.93 | 42.9 | 47.8 | 139 | 37.2 | -55.6 | -0.61 | | 20.95 | 34.20
CGV-G_c | 0.38 | 31.8 | 34.4 | 74 | 45.3 | 94.0 | -0.61 | | 18.27 | 15.73
CGV-H_a | 4.63 | 73.3 | 66.0 | | | | -0.50 | 8.45 | 4.01 | -
CGV-H_b | 4.69 | 73.6 | 68.7 | 199 | 145.5 | 164.4 | -0.50 | | 17.65 | -
CGV-H_c | 0.69 | 38.8 | 38.0 | 153 | 151.8 | -89.4 | -0.50 | | 8.84 | -
CGV-H_d | 0.28 | 28.6 | 28.4 | 92 | 111.7 | -71.6 | -0.50 | | 43.94 | -
CGV-H_e | 0.10 | 20.6 | 27.9 | 33 | 147.3 | 67.0 | -0.50 | | 67.46 | -
CGV-H_f | 0.10 | 20.5 | 22.7 | 86 | 142.1 | -76.8 | -0.50 | | 54.10 | -
Object name (1). Virial mass (2). Virial radius (3). Maximum rotational
velocity (4). Position relative to most massive system (5,6,7). Density
contrast at halo position (smoothed with $1~{}h^{-1}{\rm Mpc}$ Gaussian
filter) (8). Time at which the last major merger took place (9). Angle between
the angular momentum axis of the halo and the normal of the wall (10). Angle
between the angular momentum axis of the halo and the filament. (11)
### 2.3 Analysis of the environment
We plan to investigate the formation and evolution of the eight CGV systems
within the context of the large scale environment in which they reside. In
doing so we use the NEXUS+ method (Cautun, van de Weygaert & Jones, 2013) to
identify the morphology of large scale structure around the selected haloes.
At each location within the simulation box, it determines whether it belongs
to a void or field region, a wall, a filament or a dense cluster node.
Figure 4: The spatial distribution of the haloes and subhaloes in the eight
CGV systems. The blue points show objects more massive than
$10^{9}\mbox{${h^{-1}\rm M}_{\odot}$}$, with point sizes proportional to halo
mass. Red points show haloes and subhaloes in the mass range $(0.1-1)\times
10^{9}\mbox{${h^{-1}\rm M}_{\odot}$}$. The plane of the projection is along
the large scale wall in which these systems are embedded.
The NEXUS+ algorithm is a multiscale formalism that assigns the local
morphology on the basis of a scale-space analysis. It is an elaboration and
extension of the MMF algorithm introduced by Aragón-Calvo et al. (2007). It
translates a given density field into a scale-space representation by
smoothing the field on a range of scales. The morphology signature at each of
the scales is inferred from the eigenvalues of the Hessian of the density
field. The final morphology is determined by selecting the scale which yields
the maximum signature value. To discard spurious detections, we use a set of
physical criteria to set thresholds for significant morphology signatures.
For a detailed description of the NEXUS+ algorithm, along with a comparison
with other Cosmic Web detection algorithms, we refer to Cautun, van de
Weygaert & Jones (2013). The method involves the following sequel of key
steps:
1. 1.
Application of the Log-Gaussian filter of width $R_{n}$ to the density field.
2. 2.
Calculation of the Hessian matrix eigenvalues for the filtered density field.
3. 3.
Assigning to each point a cluster, filament and wall signature on the basis of
the three Hessian eigenvalues computed in the previous step.
4. 4.
Repetition of steps (i) to (iii) over a range of smoothing scales
$(R_{0},R_{1},..,R_{N})$. For this analysis we filter from
$R_{0}=0.1~{}h^{-1}{\rm Mpc}$ to $4~{}h^{-1}{\rm Mpc}$, in steps
$R_{n}=R_{0}2^{n/2}$.
5. 5.
Combination of the morphology signatures at each scale to determine the final
scale independent cluster, filament and wall signature.
6. 6.
Physical criteria are used to set detection thresholds for significant values
of the morphology signatures.
Two important characteristics of NEXUS+ makes it the ideal tool for studying
filamentary and wall-like structures in lower density regions. First of all,
NEXUS+ is a scale independent method which means that it has the same
detection sensitivity for both large and thin filaments and walls. And
secondly, Cautun, van de Weygaert & Jones (2013) showed that the method picks
up even the more tenuous structures that permeate the voids. Usually these
structures have smaller densities and are less pronounced than the more
massive filaments and walls, but locally they still have a high contrast with
respect to the background and serve as pathways for emptying the voids. Both
of these two strengths are crucial for this work since the CGV haloes populate
void-like regions with very thin and tenuous filaments and walls.
## 3 Evolution of void haloes
In the CosmoGrid simulation we find a total of eight systems (see Figure 3)
adhering to the search parameters specified in section 2.2. We label these
CosmoGrid void systems CGV-A to CGV-H. These systems contains from 3 up to 6
haloes with masses larger than $10^{9}\mbox{${\rm M}_{\odot}$}$. They are
labelled by an underscore letter, eg. CGV-H_a or CGV-H_f. In Figure 1, we show
the locations of these systems in three mutually perpendicular projections of
the $21~{}h^{-1}{\rm Mpc}$ CosmoGrid box. Figure 3 further zooms in on the
structure of these systems by showing the density distribution in boxes of
$1~{}h^{-1}{\rm Mpc}$ surrounding the eight configurations.
At z=0, the CGV haloes have a similar appearance. Within a radius of
$500~{}h^{-1}{\rm kpc}$, the primary halo of most of the systems is the
largest object. The exceptions are the CGV-E and CGV-H systems, which have a
larger neighbouring halo. The general properties of the CGV systems are listed
in Table LABEL:Tab:CGVsystems.
Figure 4 provides an impression of the spatial distribution of the haloes in
these eight CVG systems. The principal haloes, those with a mass in excess of
$10^{9}\mbox{${\rm M}_{\odot}$}$, are represented by a blue dot whose size is
proportional to its mass. They are the haloes listed in Table
LABEL:Tab:CGVsystems. In addition, we plot the location of surrounding small
haloes with a mass in the range of $10^{8}<M<10^{9}\mbox{${\rm M}_{\odot}$}$.
While there are substantial differences between the small-scale details of the
mass distribution, we can recognize the global aspect of a filamentary
arrangement of a few dominant haloes that characterizes VGS_31. This is
particularly clear for the systems CGV-D, CGV-E, CGV-G and CGV-H.
Figure 5: Shape of the primary CGV haloes, CGV-A_a to CGV-G_a. Each track
represents the change in the shape of the halo with distance from the halo
centre, out to the virial radius (indicated with a dot). We distinguish three
interesting regions in the plot: top right ($a\approx b\approx c$, i.e.
$c/a=b/a=1$) indicates a spherical halo, bottom left ($a>b\approx c$, ie.
$c/a\approx b/a$) indicates a stretched halo (cigar shaped), bottom right
($a\approx b\gg c$, ie. $c/a\ll b/a=1$) indicates a flattened halo. We
emphasize the tracks for CGV-D_a (solid red) and CGV-G_a (dashed blue).
### 3.1 Halo Structure
To investigate the shape characteristics of the principal haloes, we evaluate
the shape of their mass distribution as a function of radius. To this end, we
measure the principal axis ratios of the mass distribution contained within a
given radius. These are obtained from the moment of inertia tensor for the
mass contained within that radius. In Figure 5 we plot the resulting run of
shape - characterized by the two axis ratios $b/a$ and $c/a$, where $a\geq
b\geq c$ \- for a range of radii smaller than the virial radius, $r<R_{\rm
vir}$. Spherical haloes would be found in the top right-hand of the figure,
with $b/a\approx c/a\approx 1$. Haloes at the bottom left-hand corner, where
$c\approx b\ll a$, resemble elongated spindles while those at the bottom
right-hand corner, with $c\ll b\approx a$, have a flattened shape.
Each halo is represented by a trail through the shape diagram, with each point
on the trail representing the shape of the halo at one particular radius.
Figure 5 emphasizes the trails of CGV-D_a (solid red) and CGV-G_a (dashed
blue), while the results for the remaining six haloes are presented in grey.
The shape of the quiescently evolving CGV-G_a halo tends towards a near-
spherical shape, as one may expect (Araya-Melo et al., 2009). By contrast, the
strongly evolving primary CGV-D_a halo has a strongly varying shape. In the
centre and near the virial radius it is largely spherical, while in between it
is more stretched.
### 3.2 Halo Assembly and Evolution
Using the merger trees of the primary CGV haloes, we investigate the evolution
and assembly history of the eight systems.
We find that only the CGV-D and CGV-H systems experienced major mergers in the
last half Hubble time, the other systems undergoing only smaller mergers. From
the CGV systems we select the two most extreme cases that we study in more
detail: CGV-D as a recently formed system and CGV-G as a system that formed
very early on. The primary halo of the CGV-D system (CGV-D_a) formed at a very
late moment from many similar-sized progenitors. It only appears as the
dominant halo around $t=10~{}\rm{Gyr}$, when it experiences its last big
merger event. By contrast, the central CGV-G halo (CGV-G_a) formed much
earlier, and did not experience any significant merger after $t=5.5$ Gyr.
In Figure 6, we show the mass accretion history of the primary haloes of both
systems. After $t=4$ Gyr, CGV-D_a shows sudden, major accretions of mass on
three occasions, at $t=6,\ 7.7\mbox{ and }9.2$ Gyr. This halo reached 50% of
its final virial mass only at $t=8.8$ Gyr. CGV-G_a, on the other hand, had
already reached 50% of its virial mass at $t=2.5$ Gyr and does not show any
large increases of mass after $t=4$ Gyr.
The detailed merger histories of systems CGV-D and CGV-G - limited to haloes
larger than $5\times 10^{7}\mbox{${h^{-1}\rm M}_{\odot}$}$ \- are shown in
Figure 7. The figure depicts the merger tree, along with a corresponding
sequence of visual images of the assembly of these systems.
The sequence of images from Figure 7 suggests that the assembly takes place
within a spatial configuration of hierarchically evolving filamentary
structures. For both systems, the many filaments that are clearly visible in
the first snapshot merge into a single, thicker filament by the second
snapshot. As the system evolves, it collects most of the mass from the
filament as it gets accreted onto the haloes. We analyse this in more detail
in section 4.
Figure 6: Mass accretion history for two selected haloes, CGV-D_a (solid red)
and CGV-G_a (dashed blue). The plot gives the mass contained within each halo
as a function of time. CGV-D_a is marked by three sudden major mass accretions
after $t=4$ Gyr, while CGV-G_a leads a quiescent life after it experience an
early major merger at $t=2.5$ Gyr.
Figure 7: The merger history of the central haloes of CGV-D (left) and CGV-G
(right). The size of the filled circles is proportional to the virial mass of
the halo. We show only haloes and subhaloes with a peak mass larger than
$5\times 10^{7}\mbox{${h^{-1}\rm M}_{\odot}$}$ that are accreted before $z=0$.
Halo D had a violent merger history, originating from many smaller systems,
whereas halo G has remained virtually unchanged since very early in its
history.
Figure 8: The relative physical distance between the main and secondary haloes
for each of the eight CGV systems. Plotted is physical distance, in kpc, as a
function of cosmic time (Gyr).
Figure 9: The relative co-moving distance between the main and secondary
haloes for each of the eight CGV systems. Plotted is physical distance, in
kpc, as a function of cosmic time (Gyr).
### 3.3 Dynamical Evolution
We use the dominant mass concentrations in each CGV system to get an
impression of the global evolution of the mass distribution around the central
halo. Using the merger tree of each void halo configuration, we obtain the
location of the main and secondary halo for each of the CGV systems. To assess
the overall dynamics, we first look at the physical dimension of the emerging
systems. Figure 9 shows the physical distance between the main and secondary
halo of each system. In all cases we see the typical development of an
overdense region: a gradual slow-down of the cosmic expansion, followed by a
turnaround into a contraction and collapse. We find that the average physical
distance between the main and secondary haloes increases from about
$100~{}{\rm kpc}$ at $t=1~{}\rm{Gyr}$ to $200~{}{\rm kpc}$ at
$t=6~{}\rm{Gyr}$. Subsequently, the systems start to contract to $100~{}{\rm
kpc}$ at $t=13.5~{}\rm{Gyr}$. The exceptions are CGV-H and CGV-G, and as well
CGV-E and CGV-F. CGV-E and CGV-F display more erratic behaviour. For a long
timespan, CVG-E hovers around the same physical size, turning around only at
$t\approx 9~{}\rm{Gyr}$. To a large extent, this is determined by the dominant
external mass concentration in the vicinity of CGV-E. Even more deviant is the
evolution of CGV-F, where we distinguish an early and a later period of
recession and approach between the principal and secondary halo. It is a
reflection of a sequence of mergers, in which the two principal haloes at an
early time merged into a halo which subsequently started its approach towards
a third halo.
The corresponding evolution of the co-moving distance between the two main
haloes of each CGV system provides complementary information on their
dynamical evolution. The evolving co-moving distance is plotted in Figure 9.
Evidently, each of these overdense void halo systems is contracting in co-
moving space. We find that the distance between main and secondary halo
decreases from about $400-600~{}h^{-1}{\rm kpc}$ at $t=1~{}\rm{Gyr}$ to its
current value at $z=0$ of less than $200~{}h^{-1}{\rm kpc}$. CGV-G, CGV-E and
CGV-F have a markedly different history than the others. A rapid decline at
early times is followed by a shallow decline over the last $10~{}\rm{Gyr}$. It
is the reflection of an early merger of haloes, followed by a more quiescent
period in which the merged haloes gradually move towards a third halo.
## 4 Large scale environment
The various VGS_31 resembling halo configurations are embedded in either walls
or filaments within the interior of a void. For our study, it is therefore of
particular interest to investigate the nature of the large-scale filamentary
and planar features in which the CGV systems reside.
Figure 10: CGV-G and its large-scale void environment. Each of the frames
shows the projected density distribution, within a $1~{}h^{-1}{\rm Mpc}$ thick
slice in a $7~{}h^{-1}{\rm Mpc}$ wide region around CGV-G. Top left: XZ plane;
top right: YZ plane; bottom left: XY plane. Bottom right: a $1~{}h^{-1}{\rm
Mpc}$ wide zoom-in onto the XY plane, centred on CGV-G. Particularly
noteworthy is the pattern of largely aligned tenuous intravoid filaments in
the YZ plane. Figure 11: The evolution of the morphology of the mass
distribution around the central halo of CGV-G. The wall-like (orange) and
filamentary (blue) features have been identified with the help of NEXUS+. The
frames show the features in a box of $5~{}h^{-1}{\rm Mpc}$ (co-moving) size
and $1~{}h^{-1}{\rm Mpc}$ thickness. Within each frame the location of CGV-G_a
is indicated by a white dot. The figure shows the evolution of the
morphological features at four redshifts: $z=3.7,z=1.6,z=0.55$ and $z=0.0$.
The first two columns correspond to edge-on orientations of the wall, with the
leftmost ones showing the filamentary evolution along the wall and the central
one that of the evolution of the wall-like features. The right-hand column
depicts the evolution of the filamentary structures within the plane of the
wall.
(a) CGV-D_a, $z=3.7$ (b) CGV-G_a, $z=3.7$
(c) $z=1.6$ (d) $z=1.6$
(e) $z=0.55$ (f) $z=0.55$
(g) $z=0$ (h) $z=0$
Figure 12: Mollweide projection of the angular dark matter density
distribution around haloes CGV-D_a (left) and CGV-G_a (right). To obtain the
sky density we projected the dark matter density within a distance of
$1~{}h^{-1}{\rm Mpc}$ from the primary halo centre. The figure depicts the
evolution of the sky density at four redshifts: $z=3.7,z=1.6,z=0.55$ and
$z=0$. The signature of a wall-like configuration is a circular mass
arrangement over the sky, that of a filamentary structure consist of two dense
spots at diametrically opposite angular positions. In the case of both CGV-D
and CGV-G, the evolution towards a wall with an intersecting filament at $z=0$
is clearly visible. Dark blue areas correspond to a mass count
$<10^{5}\mbox{${h^{-1}\rm M}_{\odot}$}$, whereas red areas correspond to a
mass count $>10^{10}\mbox{${h^{-1}\rm M}_{\odot}$}$.
We first look at the specific structural environment of one particular CGV
complex, CGV-G. Subsequently, we inspect the generic structural morphology of
the mass and halo distribution around the CGV systems. Finally, we assess the
dynamical evolution of the anisotropic mass distribution around the CGV
systems.
### 4.1 The web-like environment of CGV-G
The mass distribution within a $7~{}h^{-1}{\rm Mpc}$ box around the CGV-G halo
complex is shown in Figure 10. It depicts the projected mass distribution
along three mutually perpendicular planes. It includes a $1~{}h^{-1}{\rm Mpc}$
sized zoom-in, in the XY plane, onto the halo complex.
The global structure of the mass distribution is that of a wall extending over
the YZ plane. In the XY- and XZ-projections, the wall is seen edge-on. They
convey the impression of the coherent nature of the wall, in particular along
the ridge in the Z-direction. This is confirmed by the NEXUS+ analysis of the
morphological nature of the mass distribution, presented in Figure 11. At the
current epoch we clearly distinguish a prominent wall-like structure (orange,
lower central frame). Within the plane of the wall, the halo \- indicated by a
white dot - is located in a filament (blue, lower right-hand frame). These
findings suggest that in the immediate vicinity of the CGV systems we should
expect haloes to be aligned along the filament.
The filamentary nature of the immediate halo environment may also be inferred
from the pattern seen in the Mollweide sky projection of the surrounding dark
matter distribution. Figure 12 shows this for the dark matter distribution
around CGV-G out to a radius of $1~{}h^{-1}{\rm Mpc}$. At $z=0$, the angular
distribution is marked by the typical signature of a filament (lower right-
hand frame): two high density spots at diametrically opposite locations. These
spots indicate the angular direction of the filament in which CGV-G is
embedded. In the same figure, we also follow the sky distribution for the
CGV-D halo (lower left-hand frame). A similar pattern is seen for this halo,
although its embedding filament appears to be more tenuous and has a lower
density.
### 4.2 The wall-like environment of CGV systems
The CGV-G constellation is quite generic for void halo systems. We find that
all 8 void halo configurations are embedded in prominent walls. The void walls
have a typical thickness of around $0.4~{}h^{-1}{\rm Mpc}$. They show a strong
coherence and retain the character of a highly flattened structure out to a
distance of at least $3~{}h^{-1}{\rm Mpc}$ at each side of the CGV haloes.
Five out of the eight haloes reside in filamentary features embedded within
the surrounding walls. Most of these filaments are rather short, not longer
than $4~{}h^{-1}{\rm Mpc}$ in length, and have a diameter of around
$0.4~{}h^{-1}{\rm Mpc}$. Compared to the prominent high-density filaments of
the cosmic web on larger scales, void haloes live in very feeble structures.
An additional quantitative impression of the morphology of the typical void
halo surroundings may be obtained from Figure 13. For haloes CGV-D_a (top) and
CGV-G_a (bottom), the figure plots the shape of the spatial distribution of
neighbouring haloes larger than $10^{8}\mbox{${h^{-1}\rm M}_{\odot}$}$ up to a
distance of $3500~{}h^{-1}{\rm kpc}$. In both situations we see that for close
distances of the halo, out to $<500~{}h^{-1}{\rm kpc}$, the distribution of
surrounding halo is strongly filamentary ($a>b,c$ and $c/a<b/a<0.1-0.15$).
Beyond a distance of $\approx 800~{}h^{-1}{\rm kpc}$, the distribution quickly
attains a more flattened geometry, characteristic of a wall-like configuration
($a>b>c$).
In all, we find that the environment of our selected void haloes displays the
expected behaviour for structure in underdense void regions. Since Zel’dovich’
seminal publication (Zel’dovich, 1970), we know that walls are the first
structures to emerge in the Universe. Subsequently, mass concentrations in and
around the wall tend to contract into filamentary structures. Within the
context of structure emerging out of a primordial Gaussian density field,
(Pogosyan et al., 1998) observed on purely statistical grounds that
infrastructure within underdense regions will retain a predominantly wall-like
character. Following the same reasoning, overdense regions would be expected
to be predominantly of a filamentary nature, as we indeed observe them to be.
It is reassuring that our analysis of the large scale environment of void
haloes appears to be entirely in line with the theoretical expectation of
predominantly wall-like intravoid structures. This conclusion is also
confirmed by the evolution of the CGV configurations, as we will discuss
extensively in section 4.4.
Figure 13: Shape of the neighbour halo distribution for central haloes CGV-D_a
(top) and CGV-G_a (bottom) (haloes included have a mass $M_{\rm
vir}>10^{8}\mbox{${h^{-1}\rm M}_{\odot}$}$). The shape is quantified by the
ratio of second largest axis to largest axis of the inertia tensor (b/a) and
the ratio of the smallest over the largest axis (c/a).
(a) z = 3.7 (b) z = 1.6 (c) z = 0.55 (d) z = 0.0
(e) z = 3.7 (f) z = 1.6 (g) z = 0.55 (h) z = 0.0
Figure 14: Evolution of web-like environment of CGV haloes. The density
distribution in a box of $2~{}h^{-1}{\rm Mpc}$ around each halo is shown at
four redshifts: $z=3,7,1.6,0.55$ and $z=0.0$. Top row: CGV-D. Bottom row:
CGV-G.
### 4.3 Intravoid Filaments
Within the confines of the wall surrounding the CGV-G void halo complex
(Figure 10, top right-hand frame), we find a large number of thin tenuous
filamentary features. A particularly conspicuous property of these tenuous
intravoid filaments is that they appear to be stretched and aligned along a
principal direction. It evokes the impression of a filigree of thin parallel
threads. The principal orientation of the filigree coincides with that of more
pronounced filamentary and planar features that span the extent of the void
(see Figure 1).
The phenomenon of a tenuous filigree of parallel intravoid filaments,
stretching along the principal direction of a void, is also a familiar aspect
of the mass distribution seen in many recent large scale cosmological computer
simulations. An outstanding and well-known example is that of the mass
distribution seen in the Millennium simulation (Springel et al., 2005; Park &
Lee, 2009). The pattern of aligned thin intravoid filaments is a direct
manifestation of the large scale tidal force field which so strongly
influences the overall dynamics and evolution of low-density regions (see van
de Weygaert & Bond, 2008; Platen, van de Weygaert & Jones, 2008). Because of
the restricted density deficit of voids (limited to $\delta>-1$) the
structure, shape and intravoid mass distribution are strongly influenced by
the surrounding mass distribution (Platen, van de Weygaert & Jones, 2008).
Often this is dominated by two, or even more, massive clusters at opposite
sides of a void. These are usually responsible for most of the tidal
stretching of the contracting features in the voids interior. Given the
collective tidal source, we may readily understand the parallel orientation of
the intravoid filaments.
The same external tidal force field is also responsible for directing the
filaments in the immediate surroundings of the wall. As may be appreciated
from the XZ and XY frame in Figure 10, the surrounding void filaments tend to
direct themselves towards and along the plane of the wall. Besides affecting
the anisotropic planar collapse of the wall, the tidal force field is also
instrumental in influencing the orientation of mass concentrations in the
surroundings. Walls and filaments are the result of the hierarchical assembly
of smaller scale filaments and walls. The first stage towards their eventual
merging with the large scale environment is the gradual re-orientation of the
small scale filaments and walls towards the principal plane or axis of the
dominant large scale mass concentration.
While the crowded filigree of tenuous intravoid filaments forms such a
characteristic aspect of the dark matter distribution in voids, it is quite
unlikely we may observe such filaments in the observed galaxy distribution.
Most matter in the universe finds itself in prominent large scale filaments.
Filaments with diameters larger than $2~{}h^{-1}{\rm Mpc}$ represent more than
$80\%$ of the overall mass and volume content of filament. For walls, $80\%$
of the mass and volume is represented by walls with a thickness larger than
$0.9~{}h^{-1}{\rm Mpc}$ (Cautun et. al, in preparation). The large number of
low density void filaments will have hardly sufficient matter content to form
any sizeable galaxy-sized dark halo.
Figure 15: Density and velocity field around two central haloes, CGV-A_a and
CGV-D_a. The fields are shown in two mutually perpendicular $0.5~{}h^{-1}{\rm
Mpc}$ thick central slices, the central XY plane (left) and the central YZ
plane (right). The XY plane is the plane of the wall in which the halo is
embedded. The YZ plane is the one perpendicular to that and provides the edge-
on view on the wall. The vector arrows show the velocity with respect to the
halo bulk velocity. The density levels are the same in each diagram, the
length of the vector arrows is scaled to the mean velocity in the region
around the halo (and thus differs in top and bottom row).
### 4.4 Evolution of the intravoid cosmic web
The intention of our study is to investigate the possible origin of the VGS_31
system. To this end, we have followed the evolution of the web-like void
environment of the eight CGV systems.
In Figure 14, we display the evolution of CGV-D (top) and CGV-G (bottom) and
their environment. At $z=0$, both systems are in a very similar configuration,
within a clearly defined wall-like environment. In earlier stages of
formation, we see a system consisting of a large number of thin filaments.
These filaments rapidly merge into a more substantial dark matter filament,
which is embedded in a wall-like plane. The tenuous walls and filaments get
rapidly drained of their matter content, while they merge with the surrounding
peers. By redshift $z=0.55$, only the most prominent wall remains, aside from
a few faint traces of the other sheet-like structures. By that time, the
filamentary network is nearly completely confined to the plane of the large
wall. Small tenuous filaments have been absorbed by the wall, while the ones
within the wall have merged to form ever larger filaments. In the interior of
the dominant wall we find the corresponding CGV halo systems.
Over the most recent 5 billion years, there is very little evolution of the
web-like environment of the haloes, with most of the changes being confined to
the main sheet. However, there is some variation in time-scale between the
different halo configurations. While the CGV-D system has not fully
materialized until $z=0.55$, the CGV-G system is already in place at $z=1.6$.
Interestingly, as we will notice below, this correlates with a substantial
difference between the morphological evolution of the surroundings at high
redshifts. From early times onward, CGV-G is found to be embedded in a locally
prominent wall. CGV-D, on the other hand, finds itself in the midst of a
vigorously evolving complex of small-scale walls and filaments that gradually
merge and accumulate in more substantial structures (eg. Figure 12).
The evolutionary trend of the voids infrastructure is intimately coupled to
the dynamics of the evolving mass distribution. Figure 15 correlates the
density field in and around two central CGV haloes with the corresponding
velocity field. To this end, we depict the mass and velocity field in two
mutually perpendicular $0.5~{}h^{-1}{\rm Mpc}$ slices. The XY plane is the
plane of the wall in which the halo is embedded. The YZ plane is the one
perpendicular to that and provides a edge-on view of the wall. The vector
arrows show the velocity with respect to the bulk velocity of the primary
halo.
The wall in which CGV-A is embedded still contains an intricate network of
small and thin filaments. Within the wall we observe a strong tendency for
mass to flow out of the area centred around the CGV-A haloes. In the XY plane
of the wall we recognize stronger motions along the filaments. However, the
flow pattern is dominated by the outflow from the sub-voids in the region. The
edge-on view of the YZ plane illustrates this clearly, showing the strength of
the outflow from the voids below and above the wall. In general, we recognize
the outflow in the entire region, inescapably leading to a gradual evacuation
from the region and the dissolution of the structural pattern. The mass
distribution in the environment of the CGV-D halo has a somewhat different
character. It is dominated by the presence of a massive and prominent
filament, oriented along the diagonal in the XY-plane. This filament is
embedded in a flattened planar mass concentration that also stretches along
the filament direction. We clearly observe that the CGV-D halo is
participating in a strong shear flows along the filament. The strong migration
flow along the filament stands out in the lower part of the YZ plane. In the
YZ plane we find it combines with a void outflow out of a large sub-void below
the wall, and a weaker outflow out of a less pronounced void above the wall.
Evidently, as matter continues to flow out of the sub-voids and subsequently
moves in the walls towards the filaments in their interior and at their
boundaries, we will see a gradual dissolution of the intravoid web-like
features. In an upcoming publication, we will focus in more detail on the
dynamics of the walls, filaments and voids.
Figure 16: CGV haloes and subhaloes: spatial distribution and velocities. The
figure shows the spatial distribution, in $200~{}h^{-1}{\rm kpc}$ boxes, of
haloes and subhaloes, projected onto the plane along the large scale wall in
which the systems are embedded. Blue dots: principal haloes with mass
$M>10^{9}\mbox{${h^{-1}\rm M}_{\odot}$}$. Red dots: small surrounding haloes
with masses between $10^{8}<M<10^{9}\mbox{${h^{-1}\rm M}_{\odot}$}$. The size
of the dots is proportional to the mass of the haloes. In each row, we show
the spatial distribution of the haloes (left), the total peculiar velocity
(arrow) of each of the objects (central left), the distribution and total
peculiar velocity of each of the objects perpendicular to the wall (central
right) and the velocity of the haloes/subhaloes wrt. the centre of mass of the
objects (right). We show four systems: CGV-A (top row), CGV-D (second row),
CGV-G (third row) and CGV-H (bottom row). The arrow in the top left figure
indicates a velocity of $100km/s$.
A more systematic analysis of the structural morphology around the CGV haloes
confirms the visual impression of the evolving system of filaments.
Particularly telling is the observed evolution of the (Mollweide) sky
projection of evolving mass distribution around the CGV halo systems. Figure
12 shows how the dark matter sky configuration around primary haloes CGV-D_a
(left row) and CGV-G_a (right row) evolves from redshift z=3.7 to the present
epoch, $z=0$. In both cases, we recognize a circular ring of matter around the
sky, the archetypical signature of the wall-like arrangement of the
surrounding mass distribution at high redshifts (z=3.7 and z=1.6).
Towards later times we observe the gradual evacuation of matter out of the
main body of the wall, and its accumulation at the two diametrically opposite
spots indicating the direction of the filament in which the haloes are
located. In other words, the evolutionary sequence reveals the draining of
matter from the main plane of the wall towards its dominant filamentary spine.
In particular the evolving CGV-D environment provides a nice illustration of
how this process is accompanied by a gradual merging of thin tenuous walls and
filaments into a dominant planar structure (cf. the distribution at z=1.6 with
z=0.55). At $z=3.7$, we cannot yet recognize a coherent wall. Instead, the
”spiderlike” pattern on one hemisphere is that of a plethora of small-scale
incoherent planar features that subsequently merge and contract into a solid
wall, via an intermediate stage marked by two planar structures (z=1.6). The
situation is somewhat different for CGV-G, which even at a high redshift is
already embedded in a solid wall marking a coherent circle over the sky
projection.
With the help of the NEXUS+ technique, we systematically analyse the evolution
of the morphology and composition of the large-scale mass distribution. Figure
11 shows the evolution of the filamentary and wall-like network around CGV-G.
Proceeding from z=2.4, the central row confirms the dramatic evolution of the
wall-like structures around the system. At high redshift the region around the
halo is dominated by a large wall, the one we recognized in the Mollweide sky
projection of Figure 12. Perpendicular to the dominant wall, we find the
presence of numerous additional sheets. However, these tend to be very tenuous
and rapidly merge with the more prominent wall. The entire planar complex has
condensed out by z=1.6. When assessing the evolution of the corresponding
filaments, we find that their concentration towards the plane of the wall is
keeping pace with the contraction of the major wall. This is clearly borne out
by the left-hand row of Figure 11, which shows the filamentary features
visible at the edge-on orientation of the wall. Within the plane of the wall,
on the other hand, we find that there is a dynamically evolving system of
intra-wall filaments. It defines an intricate network of small filaments at
high redshifts, especially prominent in the plane of the large wall and
somewhat less pronounced perpendicular to this wall. At later times the
filamentary network retracts to only a few pronounced filaments, with the
CGV-G system solidly located within the locally dominant filament within the
wall. The filaments at later times are especially pronounced at the
intersection of two or more walls.
We find that the structural evolution shown in Figure 11 is archetypical for
all eight void halo systems. All systems begin their evolution in a wall, and
within the wall in clearly outlined filaments. By $z=0.55$, these structures
are the only noticeable web-like features left in the immediate surroundings
of the haloes. At later times, the morphology of the large scale distribution
hardly evolves any more. The principal difference between the eight void
systems is their morphological affiliation at later time. At $z=0$ not all are
located in a filament. Some of these systems are exclusively located in the
main wall, while others find themselves within a remaining filamentary
condensation. In other words, void haloes always find themselves within
intravoid walls, but not necessarily within intravoid filaments.
Figure 17: Evolution of mean separation (top frame) and harmonic radius
(bottom frame) of the CGV systems. Plotted are $r_{msep}$ and $R_{harm}$, in
co-moving units, against cosmic time (in Gyr). Each CGV system is represented
by a different line character, tabulated in the left bottom corner of the top
frame.
Figure 18: Mass growth of CGV-A haloes and environment. Plotted are the growth
of mass $M$ against cosmic time $t$. Solid blue line: central CGV-A halo.
Dashed blue line: sum of mass in all three CGV-A haloes. Red lines: dark
matter mass growth spherical region centred on centre of mass CGV-A system
(excluding mass in haloes). Dotted red line: spherical region with radius
$100~{}h^{-1}{\rm kpc}$. Dashed red line: spherical region with radius
$150~{}h^{-1}{\rm kpc}$. Solid red line: spherical region of radius
$200~{}h^{-1}{\rm kpc}$. Note that over the past 3 Gyr, the haloes represent
the major share of mass in the region.
## 5 CGV halo configurations and VGS_31: a comparison
Following our investigation of the CGV void haloes and the intravoid filaments
in which they reside, we assess the possible dynamical and evolutionary status
of a system like VGS_31 (see sect. 2.2, Beygu et al. (2013)).
A visual inspection of the spatial configuration of haloes and subhaloes in
and around the CGV systems is presented in Figure 16. The blue dots are the
principal haloes with mass $M>10^{9}\mbox{${\rm M}_{\odot}$}$, the red dots
are small surrounding haloes whose masses range between
$10^{8}<M<10^{9}\mbox{${\rm M}_{\odot}$}$. The location of the primary halo is
taken as the origin of the coordinates.
Figure 19: Energy of evolving CGV halo systems. Left-hand column: time
evolution of ”Halo system” kinetic energy, potential energy and total energy
(see text for definition). Energy is plotted in units of $E_{eq}=2\times
10^{58}$ erg, the potential energy of a $2\times 10^{12}\mbox{${\rm
M}_{\odot}$}$ halo. Solid line: total energy $E_{tot}$. Dashed line: kinetic
energy $E_{kin}$, dotted line: potential energy $E_{pot}$. Right-hand column:
evolution of virial ratio ${\cal V}$ (see Eqn. 5). Top row: CGV-A. 2nd row:
CGV-D. 3rd row: CGV-G. Bottom row: CGV-H.
In four halo systems - CGV-D, CGV-E, CGV-G and CGV-H - the principal haloes
have arranged themselves in a conspicuous elongated configuration, much
resembling the situation of the VGS_31 system. On the other hand, we have not
found a configuration consisting of two massive primary haloes accompanied by
one or two minor haloes. In this respect, none of the eight CGV systems
resembles VGS_31. Instead, most systems appear to comprise one dominant
principal halo and a few accompanying ones that are less massive.
When considering the distribution of the minor haloes around the CGV systems,
we find that they tend to follow the spatial pattern defined by the major
haloes. The CGV-H system is different: the minor haloes have a much wider and
more random distribution than the more massive ones that are arranged in a
filamentary configuration. Overall, however, we do not expect a large number
of smaller haloes in the vicinity of these systems.
Part of the systems are moving with a substantial coherent velocity flow along
the walls or filaments in which they are embedded. When inspecting the central
row of Figure 16, we clearly recognize this with the CGV-A and CGV-D systems.
Interestingly, they also turn out to be the systems that are undergoing the
most active evolution. The latter obviously correlates with a strong evolution
of the surrounding mass distribution.
### 5.1 Size evolution CGV systems
To assess whether the systems have recently formed, we have determined the
mean separation and harmonic radius of the CGV systems,
$\displaystyle r_{msep}\,=\,{\displaystyle 1\over\displaystyle
N}\,\sum_{i,j;i\neq j}|r_{ij}|$ $\displaystyle{\displaystyle
1\over\displaystyle r_{h}}\,=\,{\displaystyle 1\over\displaystyle
N}\,\sum_{i,j;i\neq j}{\displaystyle 1\over\displaystyle|r_{ij}|}$ (1)
While the mean separation is sensitive to outliers and represents a measure
for the overall size of the entire halo system, the harmonic radius of the
system quantifies the size of the inner core of the halo system. The evolution
of the (co-moving) mean separation and in particular the harmonic radius,
shown in Figure 18, reflect the gradual contraction of the systems. While
CGV-B and CGV-F show a strong contraction over the past 1-2 Gyrs, the overall
size of the other systems does not change strongly. This contrasts to the
evolution of the core region. As the lower frame of Figure 18 shows, in most
systems we see a strong and marked evolution over the past 2 to 3 Gyrs,
leading to a contraction to a size considerably less than 100 $~{}h^{-1}{\rm
kpc}$. The haloes in the core will therefore have interacted strongly,
involving either infall of small haloes, mergers of major ones and certainly
strong tidal influences on each other.
### 5.2 Energy considerations
One of the remaining issues concerns the level to which the haloes of the CGV
systems are gravitationally bound. In this respect, we should first evaluate
the fraction of matter contained in the haloes. Figure 18 plots the growth of
mass in a region around the central CGV-A halo. The red lines show the
developing dark matter mass content in a spherical region of radius $100$,
$150$ and $200~{}h^{-1}{\rm kpc}$ around the central halo (excluding the mass
in the haloes themselves). In addition, the figure plots the halo mass
evolution. The solid blue line depicts the growing mass of the central halo,
the dashed blue line is the sum of the mass of the three main CGV-A haloes.
As time proceeds, we see that a larger and larger fraction of mass in the
environment of the CGV-A system gets absorbed by the haloes. At the current
epoch, most of the mass within $100~{}h^{-1}{\rm kpc}$ and $150~{}h^{-1}{\rm
kpc}$ is concentrated in those haloes. Assuming that we may therefore
approximate the kinetic and potential energy of the region by that only
involving the mass in the haloes, we may get an impression of in how far the
halo system is gravitationally bound and tends towards a virial equilibrium.
To this end, we make a rough estimate of the energy content of the halo
system. We approximate the kinetic and potential energy by considering each of
the haloes as point masses with mass $m_{i}$, location ${\vec{r}}_{i}$ and
velocity ${\vec{v}}_{i}$. Note that by doing so we ignore the contribution of
the more diffusely distributed dark matter in the same region, which at
earlier times is dynamically dominant but gradually decreases in importance
(see Figure 18). Also, it ignores the contribution of the surrounding mass
distribution to the potential energy. The kinetic energy of the system of N
CGV haloes, wrt. its centre of mass, is
$E_{K}\,=\,\frac{1}{2}\,\sum_{i=1}^{N}\,m_{i}({\vec{v}}_{i}-{\vec{v}_{CM}})^{2}\,,$
(2)
while the potential energy of the system is computed from
$E_{G}\,=\,-\sum_{i=1}^{N}\sum_{j=1}^{N}\,\frac{Gm_{i}m_{j}}{|{\vec{r}}_{i}-{\vec{r}}_{j}|}\,.$
(3)
In the left-hand column of Figure 19 we plot the evolution of the kinetic,
potential and total energy,
$E_{tot}\,=\,E_{K}+E_{G}\,$ (4)
of four halo systems (CGV-A, CGV-D, CGV-G and CGV-H). The energy is plotted in
units of $E_{eq}=2\times 10^{58}$ erg, which is approximately the potential
energy of $2\times 10^{12}\mbox{${\rm M}_{\odot}$}$ haloes at 1Mpc distance.
We see that half of the systems have a rather quiescent evolution. Of these,
CGV-H strongly and CGV-H marginally gravitationally bound. A far more
interesting and violent evolution of the energy content of the halo systems
CGV-A and CGV-D. Both involve an active and violent merger history, marked by
a continuous accretion of minor objects and a few major mergers. In particular
the major mergers are accompanied by a strong dip in the potential and binding
energy.
To get an impression of the corresponding energy stability, we plot the
evolution of the virial ratio,
${\cal V}\,=\,\frac{2E_{K}}{E_{G}}\,,$ (5)
in the second column of Figure 19. For a fully virialised object, ${\cal
V}=1$. While the computed ${\cal V}$ parameter only provides an impression of
the energy state of the systems, it does confirm the impression that CGV-H and
CGV-G are halo systems that are in largely in equilibrium. At the same time,
the same diagrams for the CGV-A and CGV-D systems reflect their violent
history. This appears to continue up to recent times.
### 5.3 The origin of VGS_31
Translating the CGV systems to VGS_31, we note that all haloes detected at
$z=0$, are local to their environment. They, and their progenitors, were never
further removed than $330~{}{\rm kpc}$ from the main halo. Even if so far
removed, we find that the distance between the haloes rapidly decreased at
early times. We therefore conclude that the galaxies in VGS_31 originated in
the same region, and originally were probably located in the same proto-wall,
and possibly even proto-filament. In other words, the galaxies in the VGS_31
system did not meet just recently, but have been relatively close to each
other all along their evolution. It answers our question whether VGS_31 might
consist of filamentary fragments that only recently assembled.
Moreover, the strong evolution of the several CGV halo cores is an indication
for the fact that the two dominant galaxies VGS_31 - VGS_31a and VGS_31b, may
recently have undergone strong interactions as indeed their appearance
confirms. It would imply that the disturbed nature of the galaxies of VGS_31
is a result of recent interactions between the galaxies. On the other hand,
other CGV systems had a rather quiescent history. If VGS_31 would correspond
to one of these systems, we may not have expected the marks of recent
interaction that we see in VGS_31a and VGS_31b.
## 6 Discussion & Conclusions
In this study, we have investigated the formation history of dark matter halo
systems resembling the filamentary void galaxy system VGS_31 (Beygu et al.,
2013). The VGS_31 system is a 120kpc long elongated configuration of 3
galaxies found in the Void Galaxy Survey (Kreckel et al., 2011). In the
CosmoGrid simulation we looked for systems of dark haloes that would resemble
the VGS_31 system. To this end, we invoked a set of five criteria. In total,
eight systems were identified, CGV-A to CGV-H.
The $2048^{3}$ particle CosmoGrid simulation has a rather limited volume,
$V=21~{}h^{-1}{\rm Mpc}^{3}$, but a very high spatial resolution. While its
limited size impedes statistically viable results on large scale clustering as
its volume is not representative for the universe, its high mass resolution
renders it ideal for high-resolution case studies such as the one described in
this study.
While the CosmoGrid simulation is a pure dark matter simulation, a more direct
comparison with the HI observations of VGS_31 will have to involve
cosmological hydro simulations that include gas, stars, and radiative
processes. Nonetheless, as galaxies will form in the larger dark matter haloes
and gaseous filaments will coincide with the more substantial dark matter
filaments, our study provides a good impression of the expected galaxy
configurations in voids. Nonetheless, it is good to realize that most of the
intricate structure seen in our simulations would contain too small amounts of
gas to be observed.
For each of the CGV systems we examined the formation history, the merging
tree, and the morphology of the large scale environment. In our presentation,
we focus on the two systems that represent the extremes of the VGS_31
resembling halo configurations. System CGV-G formed very early in the
simulation and remained virtually unchanged over the past 10 Gyr. CGV-D, on
the other hand, formed only recently and has been undergoing mergers even
until $z=0$.
We find that all CGV systems are located in prominent intra-void walls, whose
thickness is in the order of $0.4~{}h^{-1}{\rm Mpc}$. Five halo complexes are
located within filaments embedded in the intra-void wall. In all situations
the filamentary features had formed early on, and were largely in place at
$z\approx 1.6$. These intra-void filaments are short and thin, with lengths
less than $4~{}h^{-1}{\rm Mpc}$ and diameters of ${\sim}0.4~{}h^{-1}{\rm
Mpc}$.
The spatial distribution of dark matter haloes resembles that of the dark
matter. We see the same hybrid filament-wall configuration as observed in the
dark matter distribution. Close to the main halo, within a distance smaller
than $700~{}h^{-1}{\rm kpc}$, the neighbouring haloes are predominantly
distributed along a filament. On larger scales, up to $\approx
3.5~{}h^{-1}{\rm Mpc}$, the haloes are located in a flattened wall-like
structure.
In addition to our focus on the evolving dark matter halo configurations, we
also studied the morphology and evolution of the intricate filament-wall
network in voids. Our study shows the prominence of walls in the typical void
infrastructure. Unlike the larger scale overdense filaments, intra-void
filament are far less outstanding with respect to the walls in which they are
embedded.
What about VGS_31? Our study implies it belongs to a group of galaxies that
was formed in the same (proto)filament and has undergone a rather active life
over the last few Gigayears. The galaxies in the VGS_31 system did not meet
just recently, but have been relatively close to each other all along their
evolution. We also find it is not likely VGS_31 will have many smaller haloes
in its vicinity. The fact that we find quite a diversity amongst the CGV
systems also indicates that VGS_31 may not be typical for groups of galaxies
in voids.
## Acknowledgements
We thank Katherine Kreckel, Jacqueline van Gorkom and Thijs van der Hulst for
discussions within the context of the VGS project. We also gratefully
acknowledge many helpful and encouraging discussions with Bernard Jones,
Sergei Shandarin, Johan Hidding and Patrick Bos. Furthermore, we thank Peter
Behroozi, Dan Caputo, Arjen van Elteren, Inti Pelupessy and Nathan de Vries
for their assistance and useful suggestions. Finally, we thank the anonymous
referee for his or her helpful comments.
This work was supported by NWO (grants IsFast [#643.000.803], VICI
[#639.073.803], LGM [#612.071.503] and AMUSE [#614.061.608]), NCF (grants
[#SH-095-08] and [#SH-187-10]), NOVA and the LKBF in the Netherlands. RvdW
acknowledges support by the John Templeton Foundation, grant nr. FP05136-O.
The CosmoGrid simulations were partially carried out on Cray XT4 at Center for
Computational Astrophysics, CfCA, of National Astronomical Observatory of
Japan; Huygens at the Dutch National High Performance Computing and e-Science
Support Center, SURFsara (The Netherlands); HECToR at the Edinburgh Parallel
Computing Centre (United Kingdom) and Louhi at IT Center for Science in Espoo
(Finland).
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|
arxiv-papers
| 2013-07-26T22:18:57 |
2024-09-04T02:49:48.542661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Steven Rieder, Rien van de Weygaert, Marius Cautun, Burcu Beygu and\n Simon Portegies Zwart",
"submitter": "Steven Rieder",
"url": "https://arxiv.org/abs/1307.7182"
}
|
1307.7292
|
# Physics at a High-Luminosity LHC with ATLAS
The ATLAS Collaboration
(July 26, 2013
Minor Revision: July 31, 2013)
††journal: the Snowmass Community Planning Study
Revised references with respect to the version of July 26, 2013
100pt ATL-PHYS-PUB-2013-007 Submitted as input to The physics accessible at
the high-luminosity phase of the LHC extends well beyond that of the earlier
LHC program. Selected topics, spanning from Higgs boson studies to new
particle searches and rare top quark decays, are presented in this document.
They illustrate the substantially enhanced physics reach with an increased
integrated luminosity of $3000\,\mbox{fb${}^{-1}$}$, and motivate the planned
upgrades of the LHC machine and ATLAS detector.
## Foreword to the ATLAS and CMS contributions to the DPF Snowmass Process
The ATLAS and CMS Collaborations
In 2012 the CMS and ATLAS experiments at CERN discovered a Higgs boson with a
mass of 125-126 GeV. This opened a new chapter in the history of particle
physics. In this brief cover note, the main priorities of the ATLAS and CMS
Collaborations in this new era are presented. Further adjustments to these
priorities may occur as detailed studies of this particle and searches for new
physics are extended into new realms at higher energies in the future.
The Higgs discovery was anchored by the final states that afforded the best
mass resolution, namely $H\rightarrow\gamma\gamma$ and $H\rightarrow ZZ$
($4e$, $4\mu$ or $2e2\mu$). These modes placed stringent requirements on
detector design and performance. Indeed, the ability to search for the SM
Higgs boson over the fully allowed mass range played a crucial role in the
conceptual design and benchmarking of the experiments and also resulted in
excellent sensitivity to a wide array of signals of new physics at the TeV
energy scale. Remarkably, the recent discovery came at half the LHC design
energy, much more severe pileup, and one third of the integrated luminosity
that was originally judged necessary. This demonstrates the great value of a
bold early conceptual design, a systematic program of development and
construction, and a detailed understanding of detector performance, in
confronting challenging physics goals.
With data taken in coming years at or near to the design energy of 14 TeV, a
broader picture of physics at the TeV scale will emerge with implications for
the future of the energy frontier program. Amongst the essential inputs will
be precision measurements of the properties of the Higgs boson and direct
searches for new physics that will make significant inroads into new
territory. For the foreseeable future the LHC together with CMS and ATLAS will
be the only facility able to carry out these studies. This program is very
challenging for the experiments because it requires accurate reconstruction
and identification of physics objects (leptons including the $\tau$, heavy
flavor tagging, photons, jets, missing transverse energy) from relatively low
to very high transverse momenta extending to large rapidity (e.g. to
characterize events from vector boson fusion). To retain and extend these
capabilities to higher luminosities in the 2020s, existing systems need to be
upgraded or replaced. This will require a vision as ambitious as that of the
original LHC program, in particular ensuring that sufficient resources, both
financial and human, are made available in a timely fashion; for R&D in the
short term, for prototyping in close liaison with industry in the medium term,
and further down the line for construction.
To realize the full physics potential of the LHC, it is essential that the
High Luminosity (HL-LHC) upgrade to the accelerator be carried out. However,
the planned instantaneous luminosity of $\sim 5\times
10^{34}\,\text{cm}^{-2}\text{s}^{-1}$ is well beyond its original design and
thus the capabilities of the current experiments. This necessitates
upgrades/replacements explicitly targeted towards the broad program of physics
mentioned above.
The European Strategy for Particle Physics, formally adopted by the CERN
Council in May 2013, states that the “top priority should be the exploitation
of the full potential of the LHC, including the high-luminosity upgrade of the
machine and detectors with a view to collecting ten times more data than in
the initial design, by around 2030.” The European Strategy clearly recognizes
that “the scale of the facilities required by particle physics is resulting in
the globalisation of the field.”
ATLAS and CMS, as worldwide collaborations, are fully committed to engage all
international partners to deliver this program. The U.S. has made
contributions critical to the success of the CMS and ATLAS experiments to
date. The success of future data-taking, and the detector upgrade programs,
will rely on a continuing strong engagement of U.S. groups in the two
experiments to assure the continued success of the high energy frontier
program.
In summary:
The highest priority in particle physics should be to exploit fully the
physics potential of the LHC. To achieve this goal, ATLAS and CMS place the
highest priority on securing the resources needed to achieve the following
goals:
* •
Upgrade/replace selected elements of the apparatus and associated readout,
trigger, data acquisition and computing systems in order to optimally exploit
fully the phase of LHC running above the original design luminosity over the
next 10 years (to LHC long-shutdown 3, “LS3”);
* •
Prepare, prototype and construct the necessary upgrades/replacements of
detectors to operate and optimally exploit the phase of running at
instantaneous luminosities in excess of $5\times
10^{34}\text{cm}^{-2}\text{s}^{-1}$ in the roughly ten year period following
LS3.
CMS and ATLAS strongly recommend that resources be allocated for the HL-LHC to
enable the LHC to operate at luminosities significantly higher than the
original design.
## 1 Introduction
From 2015 to 2017, the ATLAS experiment will collect about 100 fb-1 of data,
with a peak instantaneous luminosity of $10^{34}\text{cm}^{-2}\text{s}^{-1}$,
and at a center-of-mass energy between 13 and 14 TeV, nearly twice the energy
of the 2012 LHC running. Following this so-called “Run 2,” the accelerator
will be upgraded to deliver two to three times the instantaneous luminosity at
$\sqrt{s}$=14 TeV and the ATLAS detector will undergo the “Phase-I” upgrade to
maintain the detector capabilities at the increased luminosities. A total of
300-350 fb-1 of data is expected to be collected by the end of Run 3 in 2021.
The final focus magnets in the interaction regions will begin to suffer
radiation damage by this stage, and a Phase-II upgrade to the LHC is proposed
to provide an instantaneous luminosity of $5\times
10^{34}\text{cm}^{-2}\text{s}^{-1}$ beginning in 2023. The ATLAS experiment
will also require upgrades for this High Luminosity LHC (HL-LHC) program, to
maintain its capabilities at the planned instantaneous luminosity, which
corresponds to an a average of 140 interactions per crossing. The ATLAS Phase-
II upgrades for HL-LHC are described in the Letter of Intent [1] and are
summarized below. A specific subset of instrumentation studies being performed
in the U.S. is given in a separate contribution to the Snowmass Community
Planning Study [2].
Hadron colliders have a major role as exploration and discovery machines, and
the HL-LHC is required to fully exploit the opportunities at the LHC for
discovery of physics beyond the Standard Model (“BSM physics”). Precision
studies of the newly discovered Higgs boson, which may provide a portal to BSM
physics, and direct searches for additional Higgs particles, are a very high
priority at the LHC and the HL-LHC provides a significant increase in
discovery reach for new Higgs particles and sensitivity to non-Standard Model
effects in Higgs boson couplings. These opportunities also exist in other
direct searches for BSM physics, such as supersymmetry (SUSY) or new heavy
gauge bosons ($Z^{\prime}$ and $W^{\prime}$).
In this Whitepaper we summarize studies that examine the physics capabilities
of ATLAS at the HL-LHC for a number of key physics processes. These studies
were started in the context of developing the European Strategy for Particle
Physics [3, 4], a process that affirmed the HL-LHC program as a top priority
in particle physics. This note summarizes both the results used as input for
the European Strategy and results that have been updated since then. We
contrast the projected results at two benchmark values of integrated
luminosity: the 300 fb-1 expected by the end of Run 3, and the 3000 fb-1
expected to be delivered by the HL-LHC. These projections are based on
extrapolating the current detector performance from the current pile-up
conditions in the data of an average number, $\mu$, of 20 interactions per
crossing, and from simulation of pile-up for Run 3 with up to $\mu=69$, to the
conditions at HL-LHC of $\mu=140$. The parameterizations used for the
extrapolation are described in detail in Ref. [5]. These parameterizations
provide rather conservative estimates of the reach and precision of
measurements. Except where otherwise noted, they do not include improvements
due to new techniques, improved understanding of backgrounds, or reduced
theoretical uncertainties.
## 2 Detector Requirements and Upgrade Designs
The promise of the rich physics program at the HL-LHC cannot be fulfilled
without essential improvements to the ATLAS apparatus. In particular, the
studies summarized below depend on robust tracking peformance, moderate
trigger rates for single leptons, good resolution on missing transverse energy
measurements, and enhanced jet tagging and $\tau$ identification. The ATLAS
Phase-II upgrades are intended to maintain and, in some cases, improve the
detector performance during high-luminosity operation.
The current ATLAS detector was designed to operate efficiently for an
integrated luminosity of $300\,\mbox{fb${}^{-1}$}$ and a pileup rate of 20-25
events per bunch crossing at the design luminosity of $1\times
10^{34}\,\text{cm}^{-2}\text{s}^{-1}$. The HL-LHC will feature integrated
luminosities and pileup rates well beyond these values.
The ATLAS Inner Detector will need to be replaced as the accumulated radiation
dose starts to damage the silicon detectors, and the occupancy in some regions
of the silicon and in the straw tubes will be unacceptably high. An all-
silicon inner tracker is proposed, using high-granularity radiation-hard
sensors that can withstand the particle fluences expected during the high-
luminosity run. The new tracker will provide a sufficient number of space
points for track reconstruction for ATLAS to maintain high tracking efficiency
and a low fake rate in the increased pileup environment. The ATLAS
calorimeters will be upgraded with new front-end electronics that make it
possible to digitize the response in every cell, so that the digitized data
can be used in fast trigger algorithms. In a similar way, the muon
spectrometer will be upgraded with new front-end electronics to read out the
chambers in time for a fast trigger decision.
The new trigger architecture is intended to operate with a 500-kHz Level-0
decision that uses the information from the fast readouts of the muon
spectrometer and calorimeters before adding tracking information at Level-1.
ATLAS submitted its Letter of Intent for the Phase-II Upgrade [1] in March
2013, and detailed Technical Design Reports are being prepared for the various
detector systems to be upgraded.
## 3 Parameterization of Expected Performance
The studies of the ATLAS high-luminosity physics program are based on a set of
performance assumptions for the reconstructed physics objects [5]. These
parameterizations leverage the excellent understanding of the current ATLAS
detector and recent full simulations of the upgraded systems. For the most
part, they are based on the Run 1 detector with conservative realistic
assumptions on the pileup dependence. The missing transverse momentum
resolution is extrapolated from studies with 7 and 8 TeV data and full
simulation. It is parameterized as a function of the number of pileup events,
which naturally increase at high instantaneous luminosity. These results are
quite conservative for the most part. For example, a comparison of the
b-tagging performance assumption and the results from full simulation is shown
in Fig. 1, in which the upgraded performance is shown to be better than the
performance inferred from the current b-tagging with high pileup, due to the
inner tracker improvements. In particular, a b-tagging algorithm that is 75%
efficient was assumed to to have a light-jet rejection factor of 30 for the
physics studies presented here, when in fact a full simulation study points to
a light-jet rejection factor of approximately 120.
Figure 1: Comparison of the ATLAS b-tagging performance parameterization as a
function of pileup (left) and the b-tagging performance in full simulation for
various pileup scenarios and detector configurations (right). The IP3D+SV1
tagging algorithm uses a combination of 3-dimensional impact parameter
likelihood and secondary vertexing to achieve high performance, especially
when the Insertable B-Layer pixel detector (IBL) or proposed all-silicon Inner
Tracker (ITK) are used.
## 4 Measurements of the Higgs boson
With the discovery of a Higgs boson [6, 7] last summer, a major focus of the
LHC program has become the measurement of the properties of this new particle.
The resonance was termed a “Higgs-like” boson when only its mass was known
with any precision. After another year of study, $J^{P}=0^{+}$ is strongly
favored [8, 9]. With limited precision, the new particle’s couplings agree
with SM expectations in those channels for which they have been measured [10].
The ratio of the combined signal strength to the Standard Model value is now
measured by ATLAS to be $1.33\pm 0.21$. There is no significant indication of
a deviation from the properties of a SM Higgs boson, but the precision of the
coupling measurements leaves room for BSM physics, for which models typically
predict deviations from the SM couplings. However, the deviations can be
arbitrarily small [11], as, for example, in SUSY scenarios where the
additional Higgs particles are very heavy. Thus it is of the utmost importance
to measure the couplings with increased precision, and to search directly for
additional Higgs particles.
### 4.1 Higgs boson couplings
Because the mass of the observed Higgs boson is approximately 125 GeV, a large
variety of decay channels are open to investigation at the LHC. The LHC
experiments measure the product of the production cross section and the
branching fraction into a particular final state. In order to extract the
Higgs boson couplings and test the Standard Model predictions, fits to the
measured signal strengths are done using all relevant channels [10]. Achieving
the best sensitivity to potential non-Standard Model Higgs boson couplings
requires precision measurements of as many different Higgs production and
decay channels as possible.
Current projections to the HL-LHC are based primarily on those channels that
are measured in the current dataset, with the addition of a few key rare
channels that can be accessed only at the HL-LHC [4]. The current analyses
measure Higgs production through both the gluon-fusion and vector boson fusion
(VBF) channels into final states $\gamma\gamma$, $ZZ^{*}$, and $WW^{*}$, and
good progress is being made towards measurements in $\tau^{+}\tau^{-}$ and
$b\bar{b}$. The luminosity of the HL-LHC will provide improved statistical
precision for already established channels and allow rare Higgs boson
production and decay modes to be studied and measured with substantially
improved precision compared to the measurements that will be made by ATLAS
with about 300 fb-1 of data (Run 3).
Changes to the trigger and to the photon and lepton selections that are needed
at high-luminosity to keep rates in check are taken into account. For the VBF
jet selection, the cuts were tightened to reduce the expected fake rate
induced by pile-up to below 1% of the jet activity from background processes.
The following channels, which are already studied in the current 7 and 8 TeV
datasets, are now evaluated for the 14 TeV HL-LHC dataset:
* •
$H\to\gamma\gamma$ in the 0-, 1-, and 2-jet final states. The analysis is
carried out analogously to Ref. [6].
* •
Inclusive $H\to ZZ^{*}\to 4\ell$, following a selection close to that in Ref.
[6].
* •
$H\to WW^{*}\to\ell\nu\,\ell\nu$ in the 0-jet and the 2-jet final state, the
latter with a VBF selection. The analysis follows closely that of Ref. [6].
* •
$H\to\mathrm{\tau^{+}\tau^{-}}$ in the 2-jet final state with a VBF selection
as in Ref. [12].
The $WW^{*}$ and $\tau^{+}\tau^{-}$ channels are challenging as they require a
detailed understanding of the various backgrounds. The ultimate precision will
depend on how well these backgrounds can be constrained, in situ, using data.
For this study it was assumed that the background understanding will not
improve beyond what was achieved by summer 2012. The projections are therefore
rather pessimistic. For example, for the $WW^{*}$ channel the background
uncertainty is already significantly improved using the full 2012 dataset,
primarily due to improved analysis techniques [10], resulting in a precision
on $\mu$ of $\approx 30$%. These improvements have not yet been propagated
into the current study, and therefore it should be possible to improve on the
quoted precision of 29%.
To exploit the projected 3000 fb-1 provided by the HL-LHC, several additional,
relatively rare, channels with Higgs boson decays into the high-resolution
final states $H\to\gamma\gamma$ and $H\to\mu\mu$ are studied:
* •
$t\bar{t}H,H\to\gamma\gamma$ and $H\to\mu\mu$.
* •
$WH/ZH,H\to\gamma\gamma$.
* •
Inclusive $H\to\mu\mu$.
The $t\bar{t}H$ and $WH/ZH$ $\gamma\gamma$ channels above have a low signal
rate at the LHC, but one can expect to observe more than 100 signal events
with the HL-LHC. The selection of the diphoton system is done in the same way
as for the inclusive $H\to\gamma\gamma$ channel. In addition, 1- and 2-lepton
selections, dilepton mass cuts and different jet requirements are used to
separate the $WH$, $ZH$ and $t\overline{t}H$ initial states from each other
and from the background processes. The $t\overline{t}H$ initial state gives
the cleanest signal with a signal-to-background ratio of $\sim$20%, to be
compared to $\sim$10% for $ZH$ and $\sim$2% for $WH$. The $H\to\mu\mu$ decay
will be measured for the first time in the Run 3, 300 fb-1 dataset, but only
with very limited precision. The HL-LHC will allow a measurement in the
inclusive channel of better than 20% and in $t\bar{t}H,H\to\mu\mu$ of just
over 20%. The expected $H\to\gamma\gamma$ signal in $t\overline{t}H$ for 3000
fb-1 is shown in Fig. 2, and the inclusive $H\to\mu\mu$ expectation is shown
in Fig. 3.
It is interesting to measure the ratio of third-generation to second-
generation couplings, to test the Standard Model Higgs boson couplings and the
potential for BSM effects. Studies focus on VBF production with decays to
$\tau\tau\rightarrow\ell\tau_{had}3\nu$ and
$\tau\tau\rightarrow\ell\ell^{\prime}4\nu$. A relatively precise $H\to\mu\mu$
signal-strength measurement and improved $H\to\tau\tau$ measurements provide a
significant improvement in the measurement of the ratio of partial widths into
second- and third-generation fermions.
The projected uncertainties on the signal-strength measurements and the ratios
of partial widths are summarized in Fig. 4 for the channels that have been
studied to date. The same uncertainties are given in tabular form in Tables 1
and 2. The partial-width ratio uncertainties are given as well in Table 3.
Figure 2: Expected diphoton mass distribution in the single lepton ttH channel
for $\sqrt{s}$=14 TeV and $\mathcal{L}=3000\,\text{fb}^{-1}$. Figure 3:
Distribution of the $\mu^{+}\mu^{-}$ invariant mass of the signal and
background processes generated for $\sqrt{s}$=14 TeV and
$\mathcal{L}=3000\,\text{fb}^{-1}$.
(a)
(b)
Figure 4: Summary of Higgs analysis sensitivities wth 300 fb-1and 3000 fb-1at $\sqrt{s}=14$ TeV for a SM Higgs boson with a mass of 125 GeV. Left: Uncertainty on the signal strength. For the $H\to\tau\tau$ channels the thin brown bars show the expected precision reached from extrapolating all tau-tau channels studied in the current 7 TeV and 8 TeV analysis to 300 fb-1, instead of using the dedicated studies at 300 fb-1 and 3000 fb-1 that are based only in the VBF $H\to\tau\tau$ channels. Right: Uncertainty on ratios of partial decay width fitted to all channels. The hashed areas indicate the increase of the estimated error due to current theory systematic uncertainties. | with theory systematics | without theory systematics
---|---|---
$H\rightarrow\mu\mu$ | 0.53 | 0.51
$ttH,H\rightarrow\mu\mu$ | 0.73 | 0.72
$VBF,H\rightarrow\tau\tau$ | 0.23 | 0.19
$VBF,H\rightarrow\tau\tau$ (extrap) | 0.15 | 0.11
$H\rightarrow ZZ$ | 0.16 | 0.093
$VBF,H\rightarrow WW$ | 0.67 | 0.66
$H\rightarrow WW$ | 0.29 | 0.26
$VH,H\rightarrow\gamma\gamma$ | 0.77 | 0.77
$ttH,H\rightarrow\gamma\gamma$ | 0.55 | 0.54
$VBF,H\rightarrow\gamma\gamma$ | 0.34 | 0.31
$H\rightarrow\gamma\gamma(+j)$ | 0.16 | 0.12
$H\rightarrow\gamma\gamma$ | 0.15 | 0.081
Table 1: Expected relative uncertainties on the signal strength $\mu$ for $300$ fb-1. The $H\rightarrow\tau\tau$ line labeled ‘(extrap)’ is based on an extrapolation to $300$ fb-1 from all $\tau\tau$ channels currently studied in the 7 TeV and 8 TeV analyses, whereas the other $\tau\tau$ projection is based on dedicated studies based only on the VBF production channel. | with theory systematics | without theory systematics
---|---|---
$H\rightarrow\mu\mu$ | 0.21 | 0.16
$ttH,H\rightarrow\mu\mu$ | 0.26 | 0.23
$VBF,H\rightarrow\tau\tau$ | 0.20 | 0.16
$H\rightarrow ZZ$ | 0.13 | 0.047
$VBF,H\rightarrow WW$ | 0.58 | 0.57
$H\rightarrow WW$ | 0.29 | 0.26
$VH,H\rightarrow\gamma\gamma$ | 0.25 | 0.25
$ttH,H\rightarrow\gamma\gamma$ | 0.21 | 0.17
$VBF,H\rightarrow\gamma\gamma$ | 0.16 | 0.11
$H\rightarrow\gamma\gamma(+j)$ | 0.12 | 0.054
$H\rightarrow\gamma\gamma$ | 0.13 | 0.040
Table 2: Expected relative uncertainties on the signal strength $\mu$ for $3000$ fb-1. | 300 fb-1 | 3000 fb-1
---|---|---
| w/theory uncert. | wo/theory uncert. | w/theory uncert. | wo/theory uncert.
$\Gamma_{Z}/\Gamma_{g}$ | 0.52 | 0.48 | 0.28 | 0.22
$\Gamma_{t}/\Gamma_{g}$ | 0.52 | 0.49 | 0.23 | 0.15
$\Gamma_{\tau}/\Gamma_{\mu}$ | 0.67 | 0.66 | 0.25 | 0.23
$\Gamma_{\tau}/\Gamma_{\mu}$ (extrap) | 0.59 | 0.58 | |
$\Gamma_{\mu}/\Gamma_{Z}$ | 0.45 | 0.45 | 0.14 | 0.14
$\Gamma_{\tau}/\Gamma_{Z}$ | 0.42 | 0.40 | 0.21 | 0.18
$\Gamma_{\tau}/\Gamma_{Z}$ (extrap) | 0.28 | 0.26 | |
$\Gamma_{W}/\Gamma_{Z}$ | 0.25 | 0.25 | 0.23 | 0.23
$\Gamma_{\gamma}/\Gamma_{Z}$ | 0.11 | 0.11 | 0.029 | 0.029
$\Gamma_{g}\bullet\Gamma_{Z}/\Gamma_{H}$ | 0.16 | 0.093 | 0.13 | 0.047
Table 3: Relative uncertainty on the ratio of partial widths for the
combination of Higgs analysis and coupling properties fits at 14 TeV, 300 fb-1
and 3000 fb-1, assuming a SM Higgs Boson with a mass of 125 GeV.
The ratios of partial widths shown in the right-hand panel of Fig. 4
correspond to coupling scale-factors according to
$\Gamma_{X}/\Gamma_{Y}=\kappa_{X}^{2}/\kappa_{Y}^{2}$, where $\kappa_{i}$ is
the coupling scale-factor for the Higgs coupling111In the case of gluons and
photons, these are effective couplings that include all loop effects into a
single value to $i=g,\gamma,W,Z,t,\mu,\tau$, and the Standard Model value is
$\kappa=1$ [13, 14]. The results of a minimal fit, where only two independent
scale factors are used, $\kappa_{V}$ for vector bosons and $\kappa_{F}$ for
fermions, is shown in Table 4. Significant improvement in the precision
between 300 and 3000 fb-1 is seen. The column including the theory uncertainty
assumes no improvement over today’s values, certainly a pessimistic
assessment. Fig. 5 shows the two-dimensional contours in $\kappa_{V}$ and
$\kappa_{F}$. The left-hand figure compares the projected results for 300 fb-1
with, and without, theory uncertainties included. The right-hand figure
compares the 300 fb-1 and 3000 fb-1 results with no theory uncertainties
included.
Coupling | With theory systematics | Without theory systematics
---|---|---
300 fb-1
$\kappa_{V}$ | ${}^{+5.9\%}_{-5.4\%}$ | ${}^{+3.0\%}_{-3.0\%}$
$\kappa_{F}$ | ${}^{+10.6\%}_{-9.9\%}$ | ${}^{+9.1\%}_{-8.6\%}$
3000 fb-1
$\kappa_{V}$ | ${}^{+4.6\%}_{-4.3\%}$ | ${}^{+1.9\%}_{-1.9\%}$
$\kappa_{F}$ | ${}^{+6.1\%}_{-5.7\%}$ | ${}^{+3.6\%}_{-3.6\%}$
Table 4: Results for $\kappa_{V}$ and $\kappa_{F}$ in a minimal coupling fit
at 14 TeV, 300 fb-1 and 3000 fb-1.
Figure 5: 68% and 95% confidence level (CL) likelihood contours for
$\kappa_{V}$ and $\kappa_{F}$ in a minimal coupling fit at 14 TeV. Left:
impact of the theory uncertainties for an assumed integrated luminosity of 300
fb-1. Right: results without theory uncertainties for 300 fb-1 and 3000 fb-1.
#### 4.1.1 Sensitivity to the Higgs self-coupling
An important feature of the Standard Model Higgs boson is its self-coupling.
The tri-linear self-coupling $\lambda_{HHH}$ can be measured through an
interference effect in Higgs boson pair production. At hadron colliders, the
dominant production mechanism is gluon-gluon fusion. At $\sqrt{s}=14$ TeV, the
production cross section of a pair of 125 GeV Higgs bosons is estimated at NLO
to be222The cross section is calculated using the HPAIR package [15].
Theoretical uncertainties are provided by Michael Spira in private
communication. $34^{+18\%}_{-15\%}\text{(QCD scale)}\pm 3\%\text{(PDFs)}\
\text{fb}$. Figure 6 shows the three contributing diagrams in which the last
diagram, the only one that depends on $\lambda_{HHH}$, interferes
destructively with the first two. The cross section is therefore enhanced at
lower values of $\lambda_{HHH}$. For
$\lambda_{HHH}/\lambda^{SM}_{HHH}=0~{}(2)$ the cross section is 71 (16) fb.
Studies using Higgs pair decays to $b\overline{b}\gamma\gamma$ and
$b\overline{b}W^{+}W^{-}$ are in progress.
Figure 6: Feynman diagrams for Higgs pair production.
## 5 Measurements of Vector Boson Scattering and Gauge Couplings
A major reason for expecting new particles or interactions at the TeV energy
scale has been the prediction that an untamed rise of the vector boson
scattering (VBS) cross section in the longitudinal mode would violate
unitarity at this scale. In the SM it is the Higgs boson which is responsible
for the damping of this cross section. It is important to confirm this effect
experimentally, now that one Higgs boson has been observed via direct
production and decay. Alternate models such as Technicolor and little Higgs
have been postulated which encompass TeV-scale resonances and a light scalar
particle. These and other mechanisms would modify the vector boson scattering
as long as there is a coupling of the new particles to the vector bosons.
The combination of vector boson scattering measurements, triboson production
measurements, and Higgs coupling measurements offers a comprehensive program
for exploring the gauge-Higgs sector in detail. For example, measuring vector
boson scattering precisely at high mass scales provides sensitivity to new
particles and interactions in the electroweak sector.
We summarize results from four studies quantifying the sensitivity to new
physics in this sector [16]. The specific studies are $WZ$ VBS in the three-
lepton channel, $ZZ$ VBS in the four-lepton channel, $WW$ VBS in the same-sign
dilepton channel, and $Z\gamma\gamma$ production in the dilepton plus diphoton
channel.
Unlike previous studies that focused on anomalous couplings in a unitarized
Higgsless theory [17], these studies are presented in the framework of higher-
dimension operators in an effective electroweak field theory [18]. Multiboson
production is modified by certain general dimension-6 and dimension-8
operators containing the Higgs and/or gauge boson fields. Several
representative operators have been chosen to study as benchmarks. Because
higher-dimension operators, as approximations of an underlying $UV$-safe
theory, ultimately violate unitarity at sufficiently high energy, care is
taken in these studies to select only events in a kinematic range within the
unitarity bound, $\Lambda_{UV}$. These new operators affect only triboson
production and vector boson scattering (VBS), but they do not affect other
diboson production mechanisms.
The common experimental feature in the following studies of vector boson
scattering is the presence of high-$p_{T}$ jets in the forward-backward
regions, similar to those found in Higgs production via vector boson fusion.
The absence of color exchange in the hard scattering process leads to large
rapidity intervals with no jets in the central part of the detector; however
the rapidity gap topology will be difficult to exploit due to the high level
of pileup at a high-luminosity LHC.
### 5.1 Vector Boson Scattering
The selection for VBS studies requires leptons with $p_{T}>25\,\text{GeV}$
and, to reduce non-VBS production, at least two high-$p_{T}$
($>50\,\text{GeV}$) forward jets are required with an invariant mass of the
two highest $p_{T}$ jets required to be greater than $1\,\text{TeV}$. In each
of the studies below, a particular higher dimension operator is chosen for
analysis, but in general each of the VBS channels studied have sensitivity to
each of these higher-dimension operators.
The scattering process $ZZ\rightarrow\ell\ell\ell\ell$ is sensitive to the
dimension-6 operator
${\cal L}_{\phi W}=\frac{c_{\phi W}}{\Lambda^{2}}{\rm
Tr}(W^{\mu\nu}W_{\mu\nu})\phi^{\dagger}\phi.$
Even though the fully-leptonic channel has a small cross section, it provides
a clean measurement of the $ZZ$ final state. The primary background comes from
non-VBS diboson production (‘SM ZZ QCD’ in Fig. 7). A statistical analysis of
the resulting $4\ell$ invariant mass distribution shown in Fig. 7 tests the
hypothesis of the new ${\cal L}_{\phi W}$ operator against the null (SM)
hypothesis. The discovery significance for various values of the coefficient
$\frac{c_{\phi W}}{\Lambda^{2}}$ is also shown in Fig. 7. The $5\sigma$
discovery reach increases by more than a factor of two when the integrated
luminosity changes from $300\,\mbox{fb${}^{-1}$}$ to
$3000\,\mbox{fb${}^{-1}$}$.
Figure 7: Left: The reconstructed 4-lepton invariant mass distribution in
$ZZ\to\ell\ell\ell\ell$ events. Right: The signal significance as a function
of $\frac{c_{\phi W}}{\Lambda^{2}}$ (right).
Another potential vector boson scattering channel is the $WZ$ final state. For
this channel, the dimension-8 operator
${\cal L}_{T,1}=\frac{f_{T1}}{\Lambda^{4}}{\rm
Tr}[\hat{W}_{\alpha\nu}\hat{W}^{\mu\beta}]\times{\rm
Tr}[\hat{W}_{\mu\beta}\hat{W}^{\alpha\nu}]$
is chosen for study. The $WZ$ final state benefits from a larger cross section
than the $ZZ$ channel. The invariant mass can still be reconstructed in the
fully leptonic channel by solving for the neutrino longitudinal momentum
$p_{Z}$ under a $W$ mass constraint. If all leptons have the same flavor, the
lepton pair with invariant mass closest to $m_{Z}$ is taken to be the $Z$. The
sensitivity of this analysis to new physics is included in the summary of
results in Table 5.
A third possible channel to investigate vector boson scattering is the same-
sign $W^{\pm}W^{\pm}$ final state, and the dimension-8 operator
${\cal
L}_{S,0}=\frac{f_{S0}}{\Lambda^{4}}[(D_{\mu}\phi)^{\dagger}D_{\nu}\phi)]\times[(D^{\mu}\phi)^{\dagger}D^{\nu}\phi)].$
is used. Two selected leptons must have the same charge, and the invariant
mass of the two highest-$p_{T}$ jets must be at least $1\,\text{TeV}$. The
primary backgrounds are Standard Model $WZ$ production, in which one of the
leptons from the $Z$-decay is not identified, and a small component of non-VBS
$W^{\pm}W^{\pm}$ production (‘SM ssWW QCD’). Misidentified-leptons, photon-
conversions in $W\gamma$ events, and charge-flip contributions, collectively
termed ‘mis-ID’ backgrounds, were accounted for by scaling the $WZ$ background
by a conservative factor of $\approx\\!\\!2$, taken from a study of same-sign
$WW$ production in the current ATLAS data. The statistical analysis is
performed by constructing templates of the $m_{lljj}$ distribution for
different values of $f_{S0}/\Lambda^{4}$. The distribution of $m_{lljj}$ and
the signal significance as a function of $f_{S0}/\Lambda^{4}$ are shown in
Fig. 8.
Figure 8: Left: The reconstructed 4-body mass spectrum using the two leading
leptons and jets, using the same-sign $WW\to\ell\nu\ell\nu$ VBS channel at
$pp$ center-of-mass collision energy of 14 TeV. Right: The signal significance
as a function of $f_{S0}$.
### 5.2 Gauge Boson Couplings in Triboson Production
The $Z\gamma\gamma$ mass spectrum at high mass is sensitive to BSM triboson
contributions through quartic gauge couplings. In this case, the lepton-photon
channel allows full reconstruction of the final state and the $Z\gamma\gamma$
invariant mass.
Beyond the simple $Z$ reconstruction, additional requirements that $\Delta
R(\ell,\gamma)>0.4$ and at least one $p_{T}(\gamma)>160\,{\mathrm{\
Ge\kern-1.20007ptV}}$ reduce the FSR contribution. This restricts the
measurement to a phase space that is uniquely sensitive to quartic gauge
couplings (QGC). The dominant process in the QGC-sensitive kinematic phase
space is the Standard Model $Z\gamma\gamma$ production, while the backgrounds
from $Z\gamma j$ and $Zjj$, with one or two jets misidentified as a photon,
are subdominant.
The new BSM effective operators chosen to study triboson production are
$\displaystyle{\cal L}_{T,8}$ $\displaystyle=$
$\displaystyle\frac{f_{T8}}{\Lambda^{4}}B_{\mu\nu}B^{\mu\nu}B_{\alpha\beta}B^{\alpha\beta}$
$\displaystyle{\cal L}_{T,9}$ $\displaystyle=$
$\displaystyle\frac{f_{T9}}{\Lambda^{4}}B_{\alpha\mu}B^{\mu\beta}B_{\beta\nu}B^{\nu\alpha}.$
(1)
which are uniquely probed by final states with neutral particles. Fig. 9 shows
the reconstructed $Z\gamma\gamma$ mass spectrum and expected discovery
significance for the ${\cal L}_{T,8}$ dimension-8 electroweak operator.
Figure 9: Left: Reconstructed mass spectrum for the charged leptons and
photons in selected $Z\gamma\gamma$ events. Right: The signal significance as
a function of $f_{T9}/\Lambda^{4}$.
### 5.3 Summary of Multiboson Studies
The higher-luminosity HL-LHC dataset increases the discovery range for these
new higher-dimension electroweak operators by more than a factor of two, as
shown in Table 5. If new physics in the electroweak sector is discovered in
the $300\,\mbox{fb${}^{-1}$}$ dataset, then the coefficients on the new
operators can be measured with 5% precision in the $3000\,\mbox{fb${}^{-1}$}$
dataset.
Parameter | dimension | channel | $\Lambda_{UV}$ [TeV] | 300 fb-1 | 3000 fb-1
---|---|---|---|---|---
$5\sigma$ | 95% CL | $5\sigma$ | 95% CL
$c_{\phi W}/\Lambda^{2}$ | 6 | $ZZ$ | 1.9 | 34 TeV-2 | 20 TeV-2 | 16 TeV-2 | 9.3 TeV-2
$f_{S0}/\Lambda^{4}$ | 8 | $W^{\pm}W^{\pm}$ | 2.0 | 10 TeV-4 | 6.8 TeV-4 | 4.5 TeV-4 | 0.8 TeV-4
$f_{T1}/\Lambda^{4}$ | 8 | $WZ$ | 3.7 | 1.3 TeV-4 | 0.7 TeV-4 | 0.6 TeV-4 | 0.3 TeV-4
$f_{T8}/\Lambda^{4}$ | 8 | $Z\gamma\gamma$ | 12 | 0.9 TeV-4 | 0.5 TeV-4 | 0.4 TeV-4 | 0.2 TeV-4
$f_{T9}/\Lambda^{4}$ | 8 | $Z\gamma\gamma$ | 13 | 2.0 TeV-4 | 0.9 TeV-4 | 0.7 TeV-4 | 0.3 TeV-4
Table 5: $5\sigma$-significance discovery values and 95% CL limits for
coefficients of higher-dimension electroweak operators. $\Lambda_{UV}$ is the
unitarity violation bound corresponding to the sensitivity with 3000 fb-1 of
integrated luminosity.
## 6 Searches for New Particles Predicted by Theories of Supersymmetry
Supersymmetry (SUSY) is an extended symmetry relating fermions and bosons. In
theories of supersymmetry, every SM boson (fermion) has a supersymmetric
fermion (boson) partner. Extending the sensitivity of the ATLAS experiment to
these new particles is one of the key aspects of the HL-LHC physics program.
In $R$-parity conserving supersymmetric extensions of the SM, SUSY particles
are produced in pairs, either through strong or weak production, and these
particles decay in a cascade of SUSY and SM particles. The lightest
supersymmetric particle (LSP) is stable in these $R$-parity conserving
extensions. As a result, the searches for evidence of SUSY particle production
focus on experimental signatures with large missing transverse momentum from
undetected LSPs.
A high-luminosity dataset benefits especially the searches for particles
produced in small cross section interactions or in signatures with small
branching fractions. Three representative searches and their potential for
discovery with a $3000\,\mbox{fb${}^{-1}$}$ dataset are presented in the
following subsections [19]. These results are only indicative of future
discovery prospects, and in fact are understood to be fairly conservative,
since they depend on conservative performance assumptions and analysis
strategies.
### 6.1 Direct Production of Weak Gauginos
Weak gauginos can be produced in decays of squarks and gluinos or directly in
weak production. For weak gaugino masses of several hundred GeV, as expected
from naturalness arguments [20], the weak production cross section is rather
small, ranging from $10^{-2}$ to $10\,\text{pb}$, and a dataset corresponding
to high integrated luminosity is necessary to achieve sensitivity to high-mass
weak gaugino production. Results with the 2012 data exclude charginos masses
of 300 to $600\,\text{GeV}$ for small LSP masses, depending on whether
sleptons are present in the decay chain. For LSP masses greater than
$100\,\text{GeV}$ there are currently no constraints from the LHC if the
sleptons are heavy .
The weak gauginos can decay via $\tilde{\chi}_{2}^{0}\rightarrow
Z\tilde{\chi}_{1}^{0}$ or $\tilde{\chi}^{\pm}_{1}\rightarrow
W^{\pm}\tilde{\chi}^{0}_{1}$, and both of these decays lead to a final state
with three leptons and large missing transverse momentum. SM background for
this final state is dominated by the irreducible $WZ$ process, even with a
high missing transverse momentum requirement of $150\,\text{GeV}$. Boosted
decision trees can be trained to use kinematic variables, such as the leptons′
transverse momenta, the $p_{T}$ of the Z-boson candidate, the summed $E_{T}$
in the event, and the transverse mass $m_{T}$ of the lepton from the $W$ and
the missing transverse momentum.
The expected sensitivity for the search is calculated using a simplified model
in which the $\tilde{\chi}^{0}_{2}$ and $\tilde{\chi}^{\pm}_{1}$ are nearly
degenerate in mass. With a ten-fold increase in integrated luminosity from 300
to $3000\,\mbox{fb${}^{-1}$}$, the discovery reach extends to chargino masses
above $800\,{\mathrm{\ Ge\kern-1.20007ptV}}$, to be compared with the reach of
$350\,{\mathrm{\ Ge\kern-1.20007ptV}}$ from the smaller dataset. The extended
discovery reach and comparison are shown in Fig. 10.
Figure 10: Discovery reach (solid lines) and exclusion limits (dashed lines)
for charginos and neutralinos in
$\tilde{\chi}^{\pm}_{1}\tilde{\chi}_{2}^{0}\rightarrow
W^{(\star)}\tilde{\chi}_{1}^{0}Z^{(\star)}\tilde{\chi}^{0}_{1}$ decays. The
results are shown for the $300\,\text{fb}^{-1}$ and $3000\,\text{fb}^{-1}$
datasets.
### 6.2 Direct Production of Top Squarks
Naturalness arguments lead to the conclusion that a Higgs boson mass of
$m_{H}=125\,{\mathrm{\ Ge\kern-1.20007ptV}}$ favors a light top squark mass,
less than $1\,{\mathrm{\ Te\kern-1.20007ptV}}$. A direct search for top
squarks needs to cover this allowed range of masses. The top squark pair
production cross section at $\sqrt{s}=14\,{\mathrm{\ Te\kern-1.20007ptV}}$ is
$10\,\text{fb}$ for $m_{\tilde{t}}=1\,{\mathrm{\ Te\kern-1.20007ptV}}$. For
the purpose of this study, the stops are assumed to decay either to a top
quark and the LSP ($\tilde{t}\rightarrow t+\tilde{\chi}^{0}_{1}$) or to a
bottom quark and the lightest chargino ($\tilde{t}\rightarrow
b+\tilde{\chi}^{\pm}_{1}$). The final state for the first decay is a top quark
pair in associated with large missing transverse momentum, while the final
state for the second decay is 2 $b$-jets, 2 $W$ bosons, and large missing
transverse momentum. In both cases, leptonic signatures are used to identify
the top quarks or the $W$ bosons. The 1-lepton + jet channel is sensitive to
$\tilde{t}\rightarrow t+\tilde{\chi}^{0}_{1}$, and the 2-lepton + jet channel
is sensitive to $\tilde{t}\rightarrow b+\tilde{\chi}_{1}^{\pm}$. For this
study, the event selection requirements were not reoptimized for a greater
integrated luminosity.
An increase in the integrated luminosity from 300 to
$3000\,\mbox{fb${}^{-1}$}$ results in an increase in a stop mass discovery
reach of approximately 150 GeV, up to $920\,\text{GeV}$ (see Fig. 11). This
increase covers a significant part of the top squark range favored by
naturalness arguments. In this study the same selection cuts were used for the
two luminosity values.
Figure 11: Discovery reach (solid lines) and exclusion limits (dashed lines)
for top squarks in the $\tilde{t}\rightarrow t+\tilde{\chi}^{0}_{1}$ (red) and
the $\tilde{t}\rightarrow
b+\tilde{\chi}^{\pm}_{1},\tilde{\chi}^{\pm}_{1}\rightarrow
W+\tilde{\chi}_{1}^{0}$ (green) decay modes.
### 6.3 Strong Production of Squarks and Gluinos
A high-luminosity dataset would allow the discovery reach for gluinos and
squarks to be pushed to the highest masses. Gluinos and light-flavor squarks
can be produced with a large cross section at $14\,\text{TeV}$, and the most
striking signature is still large missing transverse momentum as part of large
total effective mass. An optimized event selection for a benchmark point with
$m_{\tilde{q}}=m_{\tilde{g}}=3200\,\text{GeV}$ requires the missing transverse
momentum significance, defined as $E_{T}^{\text{miss}}/\sqrt{H_{T}}$, be
greater than $15\,\text{GeV}^{1/2}$. (The variable $H_{T}$ is defined to be
the scalar sum of the jet and lepton transverse energies and the missing
transverse momentum in the event.) Both the missing $E_{T}$ significance and
the effective mass are shown for the representative points in Fig. 12.
Figure 12: Distribution of missing $E_{T}$ significance for SM backgrounds and
two example SUSY benchmark points, normalized to 3000 fb-1(left), and
distributions of the effective mass (right), also normalized to 3000 fb-1. The
events shown in the effective mass distribution have passed the missing
$E_{T}$ significance $15$ GeV1/2 requirement, the lepton veto, and the jet
multiplicity requirement (at least 4 jets with $p_{T}>60\,\text{GeV}$).
The simple cut requirements on $H_{T}$, $M_{\text{eff}}$ and the
$E_{T}^{\text{miss}}$ significance are re-optimized for the high-luminosity
dataset of $3000\,\mbox{fb${}^{-1}$}$. An increase in integrated luminosity
from 300 to $3000\,\mbox{fb${}^{-1}$}$ results in a 400 GeV increase in the
discovery reach, as shown in Fig. 13.
Figure 13: Discovery reach and 95% CL limits in a simplified squark–gluino
model with a massless neutralino. The color scale shows the
$\sqrt{s}=14\,{\mathrm{\ Te\kern-1.20007ptV}}$ NLO cross-section. The solid
(dashed) lines show the $5\sigma$ discovery reach (95% CL exclusion limit)
with 300 fb-1 and with 3000 fb-1, respectively.
## 7 Searches for Exotic Particles and Interactions
The HL-LHC substantially increases the potential for the discovery of exotic
new phenomena. The range of possible phenomena is quite large. In this section
we discuss two benchmark exotic models of BSM physics and the expected gain in
sensitivity from the order of magnitude increase in integrated luminosity
provided by the HL-LHC.
### 7.1 Searches for $t\bar{t}$ Resonances
Strongly- and weakly-produced $t\bar{t}$ resonances provide benchmarks not
only for cascade decays containing leptons, jets (including $b$-quark jets)
and $E_{\mathrm{T}}^{\mathrm{miss}}$, but also the opportunity to study highly
boosted topologies. The sensitivity to the Kaluza-Klein gluon ($g_{KK}$) via
the process $pp\to g_{KK}\to t\bar{t}\,$ and a heavy $Z^{\prime}$ decaying to
$t\bar{t}$ at the HL-LHC is studied in both the dilepton and the lepton+jets
decay modes of the $t\bar{t}$ pair [21].
The two $t\bar{t}$ decay modes are complementary in that the lepton+jets mode
allows a more complete reconstruction of the $t\bar{t}$ invariant mass, but
suffers from more background, whereas the dilepton channel benefits from a
smaller background contribution, but a more difficult reconstruction of the
$t\bar{t}$ invariant mass. In addition, in the case of boosted $t\bar{t}$
pairs, the dilepton decay mode is less affected by the merging of top quark
decay products since the leptons are easier to identify close to a $b-$jet
than are jets from the $W$ decay. The lepton+jets mode therefore uses the
reconstructed $t\bar{t}$ invariant mass distribution, while the dilepton mode
uses the distribution of the scalar sum, $H_{T}$, of the $E_{T}$ of the two
leading leptons, two leading jets, and missing $E_{T}$. The statistical
analysis is performed by a likelihood fit of templates of these distributions,
using background plus varying amounts of signal, to the simulated data. The
$H_{T}$ and $m_{t\bar{t}}$ distributions and the resulting limits as a
function of the $g_{KK}$ pole mass for the dilepton and lepton+jets channel
are shown in Fig. 14 and Fig. 15, respectively.
The 95% CL expected limits in the absence of signal, using statistical errors
only, are shown in Table 6. The increase of a factor of ten in integrated
luminosity, from 300 to 3000 fb-1 raises the sensitivity to high-mass
$t\bar{t}$ resonances by up to 2.4 TeV.
Figure 14: Left: The reconstructed resonance $H_{T}$ spectrum for the
$g_{KK}\to t\bar{t}$ search in the dilepton channel with
$3000\,\mbox{fb${}^{-1}$}$ for $pp$ collisions at $\sqrt{s}=14{\mathrm{\
Te\kern-1.20007ptV}}$. The highest-$H_{T}$ bin includes the overflow. Right:
The 95% CL limit on the cross section times branching ratio. Also shown is the
theoretical expectation for the $g_{KK}$ cross section, for a ratio of the
coupling to quarks to $g_{s}$ of -0.2, where $g_{s}=\sqrt{4\pi\alpha_{s}}$.
Figure 15: Left: The reconstructed resonance mass spectrum for the $g_{KK}\to t\bar{t}$ search in the lepton+jets channel with $3000\,\mbox{fb${}^{-1}$}$ for $pp$ collisions at $\sqrt{s}=14{\mathrm{\ Te\kern-1.20007ptV}}$. The highest-mass bin includes the overflow. Right: The 95% CL limit on the cross section times branching ratio. Also shown is the theoretical expectation for the $g_{KK}$ cross section, for a ratio of the coupling to quarks to $g_{s}$ of -0.2, where $g_{s}=\sqrt{4\pi\alpha_{s}}$. model | $300\,\mbox{fb${}^{-1}$}$ | $1000\,\mbox{fb${}^{-1}$}$ | $3000\,\mbox{fb${}^{-1}$}$
---|---|---|---
$g_{KK}$ | 4.3 (4.0) | 5.6 (4.9) | 6.7 (5.6)
$Z^{\prime}_{\rm topcolor}$ | 3.3 (1.8) | 4.5 (2.6) | 5.5 (3.2)
Table 6: Summary of the expected limits for $g_{KK}\to t\bar{t}$ and
$Z^{\prime}_{\rm topcolor}\to t\bar{t}$ searches in the lepton+jets (dilepton)
channel for $pp$ collisions at $\sqrt{s}=14{\mathrm{\ Te\kern-1.20007ptV}}$.
All limits are quoted in TeV.
### 7.2 Searches for Dilepton Resonances
For studies of the sensitivity to a $Z^{\prime}$ boson [21], the dielectron
and dimuon channels are considered separately since their momentum resolutions
scale differently with $p_{T}$ and the detector acceptances are different. The
background is dominated by the SM Drell-Yan production, while $t\bar{t}$ and
diboson backgrounds are substantially smaller. Therefore, only the Drell-Yan
background is considered in this study. There is an additional background from
non-prompt electrons due to photon conversions which needs to be suppressed in
the dielectron channel. The required rejection of this background is assumed
to be achieved with the upgraded detector.
Templates of the $m_{\ell\ell}$ spectrum are constructed for the background
plus varying amounts of signal at different resonance masses and cross
sections. The Sequential Standard Model (SSM) $Z^{\prime}_{SSM}$ boson, which
has the same fermionic couplings as the Standard Model $Z$ boson, is used as
the signal template. The $m_{\ell\ell}$ distribution, for events above 200
GeV, and the resulting limits as a function of $Z^{\prime}_{SSM}$ pole mass
are shown in Figs. 16 and 17 for the $ee$ and $\mu\mu$ channels, respectively.
The 95% CL expected limits in the absence of signal, using statistical errors
only, are shown in Table 7. The increase of a factor of ten, from 300 to 3000
fb-1 in integrated luminosity raises the sensitivity to high-mass dilepton
resonances by up to 1.3 TeV.
model | $300\,\mbox{fb${}^{-1}$}$ | $1000\,\mbox{fb${}^{-1}$}$ | $3000\,\mbox{fb${}^{-1}$}$
---|---|---|---
$Z^{\prime}_{SSM}\to ee$ | 6.5 | 7.2 | 7.8
$Z^{\prime}_{SSM}\to\mu\mu$ | 6.4 | 7.1 | 7.6
Table 7: Summary of the expected limits for $Z^{\prime}_{SSM}\to ee$ and
$Z^{\prime}_{SSM}\to\mu\mu$ searches in the Sequential Standard Model for $pp$
collisions at $\sqrt{s}=14{\mathrm{\ Te\kern-1.20007ptV}}$. All limits are
quoted in TeV.
Figure 16: Left: The reconstructed dielectron mass spectrum for the
$Z^{\prime}$ search with $3000\,\mbox{fb${}^{-1}$}$ of $pp$ collisions at
$\sqrt{s}=14{\mathrm{\ Te\kern-1.20007ptV}}$. The highest-mass bin includes
the overflow. Right: The 95% CL upper limit on the cross section times
branching ratio. Also shown is the theoretical expectation for the
$Z^{\prime}_{SSM}$ cross section.
Figure 17: Left: The reconstructed dimuon mass spectrum for the $Z^{\prime}$
search with $3000\,\mbox{fb${}^{-1}$}$ of $pp$ collisions at
$\sqrt{s}=14{\mathrm{\ Te\kern-1.20007ptV}}$. The highest-mass bin includes
the overflow. Right: he 95% CL upper limit on the cross section times
branching ratio. Also shown is the theoretical expectation for the
$Z^{\prime}_{SSM}$ cross section.
## 8 Flavor-Changing-Neutral-Currents in Top-Quark Decays
Within the Standard Model, flavor-changing-neutral-current (FCNC) decays are
forbidden at tree level due to the GIM mechanism [22], and highly suppressed
at loop level with branching fractions below $10^{-12}$ [23, 24, 25, 26],
which are inaccessible even at HL-LHC. Therefore any observation of top quark
FCNC decays would be a definite indication of new physics. FCNC decays have
been sensitively searched for in lighter quarks, placing strong constraints on
many models of BSM physics. Tests of FCNC in the top sector have only recently
become sensitive enough to probe interesting BSM phase space in which the FCNC
branching fraction can be significantly enhanced [27]. Examples of BSM models
with enhanced FCNC top decay rates are quark-singlet (QS) models, two-Higgs
doublet (2HDM) and flavor-conserving two-Higgs doublet (FC 2HDM) models, the
minimal supersymmetric model (MSSM), SUSY models with R-parity violation
($\not{R}$), the topcolor assisted technicolor model (TC2) [28], and models
with warped extra dimensions (RC) [29]. FCNC decay are sought through
$t\rightarrow q\gamma$ and $t\rightarrow qZ$ channels where $q$ is either an
up or a charm quark. Table 8 shows the Standard Model and BSM decay rates in
the various channels. The best current direct search limits are 3.2% for
$t\rightarrow\gamma q$ [30] and 0.21% for $t\rightarrow Zq$ [31].
Process | SM | QS | 2HDM | FC 2HDM | MSSM | $\not{R}$ | TC2 | RS
---|---|---|---|---|---|---|---|---
$t\rightarrow u\gamma$ | $3.7\times 10^{-16}$ | $7.5\times 10^{-9}$ | — | — | $2\times 10^{-6}$ | $1\times 10^{-6}$ | — | $\sim 10^{-11}$
$t\rightarrow uZ$ | $8.0\times 10^{-17}$ | $1.1\times 10^{-4}$ | — | — | $2\times 10^{-6}$ | $3\times 10^{-5}$ | — | $\sim 10^{-9}$
$t\rightarrow ug$ | $3.7\times 10^{-14}$ | $1.5\times 10^{-7}$ | — | — | $8\times 10^{-5}$ | $2\times 10^{-4}$ | — | $\sim 10^{-11}$
$t\rightarrow c\gamma$ | $4.6\times 10^{-14}$ | $7.5\times 10^{-9}$ | $\sim 10^{-6}$ | $\sim 10^{-9}$ | $2\times 10^{-6}$ | $1\times 10^{-6}$ | $\sim 10^{-6}$ | $\sim 10^{-9}$
$t\rightarrow cZ$ | $1.0\times 10^{-14}$ | $1.1\times 10^{-4}$ | $\sim 10^{-7}$ | $\sim 10^{-10}$ | $2\times 10^{-6}$ | $3\times 10^{-5}$ | $\sim 10^{-4}$ | $\sim 10^{-5}$
$t\rightarrow cg$ | $4.6\times 10^{-12}$ | $1.5\times 10^{-7}$ | $\sim 10^{-4}$ | $\sim 10^{-8}$ | $8.5\times 10^{-5}$ | $2\times 10^{-4}$ | $\sim 10^{-4}$ | $\sim 10^{-9}$
Table 8: Branching fractions for top FCNC decays for the Standard Model and
BSM extensions. References are given in the text.
A model-independent approach to top quark FCNC decays using an effective
Lagrangian [32, 33, 34] is used here to evaluate the sensitivity of ATLAS in
the HL-LHC era. Even if the LHC does not measure the top quark FCNC branching
ratios, it can test some of these models or constrain their parameter space,
and improve significantly the current experimental limits on the FCNC
branching ratios.
Top quark FCNC decays are sought in top quark pair production in which one top
(or anti-top) decays to the SM $Wb$ final state, while the other undergoes a
FCNC decay to $Zq$ or $\gamma q$. The sensitivity is evaluated selecting
events as in [35] for the $t\rightarrow Zq$ channel and [36] for the
$t\rightarrow\gamma q$ channel. For the $t\rightarrow\gamma q$ channel, the
dominant backgrounds are $t\bar{t}$, $Z+$jets and $W+$jets events. For the
$t\rightarrow Zq$ channel, the background is mainly composed of $t\bar{t}$,
$Z+$jets and $WZ$ events.
In the absence of FCNC decays, limits on production cross-sections are
evaluated and converted to limits on branching ratios using the SM
$t\overline{t}$ cross-section. The HL-LHC expected limits at 95% CL for the
$t\rightarrow\gamma q$ and the $t\rightarrow Zq$ channels, are in the range
between $10^{-5}$ and $10^{-4}$. Figure 18 shows the expected sensitivity in
the absence of signal, for the ${t}\to{q}\gamma$ and ${t}\to{qZ}$ channels.
Here the lines labeled ‘sequential’ correspond to a sensitivity extrapolated
from the analysis done with the 7 TeV data [35]. Those labeled ‘discriminant’
correspond to a dedicated analysis using 14 TeV TopRex Monte Carlo [37] data
and a likelihood discriminant. Further improvements could come from the use of
more sophisticated analysis discriminants.
Figure 18: The present 95% CL observed limits on the $BR({t}\to\gamma q)$ vs.
$BR({t}\to{Zq})$ plane are shown as full lines for the LEP, ZEUS, H1, D0, CDF,
ATLAS and CMS collaborations. The expected sensitivity at ATLAS is also
represented by the dashed lines. For an integrated luminosity of
$L=3000\,\mbox{fb${}^{-1}$}$ the limits range from $1.3\times 10^{-5}$ to
$2.5\times 10^{-5}$ ($4.1\times 10^{-5}$ to $7.2\times 10^{-5}$) for the
${t}\to~{}\gamma q$ (${t}\to~{}{Zq}$) decay. Limits at
$L=300\,\mbox{fb${}^{-1}$}$ are also shown.
## 9 Conclusions
Studies illustrating the physics case of a high-luminosity upgrade of the LHC
have been presented. In general, very important gains in the physics reach are
possible with the HL-LHC dataset of $3000\,\mbox{fb${}^{-1}$}$, and some
studies are only viable with this high integrated luminosity. The precision on
the production cross section times branching ratio for most Higgs boson decay
modes can be improved by a factor of two to three. Furthermore, the rare decay
mode of the Higgs boson $H\rightarrow\mu\mu$ only becomes accessible with
$3000\,\mbox{fb${}^{-1}$}$. When results from both experiments are combined,
first evidence for the Higgs self-coupling may be within reach, representing a
fundamental test of the Standard Model. In searches for new particles, the
mass reach can be increased by up to 50% with the high-luminosity dataset.
The luminosity upgrade would become even more interesting if new phenomena are
seen during the $300\,\mbox{fb${}^{-1}$}$ phase of the LHC, as the ten-fold
increase in luminosity would give access to measurements of the new physics.
To reach these goals a detector performance similar to that of the present one
is needed, however under much harsher pileup and radiation conditions than
today. The ATLAS Collaboration is committed to preparing detector upgrades
that will realize the potential of HL-LHC operations with the goal of an
integrated luminosity of $3000\,\mbox{fb${}^{-1}$}$.
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|
arxiv-papers
| 2013-07-27T18:21:58 |
2024-09-04T02:49:48.557880
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "ATLAS Collaboration",
"submitter": "Jason Nielsen",
"url": "https://arxiv.org/abs/1307.7292"
}
|
1307.7301
|
# Strain-induced effects on the magnetic and electronic properties of
epitaxial Fe1-xCoxSi thin films
P. Sinha [email protected] School of Physics & Astronomy, University of Leeds,
Leeds, LS2 9JT, UK N. A. Porter School of Physics & Astronomy, University of
Leeds, Leeds, LS2 9JT, UK C. H. Marrows [email protected] School of
Physics & Astronomy, University of Leeds, Leeds, LS2 9JT, UK
###### Abstract
We have investigated the Co-doping dependence of the structural, transport,
and magnetic properties of $\epsilon$-Fe1-xCoxSi epilayers grown by molecular
beam epitaxy on silicon (111) substrates. Low energy electron diffraction,
atomic force microscopy, X-ray diffraction, and high resolution transmission
electron microscopy studies have confirmed the growth of phase-pure, defect-
free $\epsilon$-Fe1-xCoxSi epitaxial films with a surface roughness of $\sim
1$ nm. These epilayers are strained due to lattice mismatch with the
substrate, deforming the cubic B20 lattice so that it becomes rhombohedral.
The temperature dependence of the resistivity changes as the Co concentration
is increased, being semiconducting-like for low $x$ and metallic-like for
$x\gtrsim 0.3$. The films exhibit the positive linear magnetoresistance that
is characteristic of $\epsilon$-Fe1-xCoxSi below their magnetic ordering
temperatures $T_{\mathrm{ord}}$, as well as the huge anomalous Hall effect of
order several $\mu\Omega$cm. The ordering temperatures are higher than those
observed in bulk, up to 77 K for $x=0.4$. The saturation magnetic moment of
the films varies as a function of Co doping, with a contribution of $\sim
1~{}\mu_{\rm B}$/ Co atom for $x\lesssim 0.25$. When taken in combination with
the carrier density derived from the ordinary Hall effect, this signifies a
highly spin-polarised electron gas in the low $x$, semiconducting regime.
###### pacs:
68.55.-a, 72.20.My, 73.50.Jt
## I introduction
The rich behaviour shown by ferromagnetic semiconductors arise from an
interesting interplay of their electronic density of states and magnetic
interactions within the crystal structure, offering new possibilities for
spintronics.Ohno (2010) Whilst most magnetic semiconductors to date are based
on compound or oxide materials, the transition metal monosilicides are
promising candidates in that they are based on silicon, by far the most common
commercial semiconductor. These materials crystallize in cubic B20 structure,
the $\epsilon$-phase, and which belongs to the space group $P2_{1}3$.Al-Sharif
_et al._ (2001) They are continuously miscible with each other and form an
isostructural series compounds with endmembers MnSi (a metallic helimagnet),
FeSi (a paramagnetic narrow-gap semiconductor), and CoSi (a metallic
diamagnet).Manyala _et al._ (2000) They have been studied for many years as
they exhibit wide variety of different aspects of condensed matter physics
including paramagnetic anomalies,Wertheim _et al._ (1965); Jaccarino _et
al._ (1967) strongly correlated/Kondo insulator-like behaviour,Schlesinger
_et al._ (1993); Paschen _et al._ (1997); DiTusa _et al._ (1997); Aeppli and
DiTusa (1999) non-Fermi liquid behaviour,Pfleiderer _et al._ (2001); Manyala
_et al._ (2008); Ritz _et al._ (2013a) unusual magnetoresistance,Manyala _et
al._ (2000); Onose _et al._ (2005); Porter _et al._ (2012) and helical
magnetismBeille _et al._ (1981, 1983); Uchida _et al._ (2006); Grigoriev
_et al._ (2009) with skyrmion phasesMühlbauer _et al._ (2009); Münzer _et
al._ (2010); Yu _et al._ (2010); Milde _et al._ (2013) that have associated
topological Hall effects.Lee _et al._ (2009); Neubauer _et al._ (2009); Ritz
_et al._ (2013b)
Almost all work to date on the monosilicide materials has been carried out
using bulk single crystal samples. For technological applications, thin films
that can be patterned into devices with conventional planar processing
techniques are required. Epilayers of the helimagnetic metal MnSi have been
grown by using molecular beam epitaxy (MBE) by Karhu et al.,Karhu _et al._
(2010, 2011, 2012) Li et al.,Li _et al._ (2013) and Engelke et al.Engelke
_et al._ (2012) The properties are broadly comparable to those of the bulk
material, including the presence of chiral magnetismKarhu _et al._ (2011) and
a topological Hall effect.Li _et al._ (2013) Other monosilicides have
received less attention to date. The family of alloys Fe1-xCoxSi should be of
particular interest for spintronics: whilst both endmembers are non-magnetic,
magnetic ordering is evident at almost all intermediate values of $x$.Manyala
_et al._ (2000) For low doping levels of Co in the semiconducting parent FeSi,
a magnetic semiconductor with a half-metallic state is expected.Manyala _et
al._ (2000); Guevara _et al._ (2004) Polycrystalline thin films of Fe1-xCoxSi
have been grown by pulsed laser deposition,Manyala _et al._ (2009) and
sputtering,Morley _et al._ (2011) but with properties that fall short of
those in single crystal samples due to microstructural disorder and lack of
phase purity.
Here we report on the properties of epitaxial $\epsilon$-Fe1-xCoxSi layers
grown on commercial (111) Si substrates, across the doping range $0\leq x\leq
0.5$, using the growth methods we have previously developed.Porter _et al._
(2012) The films are phase pure, with a B20 lattice that is distorted by
biaxial in-plane epitaxial strain to have a rhombohedral unit cell. Although
Fe1-xCoxSi is known to possess a helimagnetic ground state,Beille _et al._
(1981, 1983); Uchida _et al._ (2006); Grigoriev _et al._ (2009) we focus
here on the properties in fields large enough to generate a uniformly
magnetized ferromagnetic state, which are modest in size. We find that these
epilayers display the full range of properties expected of this material,
including a characteristic temperature dependence of resistivity,Onose _et
al._ (2005), positive linear magnetoresistance,Manyala _et al._ (2000); Onose
_et al._ (2005), and a very large anomalous Hall effect.Manyala _et al._
(2004) Measurements of the number of Bohr magnetons ($\mu_{\mathrm{B}}$) of
magnetic moment and electron-like carriers per Co indicate the presence of a
highly spin-polarised electron gas in the low doping ($x\lesssim 0.25$)
regime,Manyala _et al._ (2000); Morley _et al._ (2011) where the half-
metallic state is expected.Guevara _et al._ (2004) Nevertheless, the presence
of epitaxial strain, giving rise to an expanded unit cell volume, leads to
some quantitative changes, the most prominent of which is a substantial
enhancement of the magnetic ordering temperature with respect to bulk
crystals. These epilayers are suitable for patterning into nanostructures that
may find use as spin injectors into siliconMin _et al._ (2006); Appelbaum
_et al._ (2007); Huang _et al._ (2007) or exploit the chiral nature of the
magnetism at low fields in skyrmion-based devices.Kiselev _et al._ (2011);
Fert _et al._ (2013); Lin _et al._ (2013)
## II Growth and structural characterisation
The Fe1-xCoxSi thin films were prepared by simultaneous co-evaporation of Fe,
Co, and Si by MBE on a lightly n-doped silicon (111) substrates with
$2000$-$3000~{}\Omega$cm resistivity at room temperature. The level of Co-
doping $x$ of the various Fe1-xCoxSi films was determined by controlling the
individual rates of incoming flux. We adopted the growth protocol described by
Porter et al. in Ref. Porter _et al._ , 2012. The base pressure of the growth
chamber remained within the range $2.8$-$4.8\times 10^{-11}$ mbar. Prior to
the deposition of the film, the substrates were annealed at 1200∘C until a
well ordered $7\times 7$ reconstructed Si (111) surface was obtained. A low
energy electron diffraction pattern demonstrating this reconstruction is shown
in Fig. 1(c). The films were then grown by depositing a seed layer of Fe of
$\sim 5.4$ Å thickness at room temperature, followed by the deposition of a
$\sim 50$ nm thick Fe1-xCoxSi layer at a net flux rate of $\sim 0.4$ Å/s at
400 ∘C . The films were then further annealed at 400∘C for 15 minutes, before
being allowed to cool to room temperature for further characterisation.
The films grow in the (111) orientation and are $\epsilon$-phase pure, as can
be seen from the Cu $K_{\alpha}$ X-ray diffraction (XRD) spectrum shown in
Fig. 1(a). In-plane epitaxy of the Fe1-xCoxSi films is seen to be achieved by
a 30∘ in-plane rotation of the surface unit cell with respect to the Si, such
that the Fe1-xCoxSi [112̄] direction is aligned parallel to Si [11̄0],
demonstrated by the LEED pattern of a completed epilayer in Fig. 1(d). Atomic
force microscopy (AFM) was used to map the surface topography of the films: an
representative micrograph is shown in Fig. 1(b). The root mean square (rms)
roughness of the films were estimated from these images to be around 1 nm.
Figure 1: (Color online) Structural characterisation of the 50 nm thick
Fe1-xCoxSi epilayers. (a) XRD spectrum of a $x=0.5$ film, illustrating the
phase purity of the B20 structure and the (111) epitaxial orientation of the
film. (b) Atomic force micrograph of the top surface of an Fe1-xCoxSi epilayer
with $x=0.5$. (c) LEED pattern of an annealed Si (111) substrate prior to film
growth. The $7\times 7$ surface reconstruction is evident. (d) LEED pattern
from an Fe1-xCoxSi film $x=0.3$, demonstrating epitaxial growth in the (111)
orientation.
For further structural verification, high resolution transmission electron
microscopy (HRTEM) and energy dispersive X-ray analysis (EDX) were carried out
on cross-section specimens prepared by focussed ion beam (FIB). Fig. 2(a) and
(b) show the top and bottom interfaces of a Fe1-xCoxSi film with $x=0.5$. The
films look well-ordered throughout and epitaxial growth can be observed with
the orientation (111)Fe1-xCoxSi$\|$(111)Si : [112̄]Fe1-xCoxSi$\|$[11̄0]Si.
Sample cross sections were mapped with EDX which confirmed the homogeneous
distribution and chemical composition of the films. In-plane (110) lattice
parameters were determined from the HRTEM images, which we discuss below.
Figure 2: HRTEM of an Fe1-xCoxSi epilayer with $x=0.5$ on the [112] zone axis,
showing the upper (a) and lower (b) interfaces.
## III Strain characterisation
Heteroepitaxy gives rise to strained growth of films as a result of the
lattice mismatch between substrate and the film. The lattice parameter of Si
is 5.431 Å, whilst that of bulk FeSi is 4.482 Å. It is to accommodate this
large difference that the film grows with the 30∘ in-plane rotation
demonstrated above by LEED (see Fig. 1(c) and 1(d)) and HRTEM (Fig. 2(a) and
2(b)). This gives rise to an in-plane lattice mismatch of $5.6\%$ at the
interface. Inspection of the LEED patterns shows that this is relaxed to $\sim
3.7\%$ at the surface of a 50 nm thick film (see above). The heteroepitaxy
induces biaxial tensile strain in the in-plane directions of the Fe1-xCoxSi
layers, with corresponding compression in the out-of-plane direction, which
distorts the cubic B20 lattice to have a rhombohedral form.
The position of the Fe1-xCoxSi [111] and [222] Bragg peaks, obtained from
$\theta$-$2\theta$ high angle XRD scans, were used to determine the out-of-
plane [111] lattice parameter of Fe1-xCoxSi films using the Bragg law. In
order to make quantitative comparisons of our samples, we define the parameter
$a^{hkl}$, the lattice constant, assuming a cubic unit cell, that is
determined from a measured interplanar spacing $d^{hkl}$ associated with a
particular set of lattice planes $(hkl)$. A systematic decrease in out-of-
plane lattice constant, $a^{111}$ is observed with increasing Co content $x$
in the films, as shown in Fig. 3(a). The linear variation of the out-of-plane
lattice parameter with $x$ shows that Vegard’s law is followed, as is the case
in bulk crystals of this material.Shinoda (1972) However, there is also the
large in-plane lattice mismatch with the Si substrate that was discussed above
in the case of thin films. The in-plane lattice parameter $a^{110}$ at the
surface of the Fe1-xCoxSi films, shown in Fig. 3(b), varies from
$4.45\pm~{}0.02$ Å for $x=0$ to $4.64\pm 0.02$ Å for $x=0.5$, as determined
from analysis of the LEED patterns, using the ($7\times 7$) reconstructed Si
(111) pattern to provide a calibration. Overall we see that the in-plane
lattice parameter of epitaxial Fe1-xCoxSi is larger than the corresponding
out-of-plane lattice parameter and is closer to that of Si (5.431Å). The
variation with $x$ is plotted in Fig. 3(b).
Figure 3: (Color online) Strain analysis. (a) Out-of-plane lattice parameter
(LP) $a^{111}$ of Fe1-xCoxSi films based on data from XRD. (b) In-plane
lattice parameter (LP) $a^{110}$ at the surface of the film, based on data
from LEED. (c) Out-of-plane of strain in the unit cell. (d) In-plane strain in
the unit cell. (e) Rhombohedral unit cell volume as a function of $x$. (f)
Rhombohedral angle as a function of $x$. The solid lines are linear best fits,
the dashed lines are guides to the eye.
Based on data from Fig. 3(a) and 3(b), the out-of-plane compressive,
$\varepsilon_{\perp}$, and in-plane tensile, $\varepsilon_{\|}$, strains in
the crystal structure were calculated, with the results shown in Fig. 3(c) and
(d), using the following expression:
$\varepsilon^{hkl}=\frac{a^{hkl}_{\mathrm{epi}}-a^{hkl}_{\mathrm{bulk}}}{a^{hkl}_{\mathrm{bulk}}},$
(1)
where $a^{hkl}_{\mathrm{epi}}$ is the lattice parameter as measured for a
given epilayer and $a^{hkl}_{\mathrm{bulk}}$ is the corresponding lattice
parameter in the bulk (Manyala _et al._ , 2004). In both the cases strain
follows a nonlinear relationship with the Co-doping level $x$. For higher
values of $x$ the out-of-plane lattice constant is more compressed, whilst the
in-plane lattice is extended.
The different methods we have used to determine the lattice constants give
information about different parts of the film. Using the TEM images as shown
in Fig. 2(b) it is possible to determine the lattice constant of the
Fe1-xCoxSi near the Si substrate. In Fig. 4(a), we plot the unit cell face
diagonal $d^{\prime}_{110}$ for selected values of $x$ as obtained from TEM.
For $x=0$ and $x=0.2$, $d^{\prime}_{110}$ is measured to be $6.62\pm~{}0.02$ Å
and $6.66\pm~{}0.02$ Å respectively. These values are seen to match well to
the Si (112) face diagonal (6.6501 Å), which it must for heteroepitaxial
growth. Our LEED data are surface sensitive, however. Measuring
$d^{\prime}_{110}$ from our LEED patterns shows considerable variation with
$x$ (Fig. 4(a)). For $x=0$, there is a good match to the bulk value for this
crystallographic distance, if we assume a cubic crystal structure. We can
conclude from this comparison that the Fe1-xCoxSi films are strained at the Si
interface to adapt to the lattice constant of Si substrate. At greater
distances from the interface with the substrate, the lattice relaxes
throughout the 50 nm film thickness, and adapts to its own strained lattice
constant for a rhombohedral crystal structure which is somewhere in between
that of Si and the Fe1-xCoxSi cubic assumption of crystal structure. The
variation of volume strain with shear strain in Fe1-xCoxSi film is shown in
the Fig. 4(b) for various Co doping ranging from $x=0$ to $x=0.5$. The
linearity in the relationship confirms that the epitaxial strain in Fe1-xCoxSi
film changes only the angle of the unit cell as shown in Fig. 3(f) and that
there are no structural phase changes associated with the strain. Thus, even
though the strained Fe1-xCoxSi films have rhombohedral unit cell but they are
phase pure as shown in the Fig. 1(a).
Figure 4: (Color online) Epitaxial strain analysis. (a) Comparison of
evolution of unit cell face diagonal $d^{\prime}_{110}$ of Fe1-xCoxSi films as
a function of cobalt content from data obtained by LEED, TEM and theoretical
prediction. b) Variation of volume strain with shear strain for various Co
doping in Fe1-xCoxSi films. The dashed line is a straight line best fit to the
data.
Knowledge of the in-plane and out-of-plane lattice constants give a full
determination of the geometry of the rhombohedral unit cell. The volume of the
unit cell as function of $x$ is plotted in Fig. 3(e). The unit cell volume
increases in a monotonic but non-linear fashion with $x$. We have also
calculated the variation of the rhombohedral angle as a function the varying
Co doping, shown in Fig. 3(f). The angle increases from little more than $90$∘
for $x=0$ to $\sim 92$∘ for $x=0.5$. Since the in-plane strain is determined
from LEED, these values apply close to the top surface of the epilayer. These
changes in unit cell geometry induced by epitaxial strain can be expected to
give rise to modifications to various properties such as the band structure,
density of states, transport properties, magnetization and magnetic
anisotropy, which we will explore in remainder of the paper.
## IV Transport Properties
The transport properties of our Fe1-xCoxSi films were measured in a gas-flow
cryostat with a base temperature of 1.4 K capable of applying magnetic fields
of up to 8 T. The films were patterned into Hall bars which were $5~{}\mu$m
wide using optical lithography, etched by Ar ion milling, and bonded onto a
chip carrier for measurement.
Measurements of the electrical resistivity $\rho(T,H)$ of the films as a
function of temperature $T$ and magnetic field $H$ applied perpendicular to
the sample plane are shown in Fig. 5. A bias current of $30~{}\mu$A was used.
The solid lines show the $\rho(T)$ in absence of magnetic field and the dashed
lines show $\rho(T)$ in presence of an 8 T magnetic field. Fig. 5(a) shows the
resistivity variation of an FeSi film. FeSi is a narrow band-gap
semiconductor,Jaccarino _et al._ (1967) and upon decreasing the temperature
the resistivity increases reaching $3700~{}\mu\Omega$cm at 1.4K. We determined
the band-gap of the epitaxial FeSi to be $\Delta=30.1\pm 0.2$ meV using the
following relation:
$\ln\rho\propto\left(\frac{\Delta}{2k_{\mathrm{B}}T}\right),$ (2)
where $k_{B}$ is the Boltzmann constant, fitted to the high temperature data
(above about 50 K).
Doping FeSi with Co introduces electron-like carriers and a lowered
resistivity. At the opposite extreme, the $\rho(T)$ relation for the film with
$x=0.5$ has a metallic form, shown in Fig. 5(f), increasing with $T$ for all
temperatures. Intermediate values of $x$ yield hybrid $\rho(T,0)$ dependences,
with a gradual crossover from semiconductor-like to metal-like behavior as $x$
rises. For these values of $x$ the $\rho(T,0)$ curve is often non-monotonic,
combining regions with both positive and negative temperature coefficients of
resistance. The curves are similar to those measured for bulk crystals at a
qualitative level,Manyala _et al._ (2000); Onose _et al._ (2005) but differ
quantitatively.
Figure 5: (Color online) Temperature dependence of resistivity in $\sim$ 50 nm
films of Fe1-xCoxSi in magnetic fields of 0 T (solid lines) and 8 T (dashed
lines). Increasing cobalt concentration $x$ changes the temperature
coefficient of resistivity from negative (semiconductor-like) for $x=0$ to
positive (metallic-like) for $x=0.5$, with mixed behavior seen for
intermediate values of $x$. $\uparrow$ and $\downarrow$ illustrate
respectively temperatures of minima, $T_{res}$, and maxima in the resistivity.
In the intermediate doping regime ($0.15<x<0.3$), we observe some distinctive
features such as points of local maximum ($T_{\mathrm{max}}$) and minimum
($T_{\mathrm{res}}$) resistivity that vary with the degree of Co doping. For
instance, in Fig. 5(b) (for $x=0.15$) we observe a broad maximum in $\rho$
around 125 K. As the Co doping increases this maximum shifts towards higher
temperatures, reaching 175 K for $x=0.3$, then becoming less pronounced until
it vanishes for $x=0.5$. The observed broad maximum is a feature reminiscent
of the narrow band-gap semiconducting parent compound FeSi Onose _et al._
(2005). The maxima and associated temperature shift can be explained in the
framework of epitaxial strain and Co doping. Substituting Co for Fe not only
introduces volume strain (as previously shown in Fig. 4(b)), but also changes
the band structure, resulting in a broadening of bands and reduced band
gap.Forthaus _et al._ (2011) Thus, increased Co doping provides more carriers
to be available for conduction, giving rise to the hybrid semiconducting-
metallic behaviour that we see. It is the competition between the temperature
dependence of mobility, importance of thermally activated carriers
(particularly at low $x$) and the carrier concentration that gives rise to
such difference in $\rho(x,T)$. Fe1-xCoxSi films thus lose the low $T$
insulating behaviour of FeSi as $x$ rises.
As the temperature is reduced further below $T_{\mathrm{max}}$, the
resistivity decreases until a minimum ($T_{\mathrm{res}}$) is reached. This
minimum in the resistivity curve is related to the magnetic behaviour of the
films and signifies the onset of magnetic ordering in the Fe1-xCoxSi crystal
structure.Forthaus _et al._ (2011) The position of the minimum
$T_{\mathrm{res}}$ varies with Co doping and is found to follow the same trend
as the magnetic ordering temperature $T_{\mathrm{ord}}$, as we shall discuss
later in §VII. Ideally, $T_{\mathrm{res}}\approx T_{\mathrm{ord}}$, but in the
samples studied here, we find that $T_{\mathrm{res}}$ is actually slightly
higher. The value of $T_{\mathrm{res}}$ increases with increasing Co doping
and reaches the maximum value of $\sim 92$ K for $x=0.4$ before decreasing
again. The transport properties of Fe1-xCoxSi epilayers are dominated by
short-ranged ferromagnetic interactions in the crystal structure.(Onose _et
al._ , 2005) When the mean free path is of the same order as the ferromagnetic
correlation length, $T_{\mathrm{ord}}$ and $T_{\mathrm{res}}$ almost coincide,
as is the case for $x=0.1,0.5$. However, if the mean free path is longer, then
$T_{\mathrm{res}}$ is higher than $T_{\mathrm{ord}}$, as we observe for
Fe1-xCoxSi films in the range $0<x<0.5$ (and discuss later in §VII). Also this
may be due to magnetic fluctuations occurring above the ordering temperature
which may contribute to the discrepancy between the magnetic ordering
temperature and $T_{\mathrm{res}}$ (Pfleiderer _et al._ , 2001) . When the
temperature is decreased below $T_{\mathrm{res}}$, the resistivity further
increases for the Fe1-xCoxSi films with $0<x<0.5$, as pointed out in the
previous studies.(Beille _et al._ , 1983; Manyala _et al._ , 2000)
Overall we observe semiconducting behaviour of the films for low $x$ and
metallic for high $x$. This remains the case when the measurements were
performed under a $\mu_{0}H=8$ T field applied perpendicular to the sample
plane (dashed lines in Fig. 5). In the high temperature region (above $\sim
T_{\mathrm{max}}$), the resistivity is almost unchanged with field for all our
Fe1-xCoxSi films. In the lower temperature regime, after the onset of magnetic
ordering, magnetoresistance gradually rises in the semiconducting regime,
washing out any maximum $\rho(T,\mathrm{8~{}T})$. Positive magnetoresistance
is a very typical property of the Fe1-xCoxSi system, and shall be discussed in
more detail in the next section.
## V Magnetoresistance
Unlike most other ferromagnetic metals, which show negative magnetoresistance
(MR) at high fields,Raquet _et al._ (2002) Fe1-xCoxSi systems show unusual
positive MR in the form of bulk crystals and epilayers.Manyala _et al._
(2000); Onose _et al._ (2005); Porter _et al._ (2012) The high field
magnetoresistance in these Fe1-xCoxSi samples, shown in Fig. 5 for a
perpendicular field orientation, is not only linear for $x>0$ , but also
isotropic for $T<T_{\mathrm{res}}$. For an FeSi film, which is a paramagnet,
the MR has a quadratic dependence on magnetic field. Introducing Co doping to
FeSi, changes the nature of the curve from quadratic to linear at $x=0.1$,
with a large MR ratio of almost 12% in an 8 T field at 5 K.
Figure 6: (Color online) Magnetoresistance. a) MR isotherms at 5 K for
Fe1-xCoxSi films of varying Co doping $x$. b) MR ratio at 8 T and 5 K as a
function of cobalt concentration $x$.
Fig 6(a) shows the magnetoresistance ratio observed in Fe1-xCoxSi epilayers
for different Co doping for a field of 8 T at 5 K. As the Co content is
increased from $x=0.1$ to $x=0.5$, we observe that the MR remains linear at
low temperatures ($T<T_{\mathrm{res}}$), i.e. in the presence of magnetic
ordering. (As the temperature is increased the linearity of the MR is lost,
and above $T_{\mathrm{max}}$ it becomes quadratic for all our Fe1-xCoxSi
films.) The maximum magnetoresistance should be observed near the metal-
insulator transition, where there is the highest Coulomb interaction. This is
observed here for $x=0.1$, as shown in Fig. 6(b) where we observe an MR ratio
of almost 12%. The MR ratio decreases with increasing Co content up to
$x=0.3$, and then flattens off at a level of $\sim 5$% for all higher values
of $x$. The explanation of this low $T$ positive linear magnetoresistance is
contested: both quantum interference effects,Manyala _et al._ (2000) and
Zeeman splitting of the majority and minority spin bands, which reduces the
high mobility minority spin carriers and in turn increases the
resistivity,Onose _et al._ (2005) have been cited as causes.
## VI Hall effect
Hall measurements were made simultaneously with the longitudinal resistivity
measurements. As an example, the Hall resistivity $\rho_{xy}(H)$ for an
Fe1-xCoxSi thin film with $x=0.4$ is shown in Fig. 7(a) for various
temperatures. There is low field hysteresis (for fields $\mu_{0}H\lesssim 0.3$
T) and a high field linear regime. (Inset in Fig.7(a) are data measured at 5 K
showing the high field response.) The high field slope is due to the ordinary
Hall effect. This high field Hall slope, measured at 5 K for Fe1-xCoxSi films
with different values of $x$, was used to determine the type of charge carrier
and carrier density, as shown in Fig. 7(b), and was combined with the
longitudinal resistivity to give the mobility of the carriers in the film, as
shown in Fig. 7(c). In the bulk, each Co dopant contributes one conduction
electron to the electron gas over the whole $x$ range.Manyala _et al._ (2000)
The data shown in Fig. 7(b) show that there is a small shortfall in our
samples, with close to, but not quite, one electron-like carrier per Co
dopant. It is possible that there are defects in our film, too subtle to pick
up by XRD or HRTEM, that act as traps preventing all the electrons released by
the Co dopants from acting as carriers. As shown in Fig. 7(c), the mobility
$\mu$ of the charge carriers drops with increasing Co doping in the films,
which can be accounted for if the Co dopants act as scattering centres.
Figure 7: (Color online) Hall measurements. (a) Hall resistivity $\rho_{xy}$
as a function of field for Fe1-xCoxSi epilayers with $x=0.4$ for selected
temperatures. Hysteresis is observed in the extraordinary Hall effect which
diminishes at elevated temperatures. The ordinary Hall effect was extracted at
high fields above the saturation field. A measurement at 5 K is shown inset up
to higher magnetic fields. (b) Charge carrier density expressed as electrons
per formula unit inferred from measurements of the high field ordinary Hall
effect at 5 K. The dashed line illustrates the ideal case of one electron
added to the electron gas per cobalt atom. (c) Carrier mobility $\mu$ as a
function of cobalt doping $x$ at 5 K.
The hysteretic part of the the Hall signal arises due to the anomalous Hall
effect that is present in magnetically ordered materials.Nagaosa _et al._
(2010) The Hall resistivity in a ferromagnetic material is given by
$\rho_{xy}=R_{\mathrm{o}}\mu_{0}H+4\pi R_{\mathrm{s}}M,$ (3)
where $R_{\mathrm{o}}$ is the ordinary Hall coefficient and $R_{\mathrm{s}}$
is the anomalous Hall coefficient. The anomalous contribution to the Hall
resistivity $\rho_{\mathrm{AH}}=4\pi R_{\mathrm{s}}M$ was determined by
extrapolating the high field Hall slope to $H=0$, where the magnetisation is
saturated, so any topological contribution of the Hall resistivityNeubauer
_et al._ (2009); Lee _et al._ (2009) is neglected in the present analysis.
(We will discuss it elsewhere.) $\rho_{\mathrm{AH}}$ for the $x=0.4$ sample,
shown in Fig. 7(a), is as large as $2~{}\mu\Omega$cm at 5 K, and diminishes as
$T$ rises, becoming almost negligible at 100 K or beyond. As shown in Fig.
8(a), even larger values of $\rho_{\mathrm{AH}}$ can be found for lower values
of $x$. Fe1-xCoxSi layers with $x\lesssim 0.3$ have $\rho_{\mathrm{AH}}\sim
5~{}\mu\Omega$cm. The highest value we observe is $5.5~{}\mu\Omega$cm for
$x=0.25$. In Fig. 8(b) we plot anomalous Hall coefficient $R_{\mathrm{s}}$ as
a function of $x$ and observe that highest value is reached for $x=0.1$, up to
$0.67~{}\pm~{}0.04~{}cm^{3}C^{-1}$ before decreasing almost linearly to
$0.09~{}\pm~{}0.01~{}cm^{3}C^{-1}$ for $x=0.5$. The large value of
$R_{\mathrm{s}}$ observed in our epilayers is of the similar order but a
little higher than that observed in bulk Fe1-xCoxSi crystals by Manyala et
al.Manyala _et al._ (2004) This could be attributed to the strained epitaxial
structure of Fe1-xCoxSi films, in which strain increases the effective spin-
orbit coupling.
Figure 8: (Color online) Anomalous Hall effect. (a) Variation of anomalous
Hall resistivity $\rho_{\mathrm{AH}}$, and (b) anomalous Hall coefficient
$R_{\mathrm{s}}$ as a function of $x$ at 5 K for Fe1-xCoxSi films.
## VII Magnetic properties
Magnetic characterisation was carried out using a vibrating sample
magnetometer (VSM) with a sensitivity of $10^{-6}$ emu and a SQUID
magnetometer with a sensitivity of $10^{-8}$ emu. For measurements in the VSM,
several pieces of sample cut from the same wafer were stacked up to increase
the signal. The temperature dependences of the magnetisation of the films were
measured with a 10 mT field applied in the film plane, the results are shown
in Fig. 9(a). It is straightforward to determine the critical temperature for
magnetic ordering from these curves. Since Fe1-xCoxSi is helimagnetic, we
refer to an ordering temperature $T_{\mathrm{ord}}$, rather than a Curie
temperature. The values of $T_{\mathrm{ord}}$ obtained for the various films
have been plotted as a function of Co content $x$ and shown in Fig. 9(b). When
compared with corresponding data for bulk samples,Onose _et al._ (2005);
Grigoriev _et al._ (2007) we see that for our Fe1-xCoxSi epilayers
$T_{\mathrm{ord}}$ has been significantly increased, and is as high as 77 K
for the $x=0.4$ epilayer. Enhanced ordering temperatures with respect to bulk
have also been observed in MnSi epilayers by Engelke et al.Engelke _et al._
(2012)
Figure 9: (Color online) Magnetic characterisation of the Fe1-xCoxSi
epilayers.(a) Magnetisation as a function of temperature in an in-plane 10 mT
field. The Co concentration, $x$, of the films is labeled on the graph. Larger
error bars correspond to measurements by VSM. b) The ordering temperature
$T_{\mathrm{ord}}$ of the epitaxial thin films shows an enhancement magnetic
ordering temperature bulk material.Onose _et al._ (2005); Grigoriev _et al._
(2007) $T_{\mathrm{res}}$, determined as discussed in §IV, is up to 10 K
higher than $T_{\mathrm{ord}}$. The dashed lines are guide to the eye. (c) The
saturation magnetisation at 5 K, extracted from hysteresis loops of the films,
expressed in Bohr magnetons per formula unit. The value is close to
$1~{}\mu_{\mathrm{B}}$ per cobalt dopant atom (ideal relationship shown by the
dashed line), in good agreement with bulk,Manyala _et al._ (2000) for
$x\lesssim 0.25$.
We attribute this increased stability of the magnetic ordering in our
Fe1-xCoxSi epitaxial films to their epitaxial strain. As shown in Fig. 3(e),
the biaxial in-plane strain increases the unit cell volume. Studies of bulk
crystals of Fe1-xCoxSi under hydrostatic pressure show that compressing the
unit cell volume suppresses magnetic order and can even induce a quantum phase
transition in the system.Forthaus _et al._ (2011) Based on this argument, we
conclude that the epitaxial strain in these Fe1-xCoxSi systems stabilises the
magnetic order and increases $T_{\mathrm{ord}}$ for the whole range of $x$.
We determined the magnetic moment, in units of Bohr magnetons
($\mu_{\mathrm{B}}$) per formula unit(f.u.), from these hysteresis loops. The
results are plotted as a function of $x$ in Fig. 9(c). Our results are
comparable to the findings of Manyala et al. for bulk crystals,Manyala _et
al._ (2000) and largely in line with theoretical expectations.Guevara _et
al._ (2004) As found previously, we see that each Co atom contributes $\sim
1~{}\mu_{\mathrm{B}}$ up to a limit of $x\approx 0.25$. Beyond this point, the
total moment is roughly constant at $\sim 0.25~{}\mu_{\mathrm{B}}$ per formula
unit (f.u.). The dashed line in Fig. 9(c) represents the ideal result of
exactly $1~{}\mu_{\mathrm{B}}$/f.u. We can see that in the low $x$ range there
is a small excess of moment per Co above the ideal result, suggesting that the
Co dopants could be weakly magnetising nearby Fe atoms in this regime.
## VIII Discussion and Conclusions
In the early report of Manyala et al., the finding of one electron-like
carrier and one $\mu_{\mathrm{B}}$ of magnetic moment per Co atom dopant in
Fe1-xCoxSi (at least in the regime $x\lesssim 0.25$) was interpreted as
indicating the presence of a fully spin-polarised electron gas.Manyala _et
al._ (2000) This half-metallic state was retrodicted by band structure
calculations a few years later,Guevara _et al._ (2004) and its presence
explains the greater stability of the magnetic order against pressure for low
$x$ samples.Forthaus _et al._ (2011) We previously detected evidence for the
partial preservation of this state in non-phase-pure sputtered Fe1-xCoxSi
polycrystalline films.Morley _et al._ (2011)
In Fig. 10 we show the magnetic moment per electron-like carrier as a function
of $x$ for our epilayer samples. The moment is determined from the
magnetometry results in Fig. 9(c) and the number of carriers from the Hall
effect, as given in Fig. 7(b).
Figure 10: (Color online) Magnetic moment per carrier of the electron gas in
Fe1-xCoxSi as a function of cobalt doping $x$.
The data show an approximately linear decrease as the Co content $x$ rises.
For $x\gtrsim 0.25$, in the metal-like regime, the behavior is much as
expected: the moment per carrier ratio drops, falling to only about 0.5
$\mu_{\mathrm{B}}$ per electron for $x=0.5$. The decrease in the spin-
polarization for high $x$ has been previously observed and explained as being
due to local disorder in the crystal structure induced by addition of Co
atoms.Guevara _et al._ (2004); Forthaus _et al._ (2011)
In the low-doping semiconductor-like regime ($x\lesssim 0.25$), the ratio of
moment per carrier exceeds unity, arising from the small shortfall in carriers
per Co that was found in the data presented in Fig. 7(b), and slight excess
moment observed in Fig. 9(c). Physically, the underlying mechanism is not
clear. A plausible picture might be that there are a low number of Co atoms on
Si antisites or in interstitial positions, too few to be readily detected by
XRD or HRTEM, that act both as charge traps and possess local moments
exceeding 1 $\mu_{\mathrm{B}}$ (either alone or by weakly polarising
neighbouring Fe sites). More detailed studies, such as ab initio calculations,
would be required to confirm this scenario. Nevertheless, it is clear that in
this regime, we have a highly spin-polarized electron gas.
To summarize, we have grown a set of Fe1-xCoxSi epitaxial thin films, and
studied the variation in the structural, transport, and magnetic properties in
the range $0\leq x\leq 0.5$. The epilayers are $\epsilon$-phase pure, but with
a deformation of the B20 unit cell into an rhombohedral form by the epitaxial
strain. Qualitatively, the properties of our epilayer samples are similar in
many ways to those of bulk crystals. In particular, we found the metal-
insulator transition to lie in the middle of this range, with a high spin-
polarization in the semiconducting regime ($x\lesssim 0.25$). However there
are quantitative differences, the most important of which is the stabilisation
of magnetic order up to much higher temperatures than in bulk crystals. The
availability of thin films amenable to planar processing techniques is an
important step to realising spintronic devices based on the remarkable physics
of these B20-ordered materials.Jonietz _et al._ (2010); Schulz _et al._
(2012); Fert _et al._ (2013)
###### Acknowledgements.
We acknowledge support from the EPSRC (grant EP/J007110/1) and Marie Curie
Initial Training Action (ITN) Q-NET 264034. We would also like to thank Dr G.
Burnell for useful discussions, Dr M. Ward for TEM sample preparation and
imaging, and Dr D. Alba Venero for help with SQUID measurements.
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|
arxiv-papers
| 2013-07-27T20:23:01 |
2024-09-04T02:49:48.570818
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Sinha, N. A. Porter, and C. H. Marrows",
"submitter": "Prof. Christopher Marrows",
"url": "https://arxiv.org/abs/1307.7301"
}
|
1307.7390
|
# Congruence successions in compositions
Toufik Mansour Department of Mathematics, University of Haifa, 31905 Haifa,
Israel [email protected] , Mark Shattuck Department of Mathematics,
University of Tennessee, Knoxville, TN 37996 USA [email protected] and
Mark C. Wilson Department of Computer Science, University of Auckland,
Private Bag 92019 Auckland, New Zealand [email protected]
###### Abstract.
A _composition_ is a sequence of positive integers, called _parts_ , having a
fixed sum. By an _$m$ -congruence succession_, we will mean a pair of adjacent
parts $x$ and $y$ within a composition such that $x\equiv
y~{}(\text{mod}~{}m)$. Here, we consider the problem of counting the
compositions of size $n$ according to the number of $m$-congruence
successions, extending recent results concerning successions on subsets and
permutations. A general formula is obtained, which reduces in the limiting
case to the known generating function formula for the number of Carlitz
compositions. Special attention is paid to the case $m=2$, where further
enumerative results may be obtained by means of combinatorial arguments.
Finally, an asymptotic estimate is provided for the number of compositions of
size $n$ having no $m$-congruence successions.
###### Key words and phrases:
composition, parity succession, combinatorial proof, asymptotic estimate
###### 2000 Mathematics Subject Classification:
05A15, 05A19, 05A16
## 1\. Introduction
Let $n$ be a positive integer. A _composition_
$\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{d}$ of $n$ is any sequence of
positive integers whose sum is $n$. Each summand $\sigma_{i}$ is called a
_part_ of the composition. If $n,d\geq 1$, then let $\mathcal{C}_{n,d}$ denote
the set of compositions of $n$ having exactly $d$ parts and
$\mathcal{C}_{n}=\cup_{d=1}^{n}\mathcal{C}_{n,d}$. By convention, there is a
single composition of $n=0$ having zero parts.
If $m\geq 1$ and $0\leq r\leq m-1$, by an $(m,r)$-congruence succession within
a composition $\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{d}$, we will mean an
index $i$ for which $\pi_{i+1}\equiv\pi_{i}+r~{}(\text{mod}~{}m)$. An
$(m,r)$-congruence succession in which $r=0$ will be referred to as an
$m$-congruence succession, the $m=2$ case being termed a parity succession. A
_parity-alternating_ composition is one that contains no parity successions,
that is, the parts alternate between even and odd values. This concept of
parity succession for compositions extends an earlier one that was introduced
for subsets [11] and later considered on permutations [12]. The terminology is
an adaptation of an analogous usage in the study of integer sequences
$(p_{1},p_{2},\ldots)$ in which a succession refers to a pair $p_{i},p_{i+1}$
with $p_{i+1}=p_{i}+1$ (see, e.g., [1, 17, 8]). For other related problems
involving restrictions on compositions, the reader is referred to the text [7]
and such papers as [3, 6].
Enumerating finite discrete structures according to the parity of individual
elements perhaps started with the following formula of Tanny [18] for the
number $g(n,k)$ of alternating $k$-subsets of $[n]$ given by
$g(n,k)=\binom{\lfloor\frac{n+k}{2}\rfloor}{k}+\binom{\lfloor\frac{n+k-1}{2}\rfloor}{k},\qquad
1\leq k\leq n.$
This result was recently generalized to any modulus in [9] and in terms of
counting by successions in [11]. Tanimoto [16] considered a comparable version
of the problem on permutations in his investigation of signed Eulerian
numbers. There one finds the formula for the number $h(n)$ of parity-
alternating permutations of length $n$ given by
$h(n)=\frac{3+(-1)^{n}}{2}\left\lfloor\frac{n}{2}\right\rfloor!\left\lfloor\frac{n+1}{2}\right\rfloor!,$
which has been generalized in terms of succession counting in [12]; see also
[10].
In the next section, we consider the problem of counting compositions of $n$
according to the number of $(m,r)$-congruence successions, as defined above,
and derive an explicit formula for the generating function for all $m$ and $r$
(see Theorem 2 below). When $r=0$, we obtain as a corollary a relatively
simple expression for the generating function $F_{m}$ which counts
compositions according to the number of $m$-congruence successions. Letting
$m\rightarrow\infty$ and taking the variable in $F_{m}$ which marks the number
of $m$-congruence successions to be zero recovers the generating function for
the number of _Carlitz_ compositions, i.e., those having no consecutive parts
equal; see, e.g., [5].
In the third section, we obtain some enumerative results concerning the case
$m=2$. In particular, we provide a bijective proof for a related recurrence
and enumerate, in two different ways, the parity-alternating compositions of
size $n$. As a consequence, we obtain a combinatorial proof of a pair of
binomial identities which we were unable to find in the literature. In the
final section, we provide asymptotic estimates for the number of compositions
of size $n$ having no $m$-congruence successions as $n\rightarrow\infty$,
which may be extended to compositions having any fixed number of successions.
## 2\. Counting compositions by number of $(m,r)$-congruence successions
We will say that the sequence $\pi=\pi_{1}\pi_{2}\cdots\pi_{d}$ has an
$(m,r)$-congruence succession at index $i$ if
$\pi_{i+1}\equiv\pi_{i}+r~{}(\text{mod}~{}m)$, where $0\leq r\leq m-1$. We
will denote the number of $(m,r)$-congruence successions within a sequence
$\pi$ by $cl_{m,r}(\pi)$. Let $R_{m,r;a}(x,y,q)=R_{a}(x,y,q)$ be the
generating function for the number of compositions of $n$ with exactly $d$
parts whose first part is $a$ according to the statistic $cl_{m,r}$, that is,
$R_{a}(x,y,q)=\sum_{n\geq
0}\sum_{d=0}^{n}x^{n}y^{d}\left(\sum_{\pi=a\pi^{\prime}\in\mathcal{C}_{n,d}}q^{cl_{m,r}(\pi)}\right).$
Clearly, we have $R_{m+a}(x,y,q)=x^{m}R_{a}(x,y,q)$ for all $a\geq 1$. Let
$R_{m,r}(x,y,q)=R(x,y,q)=1+\sum_{a\geq 1}R_{a}(x,y,q)$. By the definitions, we
have
$R_{a}(x,y,q)=x^{a}yR(x,y,q)+x^{a}y(q-1)\sum R_{t}(x,y,q),$
for all $a\geq 1$, where the sum is taken over all positive integers $t$ such
that $t\equiv\ a+r~{}(\text{mod}~{}m)$. Hence,
$\sum_{i\geq
0}R_{im+a}(x,y,q)=\frac{x^{a}y}{1-x^{m}}R(x,y,q)+\frac{x^{a}y(q-1)}{1-x^{m}}\sum_{i\geq
0}R_{im+a+r}(x,y,q),$
if $1\leq a\leq m-r$, and
$\sum_{i\geq
0}R_{im+a}(x,y,q)=\frac{x^{a}y}{1-x^{m}}R(x,y,q)+\frac{x^{a}y(q-1)}{1-x^{m}}\sum_{i\geq
0}R_{im+a+r-m}(x,y,q),$
if $m-r+1\leq a\leq m$. The last two equalities may be expressed as
$\displaystyle G_{j}(x,y,q)$
$\displaystyle=\frac{x^{j}y}{1-x^{m}}R(x,y,q)+\frac{x^{j}y(q-1)}{1-x^{m}}G_{j+r}(x,y,q),\quad
1\leq j\leq m-r,$ (1) $\displaystyle G_{j}(x,y,q)$
$\displaystyle=\frac{x^{j}y}{1-x^{m}}R(x,y,q)+\frac{x^{j}y(q-1)}{1-x^{m}}G_{j+r-m}(x,y,q),\quad
m-r+1\leq j\leq m,$
where $G_{j}(x,y,q)=\sum_{i\geq 0}R_{im+j}(x,y,q)$.
In order to find an explicit formula for $G_{j}(x,y,q)$, we will need the
following lemma.
###### Lemma 1.
Suppose $x_{j}=a_{j}+b_{j}x_{j+r}$ for all $j=1,2,\ldots,m-r$ and
$x_{j}=a_{j}+b_{j}x_{j+r-m}$ for all $j=m-r+1,m-r+2,\ldots,m$. Let
$s=\gcd(m,r)$ and $p=m/s$. Then for all $j=1,2,\ldots,s$ and
$\ell=0,1,\ldots,p-1$, we have
$x_{j+\ell
r}=\sum_{i=\ell}^{\ell+p-1}\frac{a_{j+ir}\prod_{k=\ell}^{i-1}b_{j+kr}}{1-\prod_{k=\ell}^{\ell+p-1}b_{j+kr}},$
where $x_{j+m}=x_{j}$, $a_{j+m}=a_{j}$ and $b_{j+m}=b_{j}$.
###### Proof.
Let $j=1,2,\ldots,s$. By definition of the sequence $x_{j}$ and
$m$-periodicity, we may write
$\displaystyle x_{j}$
$\displaystyle=a_{j}+b_{j}x_{j+r}=a_{j}+b_{j}a_{j+r}+b_{j}b_{j+r}x_{j+2r}$
$\displaystyle=\cdots=a_{j}+b_{j}a_{j+r}+\cdots+b_{j}b_{j+r}\cdots
b_{j+(p-2)r}a_{j+(p-1)r}+b_{j}b_{j+r}\cdots b_{j+(p-1)r}x_{j+pr}.$
Since $pr\equiv 0~{}(\text{mod}~{}m)$, we have
$x_{j}=\sum_{i=0}^{p-1}\frac{a_{j+ir}\prod_{k=0}^{i-1}b_{j+kr}}{1-\prod_{k=0}^{p-1}b_{j+kr}}.$
More generally, for any $\ell=0,1,\ldots,p-1$,
$x_{j+\ell
r}=\sum_{i=\ell}^{\ell+p-1}\frac{a_{j+ir}\prod_{k=\ell}^{i-1}b_{j+kr}}{1-\prod_{k=\ell}^{\ell+p-1}b_{j+kr}}.$
∎
Let us denote by $\overline{t}$ the member of $\\{1,2,\ldots,m\\}$ such that
$t\equiv\overline{t}~{}(\text{mod}~{}m)$ for a positive integer $t$. When
$a_{j}=\frac{x^{j}y}{1-x^{m}}R(x,y,q)$ and $b_{j}=\frac{x^{j}y(q-1)}{1-x^{m}}$
for $1\leq j\leq m$ in Lemma 1, we get
$\displaystyle x_{j+\ell r}$
$\displaystyle=\sum_{i=\ell}^{\ell+p-1}\frac{\frac{x^{\overline{j+ir}}y}{1-x^{m}}R(x,y,q)\prod_{k=\ell}^{i-1}\frac{x^{\overline{j+kr}}y(q-1)}{1-x^{m}}}{1-\prod_{k=\ell}^{\ell+p-1}\frac{x^{\overline{j+kr}}y(q-1)}{1-x^{m}}}$
(2)
$\displaystyle=\frac{R(x,y,q)}{1-\left(\frac{y(q-1)}{1-x^{m}}\right)^{p}\prod_{k=\ell}^{\ell+p-1}x^{\overline{j+kr}}}\sum_{i=0}^{p-1}\frac{x^{\overline{j+(i+\ell)r}}y^{i+1}(q-1)^{i}\prod_{k=\ell}^{i+\ell-1}x^{\overline{j+kr}}}{(1-x^{m})^{i+1}},$
for all $j=1,2,\ldots,s$ and $\ell=0,1,\ldots,p-1$, where $s=\gcd(m,r)$ and
$p=m/s$. By (1), we have $G_{\overline{j+\ell r}}(x,y,q)=x_{j+\ell
r}=x_{\overline{j+\ell r}}$, where $x_{j+\ell r}$ is given by (2). Note that
the set of indices $j+\ell r$ for $1\leq j\leq s$ and $0\leq\ell\leq p-1$ is a
complete residue set $(\text{mod}~{}m)$. Using (2) and the fact that
$R(x,y,q)=1+\sum_{a=1}^{m}G_{a}(x,y,q)$, we obtain the following result.
###### Theorem 2.
If $m\geq 1$, $0\leq r\leq m-1$, $s=\gcd(m,r)$ and $p=m/s$, then
(3)
$R_{m,r}(x,y,q)=\frac{1}{1-\sum\limits_{j=1}^{s}\sum\limits_{\ell=0}^{p-1}\sum\limits_{i=0}^{p-1}\frac{x^{\overline{j+(i+\ell)r}}y^{i+1}(q-1)^{i}\prod_{k=\ell}^{i+\ell-1}x^{\overline{j+kr}}}{(1-x^{m})^{i+1}\left(1-\left(\frac{y(q-1)}{1-x^{m}}\right)^{p}\prod_{k=\ell}^{\ell+p-1}x^{\overline{j+kr}}\right)}}.$
Note that in general we are unable to simplify the number theoretic product
$\prod_{k=\ell}^{i+\ell-1}x^{\overline{j+kr}}$ appearing in (3).
Let us say that the sequence $\pi=\pi_{1}\pi_{2}\cdots\pi_{d}$ has an
$m$-congruence succession at index $i$ if
$\pi_{i+1}\equiv\pi_{i}~{}(\text{mod}~{}m)$ and denote the number of
$m$-congruence successions in a sequence $\pi$ by $cl_{m}(\pi)$. Let
$F_{m}(x,y,q)=\sum_{n\geq
0}\sum_{d=0}^{n}x^{n}y^{d}\left(\sum_{\pi\in\mathcal{C}_{n,d}}q^{cl_{m}(\pi)}\right).$
Taking $r=0$ in (3), and noting $s=\gcd(m,0)=m$, gives the following result.
###### Corollary 3.
If $m\geq 1$, then
(4)
$F_{m}(x,y,q)=\frac{1}{1-\sum_{a=1}^{m}\left(\frac{x^{a}y}{1-x^{m}-x^{a}y(q-1)}\right)}.$
Letting $q=0$ and $m\rightarrow\infty$ in (4) yields the generating function
for the number of compositions having no $m$-congruence successions for all
large $m$. Note that the only possible such compositions are those having no
two adjacent parts the same. Thus, we get the following formula for the
generating function which counts the Carlitz compositions according to the
number of parts.
###### Corollary 4.
We have
(5)
$F_{\infty}(x,y,0)=\frac{1}{1-\sum_{a=1}^{\infty}\frac{x^{a}y}{1+x^{a}y}}.$
Let us close this section with a few remarks.
###### Remark 1.
Letting $q=1$ in (3) gives
$R_{m,r}(x,y,1)=\frac{1}{1-\frac{y}{1-x^{m}}\sum_{j=1}^{s}\sum_{\ell=0}^{p-1}x^{\overline{j+\ell
r}}}=\frac{1}{1-\frac{y}{1-x^{m}}\sum_{a=1}^{m}x^{a}}=\frac{1-x}{1-x-xy},$
which agrees with the generating function for the number of compositions of
$n$ having $d$ parts.
###### Remark 2.
In [2], the generating function for the number $c(n,d)$ of Carlitz
compositions of $n$ having $d$ parts was obtained as
(6) $\sum_{n\geq 0}\sum_{d=0}^{n}c(n,d)x^{n}y^{d}=\frac{1}{1+\sum_{j\geq
1}\frac{(-xy)^{j}}{1-x^{j}}}.$
Note that formulas (5) and (6) are seen to be equivalent since
$\sum_{a\geq 1}\frac{x^{a}y}{1+x^{a}y}=\sum_{a\geq 1}\sum_{j\geq
1}(-1)^{j-1}x^{aj}y^{j}=\sum_{j\geq 1}(-1)^{j-1}y^{j}\sum_{a\geq
1}x^{ja}=\sum_{j\geq 1}(-1)^{j-1}\frac{(xy)^{j}}{1-x^{j}}.$
###### Remark 3.
Letting $m=1$ in (4) gives
$F_{1}(x,y,q)=\frac{1-x-xy(q-1)}{1-x-xyq}.$
This formula may also be realized directly upon noting in this case that $q$
marks the number of parts minus one in any non-empty composition, whence
$F_{1}(x,y,q)=1+\frac{1}{q}\left(\frac{1-x}{1-x-xyq}-1\right).$
## 3\. Combinatorial results
We will refer to an $m$-congruence succession when $m=2$ as a _parity
succession_ , or just a _succession_. In this section, we will provide some
combinatorial results concerning successions in compositions. Let
$F(x,y,q)=F_{2}(x,y,q)$ denote the generating function which counts the
compositions of $n$ having $d$ parts according to the number of parity
successions. Taking $m=2$ in Corollary 3 gives
(7)
$F(x,y,q)=\frac{(1-x^{2}-xy(q-1))(1-x^{2}-x^{2}y(q-1))}{(1-x^{2})^{2}-x^{3}y^{2}-xy(1-x^{2})(1+x)q+x^{3}y^{2}q^{2}}.$
Let $\mathcal{C}_{n,d,a}$ denote the subset of $\mathcal{C}_{n,d}$ whose
members contain exactly $a$ successions and let
$c(n,d,a)=|\mathcal{C}_{n,d,a}|$. Comparing coefficients of $x^{n}y^{d}q^{a}$
on both sides of (7) yields the following recurrence satisfied by the array
$c(n,d,a)$.
###### Theorem 5.
If $n\geq 4$ and $d\geq 3$, then
$\displaystyle c(n,d,$ $\displaystyle
a)=c(n-1,d-1,a-1)+2c(n-2,d,a)+c(n-2,d-1,a-1)+c(n-3,d-2,a)$ (8)
$\displaystyle-c(n-3,d-1,a-1)-c(n-3,d-2,a-2)-c(n-4,d,a)-c(n-4,d-1,a-1).$
We can also provide a combinatorial proof of (8), rewritten in the form
$\displaystyle(c(n,d,a)$ $\displaystyle-c(n-2,d,a))+c(n-3,d-2,a-2)=$
$\displaystyle(c(n-1,d-1,a-1)-c(n-3,d-1,a-1))+(c(n-2,d,a)-c(n-4,d,a))$ (9)
$\displaystyle+(c(n-2,d-1,a-1)-c(n-4,d-1,a-1))+c(n-3,d-2,a).$
To do so, let $\mathcal{B}_{n,d,a}$ denote the subset of $\mathcal{C}_{n,d,a}$
all of whose members end in a part of size $1$ or $2$. Note that for all $n$,
$d$, and $a$, we have
$|\mathcal{B}_{n,d,a}|=c(n,d,a)-c(n-2,d,a),$
by subtraction, since $c(n-2,d,a)$ counts each member of $\mathcal{C}_{n,d,a}$
whose last part is of size $3$ or more (to see this, add two to the last part
of any member of $\mathcal{C}_{n-2,d,a}$, which leaves the number of parts and
successions unchanged).
So to show (9), we define a bijection between the sets
$\mathcal{B}_{n,d,a}\cup\mathcal{C}_{n-3,d-2,a-2}\text{ and
}\mathcal{B}_{n-1,d-1,a-1}\cup\mathcal{B}_{n-2,d,a}\cup\mathcal{B}_{n-2,d-1,a-1}\cup\mathcal{C}_{n-3,d-2,a}.$
For this, we refine the sets as follows. In the subsequent definitions, $x$,
$y$, and $z$ will denote an odd number, an even number, or a number greater
than or equal three, respectively. First, let $\mathcal{B}_{n,d,a}^{(i)}$,
$1\leq i\leq 4$, denote, respectively, the subsets of $\mathcal{B}_{n,d,a}$
whose members (1) end in either $1+1$ or $x+1+2$ for some $x$, (2) end in
$y+2+1$ or $2+2$ for some $y$, (3) end in $x+2+1$ or $y+1+2$, or (4) end in
$z+1$ or $z+2$ for some $z$. Let $\mathcal{B}_{n-1,d-1,a-1}^{(i)}$, $1\leq
i\leq 3$, denote the subsets of $\mathcal{B}_{n-1,d-1,a-1}$ whose members end
in $1$, $x+2$ for some $x$, or $y+2$ for some $y$, respectively. Finally, let
$\mathcal{B}_{n-2,d-1,a-1}^{(i)}$, $1\leq i\leq 3$, denote the subsets of
$\mathcal{B}_{n-2,d-1,a-1}$ whose members end in $x+1$, $y+1$, or $2$,
respectively.
So we seek a bijection between
$\left(\cup_{i=1}^{4}\mathcal{B}_{n,d,a}^{(i)}\right)\cup\mathcal{C}_{n-3,d-2,a-2}$
and
$\left(\cup_{i=1}^{3}\mathcal{B}_{n-1,d-1,a-1}^{(i)}\right)\cup\left(\cup_{i=1}^{3}\mathcal{B}_{n-2,d-1,a-1}^{(i)}\right)\cup\mathcal{B}_{n-2,d,a}\cup\mathcal{C}_{n-3,d-2,a}.$
Simple correspondences as described below show the following:
$\displaystyle(i)$
$\displaystyle\quad|\mathcal{B}_{n,d,a}^{(1)}|=|\mathcal{B}_{n-1,d-1,a-1}^{(1)}\cup\mathcal{B}_{n-1,d-1,a-1}^{(2)}|,$
$\displaystyle(ii)$
$\displaystyle\quad|\mathcal{B}_{n,d,a}^{(2)}|=|\mathcal{B}_{n-2,d-1,a-1}^{(2)}\cup\mathcal{B}_{n-2,d-1,a-1}^{(3)}|,$
$\displaystyle(iii)$
$\displaystyle\quad|\mathcal{B}_{n,d,a}^{(3)}|=|\mathcal{C}_{n-3,d-2,a}|,$
$\displaystyle(iv)$
$\displaystyle\quad|\mathcal{B}_{n,d,a}^{(4)}|=|\mathcal{B}_{n-2,d,a}|,$
$\displaystyle(v)$
$\displaystyle\quad|\mathcal{C}_{n-3,d-2,a-2}|=|\mathcal{B}_{n-2,d-1,a-1}^{(1)}\cup\mathcal{B}_{n-1,d-1,a-1}^{(3)}|.$
For (i), we remove the right-most $1$ within a member of
$\mathcal{B}_{n,d,a}^{(1)}$, while for (ii), we remove the right-most $2$
within a member of $\mathcal{B}_{n,d,a}^{(2)}$. To show (iii), we remove the
final two parts of $\lambda\in\mathcal{B}_{n,d,a}^{(3)}$ to obtain the
composition $\lambda^{\prime}$. Note that
$\lambda^{\prime}\in\mathcal{C}_{n-3,d-2,a}$ and that the mapping
$\lambda\mapsto\lambda^{\prime}$ is reversed by adding $1+2$ or $2+1$ to a
member of $\mathcal{C}_{n-3,d-2,a}$, depending on whether the last part is
even or odd, respectively. For (iv), we subtract two from the penultimate part
of $\lambda\in\mathcal{B}_{n,d,a}^{(4)}$, which leaves the number of
successions unchanged. Finally, for (v), we add either a part of size $1$ or
$2$ to $\lambda\in\mathcal{C}_{n-3,d-1,a-2}$, depending on whether the last
part of $\lambda$ is odd or even, respectively. Combining the correspondences
used to show (i)–(v) yields the desired bijection and completes the proof. ∎
We will refer to a composition having no parity successions as _parity-
alternating_. We now wish to enumerate parity-alternating compositions having
a fixed number of parts. Setting $q=0$ in (7), and expanding, gives
$\displaystyle F($ $\displaystyle
x,y,0)=\frac{(1-x^{2}+x^{2}y)(1-x^{2}+xy)}{(1-x^{2})^{2}-x^{3}y^{2}}=\frac{\left(1+\frac{x^{2}y}{1-x^{2}}\right)\left(1+\frac{xy}{1-x^{2}}\right)}{1-\frac{x^{3}y^{2}}{(1-x^{2})^{2}}}$
$\displaystyle=\left(1+\frac{x^{2}y}{1-x^{2}}\right)\left(1+\frac{xy}{1-x^{2}}\right)\sum_{i\geq
0}\frac{x^{3i}y^{2i}}{(1-x^{2})^{2i}}$ $\displaystyle=\sum_{i\geq
0}\left(2y^{2i}\sum_{j\geq 2i-1}\binom{j}{2i-1}x^{2j-i+2}+y^{2i+1}\sum_{j\geq
2i}\binom{j}{2i}x^{2j-i+2}+y^{2i+1}\sum_{j\geq
2i}\binom{j}{2i}x^{2j-i+1}\right).$
Extracting the coefficient of $x^{n}y^{m}$ in the last expression yields the
following result.
###### Proposition 6.
If $n\geq 1$ and $d\geq 0$, then
(10)
$c(n,2d,0)=\left\\{\begin{array}[]{ll}2\binom{\frac{n+d}{2}-1}{2d-1},&\mbox{if
}n\equiv d~{}(\mbox{mod}~{}2);\\\ \\\ 0,&\mbox{otherwise},\end{array}\right.$
and
(11)
$c(n,2d+1,0)=\left\\{\begin{array}[]{ll}\binom{\frac{n+d}{2}-1}{2d},&\mbox{if
}n\equiv d~{}(\mbox{mod}~{}2);\\\ \\\
\binom{\frac{n+d-1}{2}}{2d},&\mbox{otherwise}.\end{array}\right.$
It is instructive to give combinatorial proofs of (10) and (11). For the first
formula, suppose $\lambda\in\mathcal{C}_{n,2d,0}$. Then $n$ and $d$ must have
the same parity since the parts of $\lambda$ alternate between even and odd
values. In this case, the number of possible $\lambda$ is twice the number of
integral solutions to the equation
(12) $\sum_{i=1}^{d}(x_{i}+y_{i})=n,$
where each $x_{i}$ is even, each $y_{i}$ is odd, and $x_{i},y_{i}>0$. Note
that the number of solutions to (12) is the same as the number of positive
integral solutions to $\sum_{i=1}^{d}(u_{i}+v_{i})=\frac{n+d}{2},$ which is
$\binom{\frac{n+d}{2}-1}{2d-1}$, upon letting $u_{i}=\frac{x_{i}}{2}$ and
$v_{i}=\frac{y_{i}+1}{2}$. Thus, there are $2\binom{\frac{n+d}{2}-1}{2d-1}$
members of $\mathcal{C}_{n,2d,0}$ when $n$ and $d$ have the same parity, which
gives (10).
On the other hand, note that members of $\mathcal{C}_{n,2d+1,0}$, where $n$
and $d$ are of the same parity, are synonymous with positive integral
solutions to
(13) $\sum_{i=1}^{d}(x_{i}+y_{i})+z=n,$
where the $x_{i}$ are even, the $y_{i}$ are odd, and $z$ is even. Upon adding
$1$ to each $y_{i}$, and halving, the number of such solutions is seen to be
$\binom{\frac{n+d}{2}-1}{2d}$. Similarly, there are
$\binom{\frac{n+d-1}{2}}{2d}$ members of $\mathcal{C}_{n,2d+1,0}$ when $n$ and
$d$ differ in parity, which gives (11). ∎
Let $a(n)=\sum_{d=0}^{n}c(n,d,0)$. Note that $a(n)$ counts all parity-
alternating compositions of length $n$. Taking $y=1$ and $q=0$ in (7) gives
$\sum_{n\geq 0}a(n)x^{n}=F(x,1,0)=\frac{1+x-x^{2}}{(1-x^{2})^{2}-x^{3}},$
and extracting the coefficient of $x^{n}$ yields the following result.
###### Proposition 7.
If $n\geq 4$, then
(14) $a(n)=2a(n-2)+a(n-3)-a(n-4),$
with $a(0)=a(1)=a(2)=1$ and $a(3)=3$.
We provide a combinatorial argument for recurrence (14), the initial values
being clear. Let us first define two classes of compositions. By a _type A
(colored) composition_ of $n$, we will mean one whose parts are all odd
numbers greater than $1$ in which a part of size $i$ is assigned one of
$\frac{i-1}{2}$ possible colors. A composition $\rho$ of $n$ is of _type B_ if
it is of the form $\rho=a+\lambda$, where $a$ is a part of any size (not
colored) and $\lambda$ is a composition of $n-a$ of type A. Given $n\geq 1$,
let $\mathcal{S}_{n}$ denote the (multi-) set consisting of two copies of each
composition of $n$ of type A, let $\mathcal{T}_{n}$ denote the set consisting
of all compositions of $n$ of type B, and let
$\mathcal{R}_{n}=\mathcal{S}_{n}\cup\mathcal{T}_{n}$. (For convenience, we
take $R_{0}$ to consist of the empty composition in a set by itself.)
If $k_{i}$ denotes the number of the color assigned to some part of size $i$
within a composition of type A, then replacing each part $i$ with either
$(i-2k_{i})+2k_{i}$ or $2k_{i}+(i-2k_{i})$ yields all members of
$\mathcal{C}_{n}$ having an even number of parts and no parity successions.
Doing the same for every part but the first within a composition of type $B$
(note in this case, the decomposition used for each $i$ is determined by the
parity of the first part $a$) yields all members of $\mathcal{C}_{n}$ having
an odd number of parts and no parity successions. It follows that
$|\mathcal{R}_{n}|=a(n)$.
It then remains to show that $|\mathcal{R}_{n}|$ satisfies the recurrence
(14). By a _maximal_ (colored) part of size $i$ within a member of
$\mathcal{R}_{n}$, we will mean one which has been assigned the color
$\frac{i-1}{2}$ (note that any part of size $3$ is maximal). Let
$\mathcal{R}_{n}^{\prime}$ denote the subset of $\mathcal{R}_{n}$ consisting
of all type $A$ members whose first part is maximal together with all type $B$
members whose first part is $1$ or $2$. Upon increasing the length of the
first part within a member of $\mathcal{R}_{n-2}$ by two (keeping the color
the same, if that member belongs to $\mathcal{S}_{n-2}$), one sees that
$|\mathcal{R}_{n}^{\prime}|=a(n)-a(n-2)$, by subtraction. To complete the
proof of (14), we then define a bijection between the sets
$\mathcal{R}_{n-2}\cup\mathcal{R}_{n-3}$ and
$\mathcal{R}_{n}^{\prime}\cup\mathcal{R}_{n-4}$, where $n\geq 4$.
We may assume $n\geq 5$, for the equivalence of the sets in question is clear
if $n=4$. Let $\mathcal{S}_{n}^{\prime}$ and $\mathcal{T}_{n}^{\prime}$ denote
the subsets of $\mathcal{R}_{n}^{\prime}$ consisting of its type $A$ and type
$B$ members, respectively. To complete the proof, it suffices to define
bijections between the sets $\mathcal{S}_{n-2}\cup\mathcal{S}_{n-3}$ and
$\mathcal{S}_{n}^{\prime}\cup\mathcal{S}_{n-4}$ and between the sets
$\mathcal{T}_{n-2}\cup\mathcal{T}_{n-3}$ and
$\mathcal{T}_{n}^{\prime}\cup\mathcal{T}_{n-4}$.
For the first bijection, if $\lambda\in\mathcal{S}_{n-2}$, then we either
increase or decrease the length of the first part of $\lambda$ by two,
depending on whether or not this part is maximal (if so, we also increase the
color assigned to the part by one, and if not, the color is kept the same).
Note that this yields all members of $S_{n}^{\prime}$ whose first part is at
least five as well as all members of $\mathcal{S}_{n-4}$. If
$\lambda\in\mathcal{S}_{n-3}$, then we append a colored part of size three to
the beginning of $\lambda$, which yields all members of $S_{n}^{\prime}$
starting with three.
For the second bijection, we consider cases concerning
$\lambda\in\mathcal{T}_{n-2}\cup\mathcal{T}_{n-3}$. If
$\lambda\in\mathcal{T}_{n-2}$ starts with $1$, then we increase the second
part of $\lambda$ by two (keeping the assigned color the same) to obtain
$\lambda^{*}\in\mathcal{T}_{n}^{\prime}$ starting with $1$ where the second
part is not maximal. If $\lambda\in\mathcal{T}_{n-3}$ starts with $1$, then we
replace this $1$ with $2$ and increase the second part of $\lambda$ by two
(again, keeping the assigned color the same) to obtain
$\lambda^{*}\in\mathcal{T}_{n}^{\prime}$. Combining the previous two cases
then yields all members of $\mathcal{T}_{n}^{\prime}$ whose second part is not
maximal. If $\lambda\in\mathcal{T}_{n-2}$ starts with a part $i$ of size two
or more, then we append a $2$ to the beginning of $\lambda$ if $i$ is odd and
we append a $1$ to the beginning of $\lambda$ and replace $i$ with $i+1$ if
$i$ is even. In either case, we take the second part to be maximal in the
resulting composition $\lambda^{*}$ belonging to $\mathcal{T}_{n}^{\prime}$.
Finally, if $\lambda\in\mathcal{T}_{n-3}$ starts with a part of size two or
more, then we subtract one from this part to obtain
$\lambda^{*}\in\mathcal{T}_{n-4}$. It may be verified that the composite
mapping $\lambda\mapsto\lambda^{*}$ yields the desired bijection. ∎
Summing the formulas in Proposition 6 over $d$ with $n$ fixed, using the fact
$\binom{a}{b}=\binom{a-2}{b}+2\binom{a-2}{b-1}+\binom{a-2}{b-2}$, and equating
with the result in Proposition 7 yields a combinatorial proof of the following
pair of binomial identities, which we were unable to find in the literature.
###### Corollary 8.
If $n\geq 0$, then
(15) $a(2n)=\sum_{d=0}^{\lfloor\frac{n+1}{3}\rfloor}\binom{n+d+1}{4d}$
and
(16) $a(2n+1)=\sum_{d=0}^{\lfloor\frac{n}{3}\rfloor}\binom{n+d+2}{4d+2},$
where $a(m)$ is given by (14).
Note that both sides of (15) and (16) are seen to count the parity-alternating
compositions of length $2n$ and $2n+1$, respectively, the right-hand side by
the number of parts (once one applies the identity
$\binom{a}{b}=\binom{a-2}{b}+2\binom{a-2}{b-1}+\binom{a-2}{b-2}$, which has an
easy combinatorial explanation, to the binomial coefficient). Using (14), the
binomial sums in (15) and (16) can be shown to satisfy fourth order
recurrences; see [4] for other examples of recurrent binomial sums.
## 4\. Asymptotics
We recall from (4) that $F_{m}(x,y,q)$ is a rational function. Specializing
variables we obtain $F_{m}(x,1,0)=\sum_{n\geq 0}a_{n}x^{n}$, which we have
seen in the previous section. The exact formulae given there for the
coefficient $a_{n}$ are complemented here by asymptotic results. These are
analogous to known results for Smirnov words and Carlitz compositions.
Note that $F_{m}(x,1,0)=1/H_{m}(x)$, where
$H_{m}(x)=1-\sum_{a=1}^{m}\frac{x^{a}}{1+x^{a}-x^{m}}$. Each $1+x^{a}-x^{m}$
is analytic and so its modulus over each closed disk centered at $0$ is
maximized on the boundary circle. It can be shown that when $|x|$ is fixed,
$|1+x^{a}-x^{m}|$ is maximized when $x^{a}-x^{m}$ is positive real, and
minimized when $x^{a}-x^{m}$ is negative real. Furthermore, the maximum over
$a$ of this maximum value occurs when $a=1$, and similarly for the minimum.
By Pringsheim’s theorem, there is a minimal singularity of $F_{m}$ on the
positive real axis, and this is precisely the smallest real zero $\rho_{m}$ of
$H_{m}$. Furthermore, because $F_{m}$ is not periodic, this singularity is the
unique one of that modulus. Thus $F_{m}$ is analytic in the open disk centered
at $0$ with radius $\rho_{m}$. Note that $\rho_{m}\geq 1/2$ because the
exponential growth rate of unrestricted compositions is $2$, and so our
restricted class of compositions must grow no faster. However $\rho_{m}\leq 1$
because the sum defining $H_{m}$ has value $m$ when $x=1$. Since $\rho_{m}$ is
the smallest positive real solution of
$\sum_{a=1}^{m}\frac{\rho^{a}}{1+\rho^{a}-\rho^{m}}=1,$
it follows that $\rho_{m}$ is an algebraic number of degree at most $m^{2}$.
Note that the sum defining $H_{m}$ shows that $H_{m}(x)>H_{m+1}(x)$ for all
$m$ and all $0<x<1$. Thus in fact $\rho_{m}\leq\rho_{2}<0.68$ for all $m$.
The rest of the proof should proceed according to a familiar outline: apply
Rouché’s theorem to locate the dominant singularity of $F_{m}(x,1,0)$, by
approximating $H_{m}$ with a simpler function having a unique zero inside an
appropriately chosen disk of radius $c$, where $\rho_{m}<c<1$; derive
asymptotics for the coefficients $a_{n}$ via standard singularity analysis.
This technique has been used in several similar problems, for example for
Carlitz compositions.
There are some difficulties with this approach in our case. If we attack
$F_{m}$ directly, we must derive a result for all $m$. Since $F_{m}$ is
rational with numerator and denominator of degree at most $m^{2}$, for fixed
$m$, we could consider using numerical root-finding methods, but for arbitrary
symbolic $m$ this will not work. It is intuitively clear that for sufficiently
large $m$, $F_{m}$ should be close to $F_{\infty}$ and so by using Rouché’s
theorem, we could reduce to the Carlitz case.
However, even the Carlitz case is not as easy as claimed in the literature,
and we found several unconvincing published arguments. Some authors simply
assert that $F_{\infty}$ has a single root, based on a graph of the function
on a given circle. This can be made into a proof, by approximately evaluating
$F_{\infty}$ at sufficiently many points and using an upper bound on the best
Lipschitz constant for the function, but this is somewhat unpleasant. We do
not know a way of avoiding this problem — the minimum modulus of a function on
a circle in the complex plane must be computed somehow. We use an approach
similar to that taken in [5].
The obvious approximating function to use is an initial segment with $k$ terms
of the partial sum defining $F_{m}$. However, it seems easier to use the
initial segment of the sum defining $F_{\infty}$, which we denote by $S_{k}$.
We will take $k=7$ and $c=0.7$ and denote $S_{7}$ by $h$. By using the
Jenkins-Traub algorithm as implemented in the Sage command maxima.allroots(),
we see that all roots of $h$, except the real positive root (approximately
$0.572$) have real or imaginary part with modulus more than $0.7$, so they
certainly lie outside the circle $C$ given by $|x|=c$.
To apply Rouché’s theorem, we need an upper bound for $|H_{m}-h|$ on $C$ which
is less than the lower bound for $|h|$ on $C$. We first claim that the lower
bound for $|h|$ on $C$ is at least $0.43$. This can be proved by evaluation at
sufficiently many points of $C$. Since $|h^{\prime}(z)|$ is bounded by $100$
on $C$, $N:=1000$ points certainly suffice. This is because the minimum of
$|h|$ at points of the form $0.7\exp(2\pi ij/N)$, for $0\leq j\leq N$, is more
than $0.51$ (computed using Sage), and the distance between two such points is
at most $8\times 10^{-4}$, by Taylor approximation. In fact, it seems that the
minimum indeed occurs at $x=0.7$, but this is not obvious to us.
We now compute an upper bound for $|H_{m}-h|$. To this end, we compute, when
$m\geq 7$,
$\displaystyle\left|H_{m}(x)-h(x)\right|$
$\displaystyle=\sum_{a=8}^{m}\frac{x^{a}}{1-x^{a}+x^{m}}+\left(\sum_{a=1}^{7}\frac{x^{a}}{1-x^{a}+x^{m}}-h(x)\right)$
$\displaystyle\leq\sum_{a=8}^{\infty}\frac{c^{a}}{1-c^{8}}+\left(\sum_{a=1}^{7}\frac{x^{a}}{1-x^{a}+x^{m}}-h(x)\right).$
The sum $\sum_{a=8}^{\infty}\frac{c^{a}}{1-c^{8}}$ has value less than $0.204$
when $c=0.7$. The second sum is smaller than $0.2$ which can be verified by a
similar argument to the above, by evaluating at sufficiently many points.
We still need to deal with the cases $m<7$ and these can be done directly via
inspection after computing all roots numerically as above.
The above arguments show that $\rho_{m}$ is a simple zero of $H_{m}$ and hence
a simple pole of the rational function $F_{m}(x,1,0)$. The asymptotics now
follow in the standard manner by a residue computation, and we obtain
$a_{n}\sim\rho_{m}^{-n}\frac{1}{-\rho_{m}H^{\prime}(\rho_{m})}.$
For example, $F_{2}(x,1,0)=(1+x+x^{2})/((1-x^{2})^{2}-x^{3})$ has a minimal
singularity at $\rho_{2}\approx 0.6710436067037893$, which yields the
following result.
###### Theorem 9.
We have
$a(n)\sim(0.6436)1.4902^{n}$
for large $n$, where $a(n)$ is given by (14).
For example, when $n=20$, the relative error in this approximation is already
less than $0.2\%$. The exponential rate $1/\rho_{m}$ approaches the rate for
Carlitz compositions, namely $1.750\cdots$, as $m\to\infty$.
We can in fact derive asymptotics in the multivariate case. For each $m$, it
is possible in principle to compute asymptotics in a given direction by
analysis of $F_{m}(x,y,q)$, for example, using the techniques of Pemantle and
Wilson [13]. To do this for arbitrarily large $m$ is computationally
challenging, and so in order to limit the length of this article, we give a
sketch only for $m=2$, and refer the reader to the above reference or the more
recent book [14]. In this case we have
$\displaystyle F_{2}(x,y,q)$
$\displaystyle=\left(1-\frac{xy}{1-x^{2}-xy(q-1)}+\frac{x^{2}y}{1-x^{2}-x^{2}y(q-1)}\right)^{-1}$
$\displaystyle=\frac{(1-x^{2}-xy(q-1))(1-x^{2}-x^{2}y(q-1))}{1-2x^{2}-qxy+x^{4}-qx^{2}y-x^{3}y^{2}+qx^{3}y+qx^{4}y+q^{2}x^{3}y^{2}}.$
By standard algorithms, for example as implemented in Sage’s solve command,
one can check that the partial derivatives $H_{x},H_{y},H_{q}$ never vanish
simultaneously, so that the variety defined by $H_{m}$ is smooth everywhere.
The critical point equations are readily solved by the same method. For
example, for the special case when $n=2d=4t$, where $t$ denotes the number of
congruence successions, we obtain (using the Sage package amgf [15]) the first
order asymptotic
$(0.379867842273)(15.8273658508862)^{t}/(\pi t),$
which has relative error just over $1\%$ when $n=32$ (the number of such
compositions being 54865800). Bivariate asymptotics when $q=0$, or when $y=1$,
could be derived similarly. The smoothness of the variety defined by $H_{m}$
leads quickly to Gaussian limit laws in a standard way as described in [14],
and we leave the reader to explore this further.
## References
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* [2] L. Carlitz, Restricted compositions, _Fibonacci Quart._ 14 (1976) 254–264.
* [3] P. Chinn and S. Heubach, Compositions of $n$ with no occurrence of $k$, _Congr. Numer._ 164 (2003) 33–51.
* [4] C. Elsner, On recurrence formulae for sums involving binomial coefficients, _Fibonacci Quart._ 43 (2005) 31–45.
* [5] W. M. Y. Goh and P. Hitczenko, Average number of distinct part sizes in a random Carlitz composition, _European J. Combin._ 23 (2002) 647–657.
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|
arxiv-papers
| 2013-07-28T18:13:04 |
2024-09-04T02:49:48.586295
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Toufik Mansour, Mark Shattuck, Mark C. Wilson",
"submitter": "Mark C. Wilson",
"url": "https://arxiv.org/abs/1307.7390"
}
|
1307.7439
|
# Imaging of the CO snow line in a solar nebula analog
Chunhua Qi1∗, Karin I. Öberg2∗, David J. Wilner1, Paola d’Alessio3,
Edwin Bergin4, Sean M. Andrews1, Geoffrey A. Blake5,
Michiel R. Hogerheijde6, Ewine F. van Dishoeck6,7
1Harvard-Smithsonian Center for Astrophysics
2Departments of Chemistry and Astronomy, University of Virginia
3Centro de Radioastronom a y Astrofisica, Universidad Nacional Autonoma de
Mexico
4Department of Astronomy, University of Michigan
5Division of Geological and Planetary Sciences, California Institute of
Technology
6Leiden Observatory, Leiden University
7Max Planck Institute for Extraterrestrial Physics
∗Contributed equally to this manuscript
> Planets form in the disks around young stars. Their formation efficiency and
> composition are intimately linked to the protoplanetary disk locations of
> “snow lines” of abundant volatiles. We present chemical imaging of the CO
> snow line in the disk around TW Hya, an analog of the solar nebula, using
> high spatial and spectral resolution Atacama Large Millimeter/Submillimeter
> Array (ALMA) observations of N2H+, a reactive ion present in large abundance
> only where CO is frozen out. The N2H+ emission is distributed in a large
> ring, with an inner radius that matches CO snow line model predictions. The
> extracted CO snow line radius of $\sim 30$ AU helps to assess models of the
> formation dynamics of the Solar System, when combined with measurements of
> the bulk composition of planets and comets.
Condensation fronts in protoplanetary disks, where abundant volatiles deplete
out of the gas phase and are incorporated into solids, are believed to have
played a critical role in the formation of planets in the Solar System (?, ?),
and similar “snow lines” in the disks around young stars should affect the
ongoing formation of exoplanets. Snow lines can enhance particle growth and
thus planet formation efficiencies because of 1) substantial increases in
solid mass surface densities exterior to snow line locations, 2) continuous
freeze-out of gas diffusing across the snow line (cold-head effect), 3) pile-
up of dust just inside of the snow line in pressure traps, and 4) an increased
stickiness of icy grains compared to bare ones, which favors dust coagulation
(?, ?, ?, ?, ?). Experiments and theory on these processes have been focused
on the H2O snow line, but the results should be generally applicable to snow
lines of abundant volatiles, with the exception that the “stickiness” of
different icy grain mantles varies. The locations of snow lines of the most
abundant volatiles – H2O, CO2, and CO – with respect to the planet-forming
zone may also regulate the bulk composition of planets (?). Determining snow
line locations is therefore key to probing grain growth, and thus planetesimal
and planet formation efficiencies, and elemental and molecular compositions of
planetesimals and planets forming in protoplanetary disks, including the solar
nebula.
Based on the Solar System composition and disk theory, the H2O snow line
developed at $\sim$3 Astronomical Units (AU, where 1 AU is the distance from
the Earth to the Sun) from the early Sun during the epoch of chondrite
assembly (?). In other protoplanetary disks the snow line locations are
determined by the disk midplane temperature structures, set by a time
dependent combination of the luminosity of the central star, the presence of
other heating sources, the efficiencies of dust and gas cooling, and the
intrinsic condensation temperatures of different volatiles. Because of the low
condensation temperature of CO, the CO snow line occurs at radii of 10’s of AU
around Solar-type stars: this larger size scale makes the CO transition zone
the most accessible to direct observations. The CO snow line is also important
in its own right, because CO ice is a starting point for a complex, prebiotic
chemistry (?). Also without incorporating an enhanced grain growth efficiency
beyond that expected for bare silicate dust, observations of centimeter sized
dust grains in disks, including in TW Hya (?), are difficult to reproduce in
the outer disk. Condensation of CO is very efficient below the CO freeze-out
temperature, with a sticking efficiency close to unity based on experiments
(?), and a CO condensation-based dust growth mode may thus be key to
explaining these observations.
Protoplanetary disks have evolving radial and vertical temperature gradients,
with a warmer surface where CO remains in the gaseous state throughout the
disk, even as it is frozen in the cold, dense region beyond the midplane snow
line (?). This means that the midplane snow line important for planet
formation constitutes a smaller portion of a larger condensation surface.
Because the bulk of the CO emission comes from the disk surface layers, this
presents a challenge for locating the CO midplane snow line. Its location has
been observationally identified toward only one system, the disk around HD
163296, based on (sub-)millimeter interferometric observations of multiple CO
rotational transitions and isotopologues at high spatial resolution,
interpreted through detailed modeling of the disk dust and gas physical
structure (?). An alternative approach to constrain the CO snow line,
suggested in (?) and pursued here, is to image molecular emission from a
species that is abundant only where CO is highly depleted from the gas phase.
N2H+ emission is expected to be a robust tracer of CO depletion because the
presence of gas phase CO both slows down N2H+ formation and speeds up N2H+
destruction. N2H+ forms through reactions between N2 and H${}_{3}^{+}$, but
most H${}_{3}^{+}$ will instead transfer a proton to CO as long as the more
abundant CO remains in the gas phase. The most important destruction mechanism
for N2H+ is proton transfer to a CO molecule, whereas in the absence of CO,
N2H+ is destroyed through a much slower dissociative recombination reaction.
These simple astrochemical considerations predict a correlation between N2H+
and CO depletion, or equivalently an anti-correlation between N2H+ and gas-
phase CO. The latter has been observed in many pre-stellar and protostellar
environments, confirming the basic theory (?, ?). In disks N2H+ should
therefore be present at large abundances only inside the vertical and
horizontal thermal layers where CO vapor is condensing, i.e., beyond the CO
snow line. Molecular line surveys of disks have shown that N2H+ is only
present in disks cold enough to entertain CO freeze-out (?), and marginally
resolved observations hint at a N2H+ emission offset from the stellar position
(?), in agreement with the models of disk chemistry (?). Detailed imaging of
N2H+ emission in protoplanetary disks at the scales needed to directly reveal
CO snow lines with sufficient sensitivity has previously been out of reach.
We used ALMA to obtain images of emission from the 372 GHz dust continuum and
the N2H+ $J=4-3$ line from the disk around TW Hya (Fig. 1, S1) (?). TW Hya is
the closest (54$\pm$6 pc) and as such the most intensively-studied pre-main-
sequence star with a gas-rich circumstellar disk (?, ?). Based on previous
observations of dust and CO emission, and the recent detection of HD line
emission (?), this 3–10 million year old, 0.8 M⊙ T Tauri star (spectral type
K7) is known to be surrounded by an almost face-on ($\sim$6$\degree$
inclination) massive $\sim$0.04 M⊙ gas-rich disk. The disk size in millimeter
dust is $\sim$60 AU, with a more extended ($>$ 100 AU) disk in gas and
micrometer-sized dust (?). Both the disk mass and size conforms well with
solar nebula estimates – the minimum mass of the solar nebula is 0.01 M⊙ based
on planet masses and compositions (?) – and the disk around TW Hya may thus
serve as a template for planet formation in the solar nebula. Our images show
that N2H+ emission is distributed in a ring with an inner diameter of 0.8 to
1.2 arcsec (based on visual inspection), corresponding to a physical inner
radius of 21 to 32 AU. By contrast, CO emission is detected down to radial
scales $\sim$2 AU (?). The clear difference in morphology between the N2H+ and
CO emission can be simply explained by the presence of a CO midplane snow line
at the observed inner edge of the N2H+ emission ring. The different
morphologies cannot be explained by a lack of ions in the inner disk based on
previous spatially and spectrally resolved observations of another ion, HCO+
(?). These HCO+ observations had lower sensitivity and angular resolution than
the N2H+ observations, but they are sufficient to exclude a central hole
comparable in size to that seen in N2H+.
To associate the inner edge radius of the N2H+ emission with a midplane
temperature, and thus a CO freeze-out temperature, requires a model of the
disk density and temperature structure. We adopted the model presented in (?),
updated to conform with recent observations of the accretion rate and grain
settling (Fig. S2–S4, Table S1) (?). In the context of this disk structure
model, the N2H+ inner edge location implies that N2H+ becomes abundant where
the midplane temperature drops to 16–20 K. This is in agreement with
expectations for the CO freeze-out temperature based on the outcome of the
laboratory experiments and desorption modeling by (?), who found CO
condensation/sublimation temperatures of 16–18 K under interstellar
conditions, assuming heat-up rates of 1 K per 102 to 106 years. In outer disk
midplanes, condensation temperatures are expected to at most a few degrees
higher because of a weak dependence on density (?). If CO condenses onto H2O
ice rather than existing CO ice, the condensation temperature will increase
further, but this will only affect the first few monolayers of ice and is not
expected to change the location where the majority of CO freezes out. Some CO
may also remain in the gas phase below the CO freeze-out temperature in the
presence of efficient non-thermal desorption, especially UV photodesorption
(?), but this is expected to be negligible in the disk midplane at 30 AU,
because of UV shielding by upper disk layers. UV photodesorption may affect
the vertical CO snow surface location, however, and it may thus not be
possible to describe the radial and vertical condensation surfaces by a single
freeze-out temperature.
To locate the inner edge of the N2H+ ring more quantitatively, we simulated
the N2H+ emission with a power-law column density distribution and compared
with the data. We assumed the disk material orbits the central star in
Keplerian motion, and fixed the geometric and kinematic parameters of the disk
that affect its observed spatio-kinematic behavior (?). We used the same,
updated density and temperature disk structure model (?), and assumed that the
N2H+ column density structure could be approximated as a radial power-law with
inner and outer edges, while vertically the abundance was taken to be constant
between the lower (toward midplane) and upper (toward surface) boundaries.
This approach crudely mimics the results of detailed astrochemical modeling of
disks, which shows that molecules are predominantly present in well-defined
vertical layers (?, ?), and has been used to constrain molecular abundance
structures in a number of previous studies (?, ?). The inner and outer radii,
power-law index, and column density at 100 AU were treated as free parameters.
We calculated a grid of synthetic N2H+ visibility datasets using the RATRAN
code (?) to determine the radiative transfer and molecular excitation, and
compared with the N2H+ observations. We obtained the best-fit model by
minimizing $\chi^{2}$, the weighted difference between the data and the model
with the real and imaginary part of the complex visibility measured in the
($u,v$)-plane sampled by the ALMA observations of N2H+.
Fig. 2a demonstrates that the inner radius is well constrained to 28–31 AU
(3$\sigma$). This edge determination was aided by the nearly face-on viewing
geometry, because this minimizes the impact of the detailed vertical structure
on the disk modeling outcome. Furthermore, the Keplerian kinematics of the gas
help to constrain the size scale at a level finer than the spatial resolution
implied by the synthesized beam size. As a result, the fitted inner radius is
robust to the details of the density and temperature model (?) (Table S2). In
the context of this model, the best-fit N2H+ inner radius corresponds to a CO
midplane snow line at a temperature of 17 K. Fig. 2b presents the best-fit
N2H+ column density profile together with the best-fit 13CO profile, assuming
a CO freeze-out temperature of 17 K (?) (Fig. S5, Table S3). We fit 13CO
emission (obtained with the Submillimeter Array (?)) because the main
isotopologue CO lines are optically thick. The N2H+ column density contrast
across the CO snow line is at least an order of magnitude (?). Fig. 2c shows
simulated ALMA observations of the best-fit N2H+ $J=4-3$ model, demonstrating
the excellent agreement.
Our quantitative analysis thus confirms the predictions that N2H+ traces the
snow line of the abundant volatile, CO. Furthermore, the agreement between the
quantitative analysis and the visual estimate of the N2H+ inner radius
demonstrate that N2H+ imaging is a powerful tool to determine the CO snow line
radii in disks, whose density and temperature structures have not been modeled
in detail. N2H+ imaging with ALMA may therefore be used to provide statistics
on how snow line locations depend on parameters of interest for planet
formation theory, such as the evolutionary stage of the disks.
The locations of snow lines in solar nebula analogs like TW Hya are also
important to understand the formation dynamics of the Solar System. The H2O
snow line is key to the formation of Jupiter and Saturn (?), while CH4 and CO
freeze-out enhanced the solid surface density further out in the solar nebula,
which may have contributed to the feeding zones of Uranus and Neptune (?),
depending on exactly where these ice giants formed. In the popular Nice model
for the dynamics of the young Solar System, Uranus formed at the largest
radius of all planets, at $\sim 17$ AU (?), and most comets and Kuiper Belt
objects formed further out, to $\sim 35$ AU. The plausibility of this scenario
can be assessed using the bulk compositions of these bodies together with
knowledge of the CO snow line location. In particular, Kuiper Belt objects
contain CO and the even more volatile N2 (?, ?), which implies that they must
have formed beyond the CO snow line. Comets exhibit a range of CO abundances,
some of which seem to be primordial, which suggest the CO snow line was
located in the outer part of their formation region of 15–35 AU (?). This is
consistent with the CO snow line radius that we have determined in the TW Hya
disk. However, in the context of the Nice model, this CO snow line radius is
too large for the ice giants, and suggests that their observed carbon
enrichment has a different origin than the accretion of CO ice (?). A caveat
is that H2O ice can trap CO, though this process is unlikely to be efficient
enough to explain the observations. In either case, the CO snow line locations
in solar nebula analogs like TW Hya offers independent constraints on the
early history of the Solar System.
Fig. 1: Observed images of dust, CO and N2H+ emission toward TW Hya. Left:
ALMA 372 GHz continuum map, extracted from the line free channels of the N2H+
observations. Contours mark [5, 10, 20, 40, 80, 160, 320] mJy beam-1 and the
rms is 0.2 mJy beam-1. Center: image of CO $J=3-2$ emission acquired with the
SMA (?). Contours mark [1, 2, 3, 4, 5] Jy km s-1 beam-1 and the rms is 0.1 Jy
km s-1 beam-1. Right: ALMA image of N2H+ $J=4-3$ integrated emission with a
single contour at 150 mJy km s-1 beam-1 and the rms is 10 mJy km s-1 beam-1.
The synthesized beam sizes are shown in the bottom left corner of each panel.
The red dashed circle marks the best-fit inner radius of the N2H+ ring from a
modeling of the visibilities. This inner edge traces the onset of CO freeze-
out according to astrochemical theory, and thus marks the CO snow line in the
disk midplane.
Fig. 2: Model results for the N2H+ abundance structure toward TW Hya. Upper
left: The $\chi^{2}$ fit surface for the power law index and inner radius of
the N2H+ abundance profile. Contours correspond to the 1–5 $\sigma$ errors and
the blue contour marks 3$\sigma$. Upper right: The best fit N2H+ column
density structure, shown together with the total gas column density and the
best-fit 13CO column density for CO freeze-out at 17 K. The shaded region
marks the N2H+ 1$\sigma$ detection limit. Lower panel: N2H+ observations,
simulated observations of the best-fit N2H+ model, and the imaged residuals,
calculated from the visibilities.
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Acknowledgments
We are grateful to S. Schnee for data calibration and reduction assistance.
C.Q. would like to thank the SAO Radio Telescope Data Center (RTDC) staff for
their generous computational support. C.Q., K.I.O. and D.J.W. acknowledges a
grant from NASA Origins of Solar Systems grant No. NNX11AK63. P.D.
acknowledges a grant from PAPIIT-UNAM. E.A.B. acknowledges support from NSF
Grant#1008800. This Report makes use of the following ALMA data:
ADSJAO.ALMA#2011.0.00340.S. ALMA is a partnership of ESO (representing its
member states), NSF (USA), and NINS (Japan), together with NRC (Canada) and
NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint
ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. We also make use of
the Submillimeter Array (SMA) data: project #2004-214 (PI: C. Qi). The SMA is
a joint project between the Smithsonian Astrophysical Observatory and the
Academia Sinica Institute of Astronomy and Astrophysics and is funded by the
Smithsonian Institution and the Academia Sinica.
Supporting Online Material
www.sciencemag.org/cgi/content/full/science.1239560/DC1
Materials and Methods
Table S1–S3
Figs S1–S5
References (35–52)
## Materials and Methods
### Observational details
Continuum and N2H+ line observations toward TW Hya were carried out in ALMA
band 7 (PI: C. Qi) on 19 November, 2012, with 23 to 26 antennas in the Cycle 0
compact configuration. The correlator was configured to observe four windows
with a channel spacing of $\delta\nu$= 61.04 kHz and a bandwidth of 234.375
MHz each. The windows were centered at 372.672 GHz (SPW#1), the rest frequency
of the N2H+ $J=4-3$ line, 372.421 GHz (SPW#0), 358.606 GHz (SPW#2), and
357.892 GHz (SPW#3) . The nearby quasar J1037-295 was used for phase and gain
calibration and 3C279 and J0522-364 were used as bandpass calibrators. The
primary calibrator Ceres provided a mean flux density of 0.61 Jy for the gain
calibrator J1037-295. The visibility data were reduced and calibrated in CASA
3.4.
The atmospheric transmission in the upper (372 GHz) and lower (358 GHz)
sidebands is very different due to strong absorption near 370 GHz. Therefore,
we reduced the data separately for both sidebands. The continuum visibilities
were extracted by averaging the line-free channels in SPW# 0, 1 (upper
sideband) and 2,3 (lower sideband), respectively. We carried out self-
calibration procedures on the continuum as demonstrated in the TW Hya Science
Verification Band 7 CASA Guides, which are available online
(https://almascience.nrao.edu/alma-data/science-verification/tw-hya). The
synthesized beam and RMS for the continuum maps are $0.^{\prime\prime}63\times
0.^{\prime\prime}60$ ($PA=3\degree$), 0.61 mJy beam-1 (upper sideband) and
$0.^{\prime\prime}57\times 0.^{\prime\prime}55$ ($PA=17\degree$), 0.24 mJy
beam-1 (lower sideband). The continuum peak flux densities are determined to
be 2.0097$\pm$0.0062 Jy at 372 GHz and 1.7814$\pm$ 0.0044 Jy at 358 GHz, which
agrees with previous SMA observations and ALMA science verification data (?,
?). We applied the upper sideband continuum self-calibration correction to the
N2H+ $J=4-3$ line data and subtracted the continuum emission in the visibility
domain.
The resulting synthesized beam for the N2H+ $J=4-3$ data cube is
$0.^{\prime\prime}63\times 0.^{\prime\prime}59$ ($PA=-18\degree$), and the
1$\sigma$ rms is 30 mJy beam-1 in 0.1 km s-1 velocity intervals or 8.1 mJy
beam-1 km s-1, which corresponds to a column density detection limit of
2$\times$1011 cm-2 at 17 K. Fig. S1 shows the resulting channel maps for N2H+
$J=4-3$.
### Physical model
The physical model used to interpret the ALMA observations is a steady viscous
accretion disk, heated by irradiation from the central star and mechanical
energy generated by viscous dissipation near the disk midplane (?, ?, ?, ?).
The disk model is axisymmetric, in vertical hydrostatic equilibrium, and the
viscosity follows the $\alpha$ prescription (?). Energy is transported in the
disk by radiation, convection (in regions where the Schwarzschild stability
criterion is not satisfied) and a turbulent energy flux. The penetration of
the stellar and shock generated radiation is calculated using the two first
moments of the radiative transfer equation, taking into account scattering and
absorption by dust grains. This model framework has been used to successfully
reproduce observed disk structures towards several T Tauri and Herbig Ae stars
(?, ?, ?, ?, ?, ?). Following (?, ?, ?), the model includes a tapered
exponential edge to simulate viscous spreading:
$\Sigma\sim\dot{M}\Omega_{k}/\alpha T_{c}$, with
$\alpha(R)=\alpha_{0}exp(-R/R_{c})$. Using the parameter values listed in
Table S1, this physical model provides a good fit to the broadband spectral
energy distribution (SED) of the disk of TW Hya (Fig. S2), except for the mid-
and far-IR wavelengths, which is complicated by contributions of the inner
disk wall, around 3.5–4 AU and optically thin hot dust from the inner hole,
according to (?, ?). The $\alpha_{0}$ adopted in this model is also consistent
with the upper limit on the turbulent line widths, ($<$ 40 m s-1) at $\sim$1–2
scale heights (?), and an ever lower turbulence is expected in the midplane.
Following (?), given a measured disk mass accretion rate, 2$\times$10-9
$M_{\odot}$ yr-1 (?), and the viscous $\alpha$ parameter as formulated above,
the vertical temperature and density structures are mainly regulated by the
degree of grain growth and settling. In the present model, we introduce this
effect in a parametric way. The dust is assumed to consist of two populations
of grains with different size distribution functions (?), with
$a_{max}^{small}=0.25\mu$m (as in the interstellar medium) and
$a_{max}^{big}=1$ mm (consistent with the SED slope at mm wavelengths), and
different spatial distributions such that the abundance of the large grains
increases towards the midplane. The small dust grain to gas mass ratio is
parameterized by $\zeta_{\rm small}$, which is lower than the ISM value
because a large fraction of the dust mass is contained in larger grains. The
amount of dust that is in small dust grains is parameterized by
$\epsilon=\zeta_{\rm small}/\zeta_{\rm ISM}<1$. The amount of large dust
grains (parameterized by $\epsilon_{\rm big}$) is calculated so that the total
dust mass at each radius is conserved, and $\epsilon$ is constrained by the
slope of the SED in the far-IR and sub-mm wavelength range. The maximum grain
size in the model is set to 1 mm, despite observational evidence for the
existence of cm-sized grains (?), because we found a power-law size
distribution of grains with a maximum size in cm could not fit the mm SED
slope. This is probably caused by a radial distribution difference for the mm
and cm-sized dust grains due to differential dust migration. The exclusion of
cm-sized grains should have no effect on the conclusions of this study since
their impact on the midplane temperature structure is minimal.
The surface that separates the regions where the small and large dust grain
populations dominate is parameterized by $z_{\rm big}(R)$ in terms of the
local gas scale heights, $H$. Different $z_{\rm big}$ values are expected to
result in disk structures with different thermal profiles because of changes
in the shape of the irradiation surface, which determines the fraction of
stellar emission intercepted and reprocessed by the disk. We vary $z_{\rm
big}$ between 2$H$ and 3.5$H$, which results in the different vertical disk
temperature structures seen in Fig. S3. Around the observed inner edge of the
N2H+ ring, changing $z_{\rm big}$ also changes the midplane temperature by up
to 2 K (Fig. S4). Despite the importance of $z_{\rm big}$ for vertical
temperature and density structure, Fig. S2 shows that models with different
$z_{\rm big}$ values present small variations in the SEDs, and all models fit
the observed SED satisfactorily. This is generally true for SED modeling
because of the degeneracy of the dust data with the parameter $z_{\rm big}$
(?) and the nearly face-on disk geometry makes SED modeling especially
challenging. The details of the vertical structure in the TW Hya disk are
therefore uncertain, and below we analyze the N2H+ line emission using the
full range of $z_{\rm big}$ values to explore its effect on our conclusions.
### N2H+ line modeling
We adopted the molecular abundance model introduced by (?, ?), and assumed
that the N2H+ emission originates in a vertical layer with a constant
abundance between the surface ($\sigma_{s}$) and midplane ($\sigma_{m}$)
boundaries which are represented by vertically integrated hydrogen column
densities measured from the disk surface in units of 1.59$\times$1021 cm-2. We
fix the vertical surface boundary $\sigma_{s}$ to 3.2 and the midplane
boundary $\sigma_{m}$ to 100, which simulates an emission layer close to the
midplane, in accordance with model predictions. Fitting these boundaries would
require a combination of a well constrained vertical temperature structure and
multiple N2H+ transitions and is thus outside of the scope of this study. To
test the importance of the assumed vertical structure on the inferred radial
distribution of N2H+ we simulate N2H+ visibilities for disk structure models
with the range of $z_{\rm big}$ values and disk structures shown in Figs.
S2-S4.
We model the radial distribution of N2H+ as a power law
N${}_{100}\times(r/100)^{p}$ with an inner radius $R_{in}$ and outer radius
$R_{out}$, where $N_{100}$ is the column density at 100 AU in cm-2, $r$ is the
distance from the star in AU, and $p$ is the power-law index. For each $z_{\rm
big}$ structure model, we compute a grid of synthetic N2H+ $J=4-3$ visibility
datasets over a range of $R_{out}$, $R_{in}$, $p$ and $N_{100}$ values and
compare with the observations. The best-fit model is obtained by minimizing
$\chi^{2}$, the weighted difference between the data and the model with the
real and imaginary part of the complex visibility measured in the
$(u,v)$-plane sampled by the ALMA observations. We use the two-dimensional
Monte Carlo model RATRAN (?) to calculate the radiative transfer and molecular
excitation. The collisional rate coefficients are taken from the Leiden Atomic
and Molecular Database (?).
Table S2 gives the best-fit N2H+ distribution parameters for each $z_{\rm
big}$ model, as well as the corresponding midplane temperature $T_{c}$ at the
N2H+ inner edge. The power law index of the surface density varies between
$2.4$ and $-3.6$, and the column density at the inner edge between
4$\times$1012 and 2$\times$1015 cm-2. Given the 1$\sigma$ detection limit of
2$\times$1011 cm-2, the column density contrast at the inner edge of the N2H+
ring is at least 20 and could be much larger. Across this range of models, the
inner radius varies by less than 5 AU, and the midplane temperature at the
inner radius varies by less than 1 K. The inner radius is thus well
constrained, which implies that the key feature of the N2H+ distribution
needed to constrain the CO snow line is robust with respect to the details of
the physical model assumptions. Channel maps for the best-fit model using the
fiducial $z_{\rm big}=3H$ are shown in Fig. S1 together with the observed data
and imaged residuals, demonstrating the excellent agreement.
### CO distribution
To derive the CO distribution and test the self-consistency of the fiducial
best-fit model, we also modeled two CO isotopologues which we observed with
the Submillimter Array (SMA) (?) in 2005 February 27 and April 10. The main
isotopologue was also observed, but is optically thick and therefore not
included in this analysis. The SMA receivers operated in a double-sideband
mode with an intermediate frequency (IF) band of 4–6 GHz from the local
oscillator frequency, sent over fiber optic transmission lines to 24
overlapping “chunks” of the digital correlator. The correlator was configured
to include CO, 13CO and C18O $J=2-1$ in one setting: the tuning was centered
on the CO $J=2-1$ line at 230.538 GHz in chunk S15, while the 13CO $J=2-1$ at
220.399 GHz and C18O $J=2-1$ at 219.560 GHz were simultaneously observed in
chunk 12 and 22, respectively (?). Combinations of two array configuration
(compact and extended) were used to obtain projected baselines ranging from 6
to 180 m. The observing loops used J1037-295 as the gain calibrator. The
bandpass response was calibrated using observations of 3C279. Flux calibration
was done using observations of Titan and Callisto. Routine calibration tasks
were performed using the MIR software package
(http://www.cfa.harvard.edu/$\sim$cqi/mircook.html), and imaging and
deconvolution were accomplished in MIRIAD. The integrated fluxes are reported
in Table S3. Fig. S5 shows the spatially integrated spectra of 13CO and C18O
$J=2-1$ extracted from the SMA channel maps in 8′′ square boxes centered on TW
Hya.
Following (?), CO is assumed to be present in the disk between a lower
boundary set by the CO freeze-out temperature derived from the N2H+ modeling,
and an upper boundary set by photodissociation, though the choice of upper
boundary in this case has a very small effect on the modeled emission profiles
of 13CO and C18O. The CO abundance structure was optimized using the same
procedure as for N2H+ above. Fig. S5 shows that the best-fit CO abundance
distribution fits the CO isotopologue observations well when using the
fiducial disk structure and assuming standard isotope ratios and CO freeze-out
at the N2H+ inner edge temperature of 17 K. In contrast the models with much
smaller $z_{\rm big}$ cannot reproduce the relative C18O and 13CO fluxes
without order of magnitude deviations from the cosmic isotope ratios. More
data with better sensitivity and resolution from the emission of CO and its
isotopologues and a detailed surface heating model (as suggested by (?)) are
needed to constrain the temperature structure and the $z_{\rm big}$ value in
the disk of TW Hya.
Table S1: Physical model for the disk of TW Hya Parameters | Values
---|---
Stellar and accretion properties
Spectral type | K7
Effective temperature: $T_{*}$ (K) | 4110
Estimated distance: $d$ (pc) | 54
Stellar radius: $R_{*}$ (R⊙) | 1.04
Stellar mass: $M_{*}$ (M⊙) | 0.8
Accretion rate: $\dot{M}$ ($M_{\odot}$ yr-1) | 2$\times$10-9
Disk structure properties
Disk mass: $M_{d}$ (M⊙) | 0.04
Characteristic radius: $R_{c}$ (AU) | 60
Viscosity coefficient: $\alpha_{0}$ | 0.0007
Depletion factor of the atmospheric small grains: $\epsilon^{a}$ | 0.01
$z_{\rm big}$a (Hb) | 3.0
Disk geometric and kinematic propertiesc
Inclination: $i$ (deg) | 6
Systemic velocity: $V_{\rm LSR}$ (km s-1) | 2.86
Turbulent line width: $\delta$vturb (km s-1) | 0.05
Position angle: $P.A.$ (deg) | 155
aSee definition in paper. |
bGas scale height. |
cParameters adopted from (?, ?, ?, ?). |
Table S2: N2H+ $J=4-3$ fitting resultsa $z_{\rm big}$ ($H$) | $R_{in}$ (AU) | $T_{\rm c}$b (K) | $p$ | $N_{100}$ (cm-2) | $R_{out}$ (AU)
---|---|---|---|---|---
2.0 | 25${}^{+4}_{-6}$ | 15–18 | 2.4${}^{+0.6}_{-0.3}$ | (1.4$\pm$0.2) $\times$1014 | 150$\pm$10
2.5 | 30${}^{+1}_{-3}$ | 15–16 | 0.4${}^{+0.6}_{-0.4}$ | (2.9$\pm$0.5) $\times$1014 | 140$\pm$10
3.0 | 30${}^{+1}_{-2}$ | 16–17 | $-$2.0${}^{+0.5}_{-0.7}$ | (1.6$\pm$0.3) $\times$1014 | 140$\pm$10
3.5 | 30${}^{+1}_{-4}$ | 16–18 | $-$3.6${}^{+0.6}_{-0.8}$ | (2.5$\pm$0.4) $\times$1013 | 140$\pm$10
aErrors within 3$\sigma$. | | | | |
bTemperature range based on Fig. S4. | | | | |
Table S3: TW Hya CO isotopologue observation results. Transition | Frequency (GHz) | Beam /$P.A.$ | $\int Fdv$ (Jy km s-1)
---|---|---|---
13CO $J=2-1$ | 220.399 | $2.^{\prime\prime}7\times 1.^{\prime\prime}8$ / $-$3.0° | 2.72[0.18]
C18O $J=2-1$ | 219.560 | $2.^{\prime\prime}8\times 1.^{\prime\prime}9$ / $-$1.3° | 0.68[0.18]
Fig. S1: Channel maps of the N2H+ $J=4-3$ line emission observed by ALMA from
the disk around TW Hya. The LSR velocity is indicated in the upper right of
each channel, while the synthesized beam size and orientation
($0.^{\prime\prime}63\times 0.^{\prime\prime}59$ at a position angle of
$-$18.1∘) is indicated in the lower left panel. The contours are 0.03
(1$\sigma$) $\times[3,6,9,12,15,18]$ Jy beam-1 . Fig. S2: The TW Hya SED and
the model results for $\dot{M}=2\times 10^{-9}M_{\odot}/yr$,
$\alpha_{0}=0.0007$, $\epsilon=0.01$, and $R_{c}=60$ AU. See the SED
references in (?). The new mm/submm fluxes are from the ALMA science
verification data and this paper (marked by green triangles). The different
model SED lines correspond to $z_{\rm big}/H=2-3.5$, with the fiducial 3H
model shown with a solid line, demonstrating that the SED modeling does not
provide strong constraints on this parameter. Fig. S3: Vertical temperature
and density profiles at $R=30$ AU for disk models with $\dot{M}=2\times
10^{-9}\ M_{\odot}/yr$, $\alpha_{0}=0.0007$, $\epsilon=0.01$, $R_{c}=60$ AU,
and different values of $z_{\rm big}$ values. Upper panel: Temperature versus
height at $R=30$ AU for disk models with different values of $z_{\rm big}/H=$
2.0, 2.5, 3.0, 3.5 (from left to right). The fiducial model, with $z_{\rm
big}=3H$ is shown with a solid line. Lower panel: Density versus height at
$R=30$ AU for the same models. The lines from top to bottom correspond to
models with $z_{\rm big}/H=$ 2.0, 2.5, 3.0, 3.5. Fig. S4: Midplane
temperature profiles for the TW Hya disk for different $z_{\rm big}$ values
showing the effect of $z_{\rm big}$ on the midplane temperature around the
N2H+ inner edge. Fig. S5: CO isotopologue lines toward TW Hya, observed with
the SMA (grey) and the best-fit CO modeling results using the fiducial disk
structure developed to interpret the N2H+ observations. The dashed line marks
the $V_{\rm LSR}$ toward TW Hya.
|
arxiv-papers
| 2013-07-29T02:04:45 |
2024-09-04T02:49:48.596030
|
{
"license": "Public Domain",
"authors": "Chunhua Qi, Karin I. Oberg, David J. Wilner, Paola d'Alessio, Edwin\n Bergin, Sean M. Andrews, Geoffrey A. Blake, Michiel R. Hogerheijde, Ewine F.\n van Dishoeck",
"submitter": "Chunhua Qi",
"url": "https://arxiv.org/abs/1307.7439"
}
|
1307.7480
|
# A Lattice Non-Perturbative Hamiltonian Construction of
1+1D Anomaly-Free Chiral Fermions and Bosons -
on the equivalence of the anomaly matching conditions and the boundary fully
gapping rules
Juven C. Wang [email protected] Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA Perimeter Institute for Theoretical
Physics, Waterloo, ON, N2L 2Y5, Canada Xiao-Gang Wen
[email protected] Perimeter Institute for Theoretical Physics,
Waterloo, ON, N2L 2Y5, Canada Department of Physics, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA Institute for Advanced Study,
Tsinghua University, Beijing, 100084, P. R. China
###### Abstract
A non-perturbative Hamiltonian construction of chiral fermions and bosons with
anomaly-free symmetry $G$ in 1+1D spacetime is proposed. More precisely, we
ask “whether there is a _local_ _short-range_ _finite_ quantum Hamiltonian
system realizing _onsite symmetry_ $G$ defined on a 1D spatial lattice with a
continuous time, such that its low energy physics produces a 1+1D anomaly-free
chiral matter theory of symmetry $G$?” Our answer is “yes.” In particular, we
show that the 3L-5R-4L-0R U(1) chiral fermion theory, with two left-moving
fermions of charge-3 and charge-4, and two right-moving fermions of charge-5
and charge-0 at low energy, can be put on a 1D spatial lattice where the U(1)
symmetry is realized as an onsite symmetry, if we include _properly-designed_
interactions between fermions with intermediate strength. We show how to
design such proper interactions by looking for interaction terms with extra
symmetries. In general, we show that any 1+1D U(1)-anomaly-free chiral matter
theory can be defined as a finite system on 1D lattice with onsite symmetry,
by using a quantum Hamiltonian with a continuous time, if we include properly-
designed interactions between matter fields. We comment on the new ingredients
and the differences of ours comparing to Eichten-Preskill and Chen-Giedt-
Poppitz models, and suggest modifying Chen-Giedt-Poppitz model to have
successful mirror-decoupling. As an additional remark, we show a topological
non-perturbative proof on the equivalence relation between ’t Hooft anomaly
matching conditions and the boundary fully gapping rules of U(1) symmetry.
###### Contents
1. I Introduction
2. II 3L-5R-4L-0R Chiral Fermion model
3. III From a continuum field theory to a discrete lattice model
1. III.1 Free kinetic part and the edge states of a Chern insulator
1. III.1.1 Kinetic part mapping and RG analysis
2. III.1.2 Numerical simulation for the free fermion theory with nontrivial Chern number
2. III.2 Interaction gapping terms and the strong coupling scale
4. IV Topological Non-Perturbative Proof of Anomaly Matching Conditions = Boundary Fully Gapping Rules
1. IV.1 Bulk-Edge Correspondence - 2+1D Bulk Abelian SPT by Chern-Simons theory
2. IV.2 Anomaly Matching Conditions and Effective Hall Conductance
3. IV.3 Anomaly Matching Conditions and Boundary Fully Gapping Rules
1. IV.3.1 a physical picture
2. IV.3.2 topological non-perturbative proof
3. IV.3.3 perturbative arguments
4. IV.3.4 preserved U(1)N/2 symmetry and a unique ground state
5. V General Construction of Non-Perturbative Anomaly-Free chiral matter model from SPT
6. VI Summary
7. A $C$, $P$, $T$ symmetry in the 1+1D fermion theory
8. B Ginsparg-Wilson fermions with a non-onsite U(1) symmetry as SPT edge states
1. B.1 On-site symmetry and non-onsite symmetry
2. B.2 Ginsparg-Wilson fermions and its non-onsite symmetry
9. C Proof: Boundary Fully Gapping Rules $\to$ Anomaly Matching Conditions
10. D Proof: Anomaly Matching Conditions $\to$ Boundary Fully Gapping Rules
1. D.1 Proof for fermions $K=K^{f}$
2. D.2 Proof for bosons $K=K^{b0}$
11. E More about the Proof of “Boundary Fully Gapping Rules”
1. E.1 Canonical quantization
2. E.2 Approach I: Mass gap for gapping zero energy modes
3. E.3 Mass Gap for Klein-Gordon fields and non-Chiral Luttinger liquids under sine-Gordon potential
4. E.4 Approach II: Map the anomaly-free theory with gapping terms to the decoupled non-Chiral Luttinger liquids with gapped spectrum
5. E.5 Approach III: Non-Perturbative statements of Topological Boundary Conditions, Lagrangian subspace, and the exact sequence
12. F More about Our Lattice Hamiltonian Chiral Matter Models
1. F.1 More details on our Lattice Model producing nearly-flat Chern-bands
2. F.2 Explicit lattice chiral matter models
1. F.2.1 1L-(-1R) chiral fermion model
2. F.2.2 3L-5R-4L-0R chiral fermion model and others
3. F.2.3 Chiral boson model
## I Introduction
Regulating and defining chiral fermion field theory is a very important
problem, since the standard model is one such theory.Lee:1956qn ;
Donoghue:1992dd However, the fermion-doubling problemNielsen:1980rz ;
Nielsen:1981xu ; Nielsen:1981hk ; Luscher:2000hn ; Kaplan:2009yg makes it
very difficult to define chiral fermions (in an even dimensional spacetime) on
the lattice. There is much previous research that tries to solve this famous
problem. One approach is the lattice gauge theory,Kogut (1979) which is
unsuccessful since it cannot reproduce chiral couplings between the gauge
fields and the fermions. Another approach is the domain-wall fermion.Kaplan
(1992); Shamir (1993) However, the gauge field in the domain-wall fermion
approach propagates in one-higher dimension. Another approach is the overlap-
fermion,Lüscher (1999); Neuberger (2001); Suzuki (1999); Luscher:2000hn while
the path-integral in the overlap-fermion approach may not describe a finite
quantum theory with a finite Hilbert space for a finite space-lattice. There
is also the mirror fermion approachEichten and Preskill (1986); Montvay
(1992); Bhattacharya et al. (2006); Giedt and Poppitz (2007) which starts with
a lattice model containing chiral fermions in one original _light sector_
coupled to gauge theory, _and_ its chiral conjugated as the _mirror sector_.
Then, one tries to include direct interactions or boson mediated
interactionsSmit (1986); Swift (1992) between fermions to gap out the mirror
sector only. However, the later works either fail to demonstrate Golterman et
al. (1993); Lin (1994); Chen et al. (2013a) or argue that it is almost
impossible to gap out (i.e. fully open the mass gaps of) the mirror sector
without breaking the gauge symmetry in some mirror fermion models.Banks and
Dabholkar (1992)
We realized that the previous failed lattice-gauge approaches always assume
non-interacting lattice fermions (apart from the interaction to the lattice
gauge field). In this work, we show that lattice approach actually works if we
include direct fermion-fermion interaction with appropriate strength (i.e. the
dimensionaless coupling constants are of order 1).Wen:2013oza ; Wen:2013ppa
In other words, a general framework of the mirror fermion approach actually
works for constructing a lattice chiral fermion theory, at least in 1+1D.
Specifically, any anomaly-free chiral fermion/boson field theory can be
defined as a finite quantum system on a 1D lattice where the (gauge or global)
symmetry is realized as an onsite symmetry, provided that we allow lattice
fermion/boson to have interactions, instead of being free. (Here, the “chiral”
theory here means that it “breaks parity $P$ symmetry.” Our 1+1D chiral
fermion theory breaks parity $P$ and time reversal $T$ symmetry. See Appendix
A for $C,P,T$ symmetry in 1+1D.) Our insight comes from Ref. Wen:2013oza, ;
Wen:2013ppa, , where the connection between gauge anomalies and symmetry-
protected topological (SPT) statesChen:2011pg (in one-higher dimension) is
found.
To make our readers fully appreciate our thinking, we shall firstly define our
important basic notions clearly:
$(\diamond 1)$ _Onsite symmetry_Chen:2011pg ; Chen et al. (2011) means that
the overall symmetry transformation $U(g)$ of symmetry group $G$ can be
defined as the tensor product of each single site’s symmetry transformation
$U_{i}(g)$, via $U(g)=\otimes_{i}U_{i}(g)$ with $g\in G$. _Nonsite symmetry_ :
means $U(g)_{\text{non-onsite}}\neq\otimes_{i}U_{i}(g)$.
$(\diamond 2)$ _Local Hamiltonian with short-range interactions_ means that
the non-zero amplitude of matter(fermion/boson) hopping/interactions in finite
time has a _finite_ range propagation, and cannot be an _infinite_ range.
Strictly speaking, the quasi-local _exponential decay_ (of kinetic
hopping/interactions) is _non-local_ and _not short-ranged_.
$(\diamond 3)$ _finite(-Hilbert-space) system_ means that the dimension of
Hilbert space is finite if the system has finite lattice sites (e.g. on a
cylinder).
Nielsen-Ninomiya theoremNielsen:1980rz ; Nielsen:1981xu ; Nielsen:1981hk
states that the attempt to regularize chiral fermion on a lattice as a local
_free non-interacting_ fermion model with fermion number conservation (i.e.
with U(1) symmetryU(1)sym ) has fermion-doubling problemNielsen:1980rz ;
Nielsen:1981xu ; Nielsen:1981hk ; Luscher:2000hn ; Kaplan:2009yg in an even
dimensional spacetime. To apply this no-go theorem, however, the symmetry is
assumed to be an onsite symmetry.
Ginsparg-Wilson fermion approach copes with this no-go theorem by solving
Ginsparg-Wilson(GW) relationGinsparg:1981bj ; Wilson:1974sk based on the
quasi-local Neuberger-Dirac operator,Neuberger:1997fp ; Neuberger:1998wv ;
Hernandez:1998et where _quasi-local is strictly non-local_. In this work, we
show that the quasi-localness of Neuberger-Dirac operator in the GW fermion
approach imposing a _non-onsite_Chen:2011pg ; Chen:2012hc ; Santos:2013uda
U(1) symmetry, instead of an onsite symmetry. (While here we simply summarize
the result, one can read the details of onsite and non-onsite symmetry, and
its relation to GW fermion in the Appendix B.) For our specific approach for
the mirror-fermion decoupling, we _will not_ implement the GW fermions (of
non-onsite symmetry) construction, instead, we will use a lattice fermions
with onsite symmetry but with particular properly-designed interactions.
Comparing GW fermion to our approach, we see that
* •
Ginsparg-Wilson(GW) fermion approach obtains “chiral fermions from a local
free fermion lattice model with non-onsite $\text{U}(1)$ symmetry (without
fermion doublers).” (Here one regards Ginsparg-Wilson fermion applying the
Neuberger-Dirac operator, which is strictly non-onsite and non-local.)
* •
Our approach obtains “chiral fermions from local interacting fermion lattice
model with onsite $U(1)$ symmetry (without fermion doublers), if all
$\text{U}(1)$ anomalies are canncelled.”
Also, the conventional GW fermion approach discretizes the Lagrangian/the
action on the spacetime lattice, while we use a local short-range quantum
Hamiltonian on 1D spatial lattice with a continuous time. Such a distinction
causes some difference. For example, it is known that Ginsparg-Wilson fermion
_can_ implement a single Weyl fermion for the free case without gauge field on
a 1+1D space-time-lattice due to the works of Neuberger, Lüscher, etc. Our
approach _cannot_ implement a single Weyl fermion on a 1D space-lattice within
local short-range Hamiltonian. (However, such a distinction may not be
important if we are allowed to introduce a non-local infinite-range hopping.)
black
Comparison to Eichten-Preskill and Chen-Giedt-Poppitz models: Due to the past
investigations, a majority of the high-energy lattice community believes that
the mirror-fermion decoupling (or lattice gauge approach) fails to realize
chiral fermion or chiral gauge theory. Thus one may challenge us by asking
“how our mirror-fermion decoupling model is different from Eichten-Preskill
and Chen-Giedt-Poppitz models?” And “why the recent numerical attempt of Chen-
Giedt-Poppitz fails?Chen et al. (2013a)” We now stress that, our approach
provides _properly designed fermion interaction terms_ to make things work,
due to the recent understanding to topological gapped boundary conditionsh95 ;
Kapustin:2010hk ; Wang:2012am ; Levin:2013gaa :
* •
Eichten-Preskill(EP)Eichten and Preskill (1986) propose a generic idea of the
mirror-fermion approach for the chiral gauge theory. There the _perturbative_
analysis on the _weak-coupling and strong-coupling_ expansions are used to
demonstrate possible mirror-fermion decoupling phase can exist in the phase
diagram. The action is discretized on the spacetime lattice. In EP approach,
one tries to gap out the mirror-fermions via the mass term of composite
fermions that do not break the (gauge) symmetry on lattice. The mass term of
composite fermions are actually fermion interacting terms. So in EP approach,
one tries to gap out the mirror-fermions via the direct fermion interaction
that do not break the (gauge) symmetry on lattice. However, considering only
the symmetry of the interaction is not enough. Even when the mirror sector is
anomalous, one can still add the direct fermion interaction that do not break
the (gauge) symmetry. So the presence of symmetric direct fermion interaction
may or may not be able to gap out the mirror sector. When the mirror sector is
anomaly-free, we will show in this paper, some symmetric interactions are
_helpful_ for gapping out the mirror sectors, while other symmetric
interactions are _harmful_. The key issue is to design the proper interaction
to gap out the mirror section, and considering only symmetry is not enough.
* •
Chen-Giedt-Poppitz(CGP)Chen et al. (2013a) follows the EP general framework to
deal with a 3-4-5 anomaly-free model with a single U(1) symmetry. All the U(1)
symmetry-allowed Yukawa-Higgs terms are introduced to mediate multi-fermion
interactions. The Ginsparg-Wilson fermion and the Neuberger’s overlap Dirac
operator are implemented, the fermion actions are discretized on the spacetime
lattice. Again, the interaction terms are designed only based on symmetry,
which contain both helpful and harmful terms, as we will show.
* •
Our model in general belongs to the mirror-fermion-decoupling idea. The
anomaly-free model we proposed is named as the 3L-5R-4L-0R model. Our
3L-5R-4L-0R is in-reality different from Chen-Giedt-Poppitz’s 3-4-5 model,
since we impliment:
(i) an onsite-symmetry local lattice model: Our lattice Hamiltonian is built
on 1D spatial lattice with _on-site_ U(1) symmetry. We _neither_ implement the
GW fermion _nor_ the Neuberger-Dirac operator (both have non-onsite symmetry).
(ii) a particular set of interaction terms with proper strength: Our multi-
fermion interaction terms are particularly-designed gapping terms which obey
not only the symmetry but also certain Lagrangian subgroup algebra. Those
interaction terms are called _helpful_ gapping terms, satisfying Boundary
Fully Gapping Rules. We will show that the Chen-Giedt-Poppitz’s Yukawa-Higgs
terms induce extra multi-fermion interaction terms which _do not_ satsify
Boundary Fully Gapping Rules. Those extra terms are incompatible _harmful_
terms, competing with the _helpful_ gapping terms and causing the preformed
mass gap unstable so preventing the mirror sector from being gapped out. (This
can be one of the reasons for the failure of mirror-decoupling in Ref.Chen et
al., 2013a.) We stress that, due to a _topological non-perturbative_ reason,
only a particular set of ideal interaction terms are helpful to fully gap the
mirror sector. Adding more or removing interactions can cause the mass gap
unstable thus the phase flowing to gapless states. In addition, we stress that
only when the helpful interaction terms are in a proper range, _intermediate
strength_ for dimensionless coupling of order 1, can they fully gap the mirror
sector, and yet not gap the original sector (details in Sec.III.2). Throughout
our work, when we say strong coupling for our model, we really mean
intermediate(-strong) coupling in an appropriate range. In CGP model, however,
their strong coupling may be _too strong_ (with their kinetic term neglected);
which can be another reason for the failure of mirror-decoupling.Chen et al.
(2013a)
(iii) extra symmetries: For our model, a total even number $N$ of left/right
moving Weyl fermions ($N_{L}=N_{R}=N/2$), we will add only $N/2$ helpful
gapping terms under the constraint of the Lagrangian subgroup algebra and
Boundary Fully Gapping Rules. As a result, the full symmetry of our lattice
model is U(1)N/2 (where the gapping terms break U(1)N down to U(1)N/2). For
the case of our 3L-5R-4L-0R model, the full U(1)2 symmetry has two sets of
U(1) charges, $\text{U}(1)_{\text{1st}}$ 3-5-4-0 and
$\text{U}(1)_{\text{2nd}}$ 0-4-5-3, both are anomaly-free and mixed-anomaly-
free. Although the physical consideration only requires the interaction terms
to have on-site $\text{U}(1)_{\text{1st}}$ symmetry, looking for interaction
terms with extra U(1) symmetry can help us to identify the helpful gapping
terms and design the proper lattice interactions. CGP model has only a single
$\text{U}(1)_{\text{1st}}$ symmetry. Here we suggest to improve that model by
removing all the interaction terms that break the $\text{U}(1)_{\text{2nd}}$
symmetry (thus adding all possible terms that preserve the two U(1)
symmetries) with an intermediate strength.
The plan of our paper is the following. In Sec.II we first consider a
3L-5R-4L-0R anomaly-free chiral fermion field theory model, with a full
$\text{U}(1)^{2}$ symmetry: A first 3-5-4-0 $\text{U}(1)_{\text{1st}}$
symmetry for two left-moving fermions of charge-3 and charge-4, and for two
right-moving fermions of charge-5 and charge-0. And a second 0-4-5-3
$\text{U}(1)_{\text{2nd}}$ symmetry for two left-moving fermions of charge-0
and charge-5, and for two right-moving fermions of charge-4 and charge-3. If
we wish to have a _single_ $\text{U}(1)_{\text{1st}}$ symmetry, we can weakly
break the $\text{U}(1)_{\text{2nd}}$ symmetry by adding tiny local
$\text{U}(1)_{\text{2nd}}$-symmetry breaking term.
We claim that this model can be put on the lattice with an onsite
$\text{U}(1)$ symmetry, but without fermion-doubling problem. We construct a
2+1D lattice model by simply using four layers of the zeroth Landau levels(or
more precisely, four filled bands with Chern numbersThouless:1982zz
$-1,+1,-1,+1$ on a latticeHaldane:1988zza ; Wen:1990fv ) which produces
charge-3 left-moving, charge-5 right-moving, charge-4 left-moving, charge-0
right-moving, totally four fermionic modes at low energy on one edge.
Therefore, by putting the 2D bulk spatial lattice on a cylinder with two
edges, one can leave edge states on one edge untouched so they remain chiral
and gapless, while turning on interactions to gap out the mirrored edge states
on the other edge with a large mass gap.
In Sec.III, we provide a correspondence from the continuum field theory to a
discrete lattice model. The numerical result of the chiral-$\pi$ flux square
lattice with nonzero Chern numbers, supports the free fermion part of our
model. We study the kinetic and interacting part of Hamiltonian with
dimensional scaling, energy scale and interaction strength analysis. In
Sec.IV, we justify the mirrored edge can be gapped by analytically bosonizing
the fermion theory and confirm the interaction terms obeys “the boundary fully
gapping rules.h95 ; Wang:2012am ; Levin:2013gaa ; Kapustin:2013nva ;
Wang:2013vna ; Barkeshli:2013jaa ; Barkeshli:2013yta ; Plamadeala:2013zva ;
Lu:2012dt ; Kapustin:2010hk ; Hung:2013nla ”
To consider a more general model construction, inspired by the insight of
SPT,Wen:2013oza ; Wen:2013ppa ; Chen:2011pg in Sec.IV.1, we apply the bulk-
edge correspondence between Chern-Simons theory and the chiral boson
theory.Elitzur:1989nr ; WZW ; W ; Wen:1995qn ; h95 ; Wang:2012am ;
Levin:2013gaa ; Barkeshli:2013jaa ; Lu:2012dt We refine and make connections
between the key concepts in our paper in Sec.IV.2, IV.3. These are “the
anomaly factorDonoghue:1992dd ; Fujikawa:2004cx ; 'tHooft:1979bh ;
Harvey:2005it ” and “effective Hall conductance”“ ’t Hooft anomaly matching
condition'tHooft:1979bh ; Harvey:2005it ” and “the boundary fully gapping
rules.h95 ; Wang:2012am ; Levin:2013gaa ; Barkeshli:2013jaa ; Lu:2012dt ;
Kapustin:2010hk ; Hung:2013nla ” In Sec.V, a non-perturbative lattice
definition of 1+1D anomaly-free chiral matter model is given, and many
examples of fermion/boson models are provided. These model constructions are
supported by our proof of the equivalence relations between “the anomaly
matching condition” and “the boundary fully gapping rules” in the Appendix C
and D. In Appendix A, we discuss the $C,P,T$ symmetry in an 1+1 D fermion
theory. In Appendix B, we show that GW fermions realizing its axial U(1)
symmetry by a non-onsite symmetry transformation. As the non-onsite symmetry
signals the nontrivial edge states of bulk SPT,Chen:2011pg ; Chen:2012hc ;
Santos:2013uda thus GW fermions can be regarded as gapless edge states of
some bulk fermionic SPT states, such as certain topological insulators. We
also explain why it is easy to gauge an onsite symmetry (such as our chiral
fermion model), and why it is difficult to gauge a non-onsite symmetry (such
as GW fermions). Since the lattice on-site symmetry can always be gauged, our
result suggests a non-perturbative definition of any anomaly-free chiral gauge
theory in 1+1D. In Appendix E, we provide physical, perturbative and non-
perturbative understanding on “boundary fully gapping rules.” In Appendix F,
we provide more details and examples about our lattice models. With this
overall understanding, in Sec.VI we summarize with deeper implications and
future directions.
[NOTE on usages: Here in our work, U(1) symmetry may generically imply copies
of U(1) symmetry such as U(1)M, with positive integer $M$. (Topological)
Boundary Fully Gapping Rules are defined as the rules to open the mass gaps of
the boundary states. (Topological) Gapped Boundary Conditions are defined to
specify certain boundary types which are gapped (thus topological). There are
two kinds of usages of _lattices_ here discussed in our work: one is the
Hamiltonian lattice model to simulate the chiral fermions/bosons. The other
_lattice_ is the Chern-Simons lattice structure of Hilbert space, which is a
quantized lattice due to the level/charge quantization of Chern-Simons theory.
]
## II 3L-5R-4L-0R Chiral Fermion model
The simplest chiral (Weyl) fermion field theory with U(1) symmetry in $1+1$D
is given by the action
$S_{\Psi,free}=\int
dtdx\;\text{i}\psi^{\dagger}_{L}(\partial_{t}-\partial_{x})\psi_{L}.$ (1)
However, Nielsen-Ninomiya theorem claims that such a theory cannot be put on a
lattice with unbroken onsite U(1) symmetry, due to the fermion-doubling
problem.Nielsen:1980rz ; Nielsen:1981xu ; Nielsen:1981hk While the Ginsparg-
Wilson fermion approach can still implement an anomalous single Weyl fermion
on the lattice, our approach cannot (unless we modify local Hamiltonian to
infinite-range hopping non-local Hamiltonian). As we will show, our approach
is more restricted, only limited to the anomaly-free theory. Let us instead
consider an anomaly-free 3L-5R-4L-0R chiral fermion field theory with an
action,
$S_{\Psi_{\mathop{\mathrm{A}}},free}=\int
dtdx\;\Big{(}\text{i}\psi^{\dagger}_{L,3}(\partial_{t}-\partial_{x})\psi_{L,3}+\text{i}\psi^{\dagger}_{R,5}(\partial_{t}+\partial_{x})\psi_{R,5}+\text{i}\psi^{\dagger}_{L,4}(\partial_{t}-\partial_{x})\psi_{L,4}+\text{i}\psi^{\dagger}_{R,0}(\partial_{t}+\partial_{x})\psi_{R,0}\Big{)},$
(2)
where $\psi_{L,3}$, $\psi_{R,5}$, $\psi_{L,4}$, and $\psi_{R,0}$ are
1-component Weyl spinor, carrying U(1) charges 3,5,4,0 respectively. The
subscript $L$(or $R$) indicates left(or right) moving along $-\hat{x}$(or
$+\hat{x}$). Although this theory has equal numbers of left and right moving
modes, it violates parity and time reversal symmetry, so it is a chiral
theory(details about $C,P,T$ symmetry in Appendix A). Such a chiral fermion
field theory is very special because it is free from U(1) anomaly - it
satisfies the anomaly matching conditionDonoghue:1992dd ; Fujikawa:2004cx ;
'tHooft:1979bh ; Harvey:2005it in $1+1$D, which means
$\sum_{j}q_{L,j}^{2}-q_{R,j}^{2}=3^{2}-5^{2}+4^{2}-0^{2}=0$. We ask:
Question 1: “Whether there is a _local_ _finite_ Hamiltonian realizing the
above U(1) 3-5-4-0 symmetry as an onsite symmetry with _short-range
interactions_ defined on a 1D spatial lattice with a continuous time, such
that its low energy physics produces the anomaly-free chiral fermion theory
Eq.(2)?”
Yes. We would like to show that the above chiral fermion field theory can be
put on a lattice with unbroken onsite U(1) symmetry, if we include properly-
desgined interactions between fermions. In fact, we propose that the chiral
fermion field theory in Eq.(2) appears as the low energy effective theory of
the following 2+1D lattice model on a cylinder (see Fig.1) with a properly
designed Hamiltonian. To derive such a Hamiltonian, we start from thinking the
full two-edges fermion theory with the action $S_{\Psi}$, where the
particularly chosen multi-fermion interactions
$S_{\Psi_{\mathop{\mathrm{B}}},interact}$ will be explained:
$\displaystyle S_{\Psi}$
$\displaystyle=S_{\Psi_{\mathop{\mathrm{A}}},free}+S_{\Psi_{\mathop{\mathrm{B}}},free}+S_{\Psi_{\mathop{\mathrm{B}}},interact}=\int
dt\;dx\;\bigg{(}\text{i}\bar{\Psi}_{\mathop{\mathrm{A}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{A}}}+\text{i}\bar{\Psi}_{\mathop{\mathrm{B}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{B}}}$
$\displaystyle+\tilde{g}_{1}\big{(}(\tilde{\psi}_{R,3})(\tilde{\psi}_{L,5})(\tilde{\psi}^{\dagger}_{R,4}\nabla_{x}\tilde{\psi}^{\dagger}_{R,4})(\tilde{\psi}_{R,0}\nabla_{x}\tilde{\psi}_{R,0})+\text{h.c.}\big{)}+\tilde{g}_{2}\big{(}(\tilde{\psi}_{L,3}\nabla_{x}\tilde{\psi}_{L,3})(\tilde{\psi}_{R,5}^{\dagger}\nabla_{x}\tilde{\psi}_{R,5}^{\dagger})(\tilde{\psi}_{L,4})(\tilde{\psi}_{L,0})+\text{h.c.}\big{)}\bigg{)},$
The notation for fermion fields on the edge A are
$\Psi_{\mathop{\mathrm{A}}}=(\psi_{L,3},\psi_{R,5},\psi_{L,4},\psi_{R,0})$ ,
and fermion fields on the edge B are
$\Psi_{\mathop{\mathrm{B}}}=(\tilde{\psi}_{L,5},\tilde{\psi}_{R,3},\tilde{\psi}_{L,0},\tilde{\psi}_{R,4})$.
(Here a left moving mode in $\Psi_{\mathop{\mathrm{A}}}$ corresponds to a
right moving mode in $\Psi_{\mathop{\mathrm{B}}}$ because of Landau
level/Chern band chirality, the details of lattice model will be explained.)
The gamma matrices in 1+1D are presented in terms of Pauli matrices, with
$\gamma^{0}=\sigma_{x}$, $\gamma^{1}=\text{i}\sigma_{y}$,
$\gamma^{5}\equiv\gamma^{0}\gamma^{1}=-\sigma_{z}$, and
$\Gamma^{0}=\gamma^{0}\oplus\gamma^{0}$,
$\Gamma^{1}=\gamma^{1}\oplus\gamma^{1}$,
$\Gamma^{5}\equiv\Gamma^{0}\Gamma^{1}$ and
$\bar{\Psi}_{i}\equiv\Psi_{i}\Gamma^{0}$.
In 1+1D, we can do bosonization,fermionization1 where the fermion matter
field $\Psi$ turns into bosonic phase field $\Phi$, more explicitly
$\psi_{L,3}\sim e^{\text{i}\Phi^{\mathop{\mathrm{A}}}_{3}}$, $\psi_{R,5}\sim
e^{\text{i}\Phi^{\mathop{\mathrm{A}}}_{5}}$, $\psi_{L,4}\sim
e^{\text{i}\Phi^{\mathop{\mathrm{A}}}_{4}}$, $\psi_{R,0}\sim
e^{\text{i}\Phi^{\mathop{\mathrm{A}}}_{0}}$ on A edge, $\tilde{\psi}_{R,3}\sim
e^{\text{i}\Phi^{\mathop{\mathrm{B}}}_{3}}$, $\tilde{\psi}_{L,5}\sim
e^{\text{i}\Phi^{\mathop{\mathrm{B}}}_{5}}$, $\tilde{\psi}_{R,4}\sim
e^{\text{i}\Phi^{\mathop{\mathrm{B}}}_{4}}$, $\tilde{\psi}_{L,0}\sim
e^{\text{i}\Phi^{\mathop{\mathrm{B}}}_{0}}$ on B edge, up to normal orderings
$:e^{\text{i}\Phi}:$ and prefactors,fermionization2 but the precise factor is
not of our interest since our goal is to obtain its non-perturbative lattice
realization. So Eq.(II) becomes
$\displaystyle
S_{\Phi}=S_{\Phi^{\mathop{\mathrm{A}}}_{free}}+S_{\Phi^{\mathop{\mathrm{B}}}_{free}}+S_{\Phi^{\mathop{\mathrm{B}}}_{interact}}=$
$\displaystyle\frac{1}{4\pi}\int
dtdx\big{(}K^{\mathop{\mathrm{A}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}\big{)}+\big{(}K^{\mathop{\mathrm{B}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}\big{)}\;\;\;$
(4) $\displaystyle+\int
dtdx\bigg{(}g_{1}\cos(\Phi^{\mathop{\mathrm{B}}}_{3}+\Phi^{\mathop{\mathrm{B}}}_{5}-2\Phi^{\mathop{\mathrm{B}}}_{4}+2\Phi^{\mathop{\mathrm{B}}}_{0})+g_{2}\cos(2\Phi^{\mathop{\mathrm{B}}}_{5}-2\Phi^{\mathop{\mathrm{B}}}_{5}+\Phi^{\mathop{\mathrm{B}}}_{4}+\Phi^{\mathop{\mathrm{B}}}_{0})\bigg{)}.\;\;\;\;\;\;\;$
Figure 1: 3-5-4-0 chiral fermion model: (a) The fermions carry U(1) charge
$3$,$5$,$4$,$0$ for $\psi_{L,3},$$\psi_{R,5},$$\psi_{L,4},$$\psi_{R,0}$ on the
edge A, and also for its mirrored partners
$\tilde{\psi}_{R,3},$$\tilde{\psi}_{L,5},$$\tilde{\psi}_{R,4},$$\tilde{\psi}_{L,0}$
on the edge B. We focus on the model with a periodic boundary condition along
$x$, and a finite-size length along $y$, effectively as, (b) on a cylinder.
(c) The ladder model on a cylinder with the $t$ hopping term along black
links, the $t^{\prime}$ hopping term along brown links. The shadow on the edge
B indicates the gapping terms with $G_{1},G_{2}$ couplings in Eq.(II) are
imposed.
Here $I,J$ runs over $3,5,4,0$ and
$K^{\mathop{\mathrm{A}}}_{IJ}=-K^{\mathop{\mathrm{B}}}_{IJ}=\mathop{\mathrm{diag}}(1,-1,1,-1)$
$V_{IJ}=\mathop{\mathrm{diag}}(1,1,1,1)$ are diagonal matrices. All we have to
prove is that gapping terms, the cosine terms with ${g}_{1},{g}_{2}$ coupling
can gap out all states on the edge B. First, let us understand more about the
full U(1) symmetry. What are the U(1) symmetries? They are transformations of
$\text{fermions }\psi\to\psi\cdot e^{\text{i}q\theta},\;\;\;\text{bosons
}\;\;\;\Phi\to\Phi+q\;\theta$
making the full action invariant. The original _four_ Weyl fermions have a
full U(1)4 symmetry. Under _two_ linear-indepndent interaction terms in
$S_{\Psi_{\mathop{\mathrm{B}}},interact}$ (or
$S_{\Phi^{\mathop{\mathrm{B}}}_{interact}}$), U(1)4 is broken down to U(1)2
symmetry. If we denote these $q$ as a charge vector
$\mathbf{t}=(q_{3},q_{5},q_{4},q_{0})$, we find there are such two charge
vectors $\mathbf{t}_{1}=(3,5,4,0)$ and $\mathbf{t}_{2}=(0,4,5,3)$ for
U(1)${}_{\text{1st}}$, U(1)${}_{\text{2nd}}$ symmetry respectively.
We emphasize that finding those gapping terms in this U(1)2 anomaly-free
theory is not accidental. The anomaly matching conditionDonoghue:1992dd ;
Fujikawa:2004cx ; 'tHooft:1979bh ; Harvey:2005it here is satisfied, for the
anomalies
$\sum_{j}q_{L,j}^{2}-q_{R,j}^{2}=3^{2}-5^{2}+4^{2}-0^{2}=0^{2}-4^{2}+5^{2}-3^{2}=0$,
and the mixed anomaly: $3\cdot 0-5\cdot 4+4\cdot 5-0\cdot 3=0$ which can be
formulated as
${\boxed{\mathbf{t}^{T}_{i}\cdot(K^{\mathop{\mathrm{A}}})\cdot\mathbf{t}_{j}=0}}\;,\;\;\;i,j\in\\{1,2\\}$
(5)
with the U(1) charge vector $\mathbf{t}=(3,5,4,0)$, with its transpose
$\mathbf{t}^{T}$.
On the other hand, the boundary fully gapping rules (as we will explain, and
the full details in Appendix E),h95 ; Wang:2012am ; Levin:2013gaa ; Lu:2012dt
for a theory of Eq.(4), require two gapping terms, here
$g_{1}\cos(\ell_{1}\cdot\Phi)+g_{2}\cos(\ell_{2}\cdot\Phi)$, such that self
and mutual statistical angles $\theta_{ij}$W ; Wen:1995qn defined below among
the Wilson-line operators $\ell_{i},\ell_{j}$ are zeros,
${\boxed{\theta_{ij}/(2\pi)\equiv\ell_{i}^{T}\cdot(K^{\mathop{\mathrm{B}}})^{-1}\cdot\ell_{j}=0}}\;,\;\;\;i,j\in\\{1,2\\}$
(6)
Indeed, here we have blue $\ell_{1}=(1,1,-2,2),\ell_{2}=(2,-2,1,1)$ satisfying
the rules. Thus we prove that the mirrored edge states on the edge B can be
fully gapped out.
We will prove the anomaly matching condition is equivalent to find a set of
gapping terms $g_{a}\cos(\ell_{a}\cdot\Phi)$, satisfies the boundary fully
gapping rules, detailed in Sec.IV.2, IV.3, Appendix C and D. Simply speaking,
The anomaly matching condition (Eq.(5)) in 1+1D is _equivalent_ to (an if and
only if relation) the boundary fully gapping rules (Eq.(6)) in 1+1D
boundary/2+1D bulk for an equal number of left-right moving
modes($N_{L}=N_{R}$, with central charge $c_{L}=c_{R}$).
We prove this is true at least for U(1) symmetry case, with the bulk theory is
a 2+1D SPT state and the boundary theory is in 1+1D.
We now propose a lattice Hamiltonian model for this 3L-5R-4L-0R chiral fermion
realizing Eq.(II) (thus Eq.(2) at the low energy once the Edge B is gapped
out). Importantly, we _do not_ discretize the action Eq.(II) on the spacetime
lattice. We _do not_ use Ginsparg-Wilson(GW) fermion _nor_ the Neuberger-Dirac
operator. GW and Neuberger-Dirac scheme contains _non-onsite symmetry_
(details in Appendix B) which cause the lattice _difficult to be gauged_ to
chiral gauge theory. Instead, the key step is that we implement the _on-site
symmetry_ lattice fermion model. The _free kinetic part_ is a fermion-hopping
model which has a _finite 2D bulk energy gap_ but with _gapless 1D edge
states_. This can be done by using any lattice Chern insulator.
We stress that any lattice Chern insulator with onsite-symmetry shall work,
and we design one as in Fig.1. Our full Hamiltonian with two interacting
$G_{1},G_{2}$ gapping terms is
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{q=3,5,4,0}\bigg{(}\sum_{\langle
i,j\rangle}\big{(}t_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}+\sum_{\langle\langle
i,j\rangle\rangle}\big{(}t^{\prime}_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}\bigg{)}$
$\displaystyle+$ $\displaystyle
G_{1}\sum_{j\in\mathop{\mathrm{B}}}\bigg{(}\big{(}\hat{f}_{3}(j)\big{)}^{1}\big{(}\hat{f}_{5}(j)\big{)}^{1}\big{(}\hat{f}^{\dagger}_{4}(j)_{pt.s.}\big{)}^{2}\big{(}\hat{f}_{0}(j)_{pt.s.}\big{)}^{2}+h.c.\bigg{)}+G_{2}\sum_{j\in\mathop{\mathrm{B}}}\bigg{(}\big{(}\hat{f}_{3}(j)_{pt.s.}\big{)}^{2}\big{(}\hat{f}^{\dagger}_{5}(j)_{pt.s.}\big{)}^{2}\big{(}\hat{f}_{4}(j)\big{)}^{1}\big{(}\hat{f}_{0}(j)\big{)}^{1}+h.c.\bigg{)}$
where $\sum_{j\in\mathop{\mathrm{B}}}$ sums over the lattice points on the
right boundary (the edge B in Fig.1), and the fermion operators $\hat{f}_{3}$,
$\hat{f}_{5}$, $\hat{f}_{4}$, $\hat{f}_{0}$ carry a U(1)${}_{\text{1st}}$
charge 3,5,4,0 and another U(1)${}_{\text{2nd}}$ charge 0,4,5,3 respectively.
We emphasize that this lattice model has _onsite_ U(1)2 symmetry, since this
Hamiltonian, including interaction terms, is invariant under a global
U(1)${}_{\text{1st}}$ transformation _on each site_ for any $\theta$ angle:
$\hat{f}_{3}\to\hat{f}_{3}e^{\text{i}3\theta}$,
$\hat{f}_{5}\to\hat{f}_{5}e^{\text{i}5\theta}$,
$\hat{f}_{4}\to\hat{f}_{4}e^{\text{i}4\theta}$, $\hat{f}_{0}\to\hat{f}_{0}$,
and invariant under another global U(1)${}_{\text{2nd}}$ transformation for
any $\theta$ angle: $\hat{f}_{3}\to\hat{f}_{3}$,
$\hat{f}_{5}\to\hat{f}_{5}e^{\text{i}4\theta}$,
$\hat{f}_{4}\to\hat{f}_{4}e^{\text{i}5\theta}$,
$\hat{f}_{0}\to\hat{f}_{0}e^{\text{i}3\theta}$. The U(1)${}_{\text{1st}}$
charge is the reason why it is named as 3L-5R-4L-0R model.
As for notations, $\langle i,j\rangle$ stands for nearest-neighbor hopping
along black links and $\langle\langle i,j\rangle\rangle$ stands for next-
nearest-neighbor hopping along brown links in Fig.1. Here $pt.s.$ stands for
point-splitting. For example,
$(\hat{f}_{3}(j)_{pt.s.})^{2}\equiv\hat{f}_{3}(j)\hat{f}_{3}(j+\hat{x})$. We
stress that the full Hamiltonian (including interactions) Eq.(II) is _short-
ranged and local_ , because each term only involves coupling within finite
number of neighbor sites. The hopping amplitudes $t_{ij,3}=t_{ij,4}$ and
$t^{\prime}_{ij,3}=t^{\prime}_{ij,4}$ produce bands with Chern number $-1$,
while the hopping amplitudes $t_{ij,5}=t_{ij,0}$ and
$t^{\prime}_{ij,5}=t^{\prime}_{ij,0}$ produce bands with Chern number $+1$
(see Sec.III.1.2).Thouless:1982zz ; Haldane:1988zza ; Parameswaran:2013pca ;
Tang et al.(2011) ; Sun et al.(2011) ; Neupert et al.(2011) The ground state
is obtained by filling the above four bands.
As Eq.(II) contains U(1)${}_{\text{1st}}$ and an accidental extra
U(1)${}_{\text{2nd}}$ symmetry, we shall ask:
Question 2: “Whether there is a _local_ _finite_ Hamiltonian realizing _only_
a U(1) 3-5-4-0 symmetry as an onsite symmetry with _short-range interactions_
defined on a 1D spatial lattice with a continuous time, such that its low
energy physics produces the anomaly-free chiral fermion theory Eq.(2)?”
Yes, by adding a small local perturbation to break U(1)${}_{\text{2nd}}$
0-4-5-3 symmetry, we can achieve a faithful U(1)${}_{\text{1st}}$ 3-5-4-0
symmetry chiral fermion theory of Eq.(2). For example, we can adjust Eq.(II)’s
$H\to H+\delta H$ by adding:
$\displaystyle\delta
H=G_{tiny}^{\prime}\sum_{j\in\mathop{\mathrm{B}}}\Big{(}\big{(}\hat{f}_{3}(j)_{pt.s.}\big{)}^{3}\big{(}\hat{f}^{\dagger}_{5}(j)_{pt.s.}\big{)}^{1}\big{(}\hat{f}^{\dagger}_{4}(j)\big{)}^{1}+h.c.\Big{)}$
$\displaystyle\Leftrightarrow\tilde{g}_{tiny}^{\prime}\big{(}(\tilde{\psi}_{L,3}\nabla_{x}\tilde{\psi}_{L,3}\nabla_{x}^{2}\tilde{\psi}_{L,3})(\tilde{\psi}_{R,5}^{\dagger})(\tilde{\psi}_{L,4}^{\dagger})+\text{h.c.}\big{)}$
$\displaystyle\Leftrightarrow
g_{tiny}^{\prime}\cos(3\Phi^{\mathop{\mathrm{B}}}_{3}-\Phi^{\mathop{\mathrm{B}}}_{5}-\Phi^{\mathop{\mathrm{B}}}_{4})\equiv
g_{tiny}^{\prime}\cos(\ell^{\prime}\cdot\Phi^{\mathop{\mathrm{B}}}).$ (8)
Here we have $\ell^{\prime}=(3,-1,-1,0)$. The
$g_{tiny}^{\prime}\cos(\ell^{\prime}\cdot\Phi^{\mathop{\mathrm{B}}})$ is not
designed to be a gapping term (its self and mutual statistics happen to be
nontrivial: ${\ell^{\prime
T}\cdot(K^{\mathop{\mathrm{B}}})^{-1}\cdot\ell^{\prime}\neq 0}$,
${\ell^{\prime T}\cdot(K^{\mathop{\mathrm{B}}})^{-1}\cdot\ell_{2}\neq 0}$),
but this tiny perturbation term is meant to preserve U(1)${}_{\text{1st}}$
3-5-4-0 symmetry only, thus ${\ell^{\prime
T}\cdot\mathbf{t}_{1}}={\ell^{\prime
T}\cdot(K^{\mathop{\mathrm{B}}})^{-1}\cdot\ell_{1}=0}$. We must set
$(|G_{{tiny}^{\prime}}|/|G|)\ll 1$ with $|G_{1}|\sim|G_{2}|\sim|G|$ about the
same magnitude, so that the tiny local perturbation will not destroy the mass
gap.
Without the interaction, i.e. $G_{1}=G_{2}=0$, the edge excitations of the
above four bands produce the chiral fermion theory Eq.(2) on the left edge A
and the mirror partners on the right edge B. So the total low energy effective
theory is non-chiral. In Sec.III.1.2, we will provide an explicit lattice
model for this free fermion theory.
However, by turning on the intermediate-strength interaction $G_{1},G_{2}\neq
0$, we claim the interaction terms can fully gap out the edge excitations on
the right mirrored edge B as in Fig.1. To find those gapping terms is not
accidental - it is guaranteed by our proof (see Sec.IV.2, IV.3, Appendix C and
D) of equivalence between the anomaly matching conditionDonoghue:1992dd ;
Fujikawa:2004cx ; 'tHooft:1979bh ; Harvey:2005it (as
${\mathbf{t}^{T}_{i}\cdot(K)^{-1}\cdot\mathbf{t}_{j}=0}$ of Eq.(5) ) and the
boundary fully gapping rulesh95 ; Wang:2012am ; Levin:2013gaa ;
Barkeshli:2013jaa ; Lu:2012dt ; Kapustin:2010hk ; Hung:2013nla (here
$G_{1},G_{2}$ terms can gap out the edge) in $1+1$ D. The low energy effective
theory of the interacting lattice model with only gapless states on the edge A
is the chiral fermion theory in Eq.(2). Since the width of the cylinder is
finite, the lattice model Eq.(II) is actually a 1+1D lattice model, which
gives a non-perturbative lattice definition of the chiral fermion theory
Eq.(2). Indeed, the Hamiltonian and the lattice need not to be restricted
merely to Eq.(II) and Fig.1, we stress that any on-site symmetry lattice model
produces four bands with the desired Chern numbers would work. We emphasize
again that the U(1) symmetry is realized as an onsite symmetryChen:2011pg ;
Chen et al. (2011) in our lattice model. It is easy to gauge such an onsite
U(1) symmetry (explained in Appendix B) to obtain a chiral fermion theory
coupled to a U(1) gauge field.
## III From a continuum field theory to a discrete lattice model
We now comment about the mapping from a continuum field theory of the action
Eq.(2) to a discretized space Hamiltonian Eq.(II) with a continuous time. We
_do not_ pursue _Ginsparg-Wilson scheme_ , and our gapless edge states are
_not_ derived from the discretization of spacetime action. Instead, we will
show that the Chern insulator Hamiltonian in Eq.(II) as we described can
provide essential gapless edge states for a free theory (without interactions
$G_{1},G_{2}$).
Energy and Length Scales: We consider a finite 1+1D quantum system with a
periodic length scale $L$ for the compact circle of the cylinder in Fig.1. The
finite size width of the cylinder is $w$. The lattice constant is $a$. The
mass gap we wish to generate on the mirrored edge is $\Delta_{m}$, which
causes a two-point correlator has an exponential decay:
$\langle\psi^{\dagger}(r)\psi(0)\rangle\sim\langle
e^{-\text{i}\Phi(r)}e^{\text{i}\Phi(0)}\rangle\sim\exp(-|r|/\xi)$ (9)
with a correlation length scale $\xi$. The expected length scales follow that
$a<\xi\ll w\ll L.$ (10)
The 1D system size $L$ is larger than the width $w$, the width $w$ is larger
than the correlation length $\xi$, the correlation length $\xi$ is larger than
the lattice constant $a$.
### III.1 Free kinetic part and the edge states of a Chern insulator
#### III.1.1 Kinetic part mapping and RG analysis
The kinetic part of the lattice Hamiltonian contains the nearest neighbor
hopping term $\sum_{\langle i,j\rangle}$ $\big{(}t_{ij,q}$
$\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}$ together with the next-
nearest neighbor hopping term $\sum_{\langle\langle
i,j\rangle\rangle}\big{(}t^{\prime}_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}$,
which generate the leading order field theory kinetic term via
$t_{ij}\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)\sim
a\;\text{i}\psi_{q}^{\dagger}\partial_{x}\psi_{q}+\dots,$ (11)
here hopping constants $t_{ij},t_{ij}^{\prime}$ with a dimension of energy
$[t_{ij}]=[t_{ij}^{\prime}]=1$, and $a$ is the lattice spacing with a value
$[a]=-1$. Thus, $[\hat{f}_{q}(j)]=0$ and $[\psi_{q}]=\frac{1}{2}$. The map
from
$f_{q}\to\sqrt{a}\,\psi_{q}+\dots$ (12)
contains subleading terms. Subleading terms $\dots$ potentially contain higher
derivative $\nabla^{n}_{x}\psi_{q}$ are only subleading perturbative effects
$f_{q}\to\sqrt{a}\,(\psi_{q}+\dots+a^{n}\,\alpha_{\text{small}}\nabla^{n}_{x}\psi_{q}+\dots)$
with small coefficients of the polynomial of the small lattice spacing $a$ via
$\alpha_{\text{small}}=\alpha_{\text{small}}(a)\lesssim(a/L)$. We comment that
only the leading term in the mapping is important, the full account for the
exact mapping from the fermion operator $f_{q}$ to $\psi_{q}$ is immaterial to
our model, because of two main reasons:
$\bullet$(i) Our lattice construction is based on several layers of Chern
insulators, and the chirality of each layer’s edge states are protected by a
topological number - the first Chern number $C_{1}\in\mathbb{Z}$. Such an
integer Chern number cannot be deformed by small perturbation, thus it is non-
perturbative topologically robust, hence the chirality of edge states will be
protected and will not be eliminated by small perturbations. The origin of our
_fermion chirality_ (breaking parity and time reversal) is an emergent
phenomena due to the _complex hopping_ amplitude of some hopping constant
$t_{ij}^{\prime}$ or $t_{ij}\in\mathbb{C}$. Beside, it is well-known that
Chern insulator can produce the gapless fermion energy spectrum at low energy.
More details and the energy spectrum are explicitly presented in Sec.III.1.2.
$\bullet$(ii) The properly-designed interaction effect (from boundary fully
gapping rules) is a non-perturbative topological effect (as we will show in
Sec.IV.3 and Appendix E). In addition, we can also do the weak coupling and
the strong coupling RG(renormalization group) analysis to show such
subleading-perturbation is _irrelevant_.
For weak-coupling RG analysis, we can start from the free theory fixed point,
and evaluate $\alpha_{\text{small}}\psi_{q}\dots\nabla^{n}_{x}\psi_{q}$ term,
which has a higher energy dimension than
$\psi_{q}^{\dagger}\partial_{x}\psi_{q}$, thus irrelevant at the infrared low
energy, and irrelevant to the ground state of our Hamiltonian.
For strong-coupling RG analysis at large $g_{1},g_{2}$ coupling(shown to be
the massive phase with mass gap in Sec.IV.3 and Appendix E), it is convenient
to use the bosonized language to map the fermion interaction
$U_{\text{interaction}}\big{(}\tilde{\psi}_{q},\dots,\nabla^{n}_{x}\tilde{\psi}_{q},\dots\big{)}$
of $S_{\Psi_{\mathop{\mathrm{B}}},interact}$ to boson cosine term
$g_{a}\cos(\ell_{a,I}\cdot\Phi_{I})$ of
$S_{\Phi^{\mathop{\mathrm{B}}}_{interact}}$. At the large $g$ coupling fixe
point, the boson field is pinned down at the minimum of cosine potential, we
thus will consider the dominant term as the discretized spatial lattice (a
site index $j$) and only a continuous time: $\int
dt\,\big{(}\sum_{j}\frac{1}{2}\,g\,(\ell_{a,I}\cdot\Phi_{I,j})^{2}+\dots\big{)}$.
Setting this dominant term to be a marginal operator means the scaling
dimension of $\Phi_{I,j}$ is $[\Phi_{I,j}]=1/2$ at strong coupling fixed
point. Since the kinetic term is generated by an operator:
$e^{\text{i}P_{\Phi}a}\sim e^{\text{i}a\partial_{x}\Phi}\sim
e^{\text{i}(\Phi_{j+1}-\Phi_{j})}$
where $e^{\text{i}P_{\Phi}a}$ generates the lattice translation by
$e^{\text{i}P_{\Phi}a}\Phi e^{-\text{i}P_{\Phi}a}=\Phi+a$, but
$e^{\text{i}\Phi}$ containing higher powers of irrelevant operators of
$(\Phi_{I})^{n}$ for $n>2$, thus the kinetic term is an irrelevant operator at
the strong-coupling massive fixed point.
The higher derivative term
$\alpha_{\text{small}}\psi_{q}\dots\nabla^{n}_{x}\psi_{q}$ is generated by the
further long range hopping, thus contains higher powers of
$:e^{\text{i}\Phi}:$ thus this subleading terms in Eq(12) are _further
irrelevant perturbation_ at the infrared, comparing to the dominant cosine
terms. Further details of weak, strong coupling RG are presented in Appendix
E.3.
black
#### III.1.2 Numerical simulation for the free fermion theory with nontrivial
Chern number
Follow from Sec.II and III.1.1, here we provide a concrete lattice realization
for free fermions part of Eq.(II) (with $G_{1}=G_{2}=0$), and show that the
Chern insulator provides the desired gapless fermion energy spectrum (say, a
left-moving Weyl fermion on the edge A and a right-moving Weyl fermion on the
edge B, and totally a Dirac fermion for the combined). We adopt the chiral
$\pi$-flux square lattice modelWen:1990fv in Fig.2 as an example. This
lattice model can be regarded as a free theory of 3-5-4-0 fermions of Eq.(2)
with its mirrored conjugate. We will explicitly show filling the first Chern
numberThouless:1982zz $C_{1}=-1$ band of the lattice on a cylinder would give
the edge states of a free fermion with U(1) charge $3$, similar four copies of
model together render 3-5-4-0 free fermions theory of Eq.(II).
Figure 2: Chiral $\pi$-flux square lattice: (a) A unit cell is indicated as
the shaded darker region, containing two sublattice as a black dot $a$ and a
white dot $b$. The lattice Hamiltonian has hopping constants,
$t_{1}e^{i\pi/4}$ along the black arrow direction, $t_{2}$ along dashed brown
links, $-t_{2}$ along dotted brown links. (b) Put the lattice on a cylinder.
(c) The ladder: the lattice on a cylinder with a square lattice width. The
chirality of edge state is along the direction of blue arrows.
We design hopping constants $t_{ij,3}=t_{1}e^{\text{i}\pi/4}$ along the black
arrow direction in Fig.2, and its hermitian conjugate determines
$t_{ij,3}=t_{1}e^{-\text{i}\pi/4}$ along the opposite hopping direction;
$t^{\prime}_{ij,3}=t_{2}$ along dashed brown links, $t^{\prime}_{ij,3}=-t_{2}$
along dotted brown links. The shaded blue region in Fig.2 indicates a unit
cell, containing two sublattice as a black dot $a$ and a white dot $b$. If we
put the lattice model on a torus with periodic boundary conditions for both
$x,y$ directions, then we can write the Hamiltonian in
$\mathbf{k}=(k_{x},k_{y})$ space in Brillouin zone (BZ), as
$H=\sum_{\mathbf{k}}f^{\dagger}_{\mathbf{k}}H(\mathbf{k})f_{\mathbf{k}}$,
where $f_{\mathbf{k}}=(f_{a,\mathbf{k}},f_{b,\mathbf{k}})$. For two sublattice
$a,b$, we have a generic pseudospin form of Hamiltonian $H(\mathbf{k})$,
$H(\mathbf{k})=B_{0}(\mathbf{k})+\vec{B}(\mathbf{k})\cdot\vec{\sigma}.$ (13)
$\vec{\sigma}$ are Pauli matrices $(\sigma_{x},\sigma_{y},\sigma_{z})$. In
this model $B_{0}(\mathbf{k})=0$ and
$\vec{B}=(B_{x}(\mathbf{k}),B_{y}(\mathbf{k}),B_{z}(\mathbf{k}))$ have three
components in terms of $\mathbf{k}$ and lattice constants $a_{x},a_{y}$. The
eigenenergy $\mathop{\mathrm{E}}_{\pm}$ of $H(\mathbf{k})$ provide two nearly-
flat energy bands, shown in Fig.3, from
$H(\mathbf{k})|\psi_{\pm}(\mathbf{k})\rangle=\mathop{\mathrm{E}}_{\pm}|\psi_{\pm}(\mathbf{k})\rangle$.
For the later purpose to have the least mixing between edge states on the left
edge A and right edge B on a cylinder in Fig.2(b), here we fine tune
$t_{2}/t_{1}=1/2$. For convenience, we simply set $t_{1}=1$ as the order
magnitude of $\mathop{\mathrm{E}}_{\pm}$. We set lattice constants
$a_{x}=1/2,a_{y}=1$ such that BZ has $-\pi\leq k_{x}<\pi,-\pi\leq k_{y}<\pi$.
The first Chern numberThouless:1982zz of the energy band
$|\psi_{\pm}(\mathbf{k})\rangle$ is
$C_{1}=\frac{1}{2\pi}\int_{\mathbf{k}\in\text{BZ}}d^{2}\mathbf{k}\;\epsilon^{\mu\nu}\partial_{k_{\mu}}\langle\psi(\mathbf{k})|-i\partial_{k_{\nu}}|\psi(\mathbf{k})\rangle.$
(14)
We find $C_{1,\pm}=\pm 1$ for two bands. The $C_{1,-}=-1$ lower energy band
indicates the clockwise chirality of edge states when we put the lattice on a
cylinder as in Fig.2(b). Overall it implies the chirality of the edge state on
the left edge A moving along $-\hat{x}$ direction, and on the right edge B
moving along $+\hat{x}$ direction \- the clockwise chirality as in Fig.2(b),
consistent with the earlier result $C_{1,-}=-1$ of Chern number. This edge
chirality is demonstrated in Fig.4. Details are explained in its captions and
in Appendix F.1.
Figure 3: Two nearly-flat energy bands $\mathop{\mathrm{E}}_{\pm}$ in
Brillouin zone for the kinetic hopping terms of our model Eq.(II).
The above construction is for edge states of free fermion with U(1) charge $3$
of 3L-5R-4L-0R fermion model. Add the same copy with $C_{1,-}=-1$ lower band
gives another layer of U(1) charge $4$ free fermion. For another layers of
U(1) charge $5$ and $0$, we simply adjust hopping constant $t_{ij}$ to
$t_{1}e^{-\text{i}\pi/4}$ along the black arrow direction and
$t_{1}e^{\text{i}\pi/4}$ along the opposite direction in Fig.2, which makes
$C_{1,-}=+1$. Stack four copies of chiral $\pi$-flux ladders with
$C_{1,-}=-1,+1,-1,+1$ provides the lattice model of 3-5-4-0 free fermions with
its mirrored conjugate.
The lattice model so far is an effective 1+1D non-chiral theory. We claim the
interaction terms ($G_{1},G_{2}\neq 0$) can gap out the mirrored edge states
on the edge B. The simulation including interactions can be numerically
expansive, even so on a simple ladder model. Because of higher power
interactions, one can no longer diagonalize the model in $\mathbf{k}$ space as
the case of the quadratic free-fermion Hamiltonian. For interacting case, one
may need to apply exact diagonalization in real space, or density matrix
renormalization group (DMRGWhite:1992zz ), which is powerful in 1+1D. We leave
this interacting numerical study for the lattice community or the future work.
(a) (b) (c)
Figure 4: The energy spectrum $\mathop{\mathrm{E}}(k_{x})$ and the density
matrix $\langle f^{\dagger}f\rangle$ of the chiral $\pi$-flux model on a
cylinder: (a) On a 10-sites width ($9a_{y}$-width) cylinder: The blue curves
are edge states spectrum. The black curves are for states extending in the
bulk. The chemical potential at zero energy fills eigenstates in solid curves,
and leaves eigenstates in dashed curves unfilled. (b) On the ladder, a 2-sites
width ($1a_{y}$-width) cylinder: the same as the (a)’s convention. (c) The
density $\langle f^{\dagger}f\rangle$ of the edge eigenstates (the solid blue
curve in (b)) on the ladder lattice. The dotted blue curve shows the total
density sums to 1, the darker purple curve shows $\langle
f_{\mathop{\mathrm{A}}}^{\dagger}f_{\mathop{\mathrm{A}}}\rangle$ on the left
edge A, and the lighter purple curve shows $\langle
f_{\mathop{\mathrm{B}}}^{\dagger}f_{\mathop{\mathrm{B}}}\rangle$ on the right
edge B. The dotted darker(or lighter) purple curve shows density $\langle
f_{\mathop{\mathrm{A}},a}^{\dagger}f_{\mathop{\mathrm{A}},a}\rangle$ (or
$\langle f_{\mathop{\mathrm{B}},a}^{\dagger}f_{\mathop{\mathrm{B}},a}\rangle$)
on sublattice $a$, while the dashed darker(or lighter) purple curve shows
density $\langle
f_{\mathop{\mathrm{A}},b}^{\dagger}f_{\mathop{\mathrm{A}},b}\rangle$ (or
$\langle f_{\mathop{\mathrm{B}},b}^{\dagger}f_{\mathop{\mathrm{B}},b}\rangle$)
on sublattice $b$. This edge eigenstate has the left edge A density with
majority quantum number $k_{x}<0$, and has the right edge B density with
majority quantum number $k_{x}>0$. Densities on two sublattice $a,b$ are
equally distributed as we desire.
### III.2 Interaction gapping terms and the strong coupling scale
Similar to Sec.III.1.1, for the interaction gapping terms of the Hamiltonian,
we can do the mapping based on Eq.(12), where the leading terms on the lattice
is
$\displaystyle g_{a}\cos(\ell_{a,I}\cdot\Phi_{I})$ (15)
$\displaystyle=U_{\text{interaction}}\big{(}\tilde{\psi}_{q},\dots,\nabla^{n}_{x}\tilde{\psi}_{q},\dots\big{)})$
$\displaystyle\to
U_{\text{point.split.}}\bigg{(}\hat{f}_{q}(j),\dots\big{(}\hat{f}^{n}_{q}(j)\big{)}_{pt.s.},\dots\bigg{)}$
$\displaystyle+\alpha_{\text{small}}\dots$
Again, potentially there may contain subleading pieces, such as further higher
order derivatives $\alpha_{\text{small}}\nabla^{n}_{x}\psi_{q}$ with a small
coefficient $\alpha_{\text{small}}$, or tiny mixing of the different
U(1)-charge flavors
$\alpha_{\text{small}}^{\prime}{\psi_{q_{1}}\psi_{q_{2}}\dots}$. However,
using the same RG analysis in Sec.III.1.1, at both the weak coupling and the
strong coupling fix points, we learn that those $\alpha_{\text{small}}$ terms
are only subleading-perturbative effects which are _further irrelevant
perturbation_ at the infrared comparing to the dominant piece (which is the
kinetic term for the weak $g$ coupling, but is replaced by the cosine term for
the strong $g$ coupling).
One more question to ask is: what is the scale of coupling $G$ such that the
gapping term becomes dominant and the B edge states form the mass gaps, but
maintaining (without interfering with) the gapless A edge states?
To answer this question, we first know the absolute value of energy magnitude
for each term in the desired Hamiltonian for our chiral fermion model:
$|G\text{ gapping term}|\gtrsim|t_{ij},t_{ij}^{\prime}\text{ kinetic
term}|\gg|G\text{ higher order $\nabla^{n}_{x}$ and mixing
terms}|\gg|t_{ij},t_{ij}^{\prime}\text{ higher order
}\psi_{q}\dots\nabla^{n}_{x}\psi_{q}|.$ (16)
For field theory, the gapping terms (the cosine potential term or the multi-
fermion interactions) are irrelevant for a weak $g$ coupling, this implies
that $g$ needs to be large enough. Here the $g\equiv(g_{a})/a^{2}$ really
means the dimensionless quantity $g_{a}$.
For lattice model, however, the dimensional analysis is very different. Since
the $G$ coupling of gapping terms and the hopping amplitude $t_{ij}$ both have
dimension of energy $[G]=[t_{ij}]=1$, this means that the scale of the
dimensionless quantity of $|G|/|t_{ij}|$ is important. (The
$|t_{ij}|,|t_{ij}^{\prime}|$ are about the same order of magnitude.)
Presumably we can design the lattice model under Eq.(10), $a<\xi<w<L$, such
that their ratios between each length scale are about the same. We expect the
ratio of couplings of ${|G|}$ to ${|t_{ij}|}$ is about the ratio of mass gap
${\Delta_{m}}$ to kinetic energy fluctuation ${\delta E_{k}}$ caused by
$t_{ij}$ hopping, thus _very roughly_
$\frac{|G|}{|t_{ij}|}\sim\frac{\Delta_{m}}{\delta
E_{k}}\sim\frac{(\xi)^{-1}}{(w)^{-1}}\sim\frac{w}{\xi}\sim\frac{L}{w}\sim\frac{\xi}{a}.$
(17)
We expect that the scales at strong coupling $G$ is about
$|G|\gtrsim|{t_{ij}}|\cdot\frac{\xi}{a}$ (18)
this magnitude can support our lattice chiral fermion model with mirror-
fermion decoupling. If $G$ is too much smaller than
$|{t_{ij}}|\cdot\frac{\xi}{a}$, then mirror sector stays gapless. On the other
hand, if $|G|/|{t_{ij}}|$ is too much stronger or simply
$|G|/|{t_{ij}}|\to\infty$ may cause either of two disastrous cases:
(i) Both edges would be gapped and the whole 2D plane becomes _dead without
kinetic hopping_ , if the correlation length reaches the scale of the cylinder
width: $\xi\gtrsim w$.
(ii) The B edge(say at site $n\hat{y}$) becomes completely gapped, but forms a
dead overly-high-energy 1D line decoupled from the remain lattice. The
neighbored line (along $(n-1)\hat{y}$) next to edge B experiences no
interaction thus may still form mirror gapless states near B. (This may be
another reason why CGP fails in Ref.Chen et al., 2013a due to implementing
overlarge strong coupling.)
So either the two cases caused by too much strong $|G|/|{t_{ij}}|$ is not
favorable. Only $|G|\gtrsim|{t_{ij}}|\cdot\frac{\xi}{a}$, we can have the
mirrored sector at edge $B$ gapped, meanwhile keep the chiral sector at edge
$A$ gapless. $\frac{|G|}{|{t_{ij}}|}$ is somehow larger than order 1 is what
we referred as the intermediate(-strong) coupling.
$\frac{|G|}{|{t_{ij}}|}\gtrsim O(1).$ (19)
(Our $O(1)$ means some finite values, possibly as large as $10^{4},10^{6}$,
etc, but still finite. And the kinetic term is _not_ negligible.) The sign of
$G$ coupling shall not matter, since in the cosine potential language, either
$g_{1},g_{2}$ greater or smaller than zero are related by sifting the minimum
energy vaccua of the cosine potential.
To summarize, the two key messages in Sec.III are:
$\bullet$ First, the free-kinetic hopping part of lattice model has been
simulated and there gapless energy spectra have been computed shown in
Figures. The energy spectra indeed show the gapless Weyl fermions on each
edge. So, the continuum field theory to a lattice model mapping is immaterial
to the subleading terms of Eq.(12), the physics is as good or as exact as we
expect for the free kinetic part. We comment that this lattice realization of
quantum hall-like states with chiral edges have been implemented for long in
condensed matter, dated back as early such as Haldane’s work.Haldane:1988zza
$\bullet$ Second, by adding the interaction gapping terms, the spectra will be
modified from the mirror gapless edge to the mirror gapped edge. The continuum
field theory to a lattice model mapping based on Eq.(12) for the _gapping
terms_ in Eq.(15) is as good or as exact as the _free kinetic part_ Eq.(11),
because the mapping is the same procedure as in Eq.(12). Since the subleading
correction for the free and for the interacting parts are _further irrelevant
perturbation_ at the infrared, the non-perturbative topological effect of the
gapped edge contributed from the leading terms remains.
black
In the next section, we will provide a topological non-perturbative proof to
justify that the $G_{1},G_{2}$ interaction terms can gap out mirrored edge
states, without employing numerical methods, but purely based on an analytical
derivation.
## IV Topological Non-Perturbative Proof of Anomaly Matching Conditions =
Boundary Fully Gapping Rules
As Sec.II,III prelude, we now show that Eq.(II) indeed gaps out the mirrored
edge states on the edge B in Fig.1. This proof will support the evidence that
Eq.(II) gives the non-perturbative lattice definition of the 1+1D chiral
fermion theory of Eq.(2).
In Sec.IV.1, we first provide a generic way to formulate our model, with a
insulating bulk but with gapless edge states. This can be done through so
called the bulk-edge correspondence, namely the Chern-Simons theory in the
bulk and the Wess-Zumino-Witten(WZW) model on the boundary. More specifically,
for our case with U(1) symmetry chiral matter theory, we only needs a U(1)N
rank-$N$ Abelian K matrix Chern-Simons theory in the bulk and the multiplet
chiral boson theory on the boundary. We can further fermionize the multiplet
chiral boson theory to the multiplet chiral fermion theory.
In Sec.IV.2, we provide a physical understanding between the anomaly matching
conditions and the effective Hall conductance. This intuition will be helpful
to understand the relation between the anomaly matching conditions and
Boundary Fully Gapping Rules, to be discussed in Sec.IV.3.
### IV.1 Bulk-Edge Correspondence - 2+1D Bulk Abelian SPT by Chern-Simons
theory
With our 3L-5R-4L-0R chiral fermion model in mind, below we will trace back to
fill in the background how we obtain this model from the understanding of
symmetry-protected topological states (SPT). This understanding in the end
leads to a more general construction.
We first notice that the bosonized action of the free part of chiral fermions
in Eq.(4), can be regarded as the edge states action $S_{\partial}$ of a bulk
U(1)N Abelian K matrix Chern-Simons theory $S_{bulk}$ (on a 2+1D manifold
${\mathcal{M}}$ with the 1+1D boundary ${\partial\mathcal{M}}$):
$\displaystyle S_{bulk}$ $\displaystyle=$
$\displaystyle\frac{K_{IJ}}{4\pi}\int_{\mathcal{M}}a_{I}\wedge
da_{J}=\frac{K_{IJ}}{4\pi}\int_{\mathcal{M}}dt\,d^{2}x\varepsilon^{\mu\nu\rho}a^{I}_{\mu}\partial_{\nu}a^{J}_{\rho},\;\;\;\;\;\;$
(20) $\displaystyle S_{\partial}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int_{\partial\mathcal{M}}dt\;dx\;K_{IJ}\partial_{t}\Phi_{I}\partial_{x}\Phi_{J}-V_{IJ}\partial_{x}\Phi_{I}\partial_{x}\Phi_{J}.\;\;\;\;\;\;\;$
(21)
Here $a_{\mu}$ is intrinsic 1-form gauge field from a low energy viewpoint.
Both indices $I,J$ run from $1$ to $N$. Given $K_{IJ}$ matrix, it is known the
ground state degeneracy (GSD) of this theory on the $\mathbb{T}^{2}$ torus is
$\mathop{\mathrm{GSD}}=|\det K|$.Wang:2012am ; Wen:1992uk $V_{IJ}$ is the
symmetric ‘velocity’ matrix, we can simply choose $V_{IJ}=\mathbb{I}$, without
losing generality of our argument. The U(1)N gauge transformation is $a_{I}\to
a_{I}+df_{I}$ and $\Phi_{I}\to\Phi_{I}+f_{I}$. The bulk-edge correspondence is
meant to have the gauge non-invariances of the bulk-only and the edge-only
cancel with each other, so that the total gauge invariances is achieved from
the full bulk and edge as a whole.
We will consider only an even integer $N\in 2\mathbb{Z}^{+}$. The reason is
that only such even number of edge modes, we can potentially gap out the edge
states. (For odd integer $N$, such a set of gapping interaction terms
generically _do not_ exist, so the mirror edge states remain gapless.)
To formulate 3L-5R-4L-0R fermion model, as shown in Eq.(4), we need a rank-4 K
matrix $\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}$. Generically, for a general U(1) chiral
fermion model, we can use a canonical fermionic matrix
$K^{f}_{N\times N}=\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\dots$ (22)
Such a matrix is special, because it describes a more-restricted Abelian
Chern-Simons theory with GSD$=|\det K^{f}_{N\times N}|=1$ on the
$\mathbb{T}^{2}$ torus. In the condensed matter language, the uniques GSD
implies it has no long range entanglement, and it has no intrinsic topological
order. Such a state may be wronged to be only a trivial insulator, but
actually this is recently-known to be potentially nontrivial as the symmetry-
protected topological states (SPT).
(This paragraph is for readers with interests in SPT: SPT are short-range
entangled states with onsite symmetry in the bulk.Chen:2011pg For SPT, there
is no long-range entanglement, no fractionalized quasiparticles (fractional
anyons) and no fractional statistics in the bulk.Chen:2011pg The bulk onsite
symmetry may be realized as a non-onsite symmetry on the boundary. If one
gauges the non-onsite symmetry of the boundary SPT, the boundary theory
becomes an anomalous gauge theory.Wen:2013ppa The anomalous gauge theory is
ill-defined in its own dimension, but can be defined as the boundary of the
bulk SPT. However, this understanding indicates that if the boundary theory
happens to be anomaly-free, then it can be defined non-perturbatively on the
same dimensional lattice.)
$K^{f}_{N\times N}$ matrix describe fermionic SPT states, which is described
by bulk _spin Chern-Simons theory_ of $|\det K|=1$. A spin Chern-Simons theory
only exist on the spin manifold, which has spin structure and can further
define spinor bundles.Belov:2005ze However, there are another simpler class
of SPT states, the bosonic SPT states, which is described by the canonical
form $K^{b\pm}_{N\times N}$Wang:2012am ; canonical ; Ye:2013upa with blocks
of $\bigl{(}{\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}}\bigl{)}$ and a
set of all positive(or negative) coefficients $\mathop{\mathrm{E}}_{8}$
lattices $K_{\mathop{\mathrm{E}}_{8}}$,Wang:2012am ; canonical ; Lu:2012dt ;
Plamadeala:2013zva namely,
$\displaystyle K^{b0}_{N\times N}$ $\displaystyle=$
$\displaystyle\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}\oplus\dots.$ (23) $\displaystyle
K^{b\pm}_{N\times N}$ $\displaystyle=$ $\displaystyle K^{b0}\oplus(\pm
K_{\mathop{\mathrm{E}}_{8}})\oplus(\pm
K_{\mathop{\mathrm{E}}_{8}})\oplus\dots$
The $K_{\mathop{\mathrm{E}}_{8}}$ matrix describe 8-multiplet chiral bosons
moving in the same direction, thus it cannot be gapped by adding multi-fermion
interaction among themselves. We will neglect $\mathop{\mathrm{E}}_{8}$ chiral
boson states but only focus on $K^{b0}_{N\times N}$ for the reason to consider
_only the gappable states_. The K-matrix form of Eq.(22),(23) is called the
_unimodular indefinite symmetric integral matrix_.
After fermionizing the boundary action Eq.(21) with $K^{f}_{N\times N}$
matrix, we obtain multiplet chiral fermions (with several pairs, each pair
contain left-right moving Weyl fermions forming a Dirac fermion).
$\displaystyle S_{\Psi}$
$\displaystyle=\int_{\partial\mathcal{M}}dt\;dx\;(\text{i}\bar{\Psi}_{\mathop{\mathrm{A}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{A}}}).$
(24)
with $\Gamma^{0}=\underset{j=1}{\overset{N/2}{\bigoplus}}\gamma^{0}$,
$\Gamma^{1}=\underset{j=1}{\overset{N/2}{\bigoplus}}\gamma^{1}$,
$\Gamma^{5}\equiv\Gamma^{0}\Gamma^{1}$,
$\bar{\Psi}_{i}\equiv\Psi_{i}\Gamma^{0}$ and $\gamma^{0}=\sigma_{x}$,
$\gamma^{1}=\text{i}\sigma_{y}$,
$\gamma^{5}\equiv\gamma^{0}\gamma^{1}=-\sigma_{z}$.
Symmetry transformation for the edge states-
The edge states of $K^{f}_{N\times N}$ and $K^{b0}_{N\times N}$ Chern-Simons
theory are non-chiral in the sense there are equal number of left and right
moving modes. However, we can make them with a charged ‘chirality’ respect to
a global(or external probed, or dynamical gauge) symmetry group. For the
purpose to build up our ‘chiral fermions and chiral bosons’ model with ‘charge
chirality,’ we consider the simplest possibility to couple it to a global U(1)
symmetry with a charge vector $\mathbf{t}$. (This is the same as the symmetry
charge vector of SPT statesLu:2012dt ; Ye:2013upa ; Hung:2013nla )
Chiral Bosons: For the case of multiplet chiral boson theory of Eq.(21), the
group element $g_{\theta}$ of U(1) symmetry acts on chiral fields as
$\displaystyle g_{\theta}:W^{\text{U}(1)_{\theta}}=\mathbb{I}_{N\times
N},\;\;\delta\phi^{\text{U}(1)_{\theta}}=\theta\mathbf{t},$ (25)
With the following symmetry transformation,
$\displaystyle\phi\to
W^{\text{U}(1)_{\theta}}\phi+\delta\phi^{\text{U}(1)_{\theta}}=\phi+\theta\mathbf{t}$
(26)
To derive this boundary symmetry transformation from the bulk Chern-Simons
theory via bulk-edge correspondence, we first write down the charge coupling
bulk Lagrangian term, namely
$\frac{\mathbf{q}^{I}}{2\pi}\;\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}a^{I}_{\rho}$,
where the global symmetry current
${\mathbf{q}^{I}}J^{I\mu}=\frac{\mathbf{q}^{I}}{2\pi}\;\epsilon^{\mu\nu\rho}\partial_{\nu}a^{I}_{\rho}$
is coupled to an external gauge field $A_{\mu}$. The bulk U(1)-symmetry
current ${\mathbf{q}^{I}}J^{I\mu}$ induces a boundary U(1)-symmetry current
$j^{I\mu}=\frac{\mathbf{q}^{I}}{2\pi}\;\epsilon^{\mu\nu}\partial_{\nu}\phi_{I}$.
This implies the boundary symmetry operator is
$S_{sym}=\exp(\text{i}\,\theta\,\frac{\mathbf{q}^{I}}{2\pi}\int\partial_{x}\phi_{I})$,
with an arbitrary U(1) angle $\theta$ The induced symmetry transformation on
$\phi_{I}$ is:
$\displaystyle(S_{sym})\phi_{I}(S_{sym})^{-1}=\phi_{I}-\text{i}\theta\int
dx\frac{\mathbf{q}^{l}}{2\pi}[\phi_{I},\partial_{x}\phi_{l}]$
$\displaystyle=\phi_{I}+\theta(K^{-1})_{Il}{\mathbf{q}^{l}}\equiv\phi_{I}+\theta\mathbf{t}_{I},$
(27)
here we have used the canonical commutation relation
$[\phi_{I},\partial_{x}\phi_{l}]=\text{i}\,(K^{-1})_{Il}$. Compare the two
Eq.(26),(IV.1), we learn that
$\mathbf{t}_{I}\equiv(K^{-1})_{Il}{\mathbf{q}^{l}}.$
The charge vectors $\mathbf{t}_{I}$ and ${\mathbf{q}^{l}}$ are related by an
inverse of the $K$ matrix. The generic interacting or gapping termsWang:2012am
; Levin:2013gaa ; Lu:2012dt for the multiplet chiral boson theory are the
sine-Gordon or the cosine term
$S_{\partial,\text{gap}}=\int
dt\;dx\;\sum_{a}g_{a}\cos(\ell_{a,I}\cdot\Phi_{I}).$ (28)
If we insist that $S_{\partial,\text{gap}}$ obeys U(1) symmetry, to make
Eq.(28) invariant under Eq.(IV.1), we have to impose
$\displaystyle\ell_{a,I}\cdot\Phi_{I}\to\ell_{a,I}\cdot(\Phi_{I}+\delta\phi^{\text{U}(1)_{\theta}})\text{mod}\;2\pi$
$\displaystyle\text{so}\;\;\;\boxed{\ell_{a,I}\cdot\mathbf{t}_{I}=0}$ (29)
$\displaystyle\Rightarrow\boxed{\ell_{a,I}\cdot(K^{-1})_{Il}\cdot{\mathbf{q}^{l}}=0}.$
(30)
The above generic U(1) symmetry transformation works for bosonic
$K^{b0}_{N\times N}$ as well as fermionic $K^{f}_{N\times N}$.
Chiral Fermions: In the case of fermionic $K^{f}_{N\times N}$, we will do one
more step to fermionize the multiplet chiral boson theory. Fermionize the free
kinetic part from Eq.(21) to Eq.(24), as well as the interacting cosine term:
$\displaystyle g_{a}\cos(\ell_{a,I}\cdot\Phi_{I})$
$\displaystyle\to\prod_{I=1}^{N}\tilde{g}_{a}\big{(}({\psi}_{q_{I}})(\nabla_{x}{\psi}_{q_{I}})\dots(\nabla_{x}^{|\ell_{a,I}|-1}{\psi}_{q_{I}})\big{)}^{\epsilon}$
$\displaystyle\equiv
U_{\text{interaction}}\big{(}{\psi}_{q},\dots,\nabla^{n}_{x}{\psi}_{q},\dots\big{)}$
(31)
to multi-fermion interaction. The ${\epsilon}$ is defined as the complex
conjugation operator which depends on ${\text{sgn}(\ell_{a,I})}$, the sign of
$\ell_{a,I}$. When ${\text{sgn}(\ell_{a,I})}=-1$, we define
${\psi}^{\epsilon}\equiv{\psi}^{\dagger}$ and also for the higher power
polynomial terms. Again, we absorb the normalization factor and the Klein
factors through normal ordering of bosonization into the factor
$\tilde{g}_{a}$. The precise factor is not of our concern, since our goal is a
non-perturbative lattice model. Obviously, the U(1) symmetry transformation
for fermions is
${\psi}_{q_{I}}\to{\psi}_{q_{I}}e^{\text{i}\mathbf{t}_{I}\theta}={\psi}_{q_{I}}e^{\text{i}(K^{-1})_{Il}\cdot{\mathbf{q}^{l}}.\theta}$
(32)
In summary, we have shown a framework to describe U(1) symmetry chiral
fermion/boson model using the bulk-edge correspondence, the explicit Chern-
Siomns/WZW actions are given in Eq.(20),(21),(24),(28),(IV.1), and their
symmetry realization Eq.(IV.1),(32) and constrain are given in Eq.(29),(30).
Their physical properties are tightly associated to the fermionic/bosonic SPT
states.
black
### IV.2 Anomaly Matching Conditions and Effective Hall Conductance
The bulk-edge correspondence is meant, not only to achieve the gauge
invariance by canceling the non-invariance of bulk-only and boundary-only, but
also to have the boundary anomalous current flow can be transported into the
extra dimensional bulk. This is known as Callan-Harvey effectCallan_Harvey in
high energy physics, Laughlin thought experiment,Laughlin:1981jd or simply
the quantum-hall-like state bulk-edge correspondence in condensed matter
theory.
The goal of this subsection is to provide a concrete physical understanding of
the anomaly matching conditions and effective Hall conductance :
$\bullet$ (i) The anomalous current inflowing from the boundary is transported
into the bulk. We now show that this thinking can easily derive the 1+1D U(1)
Adler-Bell-Jackiw(ABJ) anomaly, or Schwinger’s 1+1D quantum
electrodynamics(QED) anomaly.
We will focus on the U(1) chiral anomaly, which is ABJ anomalyAdler:1969gk ;
Bell:1969ts type. It is well-known that ABJ anomaly can be captured by the
anomaly factor $\mathcal{A}$ of the 1-loop polygon Feynman diagrams (see
Fig.5). The anomaly matching condition requires
$\mathcal{A}=\mathop{\mathrm{tr}}[T^{a}T^{b}T^{c}\dots]=0.$ (33)
Here $T^{a}$ is the (fundamental) representation of the global or gauge
symmetry algebra, which contributes to the vertices of 1-loop polygon Feynman
diagrams.
For example, the 3+1D chiral anomaly 1-loop triangle diagram of U(1) symmetry
in Fig.5(a) with chiral fermions on the loop gives
$\mathcal{A}=\sum(q_{L}^{3}-q_{R}^{3})$. Similarly, the 1+1D chiral anomaly
1-loop diagram of U(1) symmetry in Fig.5(b) with chiral fermions on the loop
gives $\mathcal{A}=\sum(q_{L}^{2}-q_{R}^{2})$. Here $L,R$ stand for left-
moving and right-moving modes.
(a) (b)
Figure 5: Feynman diagrams with solid lines representing chiral fermions and
wavy lines representing U(1) gauge bosons: (a) 3+1D chiral fermionic anomaly
shows $\mathcal{A}=\sum_{q}(q_{L}^{3}-q_{R}^{3})$ (b) 1+1D chiral fermionic
anomaly shows $\mathcal{A}=\sum_{q}(q_{L}^{2}-q_{R}^{2})$ Figure 6: A physical
picture illustrates how the anomalous current $J$ of the boundary theory along
$x$ direction leaks to the extended bulk system along $y$ direction. Laughlin
flux insertion $\Phi_{B}=-\oint E\cdot dL$ induces the electric $E_{x}$ field
along the $x$ direction. The effective Hall effect shows
$J_{y}=\sigma_{xy}E_{x}=\sigma_{xy}\varepsilon^{\mu\nu}\,\partial_{\mu}A_{\nu}$,
with the effective Hall conductance $\sigma_{xy}$ probed by an external U(1)
gauge field $A$. The anomaly-free condition implies no anomalous bulk current,
so $J_{y}=0$ for any flux $\Phi_{B}$ or any $E_{x}$, thus we derive the
anomaly-free condition must be $\sigma_{xy}=0$.
How to derive this anomaly matching condition from a condensed matter theory
viewpoint? Conceptually, we understand that
A $d$-dimensional anomaly free theory (which satisfies the anomaly matching
condition) means that there is no anomalous current leaking from its
$d$-dimensional spacetime (as the boundary) to an extended bulk theory of
$d+1$-dimension.
More precisely, for an 1+1D U(1) anomalous theory realization of the above
statement, we can formulate it as the boundary of a 2+1D bulk as in Fig.6 with
a Chern-Simons action ($S=\int\big{(}\frac{K}{4\pi}\;a\wedge
da+\frac{q}{2\pi}A\wedge da$)). Here the field strength $F=dA$ is equivalent
to the external U(1) flux in the Laughlin’s flux-insertion thought
experimentLaughlin:1981jd threading through the cylinder (see a precise
derivation in the Appendix of Ref.Santos:2013uda, ). Without losing
generality, let us first focus on the boundary action of Eq.(21) as a chiral
boson theory with only one edge mode. We derive its equations of motion as
$\displaystyle\partial_{\mu}\,j_{\textrm{b}}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{xy}}{2}\,\varepsilon^{\mu\nu}\,F_{\mu\nu}={\sigma_{xy}}\,\varepsilon^{\mu\nu}\,\partial_{\mu}A_{\nu}=J_{y},$
(34) $\displaystyle\partial_{\mu}\,j_{\textrm{L}}$ $\displaystyle=$
$\displaystyle\partial_{\mu}(\frac{q}{2\pi}\epsilon^{\mu\nu}\partial_{\nu}\Phi)=\partial_{\mu}(q\bar{\psi}\gamma^{\mu}P_{L}\psi)=+J_{y},\;\;\;$
(35) $\displaystyle\partial_{\mu}\,j_{\textrm{R}}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}(\frac{q}{2\pi}\epsilon^{\mu\nu}\partial_{\nu}\Phi)=\partial_{\mu}(q\bar{\psi}\gamma^{\mu}P_{R}\psi)=-J_{y}.\;\;\;$
(36)
Here we derive the Hall conductance, easily obtained from its definitive
relation $J_{y}={\sigma_{xy}}E_{x}$ in Eq.(34), asW
$\sigma_{xy}=qK^{-1}q/(2\pi).$
Here $j_{\textrm{b}}$ stands for the edge current, with a left-moving current
$j_{L}=j_{\textrm{b}}$ on one edge and a right-moving current
$j_{R}=-j_{\textrm{b}}$ on the other edge, as in Fig.6. We convert a compact
bosonic phase $\Phi$ to the fermion field $\psi$ by bosonization. We can
combine currents $j_{\textrm{L}}+j_{\textrm{R}}$ as the vector current
$j_{\textrm{V}}$, then find its U(1)V current conserved. We combine currents
$j_{\textrm{L}}-j_{\textrm{R}}$ as the axial current $j_{\textrm{A}}$, then we
obtain the famous ABJ U(1)A anomalous current in 1+1D (or Schwinger 1+1D QED
anomaly).
$\displaystyle\partial_{\mu}\,j_{\textrm{V}}^{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,(j_{\textrm{L}}^{\mu}+j_{\textrm{R}}^{\mu})=0,$
(37) $\displaystyle\partial_{\mu}\,j_{\textrm{A}}^{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\,(j_{\textrm{L}}^{\mu}-j_{\textrm{R}}^{\mu})=\sigma_{xy}\varepsilon^{\mu\nu}\,F_{\mu\nu}.$
(38)
This simply physical derivation shows that the equivalent boundary theory on
the left and right edges (living on the edge of a 2+1D U(1) Chern-Simons
theory) can combine to be a 1+1D anomalous world of Schwinger’s 1+1D QED.
In other words, when the anomaly-matching condition holds ($\mathcal{A}=0$),
then there is no anomalous leaking current into the extended bulk
theory,Callan_Harvey as in Fig.6, so no ‘effective Hall conductance’ for this
anomaly-free theory.Kao:1996ey
It is straightforward to generalize the above discussion to a rank-$N$ K
matrix Chern-Simons theory. It is easy to show that the Hall conductance in a
2+1D system for a generic $K$ matrix is (via
${\mathbf{q}_{l}}=K_{Il}\,\mathbf{t}_{I}$)
$\displaystyle\boxed{\sigma_{xy}=\frac{1}{2\pi}\mathbf{q}\cdot{K}^{-1}\cdot\mathbf{q}=\frac{1}{2\pi}\mathbf{t}\cdot{K}\cdot\mathbf{t}.}\;\;\;$
(39)
For a 2+1D fermionic system for $K^{f}$ matrix of Eq.(22),
$\displaystyle\sigma_{xy}=\frac{q^{2}}{2\pi}\mathbf{t}{(K^{f}_{N\times
N})}\mathbf{t}=\frac{1}{2\pi}\sum_{q}(q_{L}^{2}-q_{R}^{2})=\frac{1}{2\pi}\mathcal{A}.\;\;\;$
(40)
Remarkably, this physical picture demonstrates that we can reverse the logic,
starting from the ‘effective Hall conductance of the bulk system’ to derive
the anomaly factor from the relation
${\boxed{\mathcal{A}\;(\text{anomaly
factor})=2\pi\sigma_{xy}\;(\text{effective Hall conductance})}}$ (41)
And from the “no anomalous current in the bulk” means that “$\sigma_{xy}=0$”,
we can further understand “the anomaly matching condition
$\mathcal{A}=2\pi\sigma_{xy}=0$.”
For the U(1) symmetry case, we can explicitly derive the anomaly matching
condition for fermions and bosons:
Anomaly Matching Conditions for 1+1D chiral fermions with U(1) symmetry
$\mathcal{A}=2\pi\sigma_{xy}=q^{2}\mathbf{t}{(K^{f}_{N\times
N})}\mathbf{t}=\sum^{N/2}_{j=1}(q_{L,j}^{2}-q_{R,j}^{2})=0.$ (42)
Anomaly Matching Conditions for 1+1D chiral bosons with U(1) symetry
$\mathcal{A}=2\pi\sigma_{xy}=q^{2}\mathbf{t}{(K^{b0}_{N\times
N})}\mathbf{t}=\sum^{N/2}_{j=1}2q_{L,j}q_{R,j}=0.$ (43)
Here
$q\mathbf{t}\equiv(q_{L,1},q_{R,1},q_{L,2},q_{R,2},\dots,,q_{L,N/2},q_{R2,N/2})$.
(For a bosonic theory, we note that the bosonic charge for this theory is
described by non-chiral Luttinger liquids. One should identify the left and
right moving charge as $q_{L}^{\prime}\propto q_{L}+q_{R}$ and
$q_{R}^{\prime}\propto q_{L}-q_{R}$.)
### IV.3 Anomaly Matching Conditions and Boundary Fully Gapping Rules
This subsection is the main emphasis of our work, and we encourage the readers
paying extra attentions on the result presented here. We will first present a
heuristic physical argument on the rules that under what situations the
boundary states can be gapped, named as the Boundary Fully Gapping Rules. We
will then provide a _topological non-perturbative_ proof using the notion of
Lagrangian subgroup and the exact sequence, following our previous work
Ref.Wang:2012am, and the work in Ref.Kapustin:2013nva, ; Levin:2013gaa, . And
we will also provide _perturbative_ RG analysis, both for strong and weak
coupling analysis of cosine potential cases.
#### IV.3.1 a physical picture
Here is the physical intuition: To define a topological gapped boundary
conditions, it means that the energy spectrum of the edge states are gapped.
We require the gapped boundary to be stable against quantum fluctuations in
order to prevent it from flowing back to the gapless states. Such a gapped
boundary must take a stable classical values at the partition function of edge
states. From the bosonization techniques, we can map the multi-fermion
interactions to the cosine potential term $g_{a}\cos(\ell_{a}\cdot\Phi)$. From
the bulk-edge correspondence, we learn to regard the 1+1D chiral fermion/boson
theory as the edge states of a K matrix Chern-Simons theory, and further learn
that the $\ell_{a}$ vector is indeed a Wilson line operator of anyons [integer
anyons (fermions or bosons) for $\det(K)=1$ matrix (e.g. SPT states),
fractional anyons for $\det(K)>1$ (e.g. Topological Orders).] However, the
nontrivial _braiding statistics_ of anyons of $\ell_{a}$ vectors will cause
quantum fluctuations to the partition function (or the path integral)
$\mathbf{Z}_{statistics}\sim\exp[\text{i}\theta_{ab}]=\exp[\text{i}\,2\pi\,\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}].$
(44)
Here the Abelian braiding statistics angle can be derived from the effective
action between anyon vectors $\ell_{a},\ell_{b}$ by integrating out the
internal gauge field $a$ of the Chern-Simons action
$\int\big{(}\frac{1}{4\pi}K_{IJ}a_{I}\wedge
da_{J}+a\wedge*j(\ell_{a})+a\wedge*j(\ell_{b})\big{)}$. (See Fig.7). In order
to define a _classically-stable_ topological gapped boundary, we need to
stabilize the unwanted quantum fluctuations. We are forced to choose the
trivial statistics for the Wilson lines from the set of interaction terms
$g_{a}\cos(\ell_{a}\cdot\Phi)$. This requires the _trivial statistics_ rule
${\text{\bf{Rule}}\;\bf{(1)}}\;\;\;\;\;\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}=0,$
(45)
known as the Haldane null condition.h95
What else rules do we require? For a total $N$ edge modes, $N_{L}=N_{R}=N/2$
number of left/right moving free Weyl fermion modes, we need to have _at
least_ $N/2$ interaction terms to open the mass gap. This can be intuitively
understood as a pair of modes can be gapped together if it is a pair of one
left-moving to one right-moving mode. It turns out that if we include _more_
linear-independent interactions of $\ell_{a}$ than $N/2$ terms, such
$\ell_{a}$ cannot be compatible with the previous set of $N/2$ terms for a
compatible trivial mutual or self statistics $\theta_{ab}=0$. So we arrive the
Rule (2), “ _no more or no less than the exact $N/2$ interaction terms_.” And
implicitly, we must have the Rule (3), “ _$N_{L}=N_{R}=N/2$ number of
left/right moving modes_.”
So from this physical picture, we have the following rules in order to gap out
the edge states of Abelian K-matrix Chern-Simons theory:
Boundary Fully Gapping Rulesh95 ; Wang:2012am ; Levin:2013gaa ;
Barkeshli:2013jaa ; Lu:2012dt ; Hung:2013nla \- There exists a Lagrangian
subgroupLevin:2013gaa ; Barkeshli:2013jaa ; Kapustin:2010hk
$\Gamma^{\partial}\equiv\\{\sum_{a}c_{a}\ell_{a,I}|c_{a}\in\mathbb{Z}\\}$ (or
named as the boundary gapping latticeWang:2012am in $K_{N\times N}$ Abelian
Chern-Simons theory), such that giving a set of interaction terms as the
cosine potential terms $g_{a}\cos(\ell_{a}\cdot\Phi)$: (1)
$\forall\ell_{a},\ell_{b}\in\Gamma^{\partial}$, the self and mutual
statistical angles $\theta_{ab}$ are zeros among quasiparticles. Namely,
$\theta_{ab}\equiv 2\pi\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}=0.$ (46) (For $a=b$,
the self-statistical angle $\theta_{aa}/2=0$ is called the self-null
condition. And for $a\neq b$, the mutual-statistical angle $\theta_{ab}=0$ is
called the mutual-null conditions.h95 ) (2) The dimension of the lattice
$\Gamma^{\partial}$ is $N/2$, where $N$ must be an even integer. This means
the Chern-Simons lattice $\Gamma^{\partial}$ is spanned by $N/2$ linear
independent vectors of $\ell_{a}$. (3) The signature of K matrix (the number
of left moving modes $-$ the number of left moving modes) is zero. Namely
$N_{L}=N_{R}=N/2$. (4) ${\ell}_{a}\in\Gamma_{e}$, where $\Gamma_{e}$ is
composed by column vectors of K matrix, namely
$\Gamma_{e}=\\{\sum_{J}c_{J}K_{IJ}\mid c_{J}\in\mathbb{Z}\\}$. $\Gamma_{e}$ is
names as the non-fractionalized Chern-Simons lattice.Wang:2012am ; Wen:1992uk
; particle lattice
Figure 7: The braiding statistical angle $\theta_{ab}$ of two quasiparticles
$\ell_{a},\ell_{b}$, obtained from the phase gain $e^{i\theta_{ab}}$ in the
wavefunction by winding $\ell_{a}$ around $\ell_{b}$. Here the effective 2+1D
Chern-Simons action with the internal 1-form gauge field $a_{I}$ is
$\int\big{(}\frac{1}{4\pi}K_{IJ}a_{I}\wedge
da_{J}+a\wedge*j(\ell_{a})+a\wedge*j(\ell_{b})\big{)}$. One can integrate out
$a$ field to obtain the Hopf term, which coefficient as a self-statistical
angle $\ell_{a}$ is $\theta_{aa}/2\equiv\pi\ell_{a,I}K^{-1}_{IJ}\ell_{a,J}$
and the mutual-statistical angle between $\ell_{a},\ell_{b}$ is
$\theta_{ab}\equiv 2\pi\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}$.W
The Rule (4) is an extra rule, which is not of our main concern here. This
extra rule is for the ground state degeneracy(GSD) matching between the bulk
GSD and the boundary GSD while applying the cutting-glueing(or sewing)
relations, studied in Ref.Wang:2012am, . (Note that the bulk GSD is the
topological ground state degeneracy for a bulk closed manifold without
boundary, the boundary GSD is the topological GSD for a compact manifold with
gapped boundaries.) Since we have the unimodular indefinite symmetric integral
$K$ matrix of Eq.(22),(23), so Rule (4) is always true, for our chiral
fermion/boson models.
#### IV.3.2 topological non-perturbative proof
The above physical picture is suggestive, but not yet rigorous enough
mathematically. Here we will formulate some topological non-perturbative
proofs for Boundary Fully Gapping Rules, and its equivalence to the anomaly-
matching conditions for the case of U(1) symmetry. The first approach is using
the topological quantum field theory(TQFT) along the logic of
Ref.Kapustin:2010hk, . The new ingredient for us is to find _the equivalence
of the gapped boundary to the anomaly-matching conditions_. We intentionally
save the details in Appendix E, especially in E.5.
For a field theory, the boundary condition is defined by a Lagrangian
submanifold in the space of Cauchy boundary condition data on the boundary.
For a topological gapped boundary condition of a TQFT with a gauge group, we
must choose a Lagrangian subspace in the Lie algebra of the gauge group. A
subspace is Lagrangian _if and only if_ it is both isotropic and coisotropic.
Specifically, for $\mathbf{W}$ be a linear subspace of a finite-dimensional
vector space $\mathbf{V}$. Define the symplectic complement of $\mathbf{W}$ to
be the subspace $\mathbf{W}^{\perp}$ as
$\mathbf{W}^{\perp}=\\{v\in\mathbf{V}\mid\omega(v,w)=0,\;\;\;\forall
w\in\mathbf{W}\\}$ (47)
Here $\omega$ is the symplectic form, in the matrix form
$\omega=\begin{pmatrix}0&\mathbf{1}\\\ -\mathbf{1}&0\end{pmatrix}$ with $0$
and $\mathbf{1}$ are the block matrix of the zero and the identity. The
symplectic complement $\mathbf{W}^{\perp}$ satisfies:
$(\mathbf{W}^{\perp})^{\perp}=\mathbf{W}$,
$\dim\mathbf{W}+\dim\mathbf{W}^{\perp}=\dim\mathbf{V}$. We have:
$\bullet$ $\mathbf{W}$ is Lagrangian if and only if it is both isotropic and
coisotropic, namely, if and only if $\mathbf{W}=\mathbf{W}_{\perp}$. In a
finite-dimensional $\mathbf{V}$, a Lagrangian subspace $\mathbf{W}$ is an
isotropic one whose dimension is half that of $\mathbf{V}$.
Now let us focus on the K-matrix $\text{U(1)}^{N}$ Chern-Simons theory, the
symplectic form $\omega$ is given by (with the restricted $a_{\parallel,I}$ on
${\partial\mathcal{M}}$ )
$\omega=\frac{K_{IJ}}{4\pi}\int_{\mathcal{M}}(\delta a_{\parallel,I})\wedge
d(\delta a_{\parallel,J}).$ (48)
The bulk gauge group $\text{U(1)}^{N}\cong\mathbb{T}_{\Lambda}$ as the torus,
is the quotient space of $N$-dimensional vector space $\mathbf{V}$ by a
subgroup $\Lambda\cong\mathbb{Z}^{N}$. Locally the gauge field $a$ is a
1-form, which has values in the Lie algebra of $\mathbb{T}_{\Lambda}$, we can
denote this Lie algebra $\mathbf{t}_{\Lambda}$ as the vector space
$\mathbf{t}_{\Lambda}=\Lambda\otimes\mathbb{R}$.
Importantly, for _topological gapped boundary_ , $a_{\parallel,I}$ lies in a
Lagrangian subspace of $\mathbf{t}_{\Lambda}$ implies that the boundary gauge
group ($\equiv\mathbb{T}_{\Lambda_{0}}$) is a Lagrangian subgroup. We can
rephrase it in terms of the exact sequence for the vector space of Abelian
group $\Lambda\cong\mathbb{Z}^{N}$ and its subgroup $\Lambda_{0}$:
$\displaystyle
0\to\Lambda_{0}\overset{\mathbf{h}}{\to}\Lambda\to\Lambda/\Lambda_{0}\to 0.$
(49)
Here $0$ means the trivial zero-dimensional vector space and $\mathbf{h}$ is
an injective map from $\Lambda_{0}$ to $\Lambda$. We can also rephrase it in
terms of the exact sequence for the vector space of Lie algebra by
$0\to\mathbf{t}_{(\Lambda/\Lambda_{0})}^{*}\to\mathbf{t}_{\Lambda}^{*}\to\mathbf{t}_{\Lambda_{0}}^{*}\to
0$.
The generic Lagrangian subgroup condition applies to K-matrix with the above
symplectic form Eq.(48) renders three conditions on $\mathbf{W}$:
$\bullet(i)$ The subspace $\mathbf{W}$ is isotropic with respect to the
symmetric bilinear form $K$.
$\bullet(ii)$ The subspace dimension is a half of the dimension of
$\mathbf{t}_{\Lambda}$.
$\bullet(iii)$ The signature of $K$ is zero. This means that $K$ has the same
number of positive and negative eigenvalues.
Now we can examine the if and only if conditions
$\bullet(i)$,$\bullet(ii)$,$\bullet(iii)$ listed above.
For $\bullet(i)$ “The subspace is isotropic with respect to the symmetric
bilinear form $K$” to be true, we have an extra condition on the injective
${\mathbf{h}}$ matrix (${\mathbf{h}}$ with $N\times(N/2)$ components) for the
$K$ matrix:
$\displaystyle\boxed{{\mathbf{h}^{T}}K{\mathbf{h}}=0}.$ (50)
Since $K$ is invertible($\det(K)\neq 0$), by defining a
$N\times(N/2)$-component $\mathbf{L}\equiv K{\mathbf{h}}$, we have an
equivalent condition:
$\boxed{\mathbf{L}^{T}K^{-1}\mathbf{L}=0}.$ (51)
For $\bullet(ii)$, “the subspace dimension is a half of the dimension of
$\mathbf{t}_{\Lambda}$” is true if $\Lambda_{0}$ is a rank-$N/2$ integer
matrix.
For $\bullet(iii)$, “the signature of $K$ is zero” is true, because our
$K_{b0}$ and fermionic $K_{f}$ matrices implies that we have same number of
left moving modes ($N/2$) and right moving modes ($N/2$), with $N\in
2\mathbb{Z}^{+}$ an even number.
Lo and behold, these above conditions
$\bullet(i)$,$\bullet(ii)$,$\bullet(iii)$ are equivalent to the boundary full
gapping rules listed earlier. We can interpret $\bullet(i)$ as trivial
statistics by either writing in the column vector of ${\mathbf{h}}$ matrix
(${\mathbf{h}}\equiv\Big{(}\eta_{1},\eta_{2},\dots,\eta_{N/2}\Big{)}$ with
$N\times(N/2)$-components):
$\boxed{\eta_{a,I^{\prime}}K_{I^{\prime}J^{\prime}}\eta_{b,J^{\prime}}=0}.$
(52)
or writing in the column vector of ${\mathbf{L}}$ matrix
($\mathbf{L}\equiv\Big{(}\ell_{1},\ell_{2},\dots,\ell_{N/2}\Big{)}$ with
$N\times(N/2)$-components):
$\boxed{\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}=0}.$ (53)
for any
$\ell_{a},\ell_{b}\in\Gamma^{\partial}\equiv\\{\sum_{\alpha}c_{\alpha}\ell_{\alpha,I}|c_{\alpha}\in\mathbb{Z}\\}$
of boundary gapping lattice(Lagrangian subgroup). Namely,
The _boundary gapping lattice_ $\Gamma^{\partial}$ is basically the
$N/2$-dimensional vector space of a Chern-Simons lattice spanned by the
$N/2$-independent column vectors of $\mathbf{L}$ matrix
($\mathbf{L}\equiv\Big{(}\ell_{1},\ell_{2},\dots,\ell_{N/2}\Big{)}$).
Moreover, we can go a step further to relate the above rules equivalent to the
anomaly-matching conditions. By adding the corresponding cosine potential
$g_{a}\cos(\ell_{a}\cdot\Phi)$ to the edge states of U(1)N Chern-Simons
theory, we break the symmetry down to
$\text{U}(1)^{N}\to\text{U}(1)^{N/2}.$
What are the remained $\text{U}(1)^{N/2}$ symmetry? By Eq.(29), this remained
$\text{U}(1)^{N/2}$ symmetry is generated by a number of $N/2$ of
$\mathbf{t}_{b,I}$ vectors satisfying ${\ell_{a,I}\cdot\mathbf{t}_{b,I}=0}$.
We can easily construct
$\mathbf{t}_{b,I}\equiv K^{-1}_{IJ}\ell_{b,J},\;\;\;\mathbf{t}\equiv
K^{-1}\mathbf{L}$ (54)
with $N/2$ number of them (or define $\mathbf{t}$ as the linear-combination of
$\mathbf{t}_{b,I}\equiv\sum_{I^{\prime}}c_{II^{\prime}}(K^{-1}_{I^{\prime}J}\ell_{b,J})$).
It turns out that $\text{U}(1)^{N/2}$ symmetry is exactly generated by
$\mathbf{t}_{b,I}$ with $b=1,\dots,N/2$, and these remained unbroken symmetry
with $N/2$ of U(1) generators are anomaly-free and mixed anomaly-free, due to
$\boxed{\mathbf{t}_{a,I^{\prime}}K_{I^{\prime}J^{\prime}}\mathbf{t}_{b,J^{\prime}}={\ell_{a,I^{\prime}}K^{-1}_{I^{\prime}J^{\prime}}\ell_{b,J^{\prime}}}=0}.$
(55)
Indeed, $\mathbf{t}_{a}$ must be anomaly-free, because it is easily notice
that by defining an $N\times N/2$ matrix
$\mathbf{t}\equiv\Big{(}\mathbf{t}_{1},\mathbf{t}_{2},\dots,\mathbf{t}_{N/2}\Big{)}=\Big{(}\eta_{1},\eta_{2},\dots,\eta_{N/2}\Big{)}$
of Eq.(181), thus we must have:
$\displaystyle\boxed{{\mathbf{t}^{T}}K{\mathbf{t}}=0},\;\;\;\text{where
}\mathbf{t}=\mathbf{h}.$ (56)
This is exactly the anomaly factor and the effective Hall conductance
discussed in Sec.IV.2.
In summary of the above, we have provided a topological non-perturbative proof
that the Boundary Fully Gapping Rules (following Ref.Kapustin:2010hk, ), and
its extension to the equivalence relation to the anomaly-matching conditions.
We emphasize that Boundary Fully Gapping Rules provide a topological statement
on the gapped boundary conditions, which is non-perturbative, while the
anomaly-matching conditions are also non-perturbative in the sense that the
conditions hold at any energy scale, from low energy IR to high energy UV.
Thus, the equivalence between the twos is remarkable, especially that both are
_non-perturbative statements_ (namely the proof we provide is as exact as
integer number values without allowing any small perturbative expansion). Our
proof apply to a bulk U(1)N K matrix Chern-Simons theory (describing bulk
Abelian topological orders or Abelian SPT states) with boundary multiplet
chiral boson/fermion theories. More discussions can be found in Appendix C, D,
E.
#### IV.3.3 perturbative arguments
Apart from the non-perturbative proof using TQFT, we can use other well-known
techniques to show the boundary is gapped when the Boundary Fully Gapping
Rules are satisfied. Using the techniques systematically studied in
Ref.Wang:2013vna, and detailed in Appendix E.4, it is convenient to map the
$K_{N\times N}$-matrix multiplet chiral boson theory to $N/2$ copies of non-
chiral Luttinger liquids, each copy with an action
$\displaystyle\int
dt\,dx\;\Big{(}\frac{1}{4\pi}((\partial_{t}\bar{\phi}_{a}\partial_{x}\bar{\theta}_{a}+\partial_{x}\bar{\phi}_{a}\partial_{t}\bar{\theta}_{a})-V_{IJ}\partial_{x}\Phi_{I}\partial_{x}\Phi_{J})$
$\displaystyle+g\cos(\beta\;\bar{\theta}_{a})\Big{)}$ (57)
at large coupling $g$ at the low energy ground state. Notice that the mapping
sends
$\Phi\to\Phi^{\prime\prime}=(\bar{\phi}_{1},\bar{\phi}_{2},\dots,\bar{\phi}_{N/2},\bar{\theta}_{1},\bar{\theta}_{2},\dots,\bar{\theta}_{N/2})$
in a new basis, such that the cosine potential only takes one field
$\bar{\theta}_{a}$ decoupled from the full multiplet. However, this mapping
has been shown to be possible _if_ $\mathbf{L}^{T}K^{-1}\mathbf{L}=0$ is
satisfied.
When the mapping is done (in Appendix E.4), we can simply study a single copy
of non-chiral Luttinger liquids, and which, by changing of variables, is
indeed equivalent to the action of Klein-Gordon fields with a sine-Gordon
cosine potential studied by S. Coleman. We have demonstrated various ways to
show the existence of mass gap of this sine-Gordon action in Appendix E.3. For
example,
$\bullet$ For non-perturbative perspectives, there is a duality between the
quantum sine-Gordon action of bosons and the massive Thirring model of
fermions in 1+1D. In the sense, it is an integrable model, and the
Zamolodchikov formula is known and Bethe ansatz can be applicable. The mass
gap is known unambiguously at the large $g$.
$\bullet$ For perturbative arguments, we can use RG to do weak or strong
coupling expansions.
For _weak coupling_ $g$ analysis, it is known that choosing the kinetic term
as a marginal term, and the scaling dimension of the normal ordered
$[\cos(\beta\bar{\theta})]=\frac{\beta^{2}}{2}$. In the weak coupling
analysis, ${\beta^{2}}<\beta_{c}^{2}\equiv 4$ will flow to the large $g$
gapped phases (with an exponentially decaying correlator) at low energy, while
${\beta^{2}}>\beta_{c}^{2}$ will have the low energy flow to the quasi-long-
range gapless phases (with an algebraic decaying correlator) at the low energy
ground state. At $\beta=\beta_{c}$, it is known to have Berezinsky-Kosterlitz-
Thouless(BKT) transition. We find that our model satisfies
${\beta^{2}}<\beta_{c}^{2}$, shown in Appendix F.2.2, thus necessarily flows
to gapped phases, because the gapping terms can be written as
$g_{a}\cos(\bar{\theta}_{1})+g_{b}\cos(\bar{\theta}_{2})$ in the new basis,
where both ${\beta^{2}}=1<\beta_{c}^{2}$.
However, the weak coupling RG may not account the correct physics at large
$g$.
We also perform the _strong coupling_ $g$ RG analysis, by setting the pin-down
fields at large $g$ coupling of $g\cos(\beta\bar{\theta})$ with the quadratic
fluctuations as the marginal operators. We find the kinetic term changes to an
irrelevant operator. And the two-point correlator at large $g$ coupling
exponentially decays implies that our starting point is a strong-coupling
fixed point of gapped phase. Such an analysis shows _$\beta$ -independence_,
where the gapped phase is universal at _strong coupling_ $g$ regardless the
values of $\beta$ and robust against kinetic perturbation. It implies that
there is no instanton connecting different minimum vacua of large-$g$ cosine
potential for 1+1D at zero temperature for this particular action Eq.(IV.3.3).
More details in Appendix E.3.
In short, from the mapping to decoupled $N/2$-copies of non-chiral Luttinger
liquids with gapped spectra together with the anomaly-matching conditions
proved in Appendix C, D, we obtain the relations:
the U(1)N/2 anomaly-free theory
($\mathbf{q}^{T}\cdot{K}^{-1}\cdot\mathbf{q}=\mathbf{t}^{T}\cdot{K}\cdot\mathbf{t}=0$)
with gapping terms $\mathbf{L}^{T}K^{-1}\mathbf{L}=0$ satisfied.
$\updownarrow$ the $K$ matrix multiplet-chirla boson theories with gapping
terms $\mathbf{L}^{T}K^{-1}\mathbf{L}=0$ satisfied. $\downarrow$
$N/2$-decoupled-copies of non-Chiral Luttinger liquid actions with gapped
energy spectra.
$\bullet$ We can also answer other questions using _perturbative analysis_ :
(Please see Appendix E.2 for the details of calculation.)
(Q1) How can we see explicitly the formation of mass gap necessarily requiring
trivial braiding statistics among Wilson line operators (the $\ell_{a}$
vectors)?
(A1) To evaluate the mass gap, we need to know the energy gap of the lowest
energy state, namely the _zero mode_. The mode expansion of chiral boson
$\Phi$ field on a compact circular $S^{1}$ boundary of size $0\leq x<L$ is
$\Phi_{I}(x)={\phi_{0}}_{I}+K^{-1}_{IJ}P_{\phi_{J}}\frac{2\pi}{L}x+\text{i}\sum_{n\neq
0}\frac{1}{n}\alpha_{I,n}e^{-inx\frac{2\pi}{L}},$ (58)
where zero modes ${\phi_{0}}_{I}$ and winding modes $P_{\phi_{J}}$ satisfy the
commutator $[{\phi_{0}}_{I},P_{\phi_{J}}]=\text{i}\delta_{IJ}$; and the
Fourier modes satisfy generalized Kac-Moody algebra:
$[\alpha_{I,n},\alpha_{J,m}]=nK^{-1}_{IJ}\delta_{n,-m}$. A _perturbative_ way
to figure the zero mode’s mass is to learn when the zero mode ${\phi_{0}}_{I}$
can be pinned down at the minimum of cosine potential, with only quadratic
fluctuations. In that case, we can evaluate the mass by solving the simple
harmonic oscillator problem. This requires the following approximation to hold
$\displaystyle g_{a}\int_{0}^{L}dx\;\cos(\ell_{a,I}\cdot\Phi_{I})$
$\displaystyle\to
g_{a}\int_{0}^{L}dx\;\cos(\ell_{a,I}\cdot({\phi_{0}}_{I}+K^{-1}_{IJ}P_{\phi_{J}}\frac{2\pi}{L}x))$
$\displaystyle\to
g_{a}L\;\cos(\ell_{a,I}\cdot{\phi_{0}}_{I})\delta_{(\ell_{a,I}\cdot
K^{-1}_{IJ}P_{\phi_{J}},0)}.$ (59)
In the second line, one neglect the higher energetic Fourier modes; while to
have the third line to be true, it demands a commutator,
$[\ell_{a,I}{\phi_{0}}_{I},\;\ell_{a,I^{\prime}}K^{-1}_{I^{\prime}J}P_{\phi_{J}}]=0$.
Remarkably, this demands the null-condition
$\ell_{a,J}K^{-1}_{I^{\prime}J}\ell_{a,I^{\prime}}=0$, and the Kronecker delta
function restricts the Hilbert space of winding modes $P_{\phi_{J}}$ residing
on the _boundary gapping lattice_ $\Gamma^{\partial}$ due to $\ell_{a,I}\cdot
K^{-1}_{IJ}P_{\phi_{J}}=0$. Thus, we see that, even at the perturbative level,
the formation of mass gap requires trivial braiding statistics among the
$\ell_{a}$ vectors of interaction terms.
(Q2) What is the scale of the mass gap?
(A2) At the _perturbative_ level, we compute from a quantum simple harmonic
oscillator solution and find the mass gap $\Delta_{m}$ of zero modes:
$\Delta_{m}\simeq\sqrt{2\pi\,g_{a}\ell_{a,l1}\ell_{a,l2}V_{IJ}K^{-1}_{Il1}K^{-1}_{Jl2}},$
(Q3) What happens to the mass gap if we include _more_ (_incompatible_)
_interaction terms or less interaction terms_ with respect to the set of
interactions dictated by Boundary Fully Gapping Rules (adding
$\ell^{\prime}\notin\Gamma^{\partial}$, namely $\ell^{\prime}$ is not a linear
combination of column vectors of $\mathbf{L}$)?
(A3) Let us check the _stability_ of the mass gap against any _incompatible_
interaction term $\ell^{\prime}$ (which has nontrivial braiding statistics
respect to at least one of $\ell_{a}\in\Gamma^{\partial}$), by adding an extra
interaction $g^{\prime}\cos(\ell^{\prime}_{I}\cdot\Phi_{I})$ to the original
set of interactions $\sum_{a}g_{a}\cos(\ell_{a,I}\cdot\Phi_{I})$. We find that
as $\ell_{a,I}K^{-1}_{IJ}\ell^{\prime}_{J}\neq 0$ for the newly added
$\ell^{\prime}$, then the energy spectra for zero modes as well as the higher
Fourier modes have the _unstable_ form:
$E_{n}=\big{(}\sqrt{\Delta_{m}^{2}+\\#(\frac{2\pi
n}{L})^{2}+\sum_{a}\\#g_{a}\,g^{\prime}(\frac{L}{n})^{2}\dots+\dots}+\dots\big{)},$
(60)
Here $\\#$ are denoted as some numerical factors. Comparing to the case for
$g^{\prime}=0$ (without $\ell^{\prime}$ term), the energy changes from the
_stable_ form $E_{n}=\big{(}\sqrt{\Delta_{m}^{2}+\\#(\frac{2\pi
n}{L})^{2}}+\dots\big{)}$ to the _unstable_ form Eq.(60) at long-wave length
low energy ($L\to\infty$) , due to the disastrous term
$g_{a}\,g^{\prime}(\frac{L}{n})^{2}$. The energy has an infinite jump, either
from $n=0$(zero mode) to $n\neq 0$(Fourier modes), or at $L\to\infty$.
With any incompatible interaction term of $\ell^{\prime}$, the pre-formed mass
gap shows an instability. This indicates the _perturbative_ analysis may not
hold, and the zero modes cannot be pinned down at the minimum. The
consideration of instanton tunneling and talking between different minimum may
be important when $\ell_{a,I}K^{-1}_{IJ}\ell^{\prime}_{J}\neq 0$. In this
case, we expect the massive gapped phase is not stable, and the phase could be
gapless. Importantly, this can be one of the reasons why the numerical
attempts of Chen-Giedt-Poppitz model finds gapless phases instead of gapped
phases. _The immediate reason is that their Higgs terms induce many extra
interaction terms, not compatible with the (trivial braiding statistics) terms
dictated by Boundary Fully Gapping Rules. As we checked explicitly, many of
their induced terms break the U(1) ${}_{\text{2nd}}$ symmetry 0-4-5-3, which
is not compatible to the set inside $\Gamma^{\partial}$ or $\mathbf{L}$
matrix. _
#### IV.3.4 preserved U(1)N/2 symmetry and a unique ground state
We would like to discuss the symmetry of the system further. As we mention in
Sec.IV.3.2, the symmetry is broken down from
$\text{U}(1)^{N}\to\text{U}(1)^{N/2}$ by adding $N/2$ gapping terms with
$N=4$. In the case of gapping terms $\ell_{1}=(1,1,-2,2)$ and
$\ell_{2}=(2,-2,1,1)$, we can find the unbroken symmetry by Eq.(54), where the
symmetry charge vectors are $\mathbf{t}_{1}=(1,-1,-2,-2)$ and
$\mathbf{t}_{2}=(2,2,1,-1)$. The symmetry vector can have another familiar
linear combination $\mathbf{t}_{1}=(3,5,4,0)$ and $\mathbf{t}_{2}=(0,4,5,3)$,
which indeed matches to our original U(1)${}_{\text{1st}}$ 3-5-4-0 and
U(1)${}_{\text{2nd}}$ 0-4-5-3 symmetries. Similarly, the two gapping terms can
have another linear combinations: $\ell_{1}=(3,-5,4,0)$ and
$\ell_{2}=(0,4,-5,3)$. We can freely choose any linear-independent combination
set of the following,
$\displaystyle\mathbf{L}=\left(\begin{array}[]{cc}3&0\\\ -5&4\\\ 4&-5\\\
0&3\end{array}\right),\left(\begin{array}[]{cc}1&2\\\ 1&-2\\\ -2&1\\\
2&1\end{array}\right),\dots$ (69)
$\displaystyle\Longleftrightarrow\mathbf{t}=\left(\begin{array}[]{cc}3&0\\\
5&4\\\ 4&5\\\ 0&3\end{array}\right),\left(\begin{array}[]{cc}1&2\\\ -1&2\\\
-2&1\\\ -2&-1\end{array}\right),\dots.$ (78)
and we emphasize the vector space spanned by the column vectors of
$\mathbf{L}$ and $\mathbf{t}$ (the complement space of $\mathbf{L}$’s) will be
the entire 4-dimensional vector space $\mathbb{Z}^{4}$. In Appendix F.2.2, we
will provide the lattice construction for the alternative $\mathbf{L}$, see
Eq.(F.2.2).
Now we like to answer:
(Q4) Whether the $\text{U}(1)^{N/2}$ symmetry stays unbroken when the mirror
sector becomes gapped by the strong interactions?
(A4) The answer is Yes. We can check: There are two possibilities that
$\text{U}(1)^{N/2}$ symmetry is broken. One is that it is _explicitly broken_
by the interaction term. This is not true. The second possibility is that the
ground state (of our chiral fermions with the gapped mirror sector)
_spontaneously or explicitly break_ the $\text{U}(1)^{N/2}$ symmetry. This
possibility can be checked by calculating its ground state degeneracy(GSD) on
the cylinder with gapped boundary. Using the method developing in our previous
work Ref.Wang:2012am, , also in Ref.Kapustin:2013nva, ; Wang:2013vna, , we
find GSD=1, there is only a unique ground state. Because there is only one
lowest energy state, it cannot _spontaneously or explicitly break_ the
remained symmetry. The GSD is 1 as long as the $\ell_{a}$ vectors are chosen
to be the minimal vector, namely the greatest common divisor(gcd) among each
component of any $\ell_{a}$ is 1,
${|\gcd(\ell_{a,1},\ell_{a,2},\dots,\ell_{a,N/2}\Big{)}|}=1$, such that
$\ell_{a}\equiv\frac{(\ell_{a,1},\ell_{a,2},\dots,\ell_{a,N/2})}{|\gcd(\ell_{a,1},\ell_{a,2},\dots,\ell_{a,N/2})|}.$
In addition, thanks to Coleman-Mermin-Wagner theorem, there is _no spontaneous
symmetry breaking for any continuous symmetry in 1+1D, due to no Goldstone
modes in 1+1D_ , we can safely conclude that $\text{U}(1)^{N/2}$ symmetry
stays unbroken. black
To summarize the whole Sec.IV, we provide both non-perturbative and
perturbative analysis on Boundary Fully Gapping Rules. This applies to a
generic K-matrix U(1)N Abelian Chern-Simons theory with a boundary multiplet
chiral boson theory. (This generic K matrix theory describes general Abelian
topological orders including all Abelian SPT states.)
In addition, in the case when K is _unimodular indefinite symmetric integral
matrix_ , for both fermions $K=K^{f}$ and bosons $K=K^{b0}$, we have further
proved:
Theorem: The boundary fully gapping rules of 1+1D boundary/2+1D bulk with
unbroken U(1)N/2 symmetry $\leftrightarrow$ ABJ’s U(1)N/2 anomaly matching
conditions in 1+1D.
Similar to our non-perturbative algebraic result on topological gapped
boundaries, the ’t Hooft anomaly matching here is a non-perturbative
statement, being exact from IR to UV, insensitive to the energy scale.
## V General Construction of Non-Perturbative Anomaly-Free chiral matter
model from SPT
As we already had an explicit example of 3L-5R-4L-0R chiral fermion model
introduced in Sec.II,III.1.2, and we had paved the way building up tools and
notions in Sec.IV, now we are finally here to present our general model
construction. Our construction of non-perturbative anomaly-free chiral
fermions and bosons model with onsite U(1) symmetry is the following.
Step 1: We start with a $K$ matrix Chern-Simons theory as in Eq.(20),(21) for
_unimodular indefinite symmetric integral $K$ matrices_, both fermions
$K=K^{f}$ of Eq.(22) and bosons $K=K^{b0}$ of Eq.(23) (describing generic
Abelian SPT states with GSD on torus is $|\det(K)|=1$.)
Step 2: We assign charge vectors $\mathbf{t}_{a}$ of U(1) symmetry as in
Eq.(25), which satisfies the anomaly matching condition Eq.(42) for fermionic
model, or satisfies Eq.(43) for bosonic model. We can assign up to $N/2$
charge vector
$\mathbf{t}\equiv\Big{(}\mathbf{t}_{1},\mathbf{t}_{2},\dots,\mathbf{t}_{N/2}\Big{)}$
with a total U(1)N/2 symmetry with the matching
$\mathcal{A}={{\mathbf{t}^{T}}K{\mathbf{t}}=0}$ such that the model is anomaly
and mixed-anomaly free.
Step 3: In order to be a _chiral_ theory, it needs to _violate the parity
symmetry_. In our model construction, assigning $q_{L,j}\neq q_{R,j}$
generally fulfills our aims by breaking both parity and time reversal
symmetry. (See Appendix A for details.)
Step 4: By the equivalence of the anomaly matching condition and boundary
fully gapping rules(proved in Sec.IV.3.2 and Appendix C,D), our anomaly-free
theory guarantees that a proper choice of gapping terms of Eq.(28) can fully
gap out the edge states. For $N_{L}=N_{R}=N/2$ left/right Weyl fermions, there
are $N/2$ gapping terms
($\mathbf{L}\equiv\Big{(}\ell_{1},\ell_{2},\dots,\ell_{N/2}\Big{)}$), and the
U(1) symmetry can be extended to U(1)N/2 symmetry by finding the corresponding
$N/2$ charge vectors
($\mathbf{t}\equiv\Big{(}\mathbf{t}_{1},\mathbf{t}_{2},\dots,\mathbf{t}_{N/2}\Big{)}$).
The topological non-perturbative proof found in Sec.IV.3.2 guarantees the
duality relation:
$\displaystyle\boxed{\mathbf{L}^{T}\cdot
K^{-1}\cdot\mathbf{L}=0\underset{\mathbf{L}=K\mathbf{t}}{\overset{\mathbf{t}=K^{-1}\mathbf{L}}{\longleftrightarrow}}\mathbf{t}^{T}\cdot{K}\cdot\mathbf{t}=0}.$
(79)
Given $K$ as a $N\times N$-component matrix of $K^{f}$ or $K^{b0}$, we have
$\mathbf{L}$ and $\mathbf{t}$ are both $N\times(N/2)$-component matrices.
So our strategy is that constructing the bulk SPT on a 2D spatial lattice with
two edges (for example, a cylinder in Fig.1,Fig.6). The low energy edge
property of the 2D lattice model has the same continuum field
theoryfermionization1 as we had in Eq.(21), and selectively only fully
gapping out states on one mirrored edge with a large mass gap by adding
symmetry allowed gapping terms Eq.(28), while leaving the other side gapless
edge states untouched.Wen:2013ppa
In summary, we start with a chiral edge theory of SPT states with
$\cos(\ell_{I}\cdot\Phi^{B}_{I})$ gapping terms on the edge B, which action is
$\displaystyle S_{\Phi}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi}\int
dtdx\big{(}K^{\mathop{\mathrm{A}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}\big{)}\;\;\;$
(80) $\displaystyle+$ $\displaystyle\frac{1}{4\pi}\int
dtdx\big{(}K^{\mathop{\mathrm{B}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}\big{)}\;\;\;$
$\displaystyle+$ $\displaystyle\int
dtdx\;\sum_{a}g_{a}\cos(\ell_{a,I}\cdot\Phi_{I}).\;\;\;\;\;\;\;$
We fermionize the action to:
$\displaystyle S_{\Psi}$ $\displaystyle=\int
dt\;dx\;(i\bar{\Psi}_{\mathop{\mathrm{A}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{A}}}+i\bar{\Psi}_{\mathop{\mathrm{B}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{B}}}$
(81)
$\displaystyle+U_{\text{interaction}}\big{(}\tilde{\psi}_{q},\dots,\nabla^{n}_{x}\tilde{\psi}_{q},\dots\big{)}).$
with $\Gamma^{0}$, $\Gamma^{1}$, $\Gamma^{5}$ follow the notations of Eq.(24).
The gapping terms on the field theory side need to be irrelevant operators or
marginally irrelevant operators with appropriate strength (to be order 1
intermediate-strength for the dimensionless lattice coupling
$|G|/|t_{ij}|\gtrsim O(1)$), so it can gap the mirror sector, but it is weak
enough to keep the original light sector gapless.
Use several copies of Chern bands to simulate the free kinetic part of Weyl
fermions, and convert the higher-derivatives fermion interactions
$U_{\text{interaction}}$ to the point-splitting $U_{\text{point.split.}}$ term
on the lattice, we propose its corresponding lattice Hamiltonian
$\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{q}\bigg{(}\sum_{\langle
i,j\rangle}\big{(}t_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}$
$\displaystyle+$ $\displaystyle\sum_{\langle\langle
i,j\rangle\rangle}\big{(}t^{\prime}_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}\bigg{)}$
$\displaystyle+$
$\displaystyle\sum_{j\in\mathop{\mathrm{B}}}U_{\text{point.split.}}\bigg{(}\hat{f}_{q}(j),\dots\big{(}\hat{f}^{n}_{q}(j)\big{)}_{pt.s.},\dots\bigg{)}.\;\;$
Our key to avoid Nielsen-Ninomiya challengeNielsen:1980rz ; Nielsen:1981xu ;
Nielsen:1981hk is that our model has the _properly-desgined_ interactions.
We have obtained a 1+1D non-perturbative lattice Hamiltonian construction (and
realization) of anomaly-free massless chiral fermions (and chiral bosons) on
one gapless edge.
For readers with interests, In Appendix F.2, we will demonstrate a step-by-
step construction on several lattice Hamiltonian models of chiral
fermions(such as 1L-(-1R) chiral fermion model and 3L-5R-4L-0R chiral fermion
model) and chiral bosons, based on our general prescription above. In short,
such our approach is generic for constructing many lattice chiral matter
models in 1+1D.
## VI Summary
We have proposed a 1+1D lattice Hamiltonian definition of non-perturbative
anomaly-free chiral matter models with U(1) symmetry. Our 3L-5R-4L-0R fermion
model is under the framework of the mirror fermion decoupling approach.
However, some importance essences make our model distinct from the lattice
models of Eichten-PreskillEichten and Preskill (1986) and Chen-Giedt-Poppitz
3-4-5 model.Chen et al. (2013a) The differences between our and theirs are:
Onsite or non-onsite symmetry. Our model only implements onsite symmetry,
which can be easily to be gauged. While Chen-Giedt-Poppitz model implements
Ginsparg-Wilson(GW) fermion approach with non-onsite symmetry(details
explained in Appendix B). To have GW relation
$\\{D,\gamma^{5}\\}=2aD\gamma^{5}D$ to be true ($a$ is the lattice constant),
the Dirac operator is non-onsite (not strictly local) as $D(x_{1},x_{2})\sim
e^{-|x_{1}-x_{2}|/{\xi}}$ but with a distribution range $\xi$. The axial U(1)A
symmetry is modified
$\delta\psi(y)=\sum_{w}\text{i}\,\theta_{A}\hat{\gamma}_{5}(y,w)\psi(w),\;\;\;\delta\bar{\psi}(x)=\text{i}\,\theta_{A}\bar{\psi}(x){\gamma}_{5}$
with the operator $\hat{\gamma}_{5}(x,y)\equiv\gamma_{5}-2a\gamma_{5}D(x,y)$.
Since its axial U(1)A symmetry transformation contains $D$ and the Dirac
operator $D$ is non-onsite, the GW approach necessarily implements non-onsite
symmetry. GW fermion has non-onsite symmetry in the way that it cannot be
written as the tensor product structure on each site:
$U(\theta_{A})_{\text{non-onsite}}\neq\otimes_{j}U_{j}(\theta_{A})$, for
$e^{\text{i}\theta_{A}}\in\text{U}(1)_{A}$. The Neuberger-Dirac operator also
contains such a non-onsite symmetry feature. The non-onsite symmetry is the
signature property of the boundary theory of SPT states. The non-onsite
symmetry causes GW fermion diffcult to be gauged to a chiral gauge theory,
because the gauge theory is originally defined by gauging the local (on-site)
degrees of freedom.
Interaction terms. Our model has properly chosen a particular set of
interactions satisfying the Eq.(79), from the Lagrangian subgroup algebra to
define a topological gapped boundary conditions. On the other hand, Chen-
Giedt-Poppitz model proposed different kinds of interactions - all Higgs terms
obeying U(1)${}_{\text{1st}}$ 3-5-4-0 symmetry (Eq.(2.4) of Ref.Chen et al.,
2013a), including the Yukawa-Dirac term:
$\displaystyle\int
dtdx\Big{(}\mathrm{g}_{30}\psi_{L,3}^{\dagger}\psi_{R,0}\phi_{h}^{-3}+\mathrm{g}_{40}\psi_{L,4}^{\dagger}\psi_{R,0}\phi_{h}^{-4}$
$\displaystyle+\mathrm{g}_{35}\psi_{L,3}^{\dagger}\psi_{R,5}\phi_{h}^{2}+\mathrm{g}_{45}\psi_{L,4}^{\dagger}\psi_{R,5}\phi_{h}^{1}+h.c.\Big{)},$
(83)
with Higgs field $\phi_{h}(x,t)$ carrying charge $(-1)$. There are also
Yukawa-Majorana term:
$\displaystyle\int
dtdx\Big{(}\text{i}\mathrm{g}_{30}^{M}\psi_{L,3}\psi_{R,0}\phi_{h}^{3}+\text{i}\mathrm{g}_{40}^{M}\psi_{L,4}\psi_{R,0}\phi_{h}^{4}$
$\displaystyle+\text{i}\mathrm{g}_{35}^{M}\psi_{L,3}\psi_{R,5}\phi_{h}^{8}+\text{i}\mathrm{g}_{45}^{M}\psi_{L,4}\psi_{R,5}\phi_{h}^{9}+h.c.\Big{)},$
(84)
Notice that the Yukawa-Majorana coupling has an extra imaginary number i in
the front, and implicitly there is also a Pauli matrix $\sigma_{y}$ if we
write the Yukawa-Majorana term in the two-component Weyl basis.
The question is: How can we compare between interactions of ours and Ref.Chen
et al., 2013a’s? If integrating out the Higgs field $\phi_{h}$, we find that:
$(\star 1)$ Yukawa-Dirac terms of Eq.(VI) _cannot_ generate any of our multi-
fermion interactions of $\mathbf{L}$ in Eq.(69) for our 3L-5R-4L-0R model.
$(\star 2)$ Yukawa-Majorana terms of Eq.(VI) _cannot_ generate any of our
multi-fermion interactions of $\mathbf{L}$ in Eq.(69) for our 3L-5R-4L-0R
model.
$(\star 3)$ Combine Yukawa-Dirac and Yukawa-Majorana terms of Eq.(VI),(VI),
one can indeed generate the multi-fermion interactions of $\mathbf{L}$ in
Eq.(69); however, many more multi-fermion interactions outside of the
Lagrangian subgroup (not being spanned by $\mathbf{L}$) are generated. Those
extra unwanted multi-fermion interactions _do not_ obey the boundary fully
gapping rules. As we have shown in Sec.IV.3.3 and Appendix E.2, those extra
unwanted interactions induced by the Yukawa term will cause the pre-formed
mass gap unstable due to the nontrivial braiding statistics between the
interaction terms. This explains why the massless mirror sector is observed in
Ref.Chen et al., 2013a. In short, we know that Ref.Chen et al., 2013a’s
interaction terms are different from us, and know that the properly-designed
interactions are crucial, and our proposal will succeed the mirror-sector-
decoupling even if Ref.Chen et al., 2013a fails.
$\text{U}(1)^{N}\to\text{U}(1)^{N/2}\to\text{U}(1)$. We have shown that for a
given $N_{L}=N_{R}=N/2$ equal-number-left-right moving mode theory, the $N/2$
gapping terms break the symmetry from $\text{U}(1)^{N}\to\text{U}(1)^{N/2}$.
Its remained $\text{U}(1)^{N/2}$ symmetry is unbroken and mixed-anomaly free.
Is it possible to further add interactions to break $\text{U}(1)^{N/2}$ to a
smaller symmetry, such as a single U(1)? For example, breaking the
U(1)${}_{\text{2nd}}$ 0-4-5-3 of 3L-5R-4L-0R model to only a single
U(1)${}_{\text{\text{1st}}}$ 3-5-4-0 symmetry remained. We argue that it is
doable. Adding any extra explicit-symmetry-breaking term may be incompatible
to the original Lagrangian subgroup and thus potentially ruins the stability
of the mass gap. Nonetheless, as long as we add an extra interaction
term(breaking the U(1)${}_{\text{2nd}}$ symmetry), which is irrelevant
operator with a tiny coupling, it can be weak enough not driving the system to
gapless states. Thus, our setting to obtain 3-5-4-0 symmetry is still quite
different from Chen-Giedt-Poppitz where the universal strong couplings are
applied.
We show that GW fermion approach implements the non-onsite symmetry (more in
Appendix B), thus GW can avoid the fermion-doubling no-go theorem (limited to
an onsite symmetry) to obtain chiral fermion states. This realization is
consistent with what had been studied in Ref. Chen et al., 2011; Chen:2011pg,
; Santos:2013uda, . Remarkably, this also suggests that
The nontrivial edge states of SPT order,Chen:2011pg such as topological
insulatorsTI4 ; TI5 ; TI6 alike, can be obtained in its own dimension
(without the need of an extra dimension to the bulk) by implementing the non-
onsite symmetry as Ginsparg-Wilson fermion approach.
To summarize, so far we have realized (see Fig.8),
* •
Nielsen-Ninomiya theorem claims that local free chiral fermions on the lattice
with onsite (U(1) or chiralU(1)sym ) symmetry have fermion-doubling problem in
even dimensional spacetime.
* •
Gilzparg-Wilson(G-W) fermions: quasi-local free chiral fermions on the lattice
with non-onsite U(1) symmetryU(1)sym have no fermion doublers. G-W fermions
correspond to gapless edge states of a nontrivial SPT state.
* •
Our 3-5-4-0 chiral fermion and general model constructions: local interacting
chiral fermions on the lattice with onsite U(1) symmetryU(1)sym have no
fermion-doublers. Our model corresponds to unprotected gapless edge states of
a trivial SPT state (i.e. a trivial insulator).
Figure 8: Gilzparg-Wilson fermions can be viewed as putting gapless states on
the edge of a nontrivial SPT state (e.g. topological insulator). Our approach
can be viewed as putting gapless states on the edge of a trivial SPT state
(trivial insulator).
We should also clarify that, from SPT classification viewpoint, all our chiral
fermion models are in the same class of $K^{f}=({\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}})$ with $\mathbf{t}=(1,-1)$, a trivial class in the
fermionic SPT with U(1) symmetry.Lu:2012dt ; Ye:2013upa ; JWunpublished All
our chiral boson models are in the same class of
$K^{b}=({\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}})$ with
$\mathbf{t}=(1,0)$, a trivial class in the bosonic SPT with U(1)
symmetry.Lu:2012dt ; Ye:2013upa ; JWunpublished In short, we understand that
From the 2+1D bulk theory viewpoint, all our chiral matter models are
equivalent to the trivial class of SPT(trivial bulk insulator) in SPT
classification. However, the 1+1D boundary theories with different U(1) charge
vectors $\mathbf{t}$ can be regarded as different chiral matter theories on
its own 1+1D.
Proof of a Special Case and some Conjectures
At this stage we already fulfill proposing our models, on the other hand the
outcome of our proposal becomes fruitful with deeper implications. We prove
that, at least for 1+1D boundary/2+1D bulk SPT states with U(1) symmetry,
There are equivalence relations between (a) “ ’t Hooft anomaly matching
conditions satisfied”, (b) “the boundary fully gapping rules satisfied”, (c)
“the effective Hall conductance is zero,” and (d) “a bulk trivial SPT (i.e.
trivial insulator), with unprotected boundary edge states (realizing an onsite
symmetry) which can be decoupled from the bulk.”
Rigorously speaking, what we actually prove in Sec.IV.3.2 and Appendix C,D is
the equivalence of
Theorem: ABJ’s U(1) anomaly matching condition in 1+1D $\leftrightarrow$ the
boundary fully gapping rules of 1+1D boundary/2+1D bulk with unbroken U(1)
symmetry for an equal number of left-right moving Weyl-fermion
modes($N_{L}=N_{R}$, $c_{L}=c_{R}$) of 1+1D theory.
Note that some modifications are needed for more generic cases:
(i) For unbalanced left-right moving modes, the number chirality also implies
the additional _gravitational anomaly_.
(ii) For a bulk with _topological order_ (instead of pure SPT states), even if
the boundary is gappable without breaking the symmetry, there still can be
nontrivial signature on the boundary, such as degenerate ground states (with
gapped boundaries) or surface topological order. This modifies the above
specific Theorem to a more general Conjecture:
Conjecture: The anomaly matching condition in $(d+1)$D $\leftrightarrow$ the
boundary fully gapping rules of $(d+1)$D boundary/$(d+2)$D bulk with unbroken
$G$ symmetry for an equal number of left-right moving modes($N_{L}=N_{R}$) of
$(d+1)$D theory, such that the system with arbitrary gapped boundaries has _a
unique non-degenerate ground state_(GSD=1),Wang:2012am ; Kapustin:2013nva _no
surface topological order_ ,Vishwanath:2012tq _no symmetry/quantum number
fractionalization_Wang:2014tia and _without any nontrivial(anomalous)
boundary signature_.
However, for an arbitrary given theory, we do not know “all kinds of
anomalies,” and thus in principle we do not know “all anomaly matching
conditions.” However, our work reveals some deep connection between the
“anomaly matching conditions” and the “boundary fully gapping rules.”
Alternatively, if we take the following statement as a definition instead,
Proposed Definition: The anomaly matching conditions (all anomalies need to be
cancelled) for symmetry $G$ $\leftrightarrow$ the boundary fully gapping rules
without breaking symmetry $G$ and without anomalous boundary signatures under
gapped boundary.
then the Theorem and the Proposed Definition together reveal that
The only anomaly type of _a theory with an equal number of left/right-hand
Weyl fermion modes_ and only with a U(1) symmetry in 1+1D is ABJ’s U(1)
anomaly.
Arguably the most interesting future direction is to test our above conjecture
for more general cases, such as other dimensions or other symmetry groups. One
may test the above statements via the modular invarianceLevin:2013gaa ;
Sule:2013qla of boundary theory. It will also be profound to address, the
boundary fully gapping rules for non-Abelian symmetry, and the anomaly
matching condition for non-ABJ anomalyWen:2013oza ; Wen:2013ppa ;
Witten:1982fp through our proposal.
Though being numerically challenging, it will be interesting to test our
models on the lattice. Our local spatial-lattice Hamiltonian with a finite
Hilbert space, onsite symmetry and short-ranged hopping/interaction terms is
exactly a condensed matter system we can realize in the lab. It may be
possible in the future we can simulate the lattice chiral model in the
physical instant time using the condensed matter set-up in the lab (such as in
cold atoms system). Such a real-quantum-world simulation may be much faster
than any classical computer or quantum computer.
black
###### Acknowledgements.
We are grateful to John Preskill and Erich Poppitz for very helpful feedback
and generous coomunications on our work. We thank Michael Levin for important
conversations at the initial stage and for his comments on the manuscript. JW
thanks Roman Jackiw, Anton Kapustin, Thierry Giamarchi, Alexander Altland,
Sung-Sik Lee, Yanwen Shang, Duncan Haldane, Shinsei Ryu, David Senechal,
Eduardo Fradkin, Subir Sachdev, Ting-Wai Chiu, Jiunn-Wei Chen, Chenjie Wang,
and Luiz Santos for comments. JW thanks H. He, L. Cincio, R. Melko and G.
Vidal for comments on DMRG. This work is supported by NSF Grant No.
DMR-1005541, NSFC 11074140, and NSFC 11274192. It is also supported by the BMO
Financial Group and the John Templeton Foundation. Research at Perimeter
Institute is supported by the Government of Canada through Industry Canada and
by the Province of Ontario through the Ministry of Research.
Appendix
In the Appendix A, we discuss the $C,P,T$ symmetry in an 1+1 D fermion theory.
In the Appendix B, we show that Ginsparg-Wilson fermions realizing its axial
U(1) symmetry by a non-onsite symmetry transformation. In the Appendix C and D
, under the specific assumption for a $2+1$D bulk Abelian symmetric protected
topological (SPT) statesWen:2013oza ; Wen:2013ppa ; Chen:2011pg with U(1)
symmetry, we prove that
Boundary fully gapping rules (in Sec.IV.3)h95 ; Wang:2012am ; Levin:2013gaa ;
Barkeshli:2013jaa ; Lu:2012dt are sufficient and necessary conditions of the
’t Hooft anomaly matching condition (in Sec.IV.2).'tHooft:1979bh
The SPT order (explained in Sec.IV.1) are short-range entangled states with
some onsite symmetry $G$ in the bulk. For the nontrivial SPT order, the
symmetry $G$ is realized as a non-onsite symmetry on the boundary.Chen:2011pg
; Chen:2012hc ; Santos:2013uda The 1+1D edge states are protected to be
gapless as long as the symmetry $G$ is unbroken on the boundary.Chen:2011pg ;
Lu:2012dt Importantly, SPT has no long-range entanglement, so no
gravitational anomalies.Wen:2013oza ; Wen:2013ppa The only anomaly here is
the ABJ’s U(1) anomalyAdler:1969gk ; Bell:1969ts ; Donoghue:1992dd for chiral
matters.
Appendix E includes several approaches for proving boundary fully gapping
rules. In the Appendix F, we discuss the property of our Chern insulator in
details, and provide additional models of lattice chiral fermions and chiral
bosons.
## Appendix A $C$, $P$, $T$ symmetry in the 1+1D fermion theory
Here we show the charge conjugate $C$, parity $P$, time reversal $T$ symmetry
transformation for the 1+1D Dirac fermion theory. Recall that the massless
Dirac fermion Lagrangian is
$\mathcal{L}=\bar{\Psi}i\gamma^{\mu}\partial_{\mu}\Psi$. Here the Dirac
fermion field $\Psi$ can be written as a two-component spinor. For
convenience, but without losing the generality, we choose the Weyl basis, so
$\Psi=(\psi_{L},\psi_{R})$, where each component of $\psi_{L},\psi_{R}$ is a
chiral Weyl fermion with left and right chirality respectively. Specifically,
gamma matrices in the Weyl basis are
$\displaystyle\gamma^{0}=\sigma_{x},\;\;\;\gamma^{1}=i\sigma_{y},\;\;\;\gamma^{5}=\gamma^{0}\gamma^{1}=-\sigma_{z}.$
(85)
satisfies Clifford algebra $\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$,
here the signature of the Minkowski metric is $(+,-)$. The projection
operators are
$P_{L}=\frac{1-\gamma^{5}}{2}=\bigl{(}{\begin{smallmatrix}1&0\\\
0&0\end{smallmatrix}}\bigl{)},\;\;P_{R}=\frac{1+\gamma^{5}}{2}=\bigl{(}{\begin{smallmatrix}0&0\\\
0&1\end{smallmatrix}}\bigl{)},$ (86)
mapping a massless Dirac fermion to two Weyl fermions, i.e.
$\mathcal{L}=i\psi^{\dagger}_{L}(\partial_{t}-\partial_{x})\psi_{L}+i\psi^{\dagger}_{R}(\partial_{t}+\partial_{x})\psi_{R}$.
We derive the $P,T,C$ transformation on the fermion field operator
$\hat{\Psi}$ in $1+1$D, up to some overall complex phases $\eta_{P},\eta_{T}$
degree of freedom,
$\displaystyle
P\hat{\Psi}(t,\vec{x})P^{-1}=\eta_{P}\,\gamma^{0}\hat{\Psi}(t,-\vec{x}),$ (87)
$\displaystyle
T\hat{\Psi}(t,\vec{x})T^{-1}=\eta_{T}\,\gamma^{0}\hat{\Psi}(-t,\vec{x}),$ (88)
$\displaystyle
C\hat{\Psi}(t,\vec{x})C^{-1}=\gamma^{0}\gamma^{1}\hat{\Psi}^{*}(t,\vec{x}).$
(89)
We can quickly verify these transformations (which works for a massive Dirac
fermion): For the $P$ transformation, $P(t,\vec{x})P^{-1}=(t,-\vec{x})\equiv
x^{\prime\mu}$. Multiply Dirac equation by $\gamma^{0}$, one obtain
$\gamma^{0}(i\gamma^{\mu}\partial_{\mu}+m)\Psi(t,\vec{x})=(i\gamma^{\mu}\partial^{\prime}_{\mu}+m)(\gamma^{0}\Psi(t,\vec{x}))=0$.
This means we should identify
$\Psi^{\prime}(t,-\vec{x})=\gamma^{0}\Psi(t,\vec{x})$ up to a phase in the
state vector (wavefunction) form. Thus, in the operator form, we derive
$P\hat{\Psi}(t,\vec{x})P^{-1}=\hat{\Psi}^{\prime}(t,\vec{x})=\eta_{P}\,\gamma^{0}\hat{\Psi}(t,-\vec{x})$.
For the $T$ transformation, one massages the Dirac equation in terms of
Schrödinger equation form,
$i\partial_{t}\Psi(t,\vec{x})=H\Psi(t,\vec{x})=(-i\gamma^{0}\gamma^{j}\partial_{j}+m)\Psi(t,\vec{x})$,
here $\Psi(t,\vec{x})$ in the state vector form. In the time reversal form:
$i\partial_{-t}\Psi^{\prime}(-t,\vec{x})=H\Psi^{\prime}(-t,\vec{x})$, this is
$i\partial_{-t}T\Psi(t,\vec{x})=HT\Psi(t,\vec{x})$. We have $T^{-1}HT=H$ and
$T^{-1}i\partial_{-t}T=i\partial_{t}$, where $T$ is anti-unitary. $T$ can be
written as $T=UK$ with a unitary transformation part $U$ and an extra $K$ does
the complex conjugate. Then $T^{-1}HT=H$ imposes the constraints
$U^{-1}\gamma^{0}U=\gamma^{0*}$ and $U^{-1}\gamma^{j}U=-\gamma^{j*}$. In 1+1D
Weyl basis, since $\gamma^{0},\gamma^{1}$ both are reals, we conclude that
$U=\gamma^{0}$ up to a complex phase. So in the operator form,
$T\hat{\Psi}(t,\vec{x})T^{-1}=\hat{\Psi}^{\prime}(t,\vec{x})=\eta_{T}\,\gamma^{0}\hat{\Psi}(-t,\vec{x})$
For the $C$ transformation, we transform a particle to its anti-particle. This
means that we flip the charge $q$ (in the term coupled to a gauge field $A$),
which can be done by taking the complex conjugate on the Dirac equation,
$\big{[}-i\gamma^{\mu*}(\partial_{\mu}+iqA_{\mu})+m\big{]}\Psi^{*}(t,\vec{x})=0$,
where $-\gamma^{\mu*}$ satisfies Clifford algebra. We can rewrite the equation
as
$\big{[}i\gamma^{\mu}(\partial_{\mu}+iqA_{\mu})+m\big{]}\Psi_{c}(t,\vec{x})=0$,
by identifying the charge conjugate state vector as
$\Psi_{c}=M\gamma^{0}\Psi^{*}$ and imposing the constraint
$-M\gamma^{0}\gamma^{\mu*}\gamma^{0}M^{-1}=\gamma^{\mu}$. Additionally, we
already have $\gamma^{0}\gamma^{\mu}\gamma^{0}=\gamma^{\mu\dagger}$. So the
constraint reduces to $-M\gamma^{\mu T}M^{-1}=\gamma^{\mu}$. In the 1+1D Weyl
basis, we obtain $-M\gamma^{0}M^{-1}=\gamma^{0}$ and
$M\gamma^{1}M^{-1}=\gamma^{1}$. Thus, $M=\eta_{C}\,\gamma^{1}$ up to a phase,
and we derive $\Psi_{c}=\gamma^{0}\gamma^{1}\Psi^{*}$ in the state vector. In
the operator form, we obtain
$C\hat{\Psi}(t,\vec{x})C^{-1}=\hat{\Psi}_{c}(t,\vec{x})=\gamma^{0}\gamma^{1}\hat{\Psi}^{*}(t,\vec{x})$.
The important feature is that our chiral matter theory has parity $P$ and time
reversal $T$ symmetry broken. Because the symmetry transformation acting on
the state vector induces
$P\Psi=\sigma_{x}\Psi=\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}\Psi$ and
$T\Psi=i\sigma_{y}K\Psi=\bigl{(}{\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}}\bigl{)}K\Psi$. So both $P$ and $T$ exchange left-
handness, right-handness particles, i.e. $\psi_{L},\psi_{R}$ becomes
$\psi_{R},\psi_{L}$. Thus $P,T$ transformation switches left, right charge by
switching its charge carrier. If $q_{L}\neq q_{R}$, then our chiral matter
theory breaks $P$ and $T$.
Our chiral matter theory, however, does not break charge conjugate symmetry
$C$. Because the symmetry transformation acting on the state vector induces
$C\Psi=-\sigma_{z}\Psi^{*}=\bigl{(}{\begin{smallmatrix}-1&0\\\
0&1\end{smallmatrix}}\bigl{)}\Psi^{*}$, while $\psi_{L},\psi_{R}$ maintains
its left-handness, right-handness as $\psi_{L},\psi_{R}$.
## Appendix B Ginsparg-Wilson fermions with a non-onsite U(1) symmetry as SPT
edge states
We firstly review the meaning of onsite symmetry and non-onsite symmetry
transformation,Chen:2011pg ; Chen et al. (2011) and then we will demonstrate
that Ginsparg-Wilson fermions realize the U(1) symmetry in the non-onsite
symmetry manner.
### B.1 On-site symmetry and non-onsite symmetry
The onsite symmetry transformation as an operator $U(g)$, with $g\in G$ of the
symmetry group, transforms the state $|v\rangle$ globally, by $U(g)|v\rangle$.
The onsite symmetry transformation $U(g)$ must be written in the tensor
product form acting on each site $i$,Chen:2011pg ; Chen et al. (2011)
$U(g)=\otimes_{i}U_{i}(g),\ \ \ g\in G.$ (90)
For example, consider a system with only two sites. Each site with a qubit
degree of freedom (i.e. with $|0\rangle$ and $|1\rangle$ eigenstates on each
site). The state vector $|v\rangle$ for the two-sites system is
$|v\rangle=\sum_{j_{1},j_{2}}c_{j_{1},j_{2}}|j_{1}\rangle\otimes|j_{2}\rangle=\sum_{j_{1},j_{2}}c_{j_{1},j_{2}}|j_{1},j_{2}\rangle$
with $1,2$ site indices and $|j_{1}\rangle,|j_{2}\rangle$ are eigenstates
chosen among $|0\rangle,|1\rangle$.
An example for the onsite symmetry transformation can be,
$\displaystyle U_{\text{onsite}}$ $\displaystyle=$
$\displaystyle|00\rangle\langle 00|+|01\rangle\langle 01|-|10\rangle\langle
10|-|11\rangle\langle 11|$ (91) $\displaystyle=$
$\displaystyle(|0\rangle\langle 0|-|1\rangle\langle
1|)_{1}\otimes(|0\rangle\langle 0|+|1\rangle\langle 1|)_{2}$ $\displaystyle=$
$\displaystyle\otimes_{i}U_{i}(g).$
Here $U_{\text{onsite}}$ is in the tensor product form, where
$U_{1}(g)=(|0\rangle\langle 0|-|1\rangle\langle 1|)_{1}$ and
$U_{2}(g)=(|0\rangle\langle 0|+|1\rangle\langle 1|)_{2}$, again with $1,2$
subindices are site indices. Importantly, this operator does not contain non-
local information between the neighbored sites.
A non-onsite symmetry transformation $U(g)_{\text{non-onsite}}$ cannot be
expressed as a tensor product form:
$U(g)_{\text{non-onsite}}\neq\otimes_{i}U_{i}(g),\ \ \ g\in G.$ (92)
An example for the non-onsite symmetry transformation can be the $CZ$
operator,Chen et al. (2011)
$\displaystyle CZ=|00\rangle\langle 00|+|01\rangle\langle
01|+|10\rangle\langle 10|-|11\rangle\langle 11|.$
$CZ$ operator contains non-local information between the neighbored sites,
which flips the sign of the state vector if both sites $1,2$ are in the
eigenstate $|1\rangle$. One cannot achieve writing $CZ$ as a tensor product
structure.
Now let us discuss how to gauge the symmetry. Gauging an onsite symmetry
simply requires replacing the group element $g$ in the symmetry group to
$g_{i}$ with a site dependence, i.e. replacing a global symmetry to a local
(gauge) symmetry. All we need to do is,
$U(g)=\otimes_{i}U_{i}(g)\stackrel{{\scriptstyle
Gauge}}{{\Longrightarrow}}U(g_{i})=\otimes_{i}U_{i}(g_{i}),$ (93)
with $g_{i}\in G$. Following Eq.(93), it is easy to gauge such an onsite
symmetry to obtain a chiral fermion theory coupled to a gauge field. Since our
chiral matter theory is implemented with an onsite U(1) symmetry, it is easy
to gauge our chiral matter theory to be a U(1) chiral gauge theory.
On the other hand, a non-onsite symmetry transformation cannot be written as a
tensor product form. So, it is difficult (or unconventional) to gauge a non-
onsite symmetry. As we will show below Ginsparg-Wilson fermions realizing a
non-onsite symmetry, so that is why it is difficult to gauge it.
### B.2 Ginsparg-Wilson fermions and its non-onsite symmetry
Below we attempt to show that Ginsparg-Wilson(G-W) fermions realizing the
symmetry transformation by the non-onsite manner. Follow the notation of
Ref.Fujikawa:2004cx, , the generic form of the Dirac fermion $\Psi$ path
integral on the lattice (with the lattice constant $a$) is
$\int\mathcal{D}{\bar{\Psi}}\mathcal{D}{\Psi}\exp[a^{\text{d}_{m}}\sum_{x_{1},x_{2}}\bar{\Psi}(x_{1})D(x_{1},x_{2})\Psi(x_{2})].$
(94)
Here the exponent $\text{d}_{m}$ is the dimension of the spacetime. For
example, the action of G-W fermions with Wilson term (the term with the front
coefficient $r$) can be written as:
$\displaystyle S_{\Psi}$ $\displaystyle=$ $\displaystyle
a^{\text{d}_{m}}\Big{(}\sum_{x,\mu}\frac{\text{i}}{2a}(\bar{\psi}(x)\gamma^{\mu}U_{\mu}(x)\psi(x+a^{\mu})-\bar{\psi}(x+a^{\mu})\gamma^{\mu}U^{\dagger}_{\mu}(x)\psi(x))-m_{0}\bar{\psi}(x){\psi}(x)$
$\displaystyle+$
$\displaystyle\frac{r}{2a}\sum_{x,\mu}\bar{\psi}(x)U_{\mu}(x)\psi(x+a^{\mu})+\bar{\psi}(x+a^{\mu})U^{\dagger}_{\mu}(x)\psi(x)-2\bar{\psi}(x)\psi(x)\Big{)}.$
Here $U_{\mu}(x)\equiv\exp(iagA_{\mu})$ are the gauge field connection. At the
weak $g$ coupling, it is also fine for us simply consider $U_{\mu}(x)\simeq
1$. One can find its Fermion propagator:
$(\sum_{\mu}\frac{1}{a}\gamma^{\mu}\sin(ak_{\mu})-m_{0}-\sum_{\mu}\frac{r}{a}(1-\cos(ak_{\mu})))^{-1}.$
The G-W fermions with $r\neq 0$ kills the doubler (at $k_{\mu}=\pi/a$) by
giving a mass of order $r/a$ to it. As $a\to 0$, the doubler disappear from
the spectrum with an infinite large mass.
This Dirac operator $D(x_{1},x_{2})$ is not strictly local, but decreases
exponentially as
$D(x_{1},x_{2})\sim e^{-|x_{1}-x_{2}|/{\xi}}$ (95)
with $\xi=\text{(local range)}\cdot a$ as some localized length scale of the
Dirac operator. We call $D(x_{1},x_{2})$ as a quasi-local operator, which is
strictly _non-local_.
One successful way to treat the lattice Dirac operator is imposing the G-W
relation:Wilson:1974sk
$\\{D,\gamma^{5}\\}=2aD\gamma^{5}D.$ (96)
Thus in the continuum limit $a\to 0$, this relation becomes
$\\{{\not}D,\gamma^{5}\\}=0$. One can choose a Hermitian $\gamma^{5}$, and ask
for the Hermitian property on $\gamma^{5}D$, which is
$(\gamma^{5}D)^{\dagger}=D^{\dagger}\gamma^{5}=\gamma^{5}D$.
It can be shown that the action (in the exponent of the path integral) is
invariant under the axial U(1) chiral transformation with a $\theta_{A}$
rotation:
$\delta\psi(y)=\sum_{w}\text{i}\theta_{A}\hat{\gamma}_{5}(y,w)\psi(w),\;\;\;\delta\bar{\psi}(x)=\text{i}\theta_{A}\bar{\psi}(x){\gamma}_{5}\;\;\;$
(97)
where
$\hat{\gamma}_{5}(x,y)\equiv\gamma_{5}-2a\gamma_{5}D(x,y).$ (98)
The chiral anomaly on the lattice can be reproduced from the Jacobian $J$ of
the path integral measure:
$J=\exp[-\text{i}\theta_{A}\mathop{\mathrm{tr}}(\hat{\gamma}_{5}+{\gamma}_{5})]=\exp[-2\text{i}\theta_{A}\mathop{\mathrm{tr}}({\Gamma}_{5})]$
(99)
here ${\Gamma}_{5}(x,y)\equiv\gamma_{5}-a\gamma_{5}D(x,y)$. The chiral anomaly
follows the index theorem $\mathop{\mathrm{tr}}({\Gamma}_{5})=n_{+}-n_{-}$,
with $n_{\pm}$ counts the number of zero mode eigenstates $\psi_{j}$, with
zero eigenvalues, i.e. $\gamma_{5}D\psi_{j}=0$, where the projection is
$\gamma_{5}\psi_{j}=\pm\psi_{j}$ for $n_{\pm}$ respectively.
Note that G-W relation can be rewritten as
$\gamma^{5}D+D\hat{\gamma}^{5}=0.$ (100)
Importantly, now axial U(1)A transformation in Eq.(97) involves with
$\hat{\gamma}_{5}(x,y)$ which contains the piece of quasi-local operators
$D(x,y)\sim e^{-|x_{1}-x_{2}|/{\xi}}$. Thus, it becomes apparent that U(1)A
transformation Eq.(97) is an non-onsite symmetry which carries nonlocal
information between different sites $x_{1}$ and $x_{2}$. It is analogous to
the $CZ$ symmetry transformation in Eq.(B.1), which contains the entangled
information between neighbored sites $j_{1}$ and $j_{2}$.
Thus we have shown G-W fermions realizing axial U(1) symmetry(U(1)A symmetry)
with a non-onsite symmetry transformation. While the left and right chiral
symmetry U(1)L and U(1)R mixes between the linear combination of vector U(1)V
symmetry and axial U(1)A symmetry, so U(1)L and U(1)R have non-onsite symmetry
transformations, too. In short,
The axial U(1)A symmetry in G-W fermion is a non-onsite symmetry. Also the
left and right chiral symmetry U(1)L and U(1)R in G-W fermion are non-onsite
symmetry.
The non-onsite symmetry here indicates the nontrivial edge states of bulk
SPT,Chen:2011pg ; Chen:2012hc ; Santos:2013uda thus Ginsparg-Wilson fermions
can be regarded as gapless edge states of some bulk fermionic SPT order. With
the above analysis, we emphasize again that our approach in the main text is
different from Ginsparg-Wilson fermions - while our approach implements only
onsite symmetry, Ginsparg-Wilson fermion implements non-onsite symmetry. In
Chen-Giedt-Poppitz model,Chen et al. (2013a) the Ginsparg-Wilson fermion is
implemented.Thus this is one of the major differences between Chen-Giedt-
Poppitz and our approaches.
## Appendix C Proof: Boundary Fully Gapping Rules $\to$ Anomaly Matching
Conditions
Here we show that if boundary states can be fully gapped(there exists a
boundary gapping lattice $\Gamma^{\partial}$ satisfies boundary fully gapping
rules (1)(2)(3) in Sec.IV.3h95 ; Wang:2012am ; Levin:2013gaa ;
Barkeshli:2013jaa ; Lu:2012dt ) with U(1) symmetry unbroken, then the boundary
theory is an anomaly-free theory free from ABJ’s U(1) anomaly. This theory
satisfies the effective Hall conductance $\sigma_{xy}=0$, so the anomaly
factor $\mathcal{A}=0$ by Eq.(41) in Sec.IV.2, and illustrated in Fig.9.
Figure 9: Feynman diagrams with solid lines representing chiral fermions and
wavy lines representing U(1) gauge bosons: 1+1D chiral fermionic anomaly shows
$\mathcal{A}=\sum(q_{L}^{2}-q_{R}^{2})$. For a generic 1+1D theory with U(1)
symmetry, $\mathcal{A}=q^{2}\mathbf{t}K^{-1}\mathbf{t}$.
Importantly, for $N$ numbers of 1+1D Weyl fermions, in order to gap out the
mirrored sector, our model enforces $N\in 2\mathbb{Z}^{+}$ is an even positive
integer, and requires equal numbers of left/right moving modes
$N_{L}=N_{R}=N/2$. When there is no interaction, we have a total U(1)N
symmetry for the free theory. We will then introduce the properly-designed
gapping terms, and (if and only if) there are $N/2$ allowed gapping terms. The
total symmetry is further broken from U(1)N down to U(1)N/2 due to $N/2$
gapping terms.
The remained U(1)N/2 symmetry stays unbroken for the following reasons:
(i) The gapping terms obey the U(1)N/2 symmetry. The symmetry is thus not
explicitly broken.
(ii) In 1+1D, there is no spontaneous symmetry breaking of a continuous
symmetry (such as our U(1) symmetry) due to Coleman-Mermin-Wagner-Hohenberg
theorem.
(iii) We explicitly check the ground degeneracy of our model with a gapped
boundary has a unique ground state, following the procedure of
Ref.Wang:2012am, ; Kapustin:2013nva, . Thus, a unique ground state implies
that there is no way to have spontaneous symmetry breaking.
Below we will prove that all the remained U(1)N/2 symmetry is anomaly-free and
mixed-anomaly-free. We will prove for both fermionic and bosonic cases
together, under Chern-Simons symmetric-bilinear $K$ matrix notation, with
fermions $K=K^{f}$ and bosons $K=K^{b0}$, where $K=K^{-1}$.
Proof: There are $N/2$ linear-independent terms of $\ell_{a}$ for
$\cos(\ell_{a}\cdot\Phi)$ in the boundary gapping terms $\Gamma^{\partial}$,
for
$\\{\ell_{a}\\}=\\{\ell_{1},\ell_{2},\dots,\ell_{N/2}\\}\in\Gamma^{\partial}$.
To find the remained unbroken U(1)N/2 symmetry, we notice that we can define
charge vectors
$\mathbf{t}_{a}\equiv K^{-1}\ell_{a}$ (101)
where any $\ell_{a}\in\Gamma^{\partial}$ is allowed, and $a=1,\dots,N/2$. So
there are totally $N/2$ charge vectors. These $\mathbf{t}_{a}$ charge vectors
are linear-independent because all $\ell_{a}$ are linear-independent to each
other.
Now we show that these ${N/2}$ charged vectors $\mathbf{t}_{a}$ span the whole
unbroken U(1)N/2-symmetry. Indeed, follow the condition Eq.(29), this is true:
$\ell_{c,I}\cdot\mathbf{t}_{a}=\ell_{c}K^{-1}\ell_{a}=0$ (102)
for all $\ell_{c}\in\Gamma^{\partial}$. This proves that ${N/2}$ charged
vectors $\mathbf{t}_{a}$ are exactly the U(1)N/2-symmetry generators. We end
the proof by showing our construction is indeed an anomaly-free theory among
all U(1)N/2-symmetries or all U(1) charge vectors $\mathbf{t}_{a}$, thus we
check that they satisfy the anomaly matching conditions:
$\mathcal{A}_{(a,b)}=2\pi\sigma_{xy,(a,b)}=q^{2}\mathbf{t}_{a}K\mathbf{t}_{b}=q^{2}\ell_{a}K^{-1}\ell_{b}=0.$
(103)
Here $\ell_{a},\ell_{b}\in\\{\ell_{1},\ell_{2},\dots,\ell_{N/2}\\}$, where we
use $K=K^{-1}$. Therefore, our U(1)N/2-symmetry theory is fully anomaly-free
($\mathcal{A}_{(a,a)}=0$) and mixed anomaly-free ($\mathcal{A}_{(a,b)}=0$ for
$a\neq b$). We thus proved
Theorem: The boundary fully gapping rules of 1+1D boundary/2+1D bulk with
unbroken U(1) symmetry $\rightarrow$ ABJ’s U(1) anomaly matching condition in
1+1D.
for both fermions $K=K^{f}$ and bosons $K=K^{b0}$. (Q.E.D.)
## Appendix D Proof: Anomaly Matching Conditions $\to$ Boundary Fully Gapping
Rules
Here we show that if the boundary theory is an anomaly-free theory (free from
ABJ’s U(1) anomaly), which satisfies the anomaly factor $\mathcal{A}=0$ (i.e.
the effective Hall conductance $\sigma_{xy}=0$ in the bulk, in Sec.IV.2), then
boundary states can be fully gapped with U(1) symmetry unbroken. Given a
charged vector $\mathbf{t}$, we will prove in the specific case of U(1)
symmetry, by finding the set of boundary gapping lattice $\Gamma^{\partial}$
satisfies boundary fully gapping rules (1)(2)(3) in Sec.IV.3.h95 ; Wang:2012am
; Levin:2013gaa ; Barkeshli:2013jaa ; Lu:2012dt We denote the charge vector
as $\mathbf{t}=(t_{1},t_{2},t_{3},\dots,t_{N})$. We will prove this for
fermions $K=K^{f}$ and bosons $K=K^{b0}$ separately. Note the fact that
$K=K^{-1}$ for both $K^{f}$ and $K^{b0}$.
### D.1 Proof for fermions $K=K^{f}$
Given a $N$-component charge vector
$\displaystyle\mathbf{t}=(t_{1},t_{2},\dots,t_{N})$ (104)
of a U(1) charged anomaly-free theory satisfying $\mathcal{A}=0$, which means
$\mathbf{t}{(K^{f})^{-1}}\mathbf{t}=0$. Here the fermionic $K^{f}$ matrix is
written in this canonical form,
$K^{f}_{N\times N}=\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\dots$ (105)
We now construct $\Gamma^{\partial}$ obeying boundary fully gapping rules. We
choose
$\ell_{1}=(K^{f})\mathbf{t}$ (106)
which satisfies self-null condition $\ell_{1}(K^{f})^{-1}\ell_{1}=0$. To
complete the proof, we continue to find out a total set of
$\ell_{1},\ell_{2},\dots,\ell_{N/2}$, so $\Gamma^{\partial}$ is a dimension
$N/2$ Chern-Simons-charge lattice (Lagrangian subgroup).
For $\ell_{2}$, we choose its form as
$\ell_{2}=(\ell_{2,1},\ell_{2,1},\ell_{2,3},\ell_{2,3},0,\dots,0)$ (107)
where even component of $\ell_{2}$ duplicates its odd component value, to
satisfy $\ell_{2}(K^{f})^{-1}\ell_{1}=\ell_{2}(K^{f})^{-1}\ell_{2}=0$. The
second constraint is automatically true for our choice of $\ell_{2}$. The
first constraint is achieved by solving
$\ell_{2,1}(t_{1}-t_{2})+\ell_{2,3}(t_{3}-t_{4})=0$. We can properly choose
$\ell_{2}$ to satisfy this constraint.
For $\ell_{n}$, by mathematical induction, we choose its form as
$\ell_{n}=(\ell_{n,1},\ell_{n,1},\ell_{n,3},\ell_{n,3},\dots,\ell_{n,2n-1},\ell_{n,2n-1}0,\dots,0)$
(108)
where even component of $\ell_{n}$ duplicates its odd component value, to
satisfy
$\ell_{n}(K^{f})^{-1}\ell_{j},\;\;\;j=1,\dots,n,$ (109)
for any $n$. For $2\leq j\leq n$, the constraint is automatically true for our
choice of $\ell_{n}$ and $\ell_{j}$. For $\ell_{n}(K^{f})^{-1}\ell_{1}=0$, it
leads to the constraint:
$\ell_{n,1}(t_{1}-t_{2})+\ell_{n,3}(t_{3}-t_{4})+\dots+\ell_{n,2n-1}(t_{2n-1}-t_{2n})=0$,
we can generically choose $\ell_{n,2n-1}\neq 0$ to have a new $\ell_{n}$
independent from other $\ell_{j}$ with $1\leq j\leq n-1$.
Notice the gapping term obeys U(1) symmetry, because
$\ell_{n}\cdot\mathbf{t}=\ell_{n}(K^{f})^{-1}\ell_{1}=0$ is always true for
all $\ell_{n}$. Thus we have constructed a dimension $N/2$ Lagrangian subgroup
$\Gamma^{\partial}=\\{\ell_{1},\ell_{2},\dots,\ell_{N/2}\\}$ which obeys
boundary fully gapping rules (1)(2)(3) in Sec.IV.3. (Q.E.D.)
### D.2 Proof for bosons $K=K^{b0}$
Similar to the proof of fermion, we start with a given $N$-component charge
vector $\mathbf{t}$,
$\displaystyle\mathbf{t}=(t_{1},t_{2},\dots,t_{N}),$ (110)
of a U(1) charged anomaly-free theory satisfying $\mathcal{A}=0$, which means
$\mathbf{t}{(K^{b0})^{-1}}\mathbf{t}=0$.
Here the bosonic $K^{b0}$ matrix is written in this canonical form,
$K^{b0}_{N\times N}=\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}\oplus\dots$ (111)
We now construct $\Gamma^{\partial}$ obeying boundary fully gapping rules. We
choose
$\ell_{1}=(K^{b0})\mathbf{t}$ (112)
which satisfies self-null condition $\ell_{1}(K^{b0})^{-1}\ell_{1}=0$. To
complete the proof, we continue to find out a total set of
$\ell_{1},\ell_{2},\dots,\ell_{N/2}$, so $\Gamma^{\partial}$ is a dimension
$N/2$ Chern-Simons-charge lattice (Lagrangian subgroup).
For $\ell_{2}$, we choose its form as
$\ell_{2}=(\ell_{2,1},0,\ell_{2,3},0,\dots,0)$ (113)
where even components of $\ell_{2}$ are zeros, to satisfy
$\ell_{2}(K^{b0})^{-1}\ell_{1}=\ell_{2}(K^{b0})^{-1}\ell_{2}=0$. The second
constraint is automatically true for our choice of $\ell_{2}$. The first
constraint is achieved by $\ell_{2,1}(t_{1})+\ell_{2,3}(t_{3})=0$. We can
properly choose $\ell_{2}$ to satisfy this constraint.
For $\ell_{n}$, by mathematical induction, we choose its form as
$\ell_{n}=(\ell_{n,1},0,\ell_{n,3},0,\dots,\ell_{n,2n-1},0,\dots,0)$ (114)
where even components of $\ell_{n}$ are zeros, to satisfy
$\ell_{n}(K^{b0})^{-1}\ell_{j},\;\;\;j=1,\dots,n,$ (115)
for any $n$. For $2\leq j\leq n$, the constraint is automatically true for our
choice of $\ell_{n}$ and $\ell_{j}$. For $\ell_{n}(K^{b0})^{-1}\ell_{1}=0$, it
leads to the constraint:
$\ell_{n,1}(t_{1})+\ell_{n,3}(t_{3})+\dots\ell_{n,2n-1}(t_{2n-1})=0$, we can
generically choose $\ell_{n,2n-1}\neq 0$ to have a new $\ell_{n}$ independent
from other $\ell_{j}$ with $1\leq j\leq n-1$.
Notice the gapping term obeys U(1) symmetry, because
$\ell_{n}\cdot\mathbf{t}=\ell_{n}(K^{b0})^{-1}\ell_{1}=0$ is always true for
all $\ell_{n}$. Thus we have constructed a dimension $N/2$ Lagrangian subgroup
$\Gamma^{\partial}=\\{\ell_{1},\ell_{2},\dots,\ell_{N/2}\\}$ which obeys
boundary fully gapping rules (1)(2)(3) in Sec.IV.3. (Q.E.D.)
Theorem: ABJ’s U(1) anomaly matching condition in 1+1D $\rightarrow$ the
boundary fully gapping rules of 1+1D boundary/2+1D bulk with unbroken U(1)
symmetry.
We emphasize again that although we start with a single U(1) anomaly-free
theory (aiming for a single U(1)-symmetry), it turns out that the full
symmetry after adding interacting gapping terms will result in a theory with
an enhanced total U(1)N/2 symmetry. The $N/2$ number of gapping terms break a
total U(1)N symmetry (for $N$ free Weyl fermions) down to U(1)N/2 symmetry.
The derivation follows directly from the statement in Appendix C, which we
shall not repeat it.
We comment that our proofs in Appendix C and D are algebraic and topological,
thus it is a non-perturbative result (instead of a perturbative result in the
sense of doing weak or strong coupling expansions).
## Appendix E More about the Proof of “Boundary Fully Gapping Rules”
This section aims to demonstrate that the Boundary Fully Gapping Rules used
throughout our work (and also used in Ref), indeed can gap the edge states. We
discuss this proof here to make our work self-contained and to further
convince the readers.
### E.1 Canonical quantization
Here we set up the canonical quantization of the bosonic field $\phi_{I}$ for
a multiplet chiral boson theory of Eq.(21) on a 1+1D spacetime, with a spatial
$S^{1}$ compact circle. The canonical quantization means that imposing a
commutation relation between $\phi_{I}$ and its conjugate momentum field
$\Pi_{I}(x)=\frac{\delta{L}}{\delta(\partial_{t}\phi_{I})}=\frac{1}{2\pi}K_{IJ}\partial_{x}\phi_{J}$.
Since $\phi_{I}$ is a compact phase of a matter field, its bosonization
contains both zero mode ${\phi_{0}}_{I}$ and winding momentum $P_{\phi_{J}}$,
in addition to Fourier modes $\alpha_{I,n}$:Wang:2012am
$\Phi_{I}(x)={\phi_{0}}_{I}+K^{-1}_{IJ}P_{\phi_{J}}\frac{2\pi}{L}x+\text{i}\sum_{n\neq
0}\frac{1}{n}\alpha_{I,n}e^{-inx\frac{2\pi}{L}}.$ (116)
The periodic boundary has a size of length $0\leq x<L$, with $x$ identified
with $x+L$. We impose the commutation relation for zero modes and winding
modes, and generalized Kac-Moody algebra for Fourier modes:
$[{\phi_{0}}_{I},P_{\phi_{J}}]=\text{i}\delta_{IJ},\;\;[\alpha_{I,n},\alpha_{J,m}]=nK^{-1}_{IJ}\delta_{n,-m}.$
(117)
Consequently, the commutation relations for the canonical quantized fields
are:
$\displaystyle[\phi_{I}(x_{1}),K_{I^{\prime}J}\partial_{x}\phi_{J}(x_{2})]$
$\displaystyle=$
$\displaystyle{2\pi}\text{i}\delta_{II^{\prime}}\delta(x_{1}-x_{2}),$ (118)
$\displaystyle\;\;[\phi_{I}(x_{1}),\Pi_{J}(x_{2})]$ $\displaystyle=$
$\displaystyle\text{i}\delta_{IJ}\delta(x_{1}-x_{2}).$ (119)
### E.2 Approach I: Mass gap for gapping zero energy modes
We provide the first approach to show that the anomaly-free edge states can be
gapped under the properly-designed gapping terms. Here we explicitly calculate
the mass gap for the zero energy mode and its higher excitations. The generic
theory is
$\displaystyle S_{\partial}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi}\int
dt\;dx\;(K_{IJ}\partial_{t}\Phi_{I}\partial_{x}\Phi_{J}-V_{IJ}\partial_{x}\Phi_{I}\partial_{x}\Phi_{J})$
(120) $\displaystyle+$ $\displaystyle\int
dt\;dx\;\sum_{a}g_{a}\cos(\ell_{a,I}\cdot\Phi_{I}).$
We will consider the even-rank symmetric $K$ matrix, so the full edge theory
has an even number of modes and thus potentially be gappable. In the following
we shall determine under what conditions that the edge states can obtain a
mass gap. Imagining at the large coupling $g$, the $\Phi_{I}$ field get
trapped at the minimum of the cosine potential with small fluctuations. We
will perform an expansion of $\cos(\ell_{a,I}\cdot\Phi_{I})\simeq
1-\frac{1}{2}(\ell_{a,I}\cdot\Phi_{I})^{2}+\dots$ to a quadratic order and see
what it implies about the mass gap. We can diagonalize the Hamiltonian,
$H\simeq(\int^{L}_{0}dx\;V_{IJ}\partial_{x}\Phi_{I}\partial_{x}\Phi_{J})+\frac{1}{2}\sum_{a}g_{a}(\ell_{a,I}\cdot\Phi_{I})^{2}L+\dots$
(121)
under a complete $\Phi$ mode expansion, and find the energy spectra for its
eigenvalues. To summarize the result, we find that:
(E-1). _If and only if_ we include all the gapping terms allowed by Boundary
Full Gapping Rules, we can open the mass gap of zero modes($n=0$) as well as
Fourier modes(non-zero modes $n\neq 0$). Namely, the energy spectrum is in the
form of
$E_{n}=\big{(}\sqrt{\Delta^{2}+\\#(\frac{2\pi n}{L})^{2}}+\dots\big{)},$ (122)
where $\Delta$ is the mass gap. Here we emphasize the energy of Fourier
modes($n\neq 0$) behaves towards zero modes at long wave-length low energy
limit ($L\to\infty$). Such spectra become continuous at $L\to\infty$ limit,
which is the expected energy behavior.
(E-2). _If_ we include the _incompatible_ Wilson line operators, such as
$\ell$ and $\ell^{\prime}$ where $\ell K^{-1}\ell^{\prime}\neq 0$, while the
interaction terms contain _incompatible_ gapping terms
$g\cos(\ell\cdot\Phi)+g^{\prime}\cos(\ell^{\prime}\cdot\Phi)$, we find the
_unstable_ energy spectra
$E_{n}=\big{(}\sqrt{\Delta^{2}+\\#(\frac{2\pi
n}{L})^{2}+g\,g^{\prime}(\frac{L}{n})^{2}\dots+\dots}+\dots\big{)},$ (123)
The energy spectra shows an _instability_ of the system, because at low energy
limit ($L\to\infty$), the spectra become discontinuous (from $n=0$ to $n\neq
0$) and jump to infinity as long as there are incompatible gapping
terms(namely, $g\cdot g^{\prime}\neq 0$). Such disastrous behavior of
$(L/n)^{2}$ implies the quadratic expansion analysis may not account for the
whole physics. In that case, the disastrous behavior invalidates the trapping
of $\Phi$ field at a local minimum, thus invalidates the mass gap, and the
_unstable_ system potentially seeks to be _gapless phases_.
Below we demonstrate the result explicitly for the simplest rank-2 $K$ matrix,
while the case for higher rank $K$ matrix can be straightforwardly
generalized. The most general rank-2 $K$ matrix is
$K\equiv{\begin{pmatrix}k_{1}&k_{3}\\\ k_{3}&k_{2}\
\end{pmatrix}}\equiv{\begin{pmatrix}k_{1}&k_{3}\\\
k_{3}&(k_{3}^{2}-p^{2})/k_{1}\
\end{pmatrix}},\,\;\;V={\begin{pmatrix}v_{1}&v_{2}\\\ v_{2}&v_{1}\\\
\end{pmatrix}},$ (124)
while the $V$ velocity matrix is chosen to be rescaled as the above. (Actually
the $V$ matrix is immaterial to our conclusion.) Our discussion below holds
for both $k_{3}=\pm|k_{3}|$ cases. We define $k_{2}=(k_{3}^{2}-p^{2})/k_{1}$,
so that $\det(K)=-p^{2}$ We find that only when
$\sqrt{|\det(K)|}\equiv p\in\mathbb{Z},$
$p$ is an integer, we can find gapping terms allowed by Boundary Fully Gapping
Rules. (A side comment is that $\det(K)=-p^{2}$ implies its bulk can be
constructed as a _quantum double_ or a _twisted quantum double model_ on the
lattice.) For the above rank-2 K matrix, we find two independent sets,
$\\{\ell_{1}=(\ell_{1,1},\ell_{1,2})\\}$ and
$\\{\ell_{1}^{\prime}=(\ell_{1,1}^{\prime},\ell_{1,2}^{\prime})\\}$, each set
has only one $\ell$ vector. Here the $\ell$ vector is written as $\ell_{a,I}$,
with the index $a$ labeling the $a$-th (linear independent) $\ell$ vector in
the Lagrangian subgroup, and the index $I$ labeling the $I$-component of the
$\ell_{a}$ vector. Their forms are:
$\displaystyle\frac{\ell_{1,1}}{\ell_{1,2}}$ $\displaystyle=$
$\displaystyle\frac{k_{1}}{k_{3}+p}=\frac{k_{3}-p}{k_{2}},$ (125)
$\displaystyle\frac{{\ell_{1,1}^{\prime}}}{{\ell_{1,2}^{\prime}}}$
$\displaystyle=$ $\displaystyle\frac{k_{1}}{k_{3}-p}=\frac{k_{3}+p}{k_{2}}.$
(126)
We denote the cosine potentials spanned by these $\ell_{1}$,
$\ell_{1}^{\prime}$ vectors in Eq.(120) as:
$g\cos(\ell_{1}\cdot\Phi)+g^{\prime}\cos(\ell_{1}^{\prime}\cdot\Phi).$ (127)
From our understanding of Boundary Full Gapping Rules, these two $\ell_{1}$,
$\ell_{1}^{\prime}$ vectors are _not compatible to each other_. In this sense,
_we shall not include both terms if we aim to fully gap the edge states_.
Now we focus on computing the mass gap of our interests for the bosonic K
matrix $K^{b}_{2\times 2}=\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}$ and the fermionic K matrix $K^{f}_{2\times
2}=\bigl{(}{\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}}\bigl{)}$. We use
both the Hamiltonian or the Lagrangian formalism to extract the energy, for
both zero modes($n=0$) and Fourier modes(non-zero modes $n\neq 0$). For both
the Hamiltonian and Lagrangian formalisms, we obtain the consistent result for
energy gaps $E_{n}$:
1st Case: Bosonic $K^{b}_{2\times 2}=\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}$:
$E_{n}=\sqrt{2\pi(g+g^{\prime})v_{1}+(\frac{2\pi
n}{L})^{2}v_{1}^{2}+g\,g^{\prime}(\frac{L}{n})^{2}}\pm(\frac{2\pi n}{L})v_{2}$
(128)
2nd Case: Fermionic $K^{f}_{2\times 2}=\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}$:
$E_{n}=\sqrt{4\pi g(v_{1}-v_{2})+4\pi g^{\prime}(v_{1}+v_{2})+(\frac{2\pi
n}{L})^{2}(v_{1}^{2}-v_{2}^{2})+(\frac{2L}{n})^{2}g\,g^{\prime}}$ (129)
Logically, for rank-2 K matrix, we have shown that:
$\bullet$ _If_ we include the gapping terms allowed by Boundary Full Gapping
Rules, either (i) $g\neq 0,g^{\prime}=0$, or (ii) $g=0,g^{\prime}\neq 0$, then
we have the _stable_ form of the mass gap in Eq.(122). Thus we show the _if_
-statement in (E-1).
$\bullet$ _If_ we include incompatible interaction terms (here
$\ell_{1}K^{-1}\ell_{1}^{\prime}\neq 0$), such that both $g\neq 0$ and
$g^{\prime}\neq 0$, then the energy gap is of the _unstable_ form in Eq.(123).
Thus we show the statement in (E-2).
$\bullet$ Meanwhile, this (E-2) implies that if we include _more_ interaction
terms allowed by Boundary Full Gapping Rules, we have an unstable energy gap,
thus it may drive the system to the gapless states due to the instability.
Moreover, if we include _less_ interaction terms allowed by Boundary Full
Gapping Rules (i.e. if we do not include all allowed _compatible_ gapping
terms), then we cannot fully gap the edge states (For $1$-left-moving mode and
$1$-right-moving mode, we need at least $1$ interaction term to gap the edge.)
Thus we also show the _only-if_ -statement in (E-1).
This approach work for a generic even-rank $K$ matrix thus can be applicable
to show the above statements (E-1) and (E-2) hold in general. More generally,
for rank-$N$ $K$ matrix Chern-Simons theory, with the boundary $N/2$-left-
moving modes and $N/2$-right-moving modes, we need _at least and at most_
$N/2$-linear-independent interaction terms to gap the edge. If one includes
more terms than the allowed terms (such as the numerical attempt in Ref.Chen
et al., 2013a), it may _drive the system to the gapless states due to the
instability from the unwanted quantum fluctuation._ This can be one of the
reasons why Ref.Chen et al., 2013a fails to achieve gapless fermions by
gapping mirror-fermions.
3rd Case: General even-rank $K$ matrix: Here we outline another view of the
energy-gap-stability for the edge states, for a generic rank-$N$ $K$ matrix
Chern-Simons theory with multiplet-chiral-boson-theory edge states. We include
the full interacting cosine term for the lowest energy states - zero and
winding modes:
$\cos(\ell_{a,I}\cdot\Phi_{I})\to\cos(\ell_{a,I}\cdot({\phi_{0}}_{I}+K^{-1}_{IJ}P_{\phi_{J}}\frac{2\pi}{L}x)),$
(130)
while we drop the higher energy Fourier modes. (Note when $L\to\infty$, the
kinetic term $H_{kin}=\frac{(2\pi)^{2}}{4\pi
L}V_{IJ}K^{-1}_{Il1}K^{-1}_{Jl2}P_{\phi_{l1}}P_{\phi_{l2}}$ has an order
$O(1/L)$ so is negligible, thus the cosine potential Eq. (130) dominates.
Though to evaluate the mass gap, we keep both kinetic and potential terms.)
The stability of the mass gap can be understood from _under what conditions_
we can safely expand the cosine term to extract the leading quadratic terms by
only keeping the zero modes via $\cos(\ell_{a,I}\cdot\Phi_{I})\simeq
1-\frac{1}{2}(\ell_{a,I}\cdot\phi_{0I})^{2}+\dots$. (If one does not decouple
the winding mode term, there is a complicated $x$ dependence in
$P_{\phi_{J}}\frac{2\pi}{L}x$ along the $x$ integration.) The challenge for
this cosine expansion is rooted to the _non-commuting_ algebra from
$[{\phi_{0}}_{I},P_{\phi_{J}}]=\text{i}\delta_{IJ}$. This can be resolved by
requiring $\ell_{a,I}{\phi_{0}}_{I}$ and
$\ell_{a,I^{\prime}}K^{-1}_{I^{\prime}J}P_{\phi_{J}}$ _commutes_ in Eq.(130),
$\displaystyle[\ell_{a,I}{\phi_{0}}_{I},\;\ell_{a,I^{\prime}}K^{-1}_{I^{\prime}J}P_{\phi_{J}}]$
$\displaystyle=$
$\displaystyle\ell_{a,I}K^{-1}_{I^{\prime}J}\ell_{a,I^{\prime}}\;(\text{i}\delta_{IJ})$
(131) $\displaystyle=$
$\displaystyle(\text{i})(\ell_{a,J}K^{-1}_{I^{\prime}J}\ell_{a,I^{\prime}})=0.\;\;\;\;\;\;$
This is indeed the Boundary Full Gapping Rules (1), the trivial statistics
rule among the Wilson line operators for the gapping terms. Under this
_commuting condition_ (we can interpret that there is _no unwanted quantum
fluctuation_), we can thus expand Eq.(130) using the trigonometric identify
for c-numbers as
$\displaystyle\cos(\ell_{a,I}{\phi_{0}}_{I})\cos(\ell_{a,I}K^{-1}_{IJ}P_{\phi_{J}}\frac{2\pi}{L}x)$
$\displaystyle-\sin(\ell_{a,I}{\phi_{0}}_{I})\sin(\ell_{a,I}K^{-1}_{IJ}P_{\phi_{J}}\frac{2\pi}{L}x)$
(132)
and then we safely integrate over $L$. Note that both $\cos(\dots x)$ and
$\sin(\dots x)$ are periodic in the region $[0,L)$, so both $x$-integration
vanish unless when $\ell_{a,I}\cdot K^{-1}_{IJ}P_{\phi_{J}}=0$ such that
$\cos(\ell_{a,I}K^{-1}_{IJ}P_{\phi_{J}}\frac{2\pi}{L}x)=1$. We thus obtain
$g_{a}\int_{0}^{L}dx\;\text{Eq}.(\ref{eq:cos})=g_{a}L\;\cos(\ell_{a,I}\cdot{\phi_{0}}_{I})\delta_{(\ell_{a,I}\cdot
K^{-1}_{IJ}P_{\phi_{J}},0)}.$ (133)
The Kronecker-delta-condition $\delta_{(\ell_{a,I}\cdot
K^{-1}_{IJ}P_{\phi_{J}},0)}=1$ implies that if and only if $\ell_{a,I}\cdot
K^{-1}_{IJ}P_{\phi_{J}}=0$. This is also consistent with the _Chern-Simons
quantized lattice_ as the Hilbert space of the ground states. Here $P_{\phi}$
forms a discrete quantized lattice because its conjugate ${\phi_{0}}$ is
periodic. This result can be applied to count the ground state degeneracy of
Chern-Simons theory of a closed manifold or a compact manifold with gapped
boundaries.Wang:2012am ; Kapustin:2013nva
In short, we have shown that when $\ell^{t}K^{-1}\ell=0$, we have the desired
the cosine potential expansion via the zero mode quadratic expansion at large
$g_{a}$ coupling, $g_{a}\int_{0}^{L}dx\cos(\ell_{a,I}\cdot\Phi_{I})\simeq-
g_{a}L\frac{1}{2}(\ell_{a,I}\cdot\phi_{0I})^{2}+\dots$. The nonzero mass gaps
of zero modes can be readily shown by solving the quadratic simple harmonic
oscillators of both the kinetic and the leading-order of the potential terms:
$\frac{(2\pi)^{2}}{4\pi
L}V_{IJ}K^{-1}_{Il1}K^{-1}_{Jl2}P_{\phi_{l1}}P_{\phi_{l2}}+\sum_{a}g_{a}L\frac{1}{2}(\ell_{a,I}\cdot\phi_{0I})^{2}$
(134)
The mass gap is independent of the system size, the order one finite gap
$\Delta\simeq\sqrt{2\pi\,g_{a}\ell_{a,l1}\ell_{a,l2}V_{IJ}K^{-1}_{Il1}K^{-1}_{Jl2}},$
(135)
which the mass matrix can be properly diagonalized, since there are only
conjugate variables $\phi_{0I},P_{\phi,J}$ in the quadratic order.
We again find that the above statements consistent with (E-1) and (E-2) for a
generic even-rank $K$ matrix.
### E.3 Mass Gap for Klein-Gordon fields and non-Chiral Luttinger liquids
under sine-Gordon potential
First, we recall the two statements (E-3),(E-4) that:
(E-3) A _scalar boson theory_ of a Klein-Gordon action with a sine-Gordon
potential:
$S_{\partial}=\int
dt\,dx\;\frac{\kappa}{2}(\partial_{t}\varphi\partial_{t}\varphi-\partial_{x}\varphi\partial_{x}\varphi)+g\cos(\beta\varphi).$
(136)
at strong coupling $g$ can induce the mass gap for the scalar mode $\varphi$.
(E-4) A _non-chiral Luttinger liquids_(non-chiral in the sense of equal left-
right moving modes, but can have U(1)-charge-chirality with respect to a U(1)
symmetry) with $\phi$ and $\theta$ dual scalar fields with a sine-Gordon
potential for $\phi$ field:
$\displaystyle S_{\partial}$ $\displaystyle=$ $\displaystyle\int
dt\,dx\;\Big{(}\frac{1}{4\pi}((\partial_{t}\phi\partial_{x}\theta+\partial_{x}\phi\partial_{t}\theta)-V_{IJ}\partial_{x}\Phi_{I}\partial_{x}\Phi_{J})$
(137) $\displaystyle+$ $\displaystyle g\cos(\beta\;\theta)\Big{)}.$
at strong coupling $g$ can induce the mass gap for _all_ the scalar mode
$\Phi\equiv(\phi,\theta)$.
Indeed, the statement (E-3) and (E-4) are related because Eq.(136) and
Eq.(137) are identified by the canonical conjugate momentum relation:
$\partial_{t}\phi\sim\partial_{x}\theta,\;\;\;\partial_{t}\theta\sim\partial_{x}\phi,$
(138)
up to a normalization factor and up to some Euclidean time transformation.
There are immense and broad amount of literatures demonstrating (E-3),(E-4)
are true, and we recommend to look for Ref.W, ; Altland:2006si, ; Giamarchi, .
Here we summarize several aspects of these understandings for our readers:
$\bullet$1\. The (E-3)’s quantum sine-Gordon action of Eq.(136) is equivalent
to the massive Thirring model:
$S_{MT}=\int
dt\,dx(\text{i}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{\lambda}{2}(\bar{\psi}\gamma^{\mu}\psi)(\bar{\psi}\gamma_{\mu}\psi)-m\bar{\psi}\psi)$
(139)
via the identification($j^{\mu}\equiv\bar{\psi}\gamma^{\mu}\psi$):
$\displaystyle\frac{4\pi\kappa}{\beta^{2}}=1+\frac{\lambda}{\pi},\;\;j^{\mu}=\frac{-\beta}{2\pi}\epsilon^{\mu\nu}\partial_{\nu}\varphi,\;\;g\cos(\beta\varphi)=-m\bar{\psi}\psi.\;\;\;\;\;$
(140)
One can compute the induced mass $m$ of scalar field $\varphi$, from the
Zamolodchikov formula,Zamolodchikov:1995xk ; Lukyanov:1996jj which coincides
with the lightest bound state of soliton-antisoliton(the first breather),
expressed in terms of a soliton of mass $M$ via:
$\displaystyle m\sim 2M\sin(\frac{\pi}{2}\frac{\beta^{2}}{(8\pi-\beta^{2})}),$
(141)
and the soliton mass $M$ is determined by $g,\beta$:
$\displaystyle
g\sim\frac{2\,\Gamma({\frac{\beta^{2}}{8\pi}})}{\pi\,\Gamma({1-\frac{\beta^{2}}{8\pi}})}\big{(}M\frac{\sqrt{\pi}\,\Gamma(\frac{1}{2}+\frac{\beta^{2}}{2(8\pi-\beta^{2})})}{2\,\Gamma(\frac{\beta^{2}}{2(8\pi-\beta^{2})})}\big{)}^{2-\frac{\beta^{2}}{4\pi}}.\;\;\;\;\;\;\;\;\;$
(142)
On the other hand, the sine-Gordon action is an integrable model, and can be
also studied by Bethe ansatz. By all means, it is well-known that the two-
point correlator exponentially decays, indicating the mass gap exists.
$\bullet$2\. Renormalization Group(RG) analysis on the sine-Gordon model of
(E-4): It is known that the 2-dimensional XY model, neutral Coulomb gas, and
sine-Gordon model, these three models describe the same universality class (up
to some Euclidean time transformation from 1+1D to 2D). The 2-dimensional XY
model with $J=\frac{1}{8\pi^{2}\kappa}$ matches the universality class of
Eq.(136) by a Hamiltonian
$H_{xy}=-J\sum_{\langle i,j\rangle}\cos(\theta_{i}-\theta_{j}).$ (143)
Its high temperature phase(small $J$) has the exponential-decaying two-point
spin-spin correlator
$\displaystyle\langle\mathbf{S}(0)\mathbf{S}(r)\rangle\sim\langle\cos(\theta_{0}-\theta_{r})\rangle\sim(J/2)^{|r|}\sim\exp(-|r|/\xi).\;\;\;\;\;$
(144)
with the correlation length
$\xi=(\ln(2/J))^{-1}=(\ln(16\pi^{2}\kappa))^{-1}.$ (145)
This high temperature phase of XY model is dual to the high temperature phase
(small $J$) of the neutral Coulomb gas with a two dimensional logarithmic
potential energy form:
$-4\pi^{2}J\sum_{i<J}n_{i}n_{j}\ln(r_{i}-r_{j})+\dots$ (146)
where $n_{i}$ are the charge density($n_{i}=\pm 1,\pm 2,\dots$, with the
totally neutral charge), and $\dots$ are unwritten terms containing the core
energy of charges and the core energy of smooth configurations without vortex
singularity. The Coulomb gas at high T is the metallic plasma phase, the
Coulomb charge interaction is screened, thus the effective interaction becomes
_exponentially decaying_. On the other hand, at low temperature phase (large
$J$), the interaction is strong and the vortices are bound together as
dipoles.
It can be also studied from the fermionization-bosonization language. While
the four-fermion interactions via the forward scattering term and the
dispersion term can be bosonized to a free boson theory through changing the
compactified radius of bosons, the four-fermion interactions via the backward
scattering term and the Umklapp scattering term can be bosonized to induce the
cosine term, which can generate the mass gap at strong interaction(large $g$).
The above indicates that when the coupling $g$ grows, the RG flows to a
massive gapped phase, but those perturbation analysis are done by the
perturbation from the free or the weak-coupling theory. Below we provide a new
demonstration explicitly here from the strong-coupling fixed point.
$\bullet$3\. RG analysis at the strong-coupling fixed point: By assuming the
perturbation is done on any of the strong-coupling fixed point of gapped
phases (there can be more than one fixed point of massive phases), we consider
at the large coupling $g$, the scalar field is pinned down at the minimum of
cosine potential, we thus will consider the dominant term as the
$g\cos(\beta\varphi)$ on the discretized spatial lattice and only a continuous
time:
$\int dt\,\big{(}\sum_{i}\frac{1}{2}\,g\,(\varphi_{i})^{2}+\dots\big{)}$ (147)
Setting this dominant term to be a marginal operator means the scaling
dimension of $\varphi_{i}$ is
$[\varphi_{i}]=1/2.$
Any operator with $(\varphi_{i})^{n}$ for $n>2$ is an irrelevant operator. The
kinetic term can be generated by an operator:
$e^{\text{i}P_{\varphi}a}\sim e^{\text{i}a\partial_{x}\varphi}\sim
e^{\text{i}(\varphi_{i+1}-\varphi_{j})}$ (148)
where $P$ is the conjugate momentum of the zero mode $\varphi_{0}$ and $a$ is
the lattice spacing, since $e^{\text{i}P_{\varphi}a}$ generates the lattice
translation by
$e^{\text{i}P_{\varphi}a}\varphi_{0}e^{-\text{i}P_{\varphi}a}=\varphi_{0}+a.$
(149)
But the kinetic term, which contains
$e^{\text{i}(\varphi_{i+1}-\varphi_{j})}$, has an _infinite scaling dimension_
due to infinite power of $\varphi$ fields. Thus it is _irrelevant_ operator in
the sense of RG at the strong-coupling fixed point.
We should remark that this above RG analysis at the strong-coupling fixed
point shows the kinetic energy is irrelevant respect to the dominant
$g\cos(\beta\varphi)$ potential, independent to the $\beta$ value. This is
remarkable because the RG analysis around the free theory fixed point has
$\beta$ value dependence. In particular, the scaling dimensions of the normal
ordered $:\cos(\beta\varphi):$ of Eq.(136) and $:\cos(\beta\theta):$of
Eq.(137) is
$[\cos(\beta\varphi)]=\frac{\beta^{2}}{4\pi\kappa},\;\;\;\cos(\beta\theta)=\frac{\beta^{2}}{2},$
and the weak-coupling RG analysis shows that $g$ flows to a large coupling $g$
when $\frac{\beta^{2}}{4\pi\kappa}<2$, $\frac{\beta^{2}}{2}<2$. However, at
non-perturbative strong-coupling (lattice-scale) regime, it is believed that
the result is insensitive to $\beta$ value. As we have shown from the strong-
coupling fixed point analysis, we believe that the $\beta$-independence result
is correct.
To summarize, we show that such an irrelevant operator of kinetic term cannot
destroy the massive gapped phases at the strong-coupling fixed point, thus the
mass gap remains robust, independent to the $\beta$ value.
### E.4 Approach II: Map the anomaly-free theory with gapping terms to the
decoupled non-Chiral Luttinger liquids with gapped spectrum
Here we provide the second approach to show that the anomaly-free edge states
can be gapped under the properly-designed gapping terms. The key step is that
we will map the $N$-component anomaly-free theory with properly-designed
gapping terms to $N/2$-decoupled-copies of non-Chiral Luttinger liquids of the
statement (E-4), each copy has the gapped spectrum. (This key step is
logically the same as the proof in Appendix A of Ref.Wang:2013vna, .) Thus, by
the equivalence mapping, we can prove that the anomaly-free edge states can be
fully gapped. We include this proofWang:2013vna to make our claim self-
contained.
We again consider the generic theory of Eq.(120):
$\displaystyle S_{\partial}(\Phi,K,\\{\ell_{a}\\})=\frac{1}{4\pi}\int
dt\;dx\;(K_{IJ}\partial_{t}\Phi_{I}\partial_{x}\Phi_{J}$ $\displaystyle-
V_{IJ}\partial_{x}\Phi_{I}\partial_{x}\Phi_{J})+\int
dt\;dx\;\sum_{a}g_{a}\cos(\ell_{a,I}\cdot\Phi_{I}),$
where $\Phi$, $K$, $\\{\ell_{a}\\}$ are the data for this 1+1D action
$S_{\partial}(\Phi,K,\\{\ell_{a}\\})$, while the velocity matrix is not
universal and is immaterial to our discussion below. In Appendix D, we had
shown that the $N$-component anomaly-free theory guarantees the $N/2$-linear
independent gapping terms of boundary gapping lattice(Lagrangian subgroup)
$\Gamma^{\partial}$ satisfying:
$\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}=0$ (150)
for any $\ell_{a},\ell_{b}\in\Gamma^{\partial}$. In our case (both bosonic and
fermionic theory), all the $K$ is invertible due to $\det(K)\neq 0$, thus one
can define a dual vector as in Ref.Wang:2013vna, ,
$\ell_{a,I}=K_{II^{\prime}}\eta_{a,I^{\prime}}$, such that Eq.(150) becomes
$\eta_{a,I^{\prime}}K_{IJ}\eta_{b,J^{\prime}}=0.$ (151)
The data of action becomes $S_{\partial}(\Phi,K,\\{\ell_{a}\\})\to
S_{\partial}(\Phi,K,\\{\eta_{a}\\})$. In our proof, we will stick to the data
$S_{\partial}(\Phi,K,\\{\ell_{a}\\})$. We can construct a
$N\times(N/2)$-component matrix $\mathbf{L}$:
$\mathbf{L}\equiv\Big{(}\ell_{1},\ell_{2},\dots,\ell_{N/2}\Big{)}$ (152)
with $N/2$ column vectors, and each column vector is
$\ell_{1},\ell_{2},\dots,\ell_{N/2}$. We can write $\mathbf{L}$ base on the
Smith normal form, so $\mathbf{L}=VDW$, with $V$ is a $N\times N$ integer
matrix and $W$ is a $(N/2)\times(N/2)$ integer matrix. Both $V$ and $W$ have a
determinant $\det(V)=\det(W)=1$. The $D$ is a $N\times(N/2)$ integer matrix:
$D\equiv\begin{pmatrix}\bar{D}\\\
0\end{pmatrix}\equiv\begin{pmatrix}d_{1}&0&\dots&0\\\ 0&d_{2}&\dots&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&\dots&d_{N/2}\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&\vdots&0\end{pmatrix},$ (153)
with $\bar{D}$ is a diagonal integer matrix. Since $\mathbf{L}$ has
$N/2$-linear-independent column vectors, thus $\det(\bar{D})\neq 0$, and all
entries of $\bar{D}$ are nonzero.
1st Mapping \- We do a change of variables:
$\displaystyle\Phi^{\prime}$ $\displaystyle=$ $\displaystyle V^{T}\Phi$
$\displaystyle\ell^{\prime}$ $\displaystyle=$ $\displaystyle V^{-1}\ell$
$\displaystyle K^{\prime}$ $\displaystyle=$ $\displaystyle
V^{-1}K(V^{T})^{-1}$ $\displaystyle S_{\partial}(\Phi,K,\\{\ell_{a}\\})$
$\displaystyle\to$ $\displaystyle
S_{\partial}(\Phi^{\prime},K^{\prime},\\{\ell_{a}^{\prime}\\})\;\;\;\;$
This makes the $\mathbf{L}^{\prime}$ form simpler:
$\displaystyle\mathbf{L}^{\prime}=V^{-1}\mathbf{L}=V^{-1}(VDW)=\begin{pmatrix}\bar{D}W\\\
0\end{pmatrix}.$ (154)
Here is the key step: due to Eq.(150), we have the important equality,
$\displaystyle\boxed{\mathbf{L}^{T}K^{-1}\mathbf{L}=0},$ (155)
thus
$\displaystyle(VDW)^{T}K^{-1}VDW=0$ (156)
$\displaystyle=W^{T}D^{T}K^{\prime-1}DW=0$ (157)
$\displaystyle=(\bar{D}W,0)K^{\prime-1}\begin{pmatrix}\bar{D}W\\\
0\end{pmatrix}=0$ (158)
Hence, $K^{\prime-1}$ can be written as the following four blocks of $N\times
N$ matrices $\text{F},\text{G}$ ($\text{F},\text{G}$ can have fractional
values):
$K^{\prime-1}=\begin{pmatrix}0&\text{F}\\\
\text{F}^{T}&\text{G}\end{pmatrix},$ (159)
with $\det(\text{F})\neq 0$ and G is symmetric. Thus the _integer_
$K^{\prime}$ matrix has the form
$K^{\prime}=\begin{pmatrix}-(\text{F}^{T})^{-1}\text{G}\text{F}^{-1}&(\text{F}^{T})^{-1}\\\
\text{F}^{-1}&0\end{pmatrix}.$ (160)
We notice that,
Lemma 1: Due to $K^{\prime}$ matrix is an _integer_ matrix, the three matrices
$-(\text{F}^{T})^{-1}\text{G}\text{F}^{-1}$, $\text{F}^{-1}$ and
$(\text{F}^{T})^{-1}$ are _integer matrices_. Therefore, $\text{F},\text{G}$
can be _fractional matrices_.
2nd Mapping \- To obtain the final mapping to $N/2$-decoupled-copies of non-
chiral Luttinger liquids, we do another change of variables:
$\displaystyle\Phi^{\prime\prime}$ $\displaystyle=$ $\displaystyle
U\Phi^{\prime}$ $\displaystyle\ell^{\prime\prime}$ $\displaystyle=$
$\displaystyle(U^{-1})^{T}\ell^{\prime}$ $\displaystyle K^{\prime\prime}$
$\displaystyle=$ $\displaystyle(U^{T})^{-1}K^{\prime}(U)^{-1}$ $\displaystyle
S_{\partial}(\Phi^{\prime},K^{\prime},\\{\ell_{a}^{\prime}\\})$
$\displaystyle\to$ $\displaystyle
S_{\partial}(\Phi^{\prime\prime},K^{\prime\prime},\\{\ell_{a}^{\prime\prime}\\})\;\;\;\;$
With the goal in mind to make the new K matrix
$K^{\prime\prime}=(K^{\prime\prime})^{-1}=\begin{pmatrix}0&\mathbf{1}\\\
\mathbf{1}&0\end{pmatrix}$ and $\mathbf{1}$ is the $N\times N$ identity
matrix. This constrains $U$, and we find
$\displaystyle(K^{\prime\prime})^{-1}=U(K^{\prime})^{-1}U^{T}=\begin{pmatrix}0&\mathbf{1}\\\
\mathbf{1}&0\end{pmatrix}$ (161) $\displaystyle\Rightarrow
U=\begin{pmatrix}-\frac{1}{2}(\text{F}^{T})^{-1}\text{G}\text{F}^{-1}&(\text{F}^{T})^{-1}\\\
\mathbf{1}&0\end{pmatrix}$ (162)
Importantly, due to Lemma 1, we have $(\text{F}^{T})^{-1}$ and
$-(\text{F}^{T})^{-1}\text{G}\text{F}^{-1}$ are _integer matrices_ , so U is
at most a matrix taking half-integer values(almost an integer matrix).
In the new $\Phi^{\prime\prime}$ basis, we define the $N$-component column
vector
$\Phi^{\prime\prime}=(\bar{\phi}_{1},\bar{\phi}_{2},\dots,\bar{\phi}_{N/2},\bar{\theta}_{1},\bar{\theta}_{2},\dots,\bar{\theta}_{N/2}).$
Based on Appendix E.1, the canonical-quantization in the new basis becomes
$\displaystyle[\Phi_{I}^{\prime\prime}(x_{1}),\partial_{x}\Phi_{J}^{\prime\prime}(x_{2})]={2\pi}\text{i}({K^{\prime\prime}}^{-1})_{IJ}\delta(x_{1}-x_{2}),$
$\displaystyle[\bar{\phi}_{I}(x_{1}),\partial_{x}\bar{\phi}_{J}(x_{2})]=[\bar{\theta}_{I}(x_{1}),\partial_{x}\bar{\theta}_{J}(x_{2})]=0,\;\;$
$\displaystyle[\bar{\phi}_{I}(x_{1}),\partial_{x}\bar{\theta}_{J}(x_{2})]={2\pi}\text{i}\delta_{IJ}\delta(x_{1}-x_{2}).$
(163)
This is exactly what we aim for the decoupled non-chiral Luttinger liquids as
the form of $N/2$-copies of (E-4). However, the cosine potential in the new
basis is not yet fully decoupled due to
$\displaystyle\ell^{\prime\prime
T}\Phi^{\prime\prime}=\ell^{T}(V^{-1})^{T}(U^{-1})\Phi^{\prime\prime}$
$\displaystyle\Rightarrow\mathbf{L}^{\prime\prime
T}=\mathbf{L}^{T}(V^{-1})^{T}(U^{-1})=(W^{T}D^{T})(U^{-1})$
$\displaystyle\Rightarrow\mathbf{L}^{\prime\prime
T}=\begin{pmatrix}W^{T}\bar{D},0\end{pmatrix}\begin{pmatrix}0&\mathbf{1}\\\
\text{F}^{T}&\frac{1}{2}\text{G}\text{F}^{-1}\end{pmatrix}=\begin{pmatrix}0,W^{T}\bar{D}\end{pmatrix}.$
We obtain the cosine potential term as
$g_{a}\cos(\ell_{a,I}\cdot\Phi_{I})=g_{a}\cos(W_{Ja}d_{J}\bar{\theta}_{J}).$
(164)
If $W_{Ja}d_{J}$ is a diagonal matrix, the non-chiral Luttinger liquids are
decoupled into $N/2$-copies also in the interacting potential terms. In
general, $W_{Ja}d_{J}$ may not be diagonal, but the charge quantization and
the large coupling $g_{a}$ of the cosine potentials cause
$\sum_{J}W_{Ja}d_{J}\bar{\theta}_{J}=2\pi
n_{I},\;\;I=1,\dots,N/2,\;\;n_{I}\in\mathbb{Z}$
locked to the minimum value. Equivalently, due to both $W$ and $W^{-1}$ are
integer matrices ($\det(W)=1$), we have
$d_{J}\bar{\theta}_{J}=2\pi
n_{J}^{\prime},\;\;J=1,\dots,N/2,\;\;n_{J}^{\prime}\in\mathbb{Z}.$ (165)
The last step is to check the constraint on the $\bar{\phi}_{I}$ and
$\bar{\theta}_{J}$. The original particle number quantization constraint
changes from $\frac{1}{2\pi}\int^{L}_{0}{\partial_{x}\Phi_{I}}=\zeta_{I}$ with
an integer $\zeta_{I}\in\mathbb{Z}$, to
$\displaystyle\left\\{\begin{array}[]{l}\int^{L}_{0}\frac{{\partial_{x}\bar{\phi}_{I}}}{2\pi}=-\frac{1}{2}((\text{F}^{T})^{-1}\text{G}\text{F}^{-1}V^{T})_{Ij}\zeta_{j}+\overset{N/2}{\underset{j=1}{\sum}}(\text{F}^{T})^{-1}_{I,j}\zeta_{N/2+j}\\\
\int^{L}_{0}\frac{{\partial_{x}\bar{\theta}_{I}}}{2\pi}=\overset{N/2}{\underset{j=1}{\sum}}V^{T}_{I,I+j}\zeta_{j}\end{array}\right.$
(168)
Again, from Lemma 1, we have $(\text{F}^{T})^{-1}$ and
$-(\text{F}^{T})^{-1}\text{G}\text{F}^{-1}$ are _integer matrices_ , and $V$
is an integer matrix, so at least the particle number quantization of
$\int^{L}_{0}\frac{{\partial_{x}\bar{\phi}_{I}}}{2\pi}$ takes as multiples of
half-integer values, due to the half-integer valued matrix term
$\frac{1}{2}((\text{F}^{T})^{-1}\text{G}\text{F}^{-1}V^{T})$. Meanwhile,
$\int^{L}_{0}\frac{{\partial_{x}\bar{\theta}_{I}}}{2\pi}$ must have integer
values.
In the following, we verify that the physics at strong coupling $g$ of cosine
potentials still render the decoupled non-Chiral Luttinger liquids with
integer particle number quantization regardless a possible half-integer
quantization at Eq.(168). The reason is that, at large $g$, the cosine
potential $g_{a}\cos(W_{Ja}d_{J}\bar{\theta}_{J})$ effectively acts as
$g_{a}\cos(d_{a}\bar{\theta}_{a})$. In this way, $\bar{\theta}_{a}$ is locked,
so $\partial_{x}\bar{\theta}_{a}=0$ and that constrains
$\int^{L}_{0}\frac{{\partial_{x}\bar{\theta}_{I}}}{2\pi}=0$ with no instanton
tunneling. This limits Eq.(168)’s $\zeta_{j}=0$ for $j=1,\dots,N/2$. And
Eq.(168) at large $g$ coupling becomes
$\displaystyle\left\\{\begin{array}[]{l}\int^{L}_{0}\frac{{\partial_{x}\bar{\phi}_{I}}}{2\pi}=\overset{N/2}{\underset{j=1}{\sum}}(\text{F}^{T})^{-1}_{I,j}\zeta_{N/2+j}\in\mathbb{Z}.\\\
\int^{L}_{0}\frac{{\partial_{x}\bar{\theta}_{I}}}{2\pi}=0.\end{array}\right.$
(171)
We now conclude that, the allowed Hilbert space at large $g$ coupling is the
same as the Hilbert space of $N/2$-decoupled-copies of non-Chiral Luttinger
liquids.
Though we choose a different basis for the gapping rules than
Ref.Wang:2013vna, , we still reach the same conclusion as long as the key
criteria Eq.(155) holds. Namely, with $\mathbf{L}^{T}K^{-1}\mathbf{L}=0$, we
can derive these three equations Eq.(E.4),(165),(171), thus we have mapped the
theory with gapping terms (constrained by $\mathbf{L}^{T}K^{-1}\mathbf{L}=0$)
to the $N/2$-decoupled-copies of non-Chiral Luttinger liquids with $N/2$
number of effective decoupled gapping terms $\cos(d_{J}\bar{\theta}_{J})$ with
$J=1,\dots,N/2$. This maps to $N/2$-copies of non-Chiral Luttinger liquids
(E-4), and we have shown that each (E-4) has the gapped spectrum. We prove the
mapping:
the $K$ matrix multiplet-chirla boson theories with gapping terms
$\mathbf{L}^{T}K^{-1}\mathbf{L}=0$ $\to$ $N/2$-decoupled-copies of non-Chiral
Luttinger liquids of (E-4) with energy gapped spectra.(Q.E.D.)
Since we had shown in Appendix D that for the U(1) theory of totally even-$N$
left/right chiral Weyl fermions, only the anomaly-free theory can provide the
$N/2$-gapping terms with $\mathbf{L}^{T}K^{-1}\mathbf{L}=0$, this means that
we have established the map:
the U(1)N/2 anomaly-free theory
($\mathbf{q}\cdot{K}^{-1}\cdot\mathbf{q}=\mathbf{t}\cdot{K}\cdot\mathbf{t}=0$)
with gapping terms $\mathbf{L}^{T}K^{-1}\mathbf{L}=0$ $\to$ $N/2$-decoupled-
copies of non-Chiral Luttinger liquids of (E-4) with gapped energy spectra.
This concludes the second approach proving the 1+1D U(1)-anomaly-free theory
can be gapped by adding properly designed interacting gapping terms with
$\mathbf{L}^{T}K^{-1}\mathbf{L}=0$. (Q.E.D.)
### E.5 Approach III: Non-Perturbative statements of Topological Boundary
Conditions, Lagrangian subspace, and the exact sequence
In this subsection, from a TQFT viewpoint, we provide another non-perturbative
proof of Topological Boundary Gapping Rules (which logically follows
Ref.Kapustin:2010hk, )
$\boxed{\mathbf{L}^{T}K^{-1}\mathbf{L}=0},$ (172)
with
$\mathbf{L}\equiv\Big{(}\ell_{1},\ell_{2},\dots,\ell_{N/2}\Big{)}$ (173)
with $N/2$ column vectors, and each column vector is
$\ell_{1},\ell_{2},\dots,\ell_{N/2}$; the even-$N$-component left/right chiral
Weyl fermion theory with Topological Boundary Gapping Rules must have
$N/2$-linear independent gapping terms of Boundary Gapping Lattice(Lagrangian
subgroup) $\Gamma^{\partial}$ satisfying: $\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}=0$
for any $\ell_{a},\ell_{b}\in\Gamma^{\partial}$.
Here is the general idea: For any field theory, a boundary condition is
defined by a Lagrangian submanifold in the space of Cauchy boundary condition
data on the boundary. If we want a boundary condition which is topological
(namely with a mass gap without gapless modes), then importantly it must treat
all directions on the boundary in the equivalent way. So, for a gauge theory,
we end up choosing a Lagrangian subspace in the Lie algebra of the gauge
group. A subspace is Lagrangian _if and only if_ it is both isotropic and
coisotropic. For a finite-dimensional vector space $\mathbf{V}$, a Lagrangian
subspace is an isotropic one whose dimension is half that of the vector space.
More precisely, for $\mathbf{W}$ be a linear subspace of a finite-dimensional
vector space $\mathbf{V}$. Define the symplectic complement of $\mathbf{W}$ to
be the subspace $\mathbf{W}^{\perp}$ as
$\mathbf{W}^{\perp}=\\{v\in\mathbf{V}\mid\omega(v,w)=0,\;\;\;\forall
w\in\mathbf{W}\\}$ (174)
Here $\omega$ is the symplectic form, in the commonly-seen matrix form is
$\omega=(\begin{smallmatrix}0&\mathbf{1}\\\ -\mathbf{1}&0\end{smallmatrix})$
with $0$ and $\mathbf{1}$ are the block matrix of the zero and the identity.
In our case, $\omega$ is related to the fermionic $K=K^{f}$ and bosonic
$K=K^{b0}$ matrices. The symplectic complement $\mathbf{W}^{\perp}$ satisfies:
$\displaystyle(\mathbf{W}^{\perp})^{\perp}=\mathbf{W},\;\;$
$\displaystyle\dim\mathbf{W}+\dim\mathbf{W}^{\perp}=\dim\mathbf{V}.$
Isotropic, coisotropic, Lagrangian means the following:
$\bullet$ $\mathbf{W}$ is isotropic if
$\mathbf{W}\subseteq\mathbf{W}_{\perp}$. This is true if and only if $\omega$
restricts to $0$ on $\mathbf{W}$.
$\bullet$ $\mathbf{W}$ is coisotropic if
$\mathbf{W}_{\perp}\subseteq\mathbf{W}$. $\mathbf{W}$ is coisotropic if and
only if $\omega$ has a nondegenerate form on the quotient space
$\mathbf{W}/\mathbf{W}_{\perp}$. Equivalently $\mathbf{W}$ is coisotropic if
and only if $\mathbf{W}_{\perp}$ is isotropic.
$\bullet$ $\mathbf{W}$ is Lagrangian if and only if it is both isotropic and
coisotropic, namely, if and only if $\mathbf{W}=\mathbf{W}_{\perp}$. In a
finite-dimensional $\mathbf{V}$, a Lagrangian subspace $\mathbf{W}$ is an
isotropic one whose dimension is half that of $\mathbf{V}$.
With this understanding, following Ref.Kapustin:2010hk, , we consider a U(1)N
Chern-Simons theory, whose bulk action is
$S_{bluk}=\frac{K_{IJ}}{4\pi}\int_{\mathcal{M}}a_{I}\wedge da_{J}.$ (175)
and the boundary action for a manifold $\mathcal{M}$ with a boundary
${\partial\mathcal{M}}$ (with the restricted $a_{\parallel,I}$ on
${\partial\mathcal{M}}$ ) is
$S_{\partial}=\delta S_{bluk}=\frac{K_{IJ}}{4\pi}\int_{\mathcal{M}}(\delta
a_{\parallel,I})\wedge da_{\parallel,J}.$ (176)
The symplectic form $\omega$ is given by the K-matrix via the differential of
this 1-form $\delta S_{bluk}$
$\omega=\frac{K_{IJ}}{4\pi}\int_{\mathcal{M}}(\delta a_{\parallel,I})\wedge
d(\delta a_{\parallel,J}).$ (177)
The gauge group U(1)N can be viewed as the torus $\mathbb{T}_{\Lambda}$, as
the quotient space of $N$-dimensional vector space $\mathbf{V}$ by a subgroup
$\Lambda\cong\mathbb{Z}^{N}$. Namely
$\text{U(1)}^{N}\cong\mathbb{T}_{\Lambda}\cong(\Lambda\otimes\mathbb{R})/(2\pi\Lambda)\equiv\mathbf{t}_{\Lambda}/(2\pi\Lambda)$
(178)
Locally the gauge field $a$ is a 1-form, which has values in the Lie algebra
of $\mathbb{T}_{\Lambda}$, we will denote this Lie algebra
$\mathbf{t}_{\Lambda}$ as the vector space
$\mathbf{t}_{\Lambda}=\Lambda\otimes\mathbb{R}$.
A self-consistent boundary condition must define a Lagrangian submanifold with
respect to this symplectic form $\omega$ and must be local. (For example, the
famous chiral boson theory has $a_{\bar{z}}=0$ along the complex coordinate
$\bar{z}$. This defines a consistent boundary condition, but it is not
topological.)
In addition, a topological boundary gapping condition must be invariant in
respect of the orientation-preserving diffeomorphism of $\mathcal{M}$. A local
diffeomorphism-invariant constraint on the Lie algebra
$\mathbf{t}_{\Lambda}$-valued 1-form $a_{\parallel,I}$ demands it to live in
the subspace of $\mathbf{t}_{\Lambda}$. This corresponds to the if and only if
conditions that:
$\bullet(i)$ The subspace is isotropic with respect to the symmetric bilinear
form $K$.
$\bullet(ii)$ The subspace dimension is a half of the dimension of
$\mathbf{t}_{\Lambda}$.
$\bullet(iii)$ The signature of $K$ is zero. This means that $K$ has the same
number of positive and negative eigenvalues.
We notice that $\bullet(ii)$ is true for our boundary gapping lattice,
$\mathbf{L}\equiv\Big{(}\ell_{1},\ell_{2},\dots,\ell_{N/2}\Big{)}$, where
there are $N/2$-linear independent gapping terms. And $\bullet(iii)$ is true
for our bosonic $K_{b0}$ and fermionic $K_{f}$ matrices. Importantly, for
topological gapped boundary conditions, $a_{\parallel,I}$ lies in a Lagrangian
subspace of $\mathbf{t}_{\Lambda}$ implies that the boundary gauge group is a
Lagrangian subgroup. (Here we consider the boundary gauge group is connected
and continuous; one can read Section 6 of Ref.Kapustin:2010hk, for the case
of more general disconnected or discrete boundary gauge group.)
The bulk gauge group is $\mathbb{T}_{\Lambda}$, and we denote the boundary
gauge group as $\mathbb{T}_{\Lambda_{0}}$, which $\mathbb{T}_{\Lambda_{0}}$ is
a Lagrangian subgroup of $\mathbb{T}_{\Lambda}$ for topological gapped
boundary conditions.
Here the torus $\mathbb{T}_{\Lambda}$ can be decomposed into a product of
$\mathbb{T}_{\Lambda_{0}}$ and other torus. $\Lambda\cong\mathbb{Z}^{N}$
contains the subgroup $\Lambda_{0}$, and $\Lambda$ contains a direct sum of
$\Lambda_{0}$. These form an exact sequence:
$\displaystyle
0\to\Lambda_{0}\overset{\mathbf{h}}{\to}\Lambda\to\Lambda/\Lambda_{0}\to 0$
(179)
Here $0$ means the trivial Abelian group with only the identity, or the zero-
dimensional vector space. The exact sequence means that a sequence of maps
$\text{f}_{i}$ from domain $A_{i}$ to $A_{i+1}$:
$\text{f}_{i}:A_{i}\to A_{i+1}$
satisfies a relation between the image and the kernel:
$\text{Im}(A_{i})=\text{Ker}(A_{i+1}).$
Here we have $\mathbf{h}$ as an injective map from $\Lambda_{0}$ to $\Lambda$:
$\Lambda_{0}\overset{\mathbf{h}}{\to}\Lambda.$
Since $\Lambda$ is a rank-$N$ integer matrix generating a $N$-dimensional
vector space, and $\Lambda_{0}$ is a rank-$N/2$ integer matrix generating a
$N/2$-dimensional vector space; we have $\mathbf{h}$ as an integral matrix of
$N\times(N/2)$-components.
The transpose matrix $\mathbf{h}^{T}$ is an integral matrix of $(N/2)\times
N$-components. $\mathbf{h}^{T}$ is a surjective map:
$\Lambda^{*}\overset{\mathbf{h}^{T}}{\to}\Lambda_{0}^{*}.$
Some mathematical relations are
$\Lambda_{0}=H_{1}(\mathbb{T}_{\Lambda_{0}},\mathbb{Z})$ ,
$\mathop{\mathrm{Hom}}(\mathbb{T}_{\Lambda_{0}},\text{U}(1))=\Lambda_{0}^{*}$,
$\mathop{\mathrm{Hom}}(\mathbb{T}_{\Lambda},\text{U}(1))=\Lambda^{*}$. Here
$H_{1}(\mathbb{T}_{\Lambda_{0}},\mathbb{Z})$ is the first homology group of
$\mathbb{T}_{\Lambda_{0}}$ with a $\mathbb{Z}$ coefficient.
$\mathop{\mathrm{Hom}}(X,Y)$ is the set of all module homomorphisms from the
module $X$ to the module $Y$.
Furthermore, for $\mathbf{t}_{\Lambda}^{*}$ being the dual of the Lie algebra
$\mathbf{t}_{\Lambda}$, one can properly define the Topological Boundary
Conditions by restricting the values of boundary gauge fields (taking values
in Lie algebra $\mathbf{t}_{\Lambda}^{*}$ or $\mathbf{t}_{\Lambda}$), and one
can obtain the corresponding exact sequence by choosing the following
splitting of the vector space $\mathbf{t}_{\Lambda}^{*}$:Kapustin:2010hk
$\displaystyle
0\to\mathbf{t}_{(\Lambda/\Lambda_{0})}^{*}\to\mathbf{t}_{\Lambda}^{*}\to\mathbf{t}_{\Lambda_{0}}^{*}\to
0.$ (180)
Now we can examine the if and only if conditions
$\bullet(i)$,$\bullet(ii)$,$\bullet(iii)$ listed earlier in this Section E.5:
For $\bullet(ii)$, “the subspace dimension is a half of the dimension of
$\mathbf{t}_{\Lambda}$” is true, because $\Lambda_{0}$ is a rank-$N/2$ integer
matrix.
For $\bullet(iii)$, “the signature of $K$ is zero” is true, because our
$K_{b0}$ and fermionic $K_{f}$ matrices implies that we have same number of
left moving modes ($N/2$) and right moving modes ($N/2$), with $N\in
2\mathbb{Z}^{+}$ an even number.
For $\bullet(i)$ “The subspace is isotropic with respect to the symmetric
bilinear form $K$” to be true, we have an extra condition on ${\mathbf{h}}$
matrix for the $K$ matrix:
$\displaystyle\boxed{{\mathbf{h}^{T}}K{\mathbf{h}}=0}$ (181)
Since $K$ is invertible($\det(K)\neq 0$), by defining $\mathbf{L}\equiv
K{\mathbf{h}}$, we have an equivalent condition:
$\boxed{\mathbf{L}^{T}K^{-1}\mathbf{L}=0},$ (182)
These above conditions $\bullet(i)$,$\bullet(ii)$,$\bullet(iii)$ are
equivalent to the boundary full gapping rules: Either written in the column
vector of ${\mathbf{h}}$ matrix
(${\mathbf{h}}\equiv\Big{(}\eta_{1},\eta_{2},\dots,\eta_{N/2}\Big{)}$):
$\eta_{a,I^{\prime}}K_{I^{\prime}J^{\prime}}\eta_{b,J^{\prime}}=0.$ (183)
or written in the column vector of ${\mathbf{L}}$ matrix
($\mathbf{L}\equiv\Big{(}\ell_{1},\ell_{2},\dots,\ell_{N/2}\Big{)}$):
$\ell_{a,I}K^{-1}_{IJ}\ell_{b,J}=0$ (184)
for any $\ell_{a},\ell_{b}\in\Gamma^{\partial}$ of boundary gapping
lattice(Lagrangian subgroup).
To summarize, in this subsection, we provide a third approach from a non-
Perturbative TQFT viewpoint to prove that, for $\text{U}(1)^{N}$-Chern-Simons
theory, Topological Boundary Conditions hold _if and only if_ the boundary
interaction terms satisfy Topological Boundary Fully Gapping Rules.(Q.E.D.)
black
## Appendix F More about Our Lattice Hamiltonian Chiral Matter Models
### F.1 More details on our Lattice Model producing nearly-flat Chern-bands
We fill more details on our lattice model presented in Sec.III.1.2 for the
free-kinetic part. The lattice model shown in Fig.2 has two sublattice
$a$(black dots), $b$(white dots). In momentum space, we have a generic
pseudospin form of Hamiltonian $H(\mathbf{k})$,
$H(\mathbf{k})=B_{0}(\mathbf{k})+\vec{B}(\mathbf{k})\cdot\vec{\sigma}.$ (185)
$\vec{\sigma}$ are Pauli matrices $(\sigma_{x},\sigma_{y},\sigma_{z})$. In
this model $B_{0}(\mathbf{k})=0$ and $\vec{B}$ have three components:
$\displaystyle B_{x}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
2t_{1}\cos(\pi/4)(\cos(k_{x}a_{x})+\cos(k_{y}a_{y}))$ $\displaystyle
B_{y}(\mathbf{k})$ $\displaystyle=$ $\displaystyle
2t_{1}\sin(\pi/4)(\cos(k_{x}a_{x})-\cos(k_{y}a_{y}))$ (186) $\displaystyle
B_{z}(\mathbf{k})$ $\displaystyle=$
$\displaystyle-4t_{2}\sin(k_{x}a_{x})\sin(k_{y}a_{y}).$
In Fig.4(a), the energy spectrum $\mathop{\mathrm{E}}(k_{x})$ is solved from
putting the system on a 10-sites width ($9a_{y}$-width) cylinder. Indeed the
energy spectrum $\mathop{\mathrm{E}}(k_{x})$ in Fig.4(b) is as good when
putting on a smaller size system such as the ladder (Fig.2(c)). The cylinder
is periodic along $\hat{x}$ direction so $k_{x}$ momentum is a quantum number,
while $\mathop{\mathrm{E}}(k_{x})$ has real-space $y$-dependence along the
finite-width $\hat{y}$ direction. Each band of $\mathop{\mathrm{E}}(k_{x})$ in
Fig.4 is solved by exactly diagonalizing $H(k_{x},y)$ with $y$-dependence. By
filling the lower energy bands and setting the chemical potential at zero, we
have Dirac fermion dispersion at $k_{x}=\pm\pi$ for the edge state spectrum,
shown as the blue curves in Fig.4(a)(b).
In Fig.4(c), we plot the density $\langle f^{\dagger}f\rangle$ of the edge
eigenstate on the ladder (which eigenstate is the solid blue curve in
Fig.4(b)), for each of two edges A and B, and for each of two sublattice $a$
and $b$. One can fine tune $t_{2}/t_{1}$ such that the edge A and the edge B
have the least mixing. The least mixing implies that the left edge and right
edge states nearly decouple. The least mixing is very important for the
interacting $G_{1},G_{2}\neq 0$ case, so we can impose interaction terms on
the right edge B only as in Eq.(II), decoupling from the edge A. We can
explicitly make the left edge A density $\langle
f_{\mathop{\mathrm{A}}}^{\dagger}f_{\mathop{\mathrm{A}}}\rangle$ dominantly
locates in $k_{x}<0$, the right edge B density $\langle
f_{\mathop{\mathrm{B}}}^{\dagger}f_{\mathop{\mathrm{B}}}\rangle$ dominantly
locates in $k_{x}>0$. The least mixing means the eigenstate is close to the
form
$|\psi(k_{x})\rangle=|\psi_{k_{x}<0}\rangle_{A}\otimes|\psi_{k_{x}>0}\rangle_{B}$.
The fine-tuning is done with $t_{2}/t_{1}=1/2$ in our case. Interpret this
result together with Fig.4(b), we see the solid blue curve at $k_{x}<0$ has
negative velocity along $\hat{x}$ direction, and at $k_{x}>0$ has positive
velocity along $\hat{x}$ direction. Overall it implies the chirality of the
edge state on the left edge A moving along $-\hat{x}$ direction, and on the
right edge B moving along $+\hat{x}$ direction - the clockwise chirality as in
Fig.2(b), consistent with the earlier result $C_{1,-}=-1$ of Chern number.
An additional bonus for this ladder model is that the density $\langle
f^{\dagger}f\rangle$ distributes equally on two sublattice $a$ and $b$ on
either edges, shown in Fig.4(c). Thus, it will be beneficial for the
interacting model in Eq.(II) when turning on interaction terms
$G_{1},G_{2}\neq 0$, we can universally add the same interaction terms for
both sublattice $a$ and $b$.
For the free kinetic theory, all of the above can be achieved by a simple
ladder lattice, which is effectively as good as 1+1D because of finite size
width. To have mirror sector becomes gapped and decoupled without interfering
with the gapless sector, we propose to design the lattice with length scales
of Eq.(16).
### F.2 Explicit lattice chiral matter models
For model constructions, we will follow the four steps introduced earlier in
Sec.V.
#### F.2.1 1L-(-1R) chiral fermion model
The most simplest model of fermionic model suitable for our purpose is, Step
1, $K^{f}_{2\times 2}=({\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}})$ in
Eq.(20),(21). We can choose, Step 2, $\mathbf{t}=(1,-1)$, so this model
satisfies Eq.(42) as anomaly-free. It also satisfies the total U(1) charge
chirality $\sum q_{L}-\sum q_{R}=2\neq 0$ as Step 3. As Step 4, we can fully
gap out one-side of edge states by a gapping term Eq.(28) with
$\ell_{a}=(1,1)$, which preserves U(1) symmetry by Eq.(30). Written in terms
of $\mathbf{t}$ and $\mathbf{L}$ matrices:
$\mathbf{t}=\left(\begin{array}[]{cc}1\\\
-1\end{array}\right)\Longleftrightarrow\mathbf{L}=\left(\begin{array}[]{cc}1\\\
1\end{array}\right).$ (187)
Through its U(1) charge assignment $\mathbf{t}=(1,-1)$, we name this model as
1L-(-1R) chiral fermion model. It is worthwhile to go through this 1L-(-1R)
chiral fermion model in more details, where its bosonized low energy action is
$\displaystyle S_{\Phi}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi}\int
dtdx\big{(}K^{\mathop{\mathrm{A}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}\big{)}\;\;\;$
(188) $\displaystyle+$ $\displaystyle\frac{1}{4\pi}\int
dtdx\big{(}K^{\mathop{\mathrm{B}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}\big{)}\;\;\;$
$\displaystyle+$ $\displaystyle\int
dtdx\;g_{1}\cos(\Phi^{\mathop{\mathrm{B}}}_{1}+\Phi^{\mathop{\mathrm{B}}}_{-1}).\;\;\;\;\;\;\;$
Its fermionized action (following the notation as Eq.(II), with a marginal
interaction term of $g_{1}$ coupling) is
$\displaystyle S_{\Psi}$ $\displaystyle=\int
dt\;dx\;\bigg{(}\text{i}\bar{\Psi}_{\mathop{\mathrm{A}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{A}}}+\text{i}\bar{\Psi}_{\mathop{\mathrm{B}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{B}}}$
(189)
$\displaystyle+\tilde{g}_{1}\big{(}\tilde{\psi}_{R,1}\tilde{\psi}_{L,-1}+\text{h.c.}\big{)}.$
We propose that a lattice Hamiltonian below (analogue to Fig.1’s) realizes
this 1L-(-1R) chiral fermions theory non-perturbatively,
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{q=1,-1}\bigg{(}\sum_{\langle
i,j\rangle}\big{(}t_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}$
$\displaystyle+$ $\displaystyle\sum_{\langle\langle
i,j\rangle\rangle}\big{(}t^{\prime}_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}\bigg{)}$
$\displaystyle+$ $\displaystyle
G_{1}\sum_{j\in\mathop{\mathrm{B}}}\bigg{(}\big{(}\hat{f}_{1}(j)_{pt.s.}\big{)}\big{(}\hat{f}_{-1}(j)_{pt.s.}\big{)}+h.c.\bigg{)}.\;\;$
This Hamiltonian is in a perfect quadratic form, which is a welcomed old
friend to us. We can solve it exactly by writing down Bogoliubov-de
Gennes(BdG) Hamiltonian in the Nambu space form, on a cylinder (in Fig.1),
$H=\frac{1}{2}\sum_{k_{x},p_{x}}(f^{\dagger},f){\begin{pmatrix}H_{\text{kinetic}}&\mathcal{G}^{\dagger}(k_{x},p_{x})\\\
\mathcal{G}(k_{x},p_{x})&-H_{\text{kinetic}}\end{pmatrix}}{\begin{pmatrix}f\\\
f^{\dagger}\end{pmatrix}}.$ (191)
Here $f^{\dagger}=(f^{\dagger}_{1,k_{x}},f^{\dagger}_{-1,p_{x}})$,
$f=(f_{1,k_{x}},f_{-1,p_{x}})$, $H_{\text{kinetic}}$ is the hopping term and
$\mathcal{G}$ is from the $G_{1}$ interaction term. Here momentum
$k_{x},p_{x}$ (for charge 1 and -1 fermions) along the compact direction x are
good quantum numbers. Along the non-compact y direction, we use the real space
basis instead. We diagonalize this BdG Hamiltonian exactly and find out the
edge modes on the right edge B become fully gapped at large $G_{1}$. For
example, at $|G_{1}|\simeq 10000$, the edge state density on the edge B is
$\langle f^{\dagger}_{B}f_{B}\rangle\leq 5\times 10^{-8}$.JWunpublished We
also check that the low energy spectrum realizes the 1-(-1) chiral fermions on
the left edge A,JWunpublished
$\displaystyle S_{\Psi_{\mathop{\mathrm{A}}},free}$ $\displaystyle=$
$\displaystyle\int
dtdx\;\Big{(}\text{i}\psi^{\dagger}_{L,1}(\partial_{t}-\partial_{x})\psi_{L,1}$
(192) $\displaystyle+$
$\displaystyle\text{i}\psi^{\dagger}_{R,-1}(\partial_{t}+\partial_{x})\psi_{R,-1}\Big{)}.$
Thus Eq.(F.2.1) defines/realizes 1L-(-1R) chiral fermions non-perturbatively
on the lattice.
The 1L-(-1R) chiral fermion model provides a wonderful example that we can
confirm, both numerically and analytically, the mirrored fermion idea and our
model construction will work.
However, unfortunately the 1L-(-1R) chiral fermion model is not strictly a
chiral theory. In a sense that one can do a field redefinition,
$\psi_{1}\to\psi_{1},\;\;\text{and}\;\;\psi_{-1}\to\psi_{1^{\prime}}^{\dagger},$
sending the charge vector $\mathbf{t}=(1,-1)\to(1,1)$. So the model becomes a
1L-1R fermion model with one left moving mode and one right moving mode both
carry the same U(1) charge 1. Here we use $\psi_{1^{\prime}}$ to indicate
another fermion field carries the same U(1) charge as $\psi_{1}$. The 1L-1R
fermion model is obviously a non-chiral Dirac fermion theory, where the
mirrored edge states can be gapped out by forward scattering mass terms
$\tilde{g}_{1}\big{(}\tilde{\psi}_{R,1}\tilde{\psi}_{L,1^{\prime}}^{\dagger}+\text{h.c.}\big{)}$,
or the
$g_{1}\cos(\Phi^{\mathop{\mathrm{B}}}_{1}-\Phi^{\mathop{\mathrm{B}}}_{1^{\prime}})$
term in the bosonized language. Since 1L-(-1R) chiral fermion model is a
field-redifiniton of 1L-1R fermion model, it becomes apparent that we can gap
out the mirrored edge of 1L-(-1R) chiral fermion model.
It turns out that the next simplest U(1)-symmetry chiral fermion model, which
violates parityand time reversal symmetry(but strictly being chiral under any
field redefinition), is the 3L-5R-4L-0R chiral fermion model, appeared already
in Sec.II.
#### F.2.2 3L-5R-4L-0R chiral fermion model and others
We consider a rank-4 $K^{f}_{4\times 4}=({\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}})\oplus({\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}})$ in Eq.(20),(21) for Step 1. We can choose
$\mathbf{t}_{a}=(3,5,4,0)$ to construct a 3L-5R-4L-0R chiral fermion model in
Sec.II for Step 2. One can choose the gapping terms in Eq.(28) with
$\ell_{a}=(3,-5,4,0),\ell_{b}=(0,4,-5,3)$. Another U(1)${}_{\text{2nd}}$
symmetry is allowed, which is $\mathbf{t}_{b}=(0,4,5,3)$. By writing down the
chiral boson theory of Eq.(21), (28) on a cylinder with two edges A and B as
in Fig.1, it becomes a multiplet chiral boson theory with an action
$\displaystyle
S_{\Phi}=S_{\Phi^{\mathop{\mathrm{A}}}_{free}}+S_{\Phi^{\mathop{\mathrm{B}}}_{free}}+S_{\Phi^{\mathop{\mathrm{B}}}_{interact}}=$
$\displaystyle\frac{1}{4\pi}\int
dtdx\big{(}K^{\mathop{\mathrm{A}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{A}}}_{J}\big{)}+\big{(}K^{\mathop{\mathrm{B}}}_{IJ}\partial_{t}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}-V_{IJ}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{I}\partial_{x}\Phi^{\mathop{\mathrm{B}}}_{J}\big{)}\;\;\;$
(193) $\displaystyle+\int
dtdx\bigg{(}g_{1}\cos(3\Phi^{\mathop{\mathrm{B}}}_{3}-5\Phi^{\mathop{\mathrm{B}}}_{5}+4\Phi^{\mathop{\mathrm{B}}}_{4})+g_{2}\cos(4\Phi^{\mathop{\mathrm{B}}}_{5}-5\Phi^{\mathop{\mathrm{B}}}_{4}+3\Phi^{\mathop{\mathrm{B}}}_{0})\bigg{)}.\;\;\;\;\;\;\;$
After fermionizing Eq.(4) by $\Psi\sim e^{i\Phi}$, we match it to
Eq.(II).fermionization2
$\displaystyle S_{\Psi}$
$\displaystyle=S_{\Psi_{\mathop{\mathrm{A}}},free}+S_{\Psi_{\mathop{\mathrm{B}}},free}+S_{\Psi_{\mathop{\mathrm{B}}},interact}=\int
dt\;dx\;\bigg{(}\text{i}\bar{\Psi}_{\mathop{\mathrm{A}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{A}}}+\text{i}\bar{\Psi}_{\mathop{\mathrm{B}}}\Gamma^{\mu}\partial_{\mu}\Psi_{\mathop{\mathrm{B}}}$
$\displaystyle+\tilde{g}_{1}\big{(}(\tilde{\psi}_{R,3}\nabla_{x}\tilde{\psi}_{R,3}\nabla^{2}_{x}\tilde{\psi}_{R,3})(\tilde{\psi}_{L,5}^{\dagger}\nabla_{x}\tilde{\psi}_{L,5}^{\dagger}\nabla^{2}_{x}\tilde{\psi}_{L,5}^{\dagger}\nabla^{3}_{x}\tilde{\psi}_{L,5}^{\dagger}\nabla^{4}_{x}\tilde{\psi}_{L,5}^{\dagger})(\tilde{\psi}_{R,4}\nabla_{x}\tilde{\psi}_{R,4}\nabla^{2}_{x}\tilde{\psi}_{R,4}\nabla^{3}_{x}\tilde{\psi}_{R,4})+\text{h.c.}\big{)}$
$\displaystyle+\tilde{g}_{2}\big{(}(\tilde{\psi}_{L,5}\nabla_{x}\tilde{\psi}_{L,5}\nabla^{2}_{x}\tilde{\psi}_{L,5}\nabla^{3}_{x}\tilde{\psi}_{L,5})(\tilde{\psi}_{R,4}^{\dagger}\nabla_{x}\tilde{\psi}_{R,4}^{\dagger}\nabla^{2}_{x}\tilde{\psi}_{R,4}^{\dagger}\nabla^{3}_{x}\tilde{\psi}_{R,4}^{\dagger}\nabla^{4}_{x}\tilde{\psi}_{R,4}^{\dagger})(\tilde{\psi}_{L,0}\nabla_{x}\tilde{\psi}_{L,0}\nabla^{2}_{x}\tilde{\psi}_{L,0})+\text{h.c.}\big{)}\bigg{)},$
Our 3-5-4-0 fermion model satisfies Eq.(30), Eq.(42) and boundary fully
gapping rules, and also violates parity and time-reversal symmetry, so the
lattice version of the Hamiltonian
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{q=3,5,4,0}\bigg{(}\sum_{\langle
i,j\rangle}\big{(}t_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}+\sum_{\langle\langle
i,j\rangle\rangle}\big{(}t^{\prime}_{ij,q}\;\hat{f}^{\dagger}_{q}(i)\hat{f}_{q}(j)+h.c.\big{)}\bigg{)}$
$\displaystyle+$ $\displaystyle
G_{1}\sum_{j\in\mathop{\mathrm{B}}}\bigg{(}\big{(}\hat{f}_{3}(j)_{pt.s.}\big{)}^{3}\big{(}\hat{f}^{\dagger}_{5}(j)_{pt.s.}\big{)}^{5}\big{(}\hat{f}_{4}(j)_{pt.s.}\big{)}^{4}+h.c.\bigg{)}+G_{2}\sum_{j\in\mathop{\mathrm{B}}}\bigg{(}\big{(}\hat{f}_{5}(j)_{pt.s.}\big{)}^{4}\big{(}\hat{f}^{\dagger}_{4}(j)_{pt.s.}\big{)}^{5}\big{(}\hat{f}_{0}(j)_{pt.s.}\big{)}^{3}+h.c.\bigg{)},\;\;$
provides a non-perturbative anomaly-free chiral fermion model on the gapless
edge A when putting on the lattice. We notice that the choices of gapping
terms with $\ell_{a}=(3,-5,4,0),\ell_{b}=(0,4,-5,3)$ of the model in
Eq.(193),(F.2.2),(F.2.2) here are distinct from the version of gapping terms
$\ell_{a}=(1,1,-2,2)$, $\ell_{b}=(2,-2,1,1)$ of the model Eq.(II), (4), (II)
in the main text. This is rooted in the _different_ choice of basis for the
_same_ vector space of column vectors of $\mathbf{L},\mathbf{t}$ matrices, and
the dual structure shown in Eq.(69). Both ways (or other linear-independent
linear combinations) will produce a 3L-5R-4L-0R model.
In Sec.E.4, we outline that our anomaly-free chiral model can be mapped to
decoupled Luttinger liquids of Eq.(137). Here let us explicitly find out the
outcomes of mapping. Based on the Smith normal form $\mathbf{L}=VDW$ shown in
Sec.E.4, we can rewrite the gapping term matrices $\mathbf{L}$. From Eq.(164),
the original cosine term $g_{a}\cos(\ell_{a,I}\cdot\Phi_{I})$ in the old basis
will be mapped to $g_{a}\cos(W_{Ja}d_{J}\bar{\theta}_{J})$. Namely, given the
model of Eq.(193),
$\displaystyle\left(\begin{array}[]{cc}3&0\\\ -5&4\\\ 4&-5\\\
0&3\end{array}\right)=\left(\begin{array}[]{cccc}3&-1&0&1\\\ -5&3&0&-2\\\
4&-3&1&2\\\ 0&1&0&0\end{array}\right).\left(\begin{array}[]{cc}1&0\\\ 0&3\\\
0&0\\\ 0&0\end{array}\right).\left(\begin{array}[]{cc}1&1\\\
0&1\end{array}\right)$ (210) $\displaystyle\Rightarrow
g_{a}\cos(\bar{\theta}_{1})+g_{b}\cos(\bar{\theta}_{1}+3\bar{\theta}_{2}).$
(211)
On the other hand, given the model of Eq.(II), we have
$\displaystyle\left(\begin{array}[]{cc}1&2\\\ 1&-2\\\ -2&1\\\
2&1\end{array}\right)=\left(\begin{array}[]{cccc}1&2&0&-1\\\ 1&-2&0&0\\\
-2&1&1&1\\\ 2&1&0&-1\end{array}\right).\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
0&0\\\ 0&0\end{array}\right).\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right)$ (226) $\displaystyle\Rightarrow
g_{a}\cos(\bar{\theta}_{1})+g_{b}\cos(\bar{\theta}_{2})$ (227)
There are two reasons that we choose Eq.(II) for our model in the main text,
instead of Eq.(F.2.2). The first reason is that even at the weak $g$
perturbative level, Eq.(II) flows to gapped phases at IR low energy. In
Sec.IV.3.3, we have done a perturbative analysis to learn that when
${\beta^{2}}<\beta_{c}^{2}\equiv 4$ for the normal ordered scaling dimension
$[\cos(\beta\bar{\theta})]={\beta^{2}}/{2}<2$, the system will flow to the
gapped phases. We notice that it is indeed the case for our model Eq.(II) with
$\ell_{a}=(1,1,-2,2)$, $\ell_{b}=(2,-2,1,1)$, and the decoupled potentials in
the new basis $g_{a}\cos(\bar{\theta}_{1})+g_{b}\cos(\bar{\theta}_{2})$ has
${\beta^{2}}=1<\beta_{c}^{2}$. The second reason is that the interaction terms
for the model of Eq.(II) has the order of 6-body interaction among each
gapping term, which is easier to simulate than the 12-body interaction among
each gapping term for the model of Eq.(F.2.2).
We list down another three similar chiral fermion models of $K^{f}_{4\times
4}$ matrix, with different choices of $\mathbf{t}$, such as:
(i) 1L-5R-7L-5R chiral fermions: $\mathbf{t}_{a}=(1,5,7,5)$,
$\mathbf{t}_{a}=(0,3,5,4)$, with gapping terms $\ell_{a}=(1,-5,7,-5)$,
$\ell_{b}=(0,3,-5,4)$. Written in terms of $\mathbf{t}$ and $\mathbf{L}$
matrices:
$\mathbf{t}=\left(\begin{array}[]{cc}1&0\\\ 5&3\\\ 7&5\\\
5&4\end{array}\right)\Longleftrightarrow\mathbf{L}=\left(\begin{array}[]{cc}1&0\\\
-5&3\\\ 7&-5\\\ -5&4\end{array}\right).$ (228)
(ii) 1L-4R-8L-7R chiral fermions: $\mathbf{t}_{a}=(1,4,8,7)$,
$\mathbf{t}_{b}=(3,-3,-1,1)$, with gapping terms $\ell_{a}=(1,-4,8,-7)$,
$\ell_{b}=(3,3,-1,-1)$.
$\mathbf{t}=\left(\begin{array}[]{cc}1&3\\\ 4&-3\\\ 8&-1\\\
7&1\end{array}\right)\Longleftrightarrow\mathbf{L}=\left(\begin{array}[]{cc}1&3\\\
-4&3\\\ 8&-1\\\ -7&-1\end{array}\right).$ (229)
(iii) 2L-6R-9L-7R chiral fermions: $\mathbf{t}_{a}=(2,6,9,7)$,
$\mathbf{t}_{b}=(2,-2,-1,1)$ with gapping terms $\ell_{a}=(2,-6,9,-7)$,
$\ell_{b}=(2,2,-1,-1)$.
$\mathbf{t}=\left(\begin{array}[]{cc}2&2\\\ 6&-2\\\ 9&-1\\\
7&1\end{array}\right)\Longleftrightarrow\mathbf{L}=\left(\begin{array}[]{cc}2&2\\\
-6&2\\\ 9&-1\\\ -7&-1\end{array}\right).$ (230)
Indeed, there are infinite many possible models just for $K^{f}_{4\times 4}$
matrix-Chern Simons theory construction. One can also construct a higher rank
$K^{f}$ theory with infinite more models of U(1)N/2-anomaly-free chiral
fermions.
#### F.2.3 Chiral boson model
Similar to fermionic systems, we will follow the four steps introduced earlier
for bosonic systems. The most simple model of bosonic SPT suitable for our
purpose is, Step 1, $K^{b}_{2\times 2}=({\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}})$ in Eq.(20),(21). We can choose, Step 2,
$\mathbf{t}=(1,0)$, so this model satisfies Eq.(43) as anomaly-free, and
violates parity and time-reversal symmetry as Step 3. As Step 4, we can fully
gap out one-side of edge states by gapping term Eq.(28) with $\ell_{a}=(0,1)$,
which preserves U(1) symmetry by Eq.(30). Written in terms of $\mathbf{t}$ and
$\mathbf{L}$ matrices:
$\mathbf{t}=\left(\begin{array}[]{cc}1\\\
0\end{array}\right)\Longleftrightarrow\mathbf{L}=\left(\begin{array}[]{cc}0\\\
1\end{array}\right).$ (231)
For $K^{b0}_{4\times 4}=({\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}})\oplus({\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}})$, we list down two models:
(i) 2L-2R-4L-$(-1)_{R}$ chiral bosons: $\mathbf{t}_{a}=(2,2,4,-1)$,
$\mathbf{t}_{b}=(0,2,0,-1)$ with gapping terms $\ell_{a}=(2,2,-1,4)$,
$\ell_{b}=(2,0,-1,0)$.
$\mathbf{t}=\left(\begin{array}[]{cc}2&0\\\ 2&2\\\ 4&0\\\
-1&-1\end{array}\right)\Longleftrightarrow\mathbf{L}=\left(\begin{array}[]{cc}2&2\\\
2&0\\\ -1&-1\\\ 4&0\end{array}\right).$ (232)
(ii) 6L-6R-9L-$(-4)_{R}$ chiral bosons: $\mathbf{t}_{a}=(6,6,9,-4)$,
$\mathbf{t}_{b}=(0,3,0,-2)$ with gapping terms $\ell_{a}=(6,6,-4,9)$,
$\ell_{b}=(3,0,-2,0)$.
$\mathbf{t}=\left(\begin{array}[]{cc}6&0\\\ 6&3\\\ 9&0\\\
-4&-2\end{array}\right)\Longleftrightarrow\mathbf{L}=\left(\begin{array}[]{cc}6&3\\\
6&0\\\ -4&-2\\\ 9&0\end{array}\right).$ (233)
Infinite many chiral boson models can be constructed in the similar manner.
black
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|
arxiv-papers
| 2013-07-29T06:59:26 |
2024-09-04T02:49:48.610094
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Juven Wang and Xiao-Gang Wen",
"submitter": "Juven C. Wang",
"url": "https://arxiv.org/abs/1307.7480"
}
|
1307.7492
|
# A New Look at Linear (Non-?) Symplectic Ion Beam Optics in Magnets
C. Baumgarten Paul Scherrer Institute, Switzerland
[email protected]
###### Abstract
We take a new look at the details of symplectic motion in solenoid and bending
magnets and rederive known (but not always well-known) facts. We start with a
comparison of the general Lagrangian and Hamiltonian formalism of the harmonic
oscillator and analyze the relation between the canonical momenta and the
velocities (i.e. the first derivatives of the canonical coordinates). We show
that the seemingly non-symplectic transfer maps at entrance and exit of
solenoid magnets can be re-interpreted as transformations between the
canonical and the mechanical momentum, which differ by the vector potential.
In a second step we rederive the transfer matrix for charged particle motion
in bending magnets from the Lorentz force equation in cartesic coordinates. We
rediscover the geometrical and physical meaning of the local curvilinear
coordinate system. We show that analog to the case of solenoids - also the
transfer matrix of bending magnets can be interpreted as a symplectic product
of 3 non-symplectic matrices, where the entrance and exit matrices are
transformations between local cartesic and curvilinear coordinate systems.
We show that these matrices are required to compare the second moment matrices
of distributions obtained by numerical tracking in cartesic coordinates with
those that are derived by the transfer a matrix method.
Beam Optics, Particle Accelerators, Cyclotrons
###### pacs:
41.85.-p, 45.50.Dd, 29.20.dg
## I Introduction
In the course of numerical simulations of coasting beams in cyclotrons it
turned out that the eigen-emittances eigen computed from the second moment
matrices were not constant as one would expect for symplectic motion inv .
Quite obviously there was something wrong in the interpretation of the data.
In this article we trace this error back to a missing transformation. The
simulation tool OPAL opal1 ; opal2 uses global cartesic coordinates for the
integration of the equations of motion (EQOM). The transformation to local co-
moving coordinates is not always sufficient to analyze the data properly and
to compare them with second moments matrices obtained from linear transfer
matrix models like the one in Ref. sc_paper . The solution of the problem
might be trivial to (some) specialists, but due to the general context we
consider it being worth a more general discussion.
The problem that we refer to, can be briefly described by either one of the
following questions:
1. 1.
Why are the transfer matrices at the entrance and exit of solenoid magnets
considered to be non-symplectic Handbook ; Intro . Is it true after all?
2. 2.
Why is the entrance and exit of a bending magnet not considered to be non-
symplectic?
3. 3.
Is it possible to derive the transfer matrix of a bending magnet in cartesic
coordinates?
4. 4.
How do we compare particle distributions generated by cartesic tracking codes
with those generated by the transfer matrix formalism?
This work is dedicated to those readers that are not ad hoc able to give the
answer to these questions or that are at least not sure about it.
In some sense this work is a continuation of Ref. rdm_paper ; geo_paper ,
where we derived new methods to “solved problems” with the general Hamiltonian
of a two-dimensional harmonic oscillator. Here we start with the general
Lagrangian description of an harmonic oscillator and derive the Hamiltonian
from it. The comparison allows us to identify the conditions for the use of
the Lagrangian state vector compared to the Hamiltonian state vector and how
they can be transformed into each other. Next we analyze the situation in case
of solenoids and bending magnets and compare different interpretations.
Finally we apply the resulting (simple) transformation to our numerical
problem.
## II Lagrangian of the harmonic oscillator
In order to formulate the Lagrangian function ${\cal L}={\cal L}({\bf
q},{\bf\dot{q}})$ of the $n$-dimensional harmonic oscillator, we define a
state vector $\phi=({\bf q},{\bf\dot{q}})^{T}=(q_{1},q_{2},\dots
q_{n},\dot{q}_{1},\dot{q}_{2},\dots,\dot{q}_{n})^{T}$. We then write the
Lagrangian function of the harmonic oscillator in the most general way as a
quadratic form:
${\cal L}=\frac{1}{2}\,\phi^{T}\,{\bf L}\,\phi\,.$ (1)
The matrix ${\bf L}$ should be symmetric, as any antisymmetric component does
not alter the Lagrangian function ${\cal L}$ and should therefore be
physically irrelevant:
${\bf L}=\left(\begin{array}[]{cc}{\bf U}&{\bf B}\\\ {\bf B}^{T}&{\bf
M}\end{array}\right)\,,$ (2)
where the matrices ${\bf U}$ and ${\bf M}$ are symmetric. Written in
components this is
${\cal
L}=\frac{1}{2}\,\left(q_{j}\,U_{jk}\,q_{k}+2\,q_{j}\,B_{jk}\,\dot{q}_{k}+\dot{q}_{j}\,M_{jk}\,\dot{q}_{k}\right)\,.$
(3)
with the $2n\times 2n$-matrix ${\bf L}$ and the $n\times n$-matrices ${\bf
U}$, ${\bf B}$ and ${\bf M}$. The Lagrangian equations of motion (EQOM) are:
${d\over dt}{\partial L\over\partial\dot{q}_{j}}={\partial L\over\partial
q_{j}}\,.$ (4)
The derivatives are explicitely:
$\begin{array}[]{rcl}{\partial
L\over\partial\dot{q}_{j}}&=&M_{jk}\,\dot{q}_{k}+B_{kj}\,q_{k}=p_{j}\\\
{\partial L\over\partial q_{j}}&=&U_{jk}\,q_{k}+B_{jk}\,\dot{q}_{k}\\\
{\partial L\over\partial{\bf\dot{q}}}&=&{\bf M}\,{\bf\dot{q}}+{\bf
B}^{T}\,{\bf q}={\bf p}\\\ {\partial L\over\partial{\bf q}}&=&{\bf U}\,{\bf
q}+{\bf B}\,{\bf\dot{q}}\\\ \end{array}$ (5)
so that one obtains for the EQOM
${\bf M}\,{\bf\ddot{q}}={\bf U}\,{\bf q}+({\bf B}-{\bf
B}^{T})\,{\bf\dot{q}}\,.$ (6)
As well-known, any matrix ${\bf B}$ can be split into two matrices ${\bf
B}_{s}$ and ${\bf B}_{a}$, representing the symmetric and the antisymmetric
part:
$\begin{array}[]{rcl}{\bf B}_{s}=({\bf B}+{\bf B}^{T})/2\\\ {\bf B}_{a}=({\bf
B}-{\bf B}^{T})/2\\\ \end{array}$ (7)
If one compares this with Eqn. 6, one finds that the EQOM depend only on the
antisymmetric (“gyroscopic”) part ${\bf B}_{a}$ while the definition of the
canonical momentum includes all components of ${\bf B}$ YaSt ; Talman .
The number of parameters $\nu$ that can be found in the Lagrangian are the
parameters that are required to describe two symmetric $n\times n$-matrices
and an arbitrary $n\times n$-matrix:
$\nu=2\,{n\,(n+1)\over 2}+n^{2}=2\,n^{2}+n\,.$ (8)
For instance, systems with $n=2$ general degrees of freedom give $\nu=10$, for
$n=3$ this gives $n=21$. Nevertheless, with respect to the dynamics (i.e. the
EQOM), some parameters can be omitted. As already mentioned, the symmetric
part of ${\bf B}$ does not enter the EQOM and secondly, the Lagrangian
function can be multiplied by an arbitrary factor without effect on the
dynamics. This is a consequence of the fact that in the Lagrangian function
appears on both sides of Eqn. 4, such that the any scaling factor applied to
the matrix ${\bf L}$ cancels out. However such a factor – even though
irrelevant for the dynamics – changes the scale of the canonical momentum:
${\bf p}={\bf M}\,{\bf\dot{q}}+{\bf B}^{T}\,{\bf q}\,.$ (9)
In summary one finds that the EQOM derived from the above Lagrangian contain
$\nu_{d}$ dynamically relevant parameters. It equals the number of parameters
that are required to define two symmetric $n\times n$-matrices and an
antisymmetric $n\times n$-matrix, minus the scale factor:
$\nu_{d}=2\,{n\,(n+1)\over 2}+{n\,(n-1)\over 2}-1={3\,n^{2}+n-2\over 2}\,.$
(10)
For $n=2$ one finds $\nu_{d}=6$ and for $n=3$ we have $\nu_{d}=14$.
## III Relation to the Hamiltonian
The Hamilton function ${\cal H}$ is obtained by
$\begin{array}[]{rcl}{\cal H}&=&p_{k}\,\dot{q}_{k}-{\cal L}\\\ {\cal H}&=&{\bf
p}^{T}\,{\bf\dot{q}}-\frac{1}{2}\,\left({\bf q}^{T}\,{\bf U}\,{\bf q}+{\bf
q}^{T}\,{\bf B}\,{\bf\dot{q}}+{\bf\dot{q}}^{T}\,{\bf B}^{T}\,{\bf
q}+{\bf\dot{q}}^{T}\,{\bf M}\,{\bf\dot{q}}\right)\\\ \end{array}$ (11)
We assume that the mass matrix ${\bf M}$ is invertible and replace
${\bf\dot{q}}={\bf M}^{-1}\,({\bf p}-{\bf B}^{T}\,{\bf q})$. If the
Hamiltonian state vector $\psi$ is defined as $\psi=({\bf q},{\bf p})$, then
the Hamiltonian function is derived in a few steps
$\begin{array}[]{rcl}{\cal H}&=&\frac{1}{2}\,\psi^{T}\,{\bf H}\,\psi\\\
&=&\frac{1}{2}\,\left(\begin{array}[]{c}{\bf q}\\\ {\bf
p}\end{array}\right)^{T}\,\left(\begin{array}[]{cc}{\bf B}\,{\bf M}^{-1}\,{\bf
B}^{T}-{\bf U}&-{\bf B}\,{\bf M}^{-1}\\\ -{\bf M}^{-1}\,{\bf B}^{T}&{\bf
M}^{-1}\end{array}\right)\,\left(\begin{array}[]{c}{\bf q}\\\ {\bf
p}\end{array}\right)\end{array}$ (12)
The symplectic unit matrix ${\bf\gamma}_{0}$ is given (in this representation)
by
${\bf\gamma}_{0}=\left(\begin{array}[]{cc}{\bf 0}&{\bf 1}\\\ -{\bf 1}&{\bf
0}\end{array}\right)\,,$ (13)
so that the Hamiltonian EQOM are
$\begin{array}[]{rcl}\left(\begin{array}[]{c}{\bf\dot{q}}\\\
{\bf\dot{p}}\end{array}\right)&=&{\bf\gamma}_{0}\,\left(\begin{array}[]{cc}{\bf
B}\,{\bf M}^{-1}\,{\bf B}^{T}-{\bf U}&-{\bf B}\,{\bf M}^{-1}\\\ -{\bf
M}^{-1}\,{\bf B}^{T}&{\bf
M}^{-1}\end{array}\right)\,\left(\begin{array}[]{c}{\bf q}\\\ {\bf
p}\end{array}\right)\\\ &=&\left(\begin{array}[]{cc}-{\bf M}^{-1}\,{\bf
B}^{T}&{\bf M}^{-1}\\\ {\bf U}-{\bf B}\,{\bf M}^{-1}\,{\bf B}^{T}&{\bf
B}\,{\bf M}^{-1}\end{array}\right)\,\left(\begin{array}[]{c}{\bf q}\\\ {\bf
p}\end{array}\right)\\\ \end{array}$ (14)
The Hamiltonian state vector $\psi$ and the Lagrangian state vector $\phi$ are
related by $\psi={\bf Q}\,\phi$:
$\left(\begin{array}[]{c}{\bf q}\\\ {\bf p}\\\
\end{array}\right)=\left(\begin{array}[]{cc}{\bf 1}&{\bf 0}\\\ {\bf
B}^{T}&{\bf M}\\\ \end{array}\right)\,\left(\begin{array}[]{c}{\bf q}\\\
{\bf\dot{q}}\\\ \end{array}\right)$ (15)
and
$\left(\begin{array}[]{c}{\bf q}\\\ {\bf\dot{q}}\\\
\end{array}\right)=\left(\begin{array}[]{cc}{\bf 1}&{\bf 0}\\\ -{\bf
M}^{-1}\,{\bf B}^{T}&{\bf M}^{-1}\\\
\end{array}\right)\,\left(\begin{array}[]{c}{\bf q}\\\ {\bf p}\\\
\end{array}\right)$ (16)
This coordinate transformation is symplectic, if
${\bf Q}\,\gamma_{0}\,{\bf Q}^{T}=\gamma_{0}\,,$ (17)
or explicitely
$\begin{array}[]{rcl}{\bf\gamma}_{0}&=&\left(\begin{array}[]{cc}{\bf 1}&{\bf
0}\\\ {\bf B}^{T}&{\bf M}\\\
\end{array}\right)\,{\bf\gamma}_{0}\,\left(\begin{array}[]{cc}{\bf 1}&{\bf
B}\\\ {\bf 0}&{\bf M}\\\ \end{array}\right)\\\ \left(\begin{array}[]{cc}{\bf
0}&{\bf 1}\\\ -{\bf 1}&{\bf 0}\\\
\end{array}\right)&=&\left(\begin{array}[]{cc}{\bf 0}&{\bf M}\\\ -{\bf M}&{\bf
B}^{T}\,{\bf M}-{\bf M}\,{\bf B}\\\ \end{array}\right)\\\ \Rightarrow&&{\bf
M}={\bf 1}\\\ \Rightarrow&&{\bf B}^{T}={\bf B}\,,\end{array}$ (18)
i.e. it is symplectic, if (and only if) the mass matrix ${\bf M}$ equals the
unit matrix fn1 and if ${\bf B}$ is symmetric which means that no gyroscopic
forces are present. Only in this case it is legitimate to identify ${\bf p}$
and ${\bf\dot{q}}$ (up to a symplectic transformation). The first condition is
usually fulfilled, if the system describes a single particle with $n$ degrees
of freedom – instead of for example $n$ coupled particles with different
masses in a linear chain.
## IV The Solenoid Magnet
The second condition is not always fulfilled. Consider for instance the
transfer-matrix ${\bf T}$ that describes the transversal motion of a charged
particle through the fringe field of a solenoid magnet fn2 . In the coordinate
ordering used so far it is Hinterberger ; Intro :
${\bf T}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&\pm K&1&0\\\ \mp
K&0&0&1\\\ \end{array}\right)\,.$ (19)
This is a nice example for the transformation from $\phi$ to $\psi$ (or vice
versa) with non-vanishing gyroscopic terms. The matrices are formally non-
symplectic Handbook ; Intro , but it would be a misinterpretation to believe
that the (equation of) motion in the fringe fields of solenoid magnets is non-
symplectic. This is not the case. The concept of symplectic motion is based on
Hamiltonian dynamics and it presumes the use of canonical momenta. The above
transformation ${\bf T}$ is only required if one uses the state vector $\phi$
instead of $\psi$, i.e. the mechanical instead of the canonical momentum. If
this difference is not properly taken into account, the motion appears to be
non-symplectic Intro .
The gyroscopic terms of the matrix ${\bf B}_{a}$ are connected to the
(derivatives of the) vector potential as one would expect by
$\vec{p}_{can}=\vec{p}_{mech}+\vec{A}(\vec{x})$ (using units where $q=1$ and
$m=1$) Intro . In the linear 3-dimensional case one finds:
$\begin{array}[]{rcl}{\bf
B}_{a}&=&\frac{1}{2}\,\left(\begin{array}[]{ccc}0&-B_{z}&B_{y}\\\
B_{z}&0&-B_{x}\\\ -B_{y}&B_{x}&0\\\ \end{array}\right)\\\ &&\\\ \vec{A}&=&{\bf
B}\,{\bf q}={\bf B}_{s}\,{\bf
q}+\frac{1}{2}\,\left(\begin{array}[]{c}-B_{z}\,y+B_{y}\,z\\\
B_{z}\,x-B_{x}\,z\\\ -B_{y}\,x+B_{x}\,y\\\ \end{array}\right)\,,\end{array}$
(20)
which directly yields
$\begin{array}[]{rcl}\vec{\nabla}\times\vec{A}&=&(B_{x},B_{y},B_{z})^{T}\\\
&&\\\ \vec{\nabla}\cdot\vec{A}&=&Tr({\bf B})=Tr({\bf B}_{s})\,.\end{array}$
(21)
Assuming for the moment that ${\bf B}_{s}=0$ one finds with $K={B_{z}\over
2\,(B\,\rho)}$ that the matrix ${\bf T}$ corresponds exactly to the 2-dim.
transformation from $\phi$ to $\psi$ as given in Eqn. 15. This matrix needs to
be applied, since the entrance of a solenoid is a transition from the field
free region where $\psi=\phi$ to a region with gyroscopic force, where the
canonical momentum is not identical with the mechanical momentum fn3 .
The symmetric part of ${\bf B}$ represents a symplectic transformation which
is irrelevant for the dynamics expressed by the coordinates. In this sense it
is a similar to a “gauge field” that changes exclusively the canonical
momentum. The antisymmetric (“gyroscopic”) part of ${\bf B}$ is (in 3
dimensions) equivalent to the magnetic field and one can literally identify
the vector potential $\vec{A}$ with ${\bf B}\,{\bf q}$.
Indeed the misinterpretation of the matrices that describe the entrance and
the exit of solenoids magnets also leads to seemingly non-symplectic motion
inside the solenoid magnet. The transfer matrix $M_{sol}$ of the solenoid
field is in the above coordinate ordering Hinterberger :
${\bf
M}_{sol}=\left(\begin{array}[]{cccc}1&0&{L\over\alpha}\,S&{L\over\alpha}\,(C-1)\\\
0&1&{L\over\alpha}\,(1-C)&{L\over\alpha}\,S\\\ 0&0&C&-S\\\ 0&0&S&C\\\
\end{array}\right)\,,$ (22)
where $S=\sin{(\alpha)}$ and $C=\cos{(\alpha)}$, which is formally also non-
symplectic. But the product of the matrix for the entrance field ${\bf T}$
(Eqn. 19), ${\bf M}_{sol}$ and ${\bf T}^{-1}$ turns out to be symplectic.
Hence we have:
$({\bf T}\,{\bf M}_{sol}\,{\bf T}^{-1})\,\gamma_{0}\,({\bf T}\,{\bf
M}_{sol}\,{\bf T}^{-1})^{T}=\gamma_{0}\\\ $ (23)
from which one derives in a few steps:
$\begin{array}[]{rcl}{\bf T}^{-1}\,\gamma_{0}\,({\bf T}^{-1})^{T}&=&{\bf
M}_{sol}\,{\bf T}^{-1}\,\gamma_{0}\,({\bf T}^{-1})^{T}\,{\bf M}_{sol}^{T}\\\
\tilde{\gamma}_{0}&=&{\bf M}_{sol}\,\tilde{\gamma}_{0}\,{\bf
M}_{sol}^{T}\,,\end{array}$ (24)
so that one may also re-interpret the process as a transformation of the
symplectic unit matrix:
$\tilde{\gamma}_{0}={\bf T}^{-1}\,\gamma_{0}\,({\bf T}^{-1})^{T}\,.$ (25)
But in fact, what it really describes is a change of the vector potential.
## V Bending Magnets
In the previous section we developed a proper interpretation of the matrices
that describe particle motion at the entrance of a solenoid magnet. This
raises the question, if there is an analog phenomenon at the entrance of
bending magnets. In order to clarify this, we rederive the transfer matrix of
a bending magnet in the following. Again we ignore motion parallel to the
magnetic field, which is in this case the axial (i.e. transverse vertical)
motion.
Motion of charged particles in electromagnetic fields is described by the
Lorentz force equation:
${d\vec{p}\over dt}=q\,(\vec{E}+\vec{v}\times\vec{B})\,,$ (26)
written in cartesic coordinates:
$\begin{array}[]{rcl}{dp_{x}\over
dt}&=&q\,(E_{x}+v_{y}\,B_{z}-v_{z}\,B_{y})\\\ {dp_{y}\over
dt}&=&q\,(E_{y}+v_{z}\,B_{x}-v_{x}\,B_{z})\\\ {dp_{z}\over
dt}&=&q\,(E_{z}+v_{x}\,B_{y}-v_{y}\,B_{x})\,.\end{array}$ (27)
We choose the $z$-coordinate as the vertical (axial) direction so that $x$ and
$y$ and the horizontal coordinates. The motion in the median plane of a
bending magnet is then (in the absence of acceleration) described by:
$\begin{array}[]{rcl}{dp_{x}\over dt}&=&q\,v_{y}\,B_{z}\\\ {dp_{y}\over
dt}&=&-q\,v_{x}\,B_{z}\\\ \end{array}$ (28)
In a first step, we devide both equations by $m\,\gamma$, which is (in the
absence of acceleration) constant:
$\begin{array}[]{rcl}{dv_{x}\over dt}&=&{q\over m\,\gamma}\,v_{y}\,B_{z}\\\
{dv_{y}\over dt}&=&-{q\over m\,\gamma}\,v_{x}\,B_{z}\\\ \end{array}$ (29)
We consider the orbit as the trajectory of the reference particle and we aim
for a description of the motion in the vicinity of the orbit, i.e. of the
trajectories of particles with small deviations from the orbit.
We start with the state vectors of the orbit $\psi_{o}$ and of the trajectory
$\psi$ in cartesic coordinates $\psi=(x,v_{x},y,v_{y})^{T}$ fn4 :
$\begin{array}[]{rcl}{d\over dt}\,\psi&=&{\bf
F}\,\psi=\left(\begin{array}[]{cccc}0&1&0&0\\\ 0&0&0&{q\over
m\,\gamma}\,B_{z}\\\ 0&0&0&1\\\ 0&-{q\over m\,\gamma}\,B_{z}&0&0\\\
\end{array}\right)\,\psi\\\ \end{array}$ (30)
Since $B_{z}$ is the only relevant component in the median plane, we skip the
“z” from now on. Furthermore, we like to have a mathematically positive
angular velocity and hence for positive charge we need to have a negative
field $B_{z}$, so that we define $B=-B_{z}$.
A rotation in the horizontal plane is described by the following generator
matrix rdm_paper ; geo_paper :
$\begin{array}[]{rcl}{\bf
F}_{rot}&=&\omega\,\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&0&0&-1\\\
1&0&0&0\\\ 0&1&0&0\\\ \end{array}\right)\\\ \end{array}$ (31)
The coordinate transformation into the rotating frame is then done by
subtracting the rotational “force matrix” from the matrix ${\bf F}$ fn5 :
$\begin{array}[]{rcl}{d\over dt}\,\psi&=&{\bf
F}\,\psi=\left(\begin{array}[]{cccc}0&1&\omega&0\\\ 0&0&0&-{q\over
m\,\gamma}\,B+\omega\\\ -\omega&0&0&1\\\ 0&-\omega+{q\over
m\,\gamma}\,B&0&0\\\ \end{array}\right)\,\psi\\\ \end{array}$ (32)
For synchronous rotation the rotational frequency $\omega$ must equal ${q\over
m\,\gamma}\,B$, so that one obtains in the co-moving frame
$\begin{array}[]{rcl}{d\over dt}\,\psi&=&{\bf
F}\,\psi=\left(\begin{array}[]{cccc}0&1&\omega&0\\\ 0&0&0&0\\\
-\omega&0&0&1\\\ 0&0&0&0\\\ \end{array}\right)\,\psi\\\ \end{array}$ (33)
Next we consider small deviations from the orbit $\psi_{o}$ and write:
$\begin{array}[]{rcl}{d\over dt}\,\psi_{o}&=&{\bf F}_{o}\,\psi_{o}\\\ {d\over
dt}\,\psi&=&{\bf F}\,\psi\\\ {d\over dt}\,(\psi-\psi_{o})&=&{\bf F}\,\psi-{\bf
F}_{o}\,\psi_{o}\\\ {d\over dt}\,\delta\psi&=&({\bf F}-{\bf F}_{o})\,\psi+{\bf
F}_{o}\,\delta\,\psi\,.\end{array}$ (34)
Since the condition $\omega={q\over m\,\gamma}\,B$ holds only for the orbit
(but not for all trajectories), we express the deviations by a Taylor series
which we evaluate at the orbit parameters and truncate to the linear terms:
$\begin{array}[]{rcl}{1\over\gamma}&=&{1\over\gamma_{o}}-\gamma_{o}\,{v_{o}\over
c^{2}}\,(v-v_{o})={1\over\gamma_{o}}(1-\gamma_{o}^{2}\,{v_{o}^{2}\over
c^{2}}\,{v-v_{o}\over v_{o}})\\\
&=&{1\over\gamma_{o}}(1-\gamma_{o}^{2}\,{\beta_{o}^{2}}\,{\delta v\over
v_{o}})\\\ \end{array}$ (35)
and
$\begin{array}[]{rcl}B&=&B_{o}+{dB\over dx}\,(x-x_{o})\\\ &=&B_{o}\,(1+{1\over
B_{o}}{dB\over dx}\,\delta x)\\\ \end{array}$ (36)
Note that we did not include a term with ${dB\over dy}\,\delta y$, since a
field change along the longitudinal coordinate contradicts our assumption that
$\omega=\mathrm{const}$. We then find (neglecting higher order terms):
${q\,B\over m\,\gamma}\to{q\,B_{o}\over
m\,\gamma}\,(1-{\gamma^{2}\,\beta^{2}\over v_{o}}\,\delta v+{1\over
B_{o}}{dB\over dx}\,\delta x)\,.$ (37)
and hence $(\delta\,{\bf F}={\bf F}-{\bf F}_{o})$ is given by
$\delta\,{\bf F}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&0&-f\\\ 0&0&0&0\\\
0&f&0&0\\\ \end{array}\right)\,,$ (38)
where
$f={q\,B_{o}\over m\,\gamma}\,(-{\gamma^{2}\,\beta^{2}\over v_{o}}\,\delta
v+{1\over B_{o}}{dB\over dx}\,\delta x)\,.$ (39)
To this point we merely transformed into the rotating frame. The global
coordinates of the orbit in the rotating frame must be constant (but not
necessarily zero). The time derivative of the orbit must vanish in the
rotating frame, so that we expect from Eqn. LABEL:eq_rot
${\bf F}\,\psi_{o}=0\,,$ (40)
which is fulfilled by $\psi_{o}=(\rho,0,0,v_{o})^{T}$, if
$v_{o}=\omega\,\rho\,.$ (41)
This choice means that we choose $x$ to be the horizontal transverse and $y$
to be the longitudinal coordinate, from which we conclude that $v_{y}\approx
v\gg v_{x}$. Then we find (again skipping higher orders)
$\delta\,{\bf F}\,\psi=\delta\,{\bf
F}\,(\psi_{o}+\delta\psi)\approx\delta\,{\bf F}\,\psi_{o}\,,$ (42)
so that with $\delta\psi=(\delta x,v_{x},\delta y,\delta v)^{T}$ one finds
$\begin{array}[]{rcl}{d\over dt}\,\delta\psi&=&({\bf F}-{\bf
F}_{o})\,\psi_{o}+{\bf F}_{o}\,\delta\,\psi\\\ &=&\left(\begin{array}[]{c}0\\\
-v_{o}\,\omega\,(-{\gamma^{2}\,\beta^{2}\over v_{o}}\,\delta v+{1\over
B_{o}}{dB\over dx}\,\delta x)\\\ 0\\\ 0\\\ \end{array}\right)\\\
&+&\left(\begin{array}[]{cccc}0&1&\omega&0\\\ 0&0&0&0\\\ -\omega&0&0&1\\\
0&0&0&0\\\ \end{array}\right)\,\delta\,\psi\\\
&=&\left(\begin{array}[]{cccc}0&1&\omega&0\\\ -v_{o}\,\omega\,{1\over
B_{o}}{dB\over dx}&0&0&v_{o}\,\omega\,{\gamma^{2}\,\beta^{2}\over v_{o}}\\\
-\omega&0&0&1\\\ 0&0&0&0\\\
\end{array}\right)\,\delta\,\psi={\bf\tilde{F}}\,\delta\psi\\\ \end{array}$
(43)
We devide both sides by $v_{o}$ so that with ${d\psi\over ds}={1\over
v_{o}}\,{d\psi\over dt}$ we obtain
$\begin{array}[]{rcl}{d\over
ds}\,\delta\psi&=&\left(\begin{array}[]{cccc}0&{1\over v_{o}}&{\omega\over
v_{o}}&0\\\ -\omega\,{1\over B_{o}}{dB\over
dx}&0&0&\omega\,{\gamma^{2}\,\beta^{2}\over v_{o}}\\\ -{\omega\over
v_{o}}&0&0&{1\over v_{o}}\\\ 0&0&0&0\\\ \end{array}\right)\,\delta\,\psi\\\
\end{array}$ (44)
In the following we apply a sequence of 3 transformations described by
matrices ${\bf T}_{i}$, where each transformation is of the general form
$\begin{array}[]{rcl}\delta\psi&\to&{\bf T}_{i}\,\delta\,\psi\\\ {\bf
F}&\to&{\bf T}_{i}\,{\bf F}\,{\bf T}_{i}^{-1}\,,\end{array}$ (45)
where we omitted the tilde of the force matrix ${\bf F}$ for a better
readability.
The first transformation matrix ${\bf T}_{1}$ is used to scale the velocities
by $1/v_{o}$ and is given by:
${\bf T}_{1}=\mathrm{Diag}(1,{1\over v_{o}},1,{1\over v_{o}})\,,$ (46)
so that
${\bf F}=\left(\begin{array}[]{cccc}0&1&{\omega\over v_{o}}&0\\\ -{\omega\over
v_{o}}\,{1\over B_{o}}{dB\over dx}&0&0&{\omega\over
v_{o}}\,\gamma^{2}\,\beta^{2}\\\ -{\omega\over v_{o}}&0&0&1\\\ 0&0&0&0\\\
\end{array}\right)\,,$ (47)
and hence $\delta\psi$ is now given by:
$\delta\psi=(\delta x,{\delta v_{x}\over v_{o}},\delta y,{\delta v\over
v_{o}})^{T}\,.$ (48)
Due to the choice of $\psi_{o}=(x_{o},0,0,v_{o})^{T}$, $\delta x$ is the local
horizontal, $\delta y=y$ the local longitudinal coordinate and ${\delta
v_{y}\over v_{o}}\approx{\delta v\over v_{o}}={1\over\gamma^{2}}\,{\delta
p\over p}$ is the velocity deviation, so that with the field index $n_{x}$
define by $n_{x}={\rho\over B_{o}}{dB\over dx}$ and ${w\over
v_{o}}={1\over\rho}$ we obtain
${\bf F}=\left(\begin{array}[]{cccc}0&1&{1\over\rho}&0\\\
-{n_{x}\over\rho^{2}}&0&0&{\gamma^{2}\,\beta^{2}\over\rho}\\\
-{1\over\rho}&0&0&1\\\ 0&0&0&0\\\ \end{array}\right)\,.$ (49)
Next we transform from the velocity deviation to the momentum deviation using
${\bf T}_{2}$
${\bf T}_{2}=\mathrm{Diag}(1,1,1,\gamma^{2})\,.$ (50)
The result is:
${\bf F}=\left(\begin{array}[]{cccc}0&1&{1\over\rho}&0\\\
-{n_{x}\over\rho}&0&0&{\beta^{2}\over\rho}\\\
-{1\over\rho}&0&0&{1\over\gamma^{2}}\\\ 0&0&0&0\\\ \end{array}\right)\,.$ (51)
Figure 1: Transformation into curvilinear coordinate system. The trajectory
with deviation $\delta x$ in position A causes a deviation $\delta y$ in
position B, where one finds no direction difference in cartesic coordinates.
Interpreted in curvilinear (i.e. cylindrical) coordinates one has (in first
order) a direction deviation $x^{\prime}={\delta y\over\rho}$.
The last transformation ${\bf T}_{3}$ required to obtain the well-known
transfer matrix of a bending magnet, transforms from the local co-moving
cartesic system to the local co-moving curvilinear system. The transformation
is explained in Fig. 1:
${\bf T}_{3}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&{1\over\rho}&0\\\
0&0&1&0\\\ 0&0&0&1\\\ \end{array}\right)\,.$ (52)
This last transformation yields finally:
${\bf F}=\left(\begin{array}[]{cccc}0&1&0&0\\\
-{1+n_{x}\over\rho^{2}}&0&0&{1\over\rho}\\\
-{1\over\rho}&0&0&{1\over\gamma^{2}}\\\ 0&0&0&0\\\ \end{array}\right)\,.$ (53)
The (symplectic) transfer matrix ${\bf M}_{b}=\exp{({\bf F}\,s)}$ then is:
$\begin{array}[]{rcl}{\bf
M}_{b}&=&\left(\begin{array}[]{cccc}C&{\rho\,S\over\sqrt{1+n_{x}}}&0&\rho\,{1-C\over
1+n_{x}}\\\ -{\sqrt{1+n_{x}}\over\rho}\,S&C&0&{S\over\sqrt{1+n_{x}}}\\\
-{S\over\sqrt{1+n_{x}}}&-{\rho\,(1-C)\over
1+n_{x}}&1&{\rho\,S\over(1+n_{x})^{3/2}}+{s\over\gamma^{2}}-{s\over
1+n_{x}}\\\ 0&0&0&1\\\ \end{array}\right)\\\
S&=&\sin{(\alpha\,\sqrt{1+n_{x}})}\\\
C&=&\cos{(\alpha\,\sqrt{1+n_{x}})}\,,\end{array}$ (54)
where the bending angle $\alpha$ is given by $\alpha={s\over\rho}$. As in case
of the solenoid magnet, it is possible to split the transfer matrix ${\bf
M}_{b}$ into 3 parts, first the transformation into curvilinear coordinates
${\bf T}_{3}$ which then represents the fringe field (without entrance angle),
second the transfer matrix of the bending magnet “itsself” and finally the
transformation ${\bf T}_{3}^{-1}$ back to cartesic coordinates. The transfer
matrix for the bending magnet (analog to ${\bf M}_{sol}$ as given in Eqn. 22)
is the matrix exponent of the force matrix (as given by Eqn. 51) multiplied by
the pathlength $s=\alpha\,\rho$ and is explicitely given by:
$\begin{array}[]{rcl}{\bf M}_{bend}&=&\exp{({\bf F}\,s)}\\\
&=&\left(\begin{array}[]{cccc}C&{\rho\,S\over\sqrt{k}}&{S\over\sqrt{k}}&{\rho\,(1-C)\over
k}\\\ -{n_{x}\,S\over\rho\,\sqrt{k}}&{1+n_{x}\,C\over
k}&{(C-1)\,n_{x}\over\rho\,k}&X\\\ -{S\over\sqrt{k}}&{\rho\,(C-1)\over
k}&{C+n_{x}\over k}&Y\\\ 0&0&0&1\\\ \end{array}\right)\\\
X&=&{\alpha\,(\gamma^{2}-1)\,\sqrt{k}+n_{x}\,(\gamma^{2}\,S-\alpha\,\sqrt{k})\over\gamma^{2}\,k^{3/2}}\\\
Y&=&\rho\,\left({S\over k^{3/2}}+\alpha\,({1\over\gamma^{2}}-{1\over
k})\right)\\\ k&=&1+n_{x}\\\ S&=&\sin{(\alpha\,\sqrt{k})}\\\
C&=&\cos{(\alpha\,\sqrt{k})}\,,\end{array}$ (55)
where $\alpha$ is the bending angle of the magnet and $\rho$ the bending
radius or the orbit. Then one verifies from Eqn. 52 and Eqn. 54:
${\bf M}_{b}={\bf T}_{3}\,{\bf M}_{bend}\,{\bf T}_{3}^{-1}\,,$ (56)
so that the complete symplectic transfer matrix of a bending magnet may be
regarded as a product of 3 “non-symplectic” matrices, just as one finds it for
solenoids. In essence we merely applied the equation
$\exp{({\bf T}_{3}\,{\bf F}\,{\bf T}_{3}^{-1}\,s)}={\bf T}_{3}\,\exp{({\bf
F}\,s)}\,{\bf T}_{3}^{-1}\,,$ (57)
which we believe to reflect the essential difference in typical textbook
descriptions of bending magnets (left side, symplectic) and solenoids (right
side, 3 times “non-symplectic”).
In order to facilitate comparison with Sec. IV, we go back to the coordinate
ordering from Sec. II, i.e. first the coordinates and then the momenta (or
“velocities”). The matrix ${\bf T}_{3}$ is then written as
${\bf T}_{3}=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\
0&{1\over\rho}&1&0\\\ 0&0&0&1\\\ \end{array}\right)$ (58)
If we compare this with Eqn. 19 and Eqn. 15, then we find that the difference
is merely the gauge represented by a symmetric matrix ${\bf B}$ of the form
${\bf B}_{s}=\frac{1}{2}\,\left(\begin{array}[]{cc}0&-{1\over\rho}\\\
-{1\over\rho}&0\\\ \end{array}\right)\,.$ (59)
And as we derived above, a non-vanishing symmetric part of ${\bf B}$ equals a
symplectic gauge-transformation without influence on the dynamics of ${\bf q}$
and ${\bf\dot{q}}$.
## VI Application to Numerical Tracking Computations
All the above developed formalism stays academic as long as we do not refer to
a practical “problem”. In Ref. sc_paper we described an iterative method to
determine the parameters of a matched beam matrix of second moments $\sigma$
for cyclotrons with strong space charge forces. Using samples with typically
$10^{5}$ particles random , the parallel framework OPAL has been used to
simulate coasting beams in cyclotrons similar to the PSI ring machine Ring
and some results have been presented scopal .
Figure 2: Upper graph: Eigenvalues of the matrix ${\bf S}=\sigma\,\gamma_{0}$
of a Gaussian particle distribution tracked along the equilibrium orbit over
one sector of a separate sector ring cyclotron. The transformation ${\bf
T}_{3}$ has not been applied. The horizontal eigenvalues (thin solid line),
the longitudinal eigenvalue (dashed line) and the product of both (dotted
line). The thick solid line shows the magnetic field in Tesla. Lower graph:
The same figure after the transformation ${\bf T}_{3}$. The eigenvalues are
all constant along the orbit as expected for symplectic motion.
The distributions turned out to be properly matched only for a starting
position in the field free region (i.e. between sector magnets), while the
matching failed when the tracking started somewhere within the sector magnet.
A detailled analysis (including a cross check with a second tracking code
without space charge solver) suggested, that the eigen-emittances from the
distributions evaluated in cartesic coordinates where constant only in
constant field regions, but changed from valley to sector (and vice versa).
The transformation from the local cartesic to the local curvilinear coordinate
system with the matrix ${\bf T}_{3}$ as derived above solved the problem and
verified that the motion is indeed symplectic. The eigen-emittances evaluated
in local cartesic and local curvilinear coordinate systems are shown in Fig. 2
as a function of time (i.e. step-number).
## VII Summary
We investigated symplectic motion in magnetic fields using the examples of
solenoid and bending magnets. We rederived the transfer matrix of a bending
magnet starting from the Lorentz force equation in cartesic coordinates. We
found that the motion is symplectic in both types of magnets, if one takes the
proper canonical momentum into account. Furthermore it turned out that there
is no essential difference between solenoid and bending magnets, despite the
fact that they are often described differently. We also found that the
curvature ($1/\rho$) of the local coordinate system is intimately connected to
the vector potential which is (in linear approximation) given by the matrix
${\bf B}$ multiplied by the coordinates ${\bf q}$.
We applied these findings to tracking of particle distributions in cartesic
coordinates and gave the transformation between local curvilinear and local
cartesic coordinates. We showed that the motion is formally symplectic only in
local curvilinear coordinates.
## VIII Acknowledgements
We thank J.J. Yang for fruitful discussions about particle tracking with OPAL.
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* (9) C. Baumgarten; Phys. Rev. ST Accel. Beams. 15, 124001 (2012).
* (10) R. Talman: Geometric Mechanics; 2nd Ed., Wiley-VCH Weinheim, Germany, 2007.
* (11) V.A. Yakubovich and V.M. Starzhinskii: Linear Differential Equations with Periodic Coefficients; John Wiley and Sons, New York, 1975.
* (12) Frank Hinterberger, Physik der Teilchenbeschleuniger (Springer, Heidelberg 2008), 2nd ed.
* (13) C. Baumgarten; arXiv:1205.3601 .
* (14) M. Seidel et. al.; Proc of IPAC 2010, ISBN 978-92-9083-352-9, p. 1309-1313.
* (15) C. Baumgarten; European Cyclotron Progress Meeting 2012, May 9-12, Villigen, Switzerland; Slides are available under indico.psi.ch:
https://indico.psi.ch/getFile.py/access?contribId=10
&sessionId=10&resId=0&materialId=slides&confId=1146.
* (16) If one takes into account that the Lagrangian allows for a scaling factor, the mass matrix effectivly has to be proportional to a unit matrix.
* (17) We ignore the motion parallel to the magnetic field, which is in case of a solenoid the longitudinal coordinate.
* (18) Note that this transformation has no influence on the second moments of the particle displacements, i.e. the beam size (or beam envelope, respectively). But it indeed changes the eigen-emittances.
* (19) Here we choose a different coordinate ordering compared to Sec. II in order to facilitate comparison with the conventional notation and we postpone the question, if the state vector is Hamiltonian of Lagrangian.
* (20) In analogy to Einstein’s equivalence principle of a uniformly accelerated reference frame and the force of gravitation, a frame rotating at constant angular velocity is equivalent to a gyroscopic force, i.e. a “magnetic” field.
|
arxiv-papers
| 2013-07-29T08:22:22 |
2024-09-04T02:49:48.635903
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Baumgarten",
"submitter": "Christian Baumgarten",
"url": "https://arxiv.org/abs/1307.7492"
}
|
1307.7581
|
# Stochastic switching in slow-fast systems: a large fluctuation approach
Christoffer R. Heckman [email protected] Ira B. Schwartz
[email protected] U.S. Naval Research Laboratory, Code 6792
Plasma Physics Division, Nonlinear Dynamical Systems Section
Washington, DC 20375, USA
###### Abstract
In this paper we develop a perturbation method to predict the rate of
occurrence of rare events for singularly perturbed stochastic systems using a
probability density function approach. In contrast to a stochastic normal form
approach, we model rare event occurrences due to large fluctuations
probabilistically and employ a WKB ansatz to approximate their rate of
occurrence. This results in the generation of a two-point boundary value
problem that models the interaction of the state variables and the most likely
noise force required to induce a rare event. The resulting equations of motion
of describing the phenomenon are shown to be singularly perturbed. Vastly
different time scales among the variables are leveraged to reduce the
dimension and predict the dynamics on the slow manifold in a deterministic
setting. The resulting constrained equations of motion may be used to directly
compute an exponent that determines the probability of rare events.
To verify the theory, a stochastic damped Duffing oscillator with three
equilibrium points (two sinks separated by a saddle) is analyzed. The
predicted switching time between states is computed using the optimal path
that resides in an expanded phase space. We show that the exponential scaling
of the switching rate as a function of system parameters agrees well with
numerical simulations. Moreover, the dynamics of the original system and the
reduced system via center manifolds are shown to agree in an exponentially
scaling sense.
singular perturbation, stochastic differential equation, optimal path, noise,
rare event
## I Introduction
Many stochastic systems of physical interest possess dynamics which occur over
multiple time scales. These systems present unique difficulties since the
multiple time scales interact with the stochasticity to affect the dynamics,
leading to phenomena such as stochastic switching resulting from large
fluctuations. For deterministic slow-fast systems singular perturbation theory
may be applied to guide analysis, while noisy systems are better understood
through tools from statistical mechanics.
The study of slow-fast systems has recently become popular as a result of the
insight it affords into fields such as chemical reactions and electro-
mechanical systems Desroches _et al._ (2012). Due to the presence of distinct
timescales on which phenomena occur in these singularly perturbed systems, it
becomes mathematically tractable to apply perturbation methods to accurately
predict the behavior of high-order systems in terms of low order ones. This
model reduction greatly simplifies bifurcation analysis and the identification
of qualitative behaviors. The approach of perturbation methods is especially
useful because the alternative, running large-scale numerical simulations from
which one may calculate statistics, is particularly burdensome for slow-fast
systems. Such systems generally require the use of implicit numerical
integrators in order to ensure numerical stability, the use of which is
extremely computationally expensive.
Separately, stochastic systems are frequently used to model both microscale
and macroscale behaviors that are inherently noisy or simpler to visualize as
driven by randomness. Examples of these systems range from networks of sensors
in noisy environments to the control of epidemics. There are many intricacies
under investigation within this field such as finite noise effects and
stochastic resonance Gammaitoni _et al._ (1998) that provide for much lively
research, but will not be our focus in this paper. We will in particular study
the effect of small noise on the escape times for a particle in a multi-scale
potential well. To do so, we will make use of the variational theory of large
fluctuations as it applies to finding the _most probable path_ along which
noise directs a particle to escape Chan _et al._ (2008).
It is well-known that noise has a significant effect on deterministic
dynamical systems. For example, consider a given initial state in the basin of
attraction for a given attractor, which might be steady, periodic, or chaotic.
Noise can cause the trajectory to cross the deterministic basin boundary and
move into another, distinct basin of attraction Dykman (1990); Dykman _et
al._ (1992); Millonas (1996); Luchinsky _et al._ (1998). For sufficiently
small noise, basin boundary crossings usually occur near a saddle on the
boundary. However we note that for large noise, such a crossing may be
determined by the global manifold structure away from the saddle.
This paper will consider small noise effects in particular. Specifically, we
will investigate the effect of arbitrarily small noise on the escape of a
particle from a potential well. In the small noise limit, one can apply large
fluctuation theory Feynman and Hibbs (1965); Dykman (1990); Dykman _et al._
(1992); Luchinsky _et al._ (1998); also known as large deviation theory used
in white noise analysis Freidlin and Wentzell (1984); Feynman and Hibbs
(1965); E (2011), this approach enables us to determine the first passage
times in a multi-scale environment. For a vector field that exhibits dynamics
on only one timescale, it is clear how to use the theory to generate an
optimal path of escape. The theory has been applied to a variety of
Hamiltonian and Lagrangian variational problems Wentzell (1976); Hu (1987);
Dykman _et al._ (1994); Freidlin and Wentzell (1984); Graham and Tél (1984);
Maier and Stein (1993); Hamm _et al._ (1994) that do not exhibit singularly
perturbed behavior.
For slow-fast systems however, technical issues arise while determining the
projection of noise restricted to the lower dimensional manifold. Several
sample based approaches have been developed to understand dimension reduction
in systems that have well separated time scales Berglund and Gentz (2006). The
existence of a stochastic center manifold was proven in Boxler (1989) for
systems with certain spectral requirements. Non-rigorous stochastic normal
form analyses (which lead to the stochastic center manifold) were performed in
Knobloch and Wiesenfeld (1983); Namachchivaya (1990); Namachchivaya and Lin
(1991). More rigorous theoretical treatments of normal form coordinate
transformations for stochastic center manifold reduction were developed in
Arnold and Imkeller (1998); Arnold (1998); Kabanov and Pergamenshchikov
(2003). Later, another method of stochastic normal form reduction was
developed in which anticipatory convolutions (integrals into the future of the
noise processes) that appeared in the equations for the slow dynamics were
ignored Roberts (2008). This latter stochastic normal form technique was
possible because the epidemic model under study permitted certain assumptions
on the magnitude of the noise projections. The disadvantage of such
assumptions compared to probabilistic methods is that there must be guarantees
to keep stochastic solutions bounded in the past and future Forgoston and
Schwartz (2009), which we may not always have.
We will restrict our study to systems with two stable equilibria separated by
an unstable equilibrium point in phase space; the method of center manifold
approximations however is not strictly reserved for this case. This paper
begins by introducing some general theory related to slow-fast systems and
center manifold reductions. We then review large fluctuation theory and how it
applies to determining the optimal path between invariant manifolds in
stochastic systems. Next we follow many other works and apply the theory to
the example of a damped Duffing oscillator to compare. Finally we compare the
switching time estimated via large fluctuation theory with numerical results
for the example system. We note that although much work on white noise model
reduction is being done using sample-based methods and asymptotics, our
variational approach is more general in that it may include non-Gaussian noise
sources as well.
## II Theory
We consider a general $(m+n)$-dimensional dynamical system of stochastic
differential equations with two well-separated timescales and additive noise
on the slow variables:
$\displaystyle\bm{\dot{x}}$ $\displaystyle=\bm{F}(\bm{x},\bm{y})+\bm{\Phi}(t)$
(1) $\displaystyle\epsilon\bm{\dot{y}}$ $\displaystyle=\bm{G}(\bm{x},\bm{y})$
(2)
where $\bm{x}\in\mathbb{R}^{m}$, $\bm{y}\in\mathbb{R}^{n}$; $\bm{\Phi}(t)$ are
stochastic terms with characteristics depending on the application;
$\bm{F}:\mathbb{R}^{m}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ and
$\bm{G}:\mathbb{R}^{m}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ are
differentiable functions with equilibrium points at the origin, and $\epsilon$
is a small parameter. Such systems are known as singularly perturbed or slow-
fast systems Guckenheimer _et al._ (2004) with timescales separated by a
ratio of $\epsilon$. In this system, $\bm{x}$ are slow variables and $\bm{y}$
are fast variables. Rescaling $\tau=\epsilon t$ and temporarily removing the
stochastic terms results in the _layer equations_. Denoting
$(\cdot)^{\prime}=\frac{d}{d\tau}$, the deterministic part of Eqs. (1), (2)
becomes:
$\displaystyle\bm{x}^{\prime}$ $\displaystyle=\epsilon\bm{F}(\bm{x},\bm{y})$
(3) $\displaystyle\bm{y}^{\prime}$ $\displaystyle=\bm{G}(\bm{x},\bm{y})$ (4)
$\displaystyle\epsilon^{\prime}$ $\displaystyle=0.$ (5)
Note that since $\epsilon$ is treated as a state variable in Eqs. (3)–(5),
then all terms in Eq. (3) are necessarily nonlinear. If
$\bm{G}(\bm{x},\bm{y})$ has a linear part with nonzero determinant, then there
exists an $m$-dimensional center manifold tangent to the center eigenspace at
the origin. By the implicit function theorem, we may write the manifold
locally as a function
$\bm{h}:\mathbb{R}^{m}\times\mathbb{R}\rightarrow\mathbb{R}^{n}$:
$\bm{y}=\bm{h}(\bm{x},\epsilon).$ (6)
Following Carr Carr (1981), the center manifold may be approximated to
arbitrary order by a polynomial series in $\bm{x}$ and $\epsilon$. All
solutions collapse to this manifold at an exponential rate since it is
hyperbolic.
### II.1 Stochastic switching
Stochastic differential equations cannot be described by deterministic orbits
representing trajectories of a particle through phase space. Instead, other
approaches are used to qualitiatively describe the system. For instance,
sample based techniques may describe individual realizations in phase space,
or families of such realizations. Another technique is to find a probability
density function (pdf) describing the likelihood of finding a particle at a
given point and time. If the noise is Gaussian and uncorrelated in time, the
dynamics of the pdf $\rho(\bm{z},t)$, where $\bm{z}=(\bm{x};\bm{y})$ is the
concatenated vector of state variables, are governed by the Fokker-Planck
equation Gardiner (2004):
$\displaystyle\frac{\partial\rho(\bm{z},t)}{\partial t}=$
$\displaystyle-\sum_{i=1}^{m+n}\frac{\partial}{\partial
z_{i}}\left(\rho(\bm{z},t)\,\mathcal{F}_{i}\right)$
$\displaystyle+\sum_{i=1}^{m+n}\sum_{j=1}^{m+n}\frac{\partial^{2}}{\partial
z_{i}\,\partial z_{j}}\left(D_{ij}\,\rho(\bm{z},t)\right)$ (7)
where $\bm{\mathcal{F}}=(\bm{F};\bm{G})$ is the concatenated vector of
functions describing the vector field and $D_{ij}$ is a diffusion coefficient
matrix.
Equation (7) relates the time derivative of the probability density function
$\rho(\bm{z},t)$ with expressions involving spatial derivatives of the vector
field $\bm{\mathcal{F}}$. Presuming the characterization of the noise and the
vector fields are autonomous, the pdf will asymptotically approach a steady
state distribution that is independent of time. Therefore, we seek steady
state solutions to Eq. (7); that is,
$\sum_{i=1}^{m+n}\frac{\partial}{\partial
z_{i}}\left(\rho(\bm{z})\,\mathcal{F}_{i}\right)=\sum_{i=1}^{m+n}\sum_{j=1}^{m+n}\frac{\partial^{2}}{\partial
z_{i}\,\partial z_{j}}\left(D_{ij}\,\rho(\bm{z})\right).$ (8)
If the intensity for each noise term is equal and each component is
uncorrelated, then we may write $D_{ij}=D\delta_{ij}$. For the system
described in Eqs. (1), (2), it is also relevant that $D_{ij}|_{i,j>m}=0$ since
additive noise only affects the slow variables; this results in
$\sum_{i=1}^{m+n}\frac{\partial}{\partial
z_{i}}\left(\rho(\bm{z})\,\mathcal{F}_{i}\right)=D\sum_{i=1}^{m}\frac{\partial^{2}}{\partial
z_{i}^{2}}\rho(\bm{z}).$ (9)
We will now assume a certain form for the pdf that will allow us to solve Eq.
(9) keeping in mind that the goal is to analyze stochastically-induced
switching. In the small-noise limit, transitions between attractors happen
only rarely. Therefore, noise leading to a transition is considered to be in
the tail of the probability distribution that governs the amplitude of the
noise. A stochastically-induced switch is _most likely_ to occur in the
presence of a hypothesized “optimal noise,” which has a finite likelihood of
occurrence. The path that the system travels through phase space under the
influence of the optimal noise is known as the “optimal path.” Such an event
follows an exponential distribution which we will use as an ansatz to solve
Eq. (9). The WKB ansatz states that
$\rho(\bm{z})\propto\exp\left(-\frac{1}{2D}R(\bm{z})\right)$. Applying this to
the steady-state Fokker-Planck equation (9) yields the differential equation
$\sum_{i=1}^{m+n}-\frac{\partial R}{\partial
z_{i}}\mathcal{F}_{i}+2D\frac{\partial\mathcal{F}_{i}}{\partial
z_{i}}=\sum_{i=1}^{m}-D\frac{\partial R}{\partial
z_{i}^{2}}+\frac{1}{2}\left(\frac{\partial R}{\partial z_{i}}\right)^{2}.$
Since we are operating in the small-noise limit, any terms multiplied by $D$
will be small; for now, we will neglect them leaving the first order nonlinear
equation:
$\sum_{i=1}^{m+n}-\frac{\partial R}{\partial
z_{i}}\mathcal{F}_{i}=\frac{1}{2}\sum_{i=1}^{m}\left(\frac{\partial
R}{\partial z_{i}}\right)^{2}.$ (10)
In some cases, solving for $R$ in Eq. (10) is possible and would result in a
stationary pdf for Eqs. (1), (2). However, combined with the results in the
following section, we will demonstrate that not only is there a
straightforward way to tackle solving for $R$ but also that it is intimately
related with the principle of least action and the formulation of an optimal
path.
### II.2 Formulation of the Optimal Path
We wish to study the transition rates due to stochastic fluctuation between
two energy minima. Consider a system with two stable equilibrium points
$\bm{z}_{1}$ and $\bm{z}_{2}$ with a saddle point $\bm{z}_{s}$ separating
them. Since the noise intensity $D$ is small, we assume that switching between
the two states will be considered a “rare event.” The frequency of such an
event is approximately determined by the most likely noise to bring the system
from $\bm{z}_{1}$ to $\bm{z}_{s}$, i.e. the optimal noise. A realization of
the optimal noise is calculated to guide the particle to the saddle point
$\bm{z}_{s}$, which corresponds to the mean first passage time (MFPT). The
method to calculate this path makes use of Hamiltonian’s principle. One may
predict the switching rate by first finding the optimal path between the two
states in an expanded phase space which accounts for the noise and then
calculating the dynamical quantity known as the action along that path. For a
rigorous explanation of this procedure, see Dykman _et al._ (1994); Weiss
(1994).
The optimal path is the path that is of minimal action. We write the action of
the noise on the system (1), (2) as:
$\displaystyle\mathcal{R}[\bm{x},\bm{y},$
$\displaystyle\bm{\Phi},\bm{\lambda_{x}},\bm{\lambda_{y}}]=\frac{1}{2}\int\bm{\Phi}(t)\cdot\bm{\Phi}(t)dt$
$\displaystyle+\int\bm{\lambda_{x}}\cdot(\dot{\bm{x}}-\bm{F}(\bm{x},\bm{y})-\bm{\Phi}(t))dt$
$\displaystyle+\int\bm{\lambda_{y}}\cdot(\epsilon\dot{\bm{y}}-\bm{G}(\bm{x},\bm{y}))dt.$
(11)
The action integral Eq. (11) represents the total effect of noise on the
system subject to the constraints of the vector field. The first term
involving the action of the noise is derived by taking a WKB approximation of
the Chapman-Kolmogorov equation Chan _et al._ (2008) for the infinitesimal
noise events along the path for white noise. The $\bm{\lambda}$ factors are
Lagrange multipliers, and the terms multiplying them are the constraint
equations. The integral is calculated along the path for all time. We note
that Eq. (11) is a natural way to describe the effects of noise from both
white and non-Gaussian sources.
To find the functions that minimize the action, we take the first variation of
the above equation with respect to the independent variables and set them
equal to zero. This will give five sets of equations that when solved will
extremize the action $\mathcal{R}$. An example of these variational
calculations (with variation $\xi\in C^{1}$ bounded) on the action with
respect to the functions $x_{i}$ is:
$\displaystyle\frac{\delta\mathcal{R}}{\delta x_{i}}=$
$\displaystyle\int\lambda_{x_{j}}\left(\dot{\xi}-\xi\frac{\partial
F_{j}}{\partial x_{i}}\right)dt+\int\lambda_{y_{j}}\left(-\xi\frac{\partial
G_{j}}{\partial x_{i}}\right)dt$ $\displaystyle=$
$\displaystyle\int\xi\left(-\dot{\lambda}_{x_{i}}-\lambda_{x_{j}}\frac{\partial
F_{j}}{\partial x_{i}}-\lambda_{y_{j}}\frac{\partial G_{j}}{\partial
x_{i}}\right)dt=0.$ (12)
Arriving at the second equality involves integrating by parts; since the
functional derivative restricts the variations $\xi$ to be bounded, the first
term arising from integration by parts vanishes. Given Eq. (12), we have that
the function multiplying $\xi$ in the integrand must vanish; this yields the
differential equation:
$\dot{\lambda}_{x_{i}}+\lambda_{x_{j}}\frac{\partial F_{j}}{\partial
x_{i}}+\lambda_{y_{j}}\frac{\partial G_{j}}{\partial x_{i}}=0.$ (13)
In the same way, the following equations were derived for the first variation
with respect to $y_{i}$, $\lambda_{x_{i}}$, $\lambda_{y_{i}}$ and $\Phi_{i}$:
$\displaystyle\frac{\delta\mathcal{R}}{\delta y_{i}}=0$
$\displaystyle\implies\epsilon\dot{\lambda}_{y_{i}}+\lambda_{y_{j}}\frac{\partial
G_{j}}{\partial y_{i}}+\lambda_{x_{j}}\frac{\partial F_{j}}{\partial y_{i}}=0$
(14) $\displaystyle\frac{\delta\mathcal{R}}{\delta\lambda_{y_{i}}}=0$
$\displaystyle\implies\epsilon\dot{y}_{i}=G_{i}$ (15)
$\displaystyle\frac{\delta\mathcal{R}}{\delta\lambda_{x_{i}}}=0$
$\displaystyle\implies\dot{x}_{i}=F_{i}+\Phi_{i}$ (16)
$\displaystyle\frac{\delta\mathcal{R}}{\delta\Phi_{i}}=0$
$\displaystyle\implies\Phi_{i}=\lambda_{x_{i}}$ (17)
where $i=1,\dots,m$ and $i=1,\dots,n$ for the slow and fast variables and
their conjugate momenta respectively.
To make a connection with Section II.1, we will for a moment consider the
singular limit as $\epsilon\rightarrow 0$ of the vector field in Eqs.
(14)–(17). This approximation describes the behavior of a particle in the
$x_{i}$ and $\lambda_{x_{i}}$ coordinates after fast transients have died out
and yields a system known as the “slow equations.” The slow equations are:
$\displaystyle\dot{x}_{i}$ $\displaystyle=F_{i}+\lambda_{x_{i}}$ (18)
$\displaystyle\dot{\lambda}_{x_{i}}$ $\displaystyle=-\frac{\partial
F_{j}}{\partial x_{i}}\lambda_{x_{j}}.$ (19)
The slow equations represent a conservative system. To calculate the
corresponding Hamiltonian, we note that:
$\displaystyle\dot{x}_{i}$
$\displaystyle=\frac{\partial\mathcal{H}}{\partial\lambda_{x_{i}}}$
$\displaystyle\dot{\lambda}_{x_{i}}$
$\displaystyle=-\frac{\partial\mathcal{H}}{\partial x_{i}}$
where the Hamiltonian is:
$\mathcal{H}=F_{i}\lambda_{x_{i}}+\frac{1}{2}\lambda_{x_{i}}\lambda_{x_{i}}.$
(20)
Setting $\mathcal{H}=0$ in Eq. (20) verifies an intriguing relationship: if
one identifies $\lambda_{x_{i}}(\bm{x})=\frac{\partial R(\bm{x})}{\partial
x_{i}}$, Eq. (20) and Eq. (10) are equivalent for the singular case. This
confirms our earlier analysis using variational calculus and verifies that
$R(\bm{z})$ in the Eikonal approximation is indeed the action.
The probability of a rare event occurring described by that approximation is
directly proportional to the switching rate, or its inverse, the mean first
passage time. This quantity, denoted $T_{S}$ is inversely proportional to the
switching rate. Since the action will be calculated along the optimal path,
$R=\min\mathcal{R}$ and the relation to the switching time is
$T_{S}=c\exp(R/2D).$ (21)
Since the switching rate is proportional to the probability of a large
fluctuation, there is a proportionality constant $c$ that is yet to be
determined. The calculation of this prefactor is the subject of ongoing
research Dykman (2010), but is not the focus of the current work.
## III Application: the damped Duffing oscillator
To test the method, we consider a prototypical example for a double-welled
potential—the damped Duffing oscillator. A stochastic variant of this
oscillator is:
$\displaystyle\dot{x}$ $\displaystyle=y+\eta(t)$ (22)
$\displaystyle\epsilon\dot{y}$ $\displaystyle=x-x^{3}-y$ (23)
where $\epsilon$ is a small parameter and $\eta(t)$ is a noise source. We will
consider the case where $\eta(t)$ represents uncorrelated Gaussian white noise
and is defined by
$\langle\eta(t)\eta(t^{\prime})\rangle=2D\delta(t-t^{\prime}).$
The noise intensity, which controls the width of the distribution of noise, is
represented by $D=\sigma^{2}/2$ where $\sigma$ is the standard deviation of
the noise.
Applying Eqs. (14)-(17) to this system, the variational equations for the
damped Duffing oscillator in Eqs. (22)-(23) are:
$\displaystyle\dot{\lambda_{1}}$ $\displaystyle=(3x^{2}-1)\lambda_{2}$ (24)
$\displaystyle\epsilon\dot{\lambda_{2}}$
$\displaystyle=\lambda_{2}-\lambda_{1}$ (25) $\displaystyle\epsilon\dot{y}$
$\displaystyle=x-x^{3}-y$ (26) $\displaystyle\dot{x}$
$\displaystyle=y+\lambda_{1}$ (27)
Following the language of Kaper Kaper (1999), there are two limits over which
the system in Eqs. (24)–(27) may be studied. The first involves immediately
taking the limit as $\epsilon\rightarrow 0$ in the equations, while the latter
involves a rescaling of time and will be considered in the following section.
The first limit yields the slow equations; they are:
$\displaystyle\dot{\lambda_{1}}=(3x^{2}-1)\lambda_{1}$ (28)
$\displaystyle\dot{x}=x-x^{3}+\lambda_{1}.$ (29)
The critical dynamics in Eqs. (28), (29) have the equilibria
$(x,\lambda_{1})=\left\\{(\pm
1,0),(0,0),\left(\pm\frac{1}{\sqrt{3}},\mp\frac{2}{3\sqrt{3}}\right)\right\\}$.
Note that in the absence of noise, there is a path connecting the equilibria
along the $x$ axis. For nonzero noise, there is a heteroclinic connection in
the $x,\lambda_{1}$ plane between the two states which represents the optimal
path—the most likely trajectory for switching between the basins at $x=\pm 1$
and $x=0$. For this system it is possible to solve for this path explicitly
using a series of transformations. The optimal path for the $x$ coordinate
given as a solution to Eqs. (28), (29) is
$x(t)=\pm\frac{1}{\sqrt{1-A\exp(2t)}},$
where $A$ is an arbitrary coefficient to be determined by the initial
condition. Due to symmetry, it suffices to study switching between either
$x=\pm 1$ and $x=0$; we choose to examine switching from $-1$ to $0$, i.e. the
negative branch of $x(t)$. By inspection it is clear that $A<0$, otherwise
solutions would cease to exist in finite time. In calculating the action this
coefficient is irrelevant. Choosing $A=-1$ (implying $x(0)=\frac{1}{2}$)
without loss of generality results in the optimal path:
$x(t)=-\left(1+\exp(2t)\right)^{-1/2}.$ (30)
Integrating and solving for the arbitrary unknown functions, the Hamiltonian
for the slow system is:
$\mathcal{H}=(x-x^{3})\lambda_{1}+\lambda_{1}^{2}/2.$ (31)
By inspection, we find that $\mathcal{H}=0$ at both the origin and
$(x,\lambda_{1})=(\pm 1,0)$. The equation for the curve connecting the two
states is easily obtained from Eq. (31):
$\lambda_{1}(x(t))=2(x(t)^{3}-x(t)).$
One may calculate the action in the singular case by carrying out the integral
$R(x)=\int_{-1}^{0}\lambda_{1}(x)dx$. However, this would ignore the
dependence of the action on the fast variables; to approximate this influence,
we will resort to center manifold approximations.
## IV Center manifold reduction
To analyze Eqs. (24)-(27), we will apply center manifold approximations to
reduce the number of dimensions in the system. The system must first be
rescaled to be placed in a form that is amenable for this process. To obtain
the layer equations, we apply the scaling $t=\epsilon\tau$:
$\displaystyle\lambda_{1}^{\prime}$
$\displaystyle=\epsilon(3x^{2}-1)\lambda_{2}$ (32)
$\displaystyle\lambda_{2}^{\prime}$ $\displaystyle=\lambda_{2}-\lambda_{1}$
(33) $\displaystyle y^{\prime}$ $\displaystyle=x-x^{3}-y$ (34) $\displaystyle
x^{\prime}$ $\displaystyle=\epsilon(y+\lambda_{1})$ (35)
$\displaystyle\epsilon^{\prime}$ $\displaystyle=0.$ (36)
One benefit of Eqs. (32)–(36) is that it is no longer singular as $\epsilon$
vanishes. A second benefit is that Eqs. (33)–(34) involve terms that are
linear in the state variables (a space which now includes $\epsilon$) and that
all other equations are purely nonlinear. Therefore, the hypotheses of the
center manifold theorem are satisfied and center manifold reductions may be
applied to Eqs. (32)–(36) to reduce the dimensionality of the system Carr
(1981) Guckenheimer and Holmes (1997). Since the vector field Eqs. (32)–(36)
is smooth, we may assume:
$\displaystyle y$ $\displaystyle=h(x,\lambda_{1},\epsilon)$ (37)
$\displaystyle\lambda_{2}$ $\displaystyle=k(x,\lambda_{1},\epsilon)$ (38)
where $h$ and $k$ are differentiable functions of the quantities specified.
Applying the definitions in Eqs. (37), (38) to Eqs. (33)–(34) and substituting
the vector fields in Eqs. (32), (35) when applying the chain rule, we obtain a
system of two partial differential equations that may be solved for the
unknown functions that will define the center manifold. These equations are
known as the _center manifold conditions_. Beginning with the condition
resulting from Eq. (34):
$\displaystyle\left(\frac{\partial k}{\partial x}x^{\prime}+\frac{\partial
k}{\partial\lambda_{1}}\lambda_{1}^{\prime}\right)$
$\displaystyle=k(x,\lambda_{1},\epsilon)-\lambda_{1},$ (39) also for Eq. (33),
$\displaystyle\left(\frac{\partial h}{\partial x}x^{\prime}+\frac{\partial
h}{\partial\lambda_{1}}\lambda_{1}^{\prime}\right)$
$\displaystyle=x-x^{3}-h(x,\lambda_{1},\epsilon).$ (40)
In general, solving the partial differential Eqs. (40), (39) will be
difficult. However, the center manifold reduction method next calls for making
approximations for the functions $h$ and $k$ in terms of polynomials of
increasingly higher order in their dependent variables. Each variable
contributes to the order of a given term; to represent this, one may consider
each variable scaled by a parameter $\alpha$. The series is truncated at an
arbitrarily specified order in $\alpha$. Explicitly, this means:
$\displaystyle h(x,\lambda_{1},\epsilon)=c_{0}+$
$\displaystyle\alpha\left(c_{1}x+c_{2}\epsilon+c_{3}\lambda_{1}\right)+\alpha^{2}\left(c_{4}x^{2}+c_{5}x\lambda_{1}\right.$
$\displaystyle\left.+c_{6}x\epsilon+c_{7}\lambda_{1}^{2}+c_{8}\lambda_{1}\epsilon+c_{9}\epsilon^{2}\right)+\ldots$
$\displaystyle k(x,\lambda_{1},\epsilon)=d_{0}+$
$\displaystyle\alpha\left(d_{1}x+d_{2}\epsilon+d_{3}\lambda_{1}\right)+\alpha^{2}\left(d_{4}x^{2}+d_{5}x\lambda_{1}\right.$
$\displaystyle+\left.d_{6}x\epsilon+d_{7}\lambda_{1}^{2}+d_{8}\lambda_{1}\epsilon+d_{9}\epsilon^{2}\right)+\ldots.$
The center manifold expressions to fourth order in $\alpha$ and ordered by
power in $\epsilon$ are:
$\displaystyle h=$ $\displaystyle
x-x^{3}-(x+\lambda_{1}-4x^{3}-3x^{2}\lambda_{1})\epsilon$
$\displaystyle+(2x+\lambda_{1})\epsilon^{2}+\mathcal{O}(|\alpha|^{5})$ (41)
$\displaystyle k=$
$\displaystyle\lambda_{1}+\left(-\lambda_{1}+3{x}^{2}\lambda_{1}\right)\epsilon$
$\displaystyle+\left(2\lambda_{1}+6x{\lambda_{1}}^{2}\right){\epsilon}^{2}-5\lambda_{1}\epsilon^{3}+\mathcal{O}(|\alpha|^{5}).$
(42)
Note that setting $\epsilon=0$ in the expressions for $h$ and $k$ in Eqs.
(41), (42) recovers precisely the critical dynamics of Eqs. (28), (29). Since
the expressions are given in powers of $\alpha$, they do not represent an
accounting of all terms that may be present for a given high order in
$\epsilon$. However, after taking the series in $\alpha$ to high enough order,
low-order terms in $\epsilon$ stop appearing and the resulting series is
treated as one in $\epsilon$.
Numerical integration of the original system Eqs. (24)–(27) compared with its
center manifold approximation (with $y$, $\lambda_{2}$ calculated using Eqs.
(41), (42) respectively) gives remarkable agreement even at first order in
$\epsilon$. A plot of the integration is shown in Figure 1.
$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0$$0.2$$0.4$$0.6$$0.8$$x$$\lambda_{1}$ Figure
1: Phase portraits of the full system in Eqs. (24)–(27) (blue line) compared
with its center manifold approximation (red dots) for $\epsilon=0.001$.
The action of noise along the optimal path in this system is:
$\displaystyle\mathcal{R}[x,y,$
$\displaystyle\eta,\lambda_{1},\lambda_{2}]=\frac{1}{2}\int_{-\infty}^{\infty}\eta^{2}(t)dt$
$\displaystyle+\int_{-\infty}^{\infty}\lambda_{1}(\dot{x}-f(x,y)-\eta(t))dt$
$\displaystyle+\int_{-\infty}^{\infty}\lambda_{2}(\epsilon\dot{y}-g(x,y))dt$
Having established approximations for these quantities previously as series
expansions in $\epsilon$, it is merely a matter of careful substitution,
differentiation and integration to obtain an approximation to the integral to
arbitrary order in $\epsilon$. First, substitutions may be made using the
center manifold expressions $y=h(x,\lambda_{1},\epsilon)$ and
$\lambda_{2}=k(x,\lambda_{1},\epsilon)$ in Eqs. (39),(40). Second, apply the
identity along the zero-Hamiltonian curve for $\lambda(t)$ as obtained in Eq.
(31), along with $x(t)$ from Eq. (30). Differentiating and integrating as
necessary results in the expression
$\mathcal{R}=\frac{1}{2}-\frac{1}{4}\epsilon^{2}+\mathcal{O}(\epsilon^{3})$
(43)
where the leading order term is the contribution from the singular case.
## V Numerical Results
To test the predictions resulting from this method, we compared the scaling
predicted from the perturbation method with repeated stochastic simulation of
the damped Duffing oscillator for various values of $D$ and $\epsilon$. The
stochastic simulations were run using implicit numerical integration, the
details of which are outlined in the Appendix.
It is convenient to make comparisons between numerics and analytical
approximations by analyzing the logarithm of the escape time across multiple
orders of magnitude. A plot of the stochastic simulations compared against the
escape time as predicted using the perturbation method is shown in Figure 2;
the two methods agree very well. Table 1 provides a side-by-side comparison of
the scaling coefficient between the MFPT and $\epsilon$ as calculated by the
perturbation method and from linear regression of stochastically simulated
switching. The error bounds represent the standard deviation on the slope of
the regression line.
$14$$16$$18$$20$$22$$24$$26$$28$$1.5$$2$$2.5$$3$$3.5$$4$$1/D$$\log_{10}\left(T_{S}\right)$ Figure 2: Mean first passage times from a potential well varying with $\epsilon$ and $D$. Data points were computed as an ensemble average of 1000 trials. $\circ$ represents $\epsilon=1.0$, $\color[rgb]{0.847058832645416,0.160784319043159,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.847058832645416,0.160784319043159,0}\Box$ $\epsilon=0.5$, $\color[rgb]{0.749019622802734,0,0.749019622802734}\definecolor[named]{pgfstrokecolor}{rgb}{0.749019622802734,0,0.749019622802734}\times$ $\epsilon=0.2$, $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+$ $\epsilon=0.1$ and $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\ast$ $\epsilon=0.01$. Color-corresponding lines show the perturbation-predicted escape times. Lines have been shifted to allow comparison with the slopes of simulation data. | Scaling coefficient $C_{S}\times 10^{2}$
---|---
$\epsilon$ | perturbation method | stochastic simulation
0.001 | 10.86 | 10.91 $\pm$ 1.213
0.003 | 10.86 | 10.84 $\pm$ 1.370
0.01 | 10.86 | 10.79 $\pm$ 1.034
0.1 | 10.80 | 10.80 $\pm$ 1.246
0.2 | 10.64 | 10.60 $\pm$ 1.189
0.5 | 9.500 | 9.295 $\pm$ 1.107
1.0 | 5.428 | 6.469 $\pm$ 0.9437
Table 1: Comparison of scaling coefficients for MFPT between the perturbation
method and stochastic simulation. The scaling law is assumed to be
$\log_{10}(T_{S})=C_{S}(1/D)+b$, where $b$ is a constant determined by
simulation. The data above quantify the predictions and observations in Figure
2.
As shown in Table 1, the agreement between stochastic simulation and the
perturbation method is quite good and well within the standard deviation for
the slope of the regression line. However, as the timescales are brought into
alignment with one another, the center manifold approximation applied to the
slow system becomes a poor approximation for the system dynamics. This can be
confirmed visually by observing that the agreement for $\epsilon=1.0$ in
Figure 2 is not strong.
## VI Discussion
Figure 2 shows that the method fails to predict the mean first passage time if
$\epsilon$ is too large. Both $\epsilon$ and $D$ are assumed to be small for
the perturbation series and simplifications made. The magnitude of the noise
intensity $D$ may be compared with the height of the barrier through which the
particle must traverse to switch states. The approximations made do not apply
to events where noise is so significant as to typically cause a transition or
where there is little separation between the time scales. However, the process
may be applied to even higher-dimensional systems where the time scale
separation translates into a spectral gap in relaxation times.
In the regime where $D$ is large compared to the height of the barrier, the
mean first passage time will rapidly decrease. This behavior cannot be
captured by the WKB approximation ansatz; the Eikonal approximation can only
capture a linear relationship between $\log T_{S}$ and $1/D$. A method could
be developed to obtain statistics about slow-fast stochastic systems when $D$
is significant compared to the effective barrier height, and this will be left
to future work in which noise is finite and large.
These restrictions aside, the method is resilient to choices of vector field.
Despite the Duffing oscillator’s symmetry, the method has been applied to
another double-welled system with broken symmetry and has resulted in
similarly good agreement. Our test system was an unsymmetric Duffing-like
oscillator with differential equations:
$\displaystyle\dot{x}$ $\displaystyle=y+\eta(t)$ (44)
$\displaystyle\epsilon\dot{y}$ $\displaystyle=x(1+x)(2-x)-y.$ (45)
The system in Eqs. (44), (45) has two stable equilibrium points at $x=-1$ and
$x=2$ separated by a saddle at $x=0$. The method outlined in this paper gives
the approximate expression for the action of:
$\mathcal{R}=\frac{5}{6}-\frac{13}{12}\epsilon^{2}+\mathcal{O}\left(\epsilon^{3}\right).$
A comparison of numerically- and formally-generated results for the mean first
passage time in this system is provided in Figure 3. Both examples we have
carried out do not have any $\mathcal{O}(\epsilon)$ terms appearing in the
approximation to the action; this may be understood via an analogy with
function optimization. The local behavior of a function at a minimum with
respect to a parameter has no linear dependence on said parameter by
definition.
$10$$15$$20$$1$$2$$3$$4$$1/D$$\text{log}_{10}\left(T_{S}\right)$ Figure 3:
Mean first passage times from a potential well varying with $\epsilon$ and
$D$. Data points were computed as an ensemble average of 1000 trials. $\circ$
represents $\epsilon=0.5$,
$\color[rgb]{0.847058832645416,0.160784319043159,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.847058832645416,0.160784319043159,0}\Box$
$\epsilon=0.4$,
$\color[rgb]{0.749019622802734,0,0.749019622802734}\definecolor[named]{pgfstrokecolor}{rgb}{0.749019622802734,0,0.749019622802734}\times$
$\epsilon=0.2$.
A contemporary and popular approach to obtain similar results for the
occurrence of rare events uses what are known as sample-based techniques.
Throughout our approach, we have completely avoided the use of such methods.
These approaches generally have required the calculation of convolution
integrals that depend on the realization of noise for all past and future
times; while analytically tractable, this comes with some assumptions. Some of
the integrals that result from sample-base approaches must remain bounded,
putting further restrictions on the noise distribution. Such restrictions can
be challenging to rigorously justify and are at times opaque. Our approach
does not require such justifications and reaches complementary conclusions
while remaining transparent throughout the process, making it a useful and
very straightforward alternative to sample-based techniques.
Finally, the use of center manifold reductions requires considerable algebraic
manipulation that may not be tenable in all circumstances, e.g. in high
dimensional systems or those with many parameters. Such systems often have
lower-dimensional analogs which may be amenable to this analysis and thus are
within reach of this method. However, there are other approaches. For
instance, computational methods exist to minimize the action in a variety of
gradient and non-gradient systems E _et al._ (2004). These numerical
algorithms provide a new approach to verify the scaling relationships
generated by our method in theory and experiment.
## VII Conclusion
In this work, a method was developed to leverage the disparate timescales in
slow-fast stochastic systems to aid analysis and predict switching times
between attractors. The process avoided the projection of noise vectors onto
the slow manifold in favor of analyzing the noisy system via a variational
approach to find the optimal path. The damped Duffing oscillator was used as
an example of a prototypical system with two potential wells where switching
can occur as a result of large fluctuations. Using this theory, we transformed
the original 2-dimensional stochastic system into a 4-dimensional
deterministic system and proceeded to analyze the optimal path representing
the most likely noise to induce a transition. The action along this path was
crucial to determining the switching time between the two metastable states
present.
For future work, we intend to apply this theory to prescient examples of slow-
fast stochastic systems, including epidemic models with non-Gaussian noise. We
also will apply this method to systems which exhibit delayed feedback.
## VIII Acknowledgements
The authors gratefully thank Luis Mier-y-Teran Romero for helpful discussions
and his prescient insight. This research was performed while CRH held a
National Research Council Research Associateship Award at the U.S. Naval
Research Laboratory. This research is funded by the Office of Naval Research
contract F1ATA01098G001 and by Naval Research Base Program Contract
N0001412WX30002.
## Appendix A Stochastic simulations
Ordinary differential equations with multiple timescales present unique
challenges when numerically integrating to obtain a time series, including the
possibility of the system’s being “stiff.” Stiffness is a qualitative property
of a dynamical system that stymies standard (i.e. explicit) numerical
integration methods. This effect may be illustrated with a simple example;
consider the system of differential equations:
$\displaystyle\dot{\bm{x}}$
$\displaystyle=\bm{F}(\bm{x},\bm{y})+\alpha\bm{\Phi}$ (46)
$\displaystyle\epsilon\dot{\bm{y}}$ $\displaystyle=\bm{G}(\bm{x},\bm{y})$ (47)
where $\bm{x}\in\mathbb{R}^{m}$, $\bm{y}\in\mathbb{R}^{n}$, $\bm{F}$ and
$\bm{G}$ are differentiable functions, $\bm{\Phi}$ is a white noise term with
amplitude controlled by $\alpha$ and $\epsilon$ is a parameter that tunes the
separation of the timescales between the variables $\bm{x}$ and $\bm{y}$. For
the purpose of illustration, we first set $\alpha=0$. To obtain a time series
of Eqs. (46), (47) there are many numerical recipes that may be applied, the
simplest of which is Euler’s Method. Let $\mathcal{D}$ represent taking the
Jacobian of the vector field, and let $\mathcal{D}\bm{F}$, $\mathcal{D}\bm{G}$
be nonsingular. Euler’s Method calls for generating successive iterations of
the underlying function by discretizing time with a uniform step size $\nu$
and iterating the resulting map:
$\displaystyle\bm{x}_{k+1}$
$\displaystyle=\bm{x}_{k}+\nu\bm{F}(\bm{x}_{k},\bm{y}_{k})$ (48)
$\displaystyle\bm{y}_{k+1}$
$\displaystyle=\bm{y}_{k}+\frac{\nu}{\epsilon}\bm{G}(\bm{x}_{k},\bm{y}_{k})$
(49)
In general, the eigenvalues of both $\mathcal{D}\bm{F}$, $\mathcal{D}\bm{G}$
are $\mathcal{O}(1)$. Of particular concern is the factor of
$\frac{\nu}{\epsilon}$, which is generally very large. The eigenvalues of Eq.
(49) will in general be much larger than those of Eq. (48), which leads to
stiffness. This inverse relationship between $\nu$ and $\epsilon$ creates a
numerical quandary since the necessary step size to ensure stability is
$\mathcal{O}(\epsilon)$, which is arbitrarily small. For accuracy, step sizes
must be chosen much smaller than this necessary step size, further aggravating
the numerical challenges.
To circumvent this complication, implicit methods are often used to solve for
the state of the system after a time step. We now re-introduce noise by
setting $\alpha=1$ and draw the noise $\bm{W}_{k}$ at step $k$ from a Gaussian
distribution with mean 0 and standard deviation 1. Using a first-order
Milstein method, the implicit recipe used in our stochastic simulations is:
$\displaystyle\bm{x}_{k+1}$
$\displaystyle=(\bm{x}_{k},\bm{y}_{k})+\nu\bm{F}(\bm{x}_{k+1},\bm{y}_{k+1})+\sqrt{\nu}\bm{W}_{k}$
(50) $\displaystyle\bm{y}_{k+1}$
$\displaystyle=(\bm{x}_{k},\bm{y}_{k})+\frac{\nu}{\epsilon}\bm{G}(\bm{x}_{k+1},\bm{y}_{k+1})$
(51)
Solving for $(\bm{x}_{k+1},\bm{y}_{k+1})$ in Eqs. (50), (51) is an exercise in
nonlinear, multidimensional root-finding. Since we expect the system’s value
at two adjacent timesteps to be close, we may take $\nu$ arbitrarily small
such that a Newton-Raphson iterative scheme will converge to the value of
$\bm{x}_{k+1}$. While the iterative scheme converges quickly, this still
involves significant computational overhead since it in general requires the
inversion of a large matrix.
To compute the mean first passage time for the stochastic systems compared in
Fig. 2, the implicit recipe provided in Eqs. (50), (51) was used to integrate
the system until noise caused it to escape from the potential well. The first
passage time was recorded, and the system was reset. This ensemble was run for
1,000 simulations for each value of $\epsilon$ and $D$ and the mean of all
first passage times for the given parameter values was computed. The total
computation time was considerable even on a desktop PC with 8 processors using
Matlab’s parallel computing toolbox. The processing time to generate Figure 2
was over two weeks using all eight cores clocked at 2.8 GHz.
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|
arxiv-papers
| 2013-07-29T13:51:39 |
2024-09-04T02:49:48.653821
|
{
"license": "Public Domain",
"authors": "Christoffer R. Heckman, Ira B. Schwartz",
"submitter": "Christoffer Heckman",
"url": "https://arxiv.org/abs/1307.7581"
}
|
1307.7595
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-137 LHCb-PAPER-2013-039 4 September 2013
Observation of a resonance in $B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ decays at
low recoil
The LHCb collaboration†††Authors are listed on the following pages.
A broad peaking structure is observed in the dimuon spectrum of
$B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ decays in the kinematic region where
the kaon has a low recoil against the dimuon system. The structure is
consistent with interference between the $B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}$ decay and a resonance and has a statistical significance
exceeding six standard deviations. The mean and width of the resonance are
measured to be $4191^{+9}_{-8}\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}$ and
$65^{+22}_{-16}\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}$, respectively, where
the uncertainties include statistical and systematic contributions. These
measurements are compatible with the properties of the $\psi(4160)$ meson.
First observations of both the decay $B^{+}\rightarrow\psi(4160)K^{+}$ and the
subsequent decay $\psi(4160)\rightarrow\mu^{+}\mu^{-}$ are reported. The
resonant decay and the interference contribution make up 20 % of the yield for
dimuon masses above 3770${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This
contribution is larger than theoretical estimates.
Accepted by Phys. Rev. Lett.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, E.
Cowie45, D.C. Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8,
P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O.
Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M.
Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F.
Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M.
Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra
Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53,
T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H.
Gordon37, C. Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado
Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S.
Gregson46, P. Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu.
Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C.
Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N.
Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, T. Head37, V.
Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, M. Hess60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44,
R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J.
Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O.
Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U.
Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T. Palczewski27, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B.
Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8,
G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N.
Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E.
Santovetti23,k, M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M.
Savrie16,e, D. Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M.
Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A.
Semennikov30, K. Senderowska26, I. Sepp52, N. Serra39, J. Serrano6, P.
Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T.
Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires9, R.
Silva Coutinho47, M. Sirendi46, N. Skidmore45, T. Skwarnicki58, N.A. Smith51,
E. Smith54,48, J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro38, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stevenson54, S.
Stoica28, S. Stone58, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K.
Subbiah37, L. Sun56, S. Swientek9, V. Syropoulos41, M. Szczekowski27, P.
Szczypka38,37, T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F.
Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin38, S. Tolk41, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E.
Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6,
P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, D. Urner53, A.
Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, M. Van
Dijk45, R. Vazquez Gomez18, P. Vazquez Regueiro36, C. Vázquez Sierra36, S.
Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B.
Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10,
D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C.
Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J.
Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wimberley57, J. Wishahi9, W.
Wislicki27, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
The decay of the $B^{+}$ meson to the final state $K^{+}\mu^{+}\mu^{-}$
receives contributions from tree level decays and decays mediated through
virtual quantum loop processes. The tree level decays proceed through the
decay of a $B^{+}$ meson to a vector $c\overline{}c$ resonance and a $K^{+}$
meson, followed by the decay of the resonance to a pair of muons. Decays
mediated by flavour changing neutral current (FCNC) loop processes give rise
to pairs of muons with a non-resonant mass distribution. To probe
contributions to the FCNC decay from physics beyond the Standard Model (SM),
it is essential that the tree level decays are properly accounted for. In all
analyses of the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay, from
discovery [1] to the latest most accurate measurement [2], this has been done
by placing a veto on the regions of dimuon mass, $m_{\mu^{+}\mu^{-}}$,
dominated by the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ resonances. In the low recoil region, corresponding to a dimuon
mass above the open charm threshold, theoretical predictions of the decay rate
can be obtained with an operator product expansion (OPE) [3] in which the
$c\overline{}c$ contribution and other hadronic effects are treated as
effective interactions.
Nearly all available information about the ${J^{PC}}=1^{--}$ charmonium
resonances above the open charm threshold, where the resonances are wide as
decays to $D$${}^{(*)}\overline{D}{{}^{(*)}}$ are allowed, comes from
measurements of the cross-section ratio of $e^{+}e^{-}\rightarrow{\rm
hadrons}$ relative to $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$. Among these
analyses, only that of the BES collaboration in Ref. [4] takes interference
and strong phase differences between the different resonances into account.
The broad and overlapping nature of these resonances means that they cannot be
excluded by vetoes on the dimuon mass in an efficient way, and a more
sophisticated treatment is required.
This Letter describes a measurement of a broad peaking structure in the low
recoil region of the $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decay, based on
data corresponding to an integrated luminosity of 3$\mbox{\,fb}^{-1}$ taken
with the LHCb detector at a centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$ in 2011 and $8\mathrm{\,Te\kern-1.00006ptV}$
in 2012. Fits to the dimuon mass spectrum are performed, where one or several
resonances are allowed to interfere with the non-resonant $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ signal, and their parameters determined. The inclusion of
charge conjugated processes is implied throughout this Letter.
The LHCb detector [5] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4 % at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6 % at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors.
Muons are identified by a system composed of alternating layers of iron and
multiwire proportional chambers. Simulated events used in this analysis are
produced using the software described in Refs. [6, 7, 8, 9, 10,
*Agostinelli:2002hh, 12].
Candidates are required to pass a two stage trigger system [13]. In the
initial hardware stage, candidate events are selected with at least one muon
with transverse momentum, $\mbox{$p_{\rm
T}$}>1.48\,(1.76){\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in 2011 (2012). In the
subsequent software stage, at least one of the final state particles is
required to have both $\mbox{$p_{\rm
T}$}>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter larger than
$100\,\upmu\rm m$ with respect to all of the primary $pp$ interaction vertices
(PVs) in the event. Finally, a multivariate algorithm [14] is used for the
identification of secondary vertices consistent with the decay of a $b$ hadron
with muons in the final state.
The selection of the $K^{+}\mu^{+}\mu^{-}$ final state is made in two steps.
Candidates are required to pass an initial selection, which reduces the data
sample to a manageable level, followed by a multivariate selection. The
dominant background is of a combinatorial nature, where two correctly
identified muons from different heavy flavour hadron decays are combined with
a kaon from either of those decays. This category of background has no peaking
structure in either the dimuon mass or the $K^{+}\mu^{+}\mu^{-}$ mass. The
signal region is defined as
$5240<m_{K^{+}\mu^{+}\mu^{-}}<5320{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and the sideband region as
$5350<m_{K^{+}\mu^{+}\mu^{-}}<5500{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The sideband below the $B^{+}$ mass is not used as it contains backgrounds
from partially reconstructed decays, which do not contaminate the signal
region.
The initial selection requires: $\chi^{2}_{\rm IP}>9$ for all final state
particles, where $\chi^{2}_{\rm IP}$ is defined as the minimum change in
$\chi^{2}$ when the particle is included in a vertex fit to any of the PVs in
the event; that the muons are positively identified in the muon system; and
that the dimuon vertex has a vertex fit $\chi^{2}<9$. In addition, based on
the lowest $\chi^{2}_{\rm IP}$ of the $B^{+}$ candidate, an associated PV is
chosen. For this PV it is required that: the $B^{+}$ candidate has
$\chi^{2}_{\rm IP}<16$; the vertex fit $\chi^{2}$ must increase by more than
$121$ when including the $B^{+}$ candidate daughters; and the angle between
the $B^{+}$ candidate momentum and the direction from the PV to the decay
vertex should be below 14$\rm\,mrad$. Finally, the $B^{+}$ candidate is
required to have a vertex fit $\chi^{2}<24$ (with three degrees of freedom).
The multivariate selection is based on a boosted decision tree (BDT) [15] with
the AdaBoost algorithm[16] to separate signal from background. It is trained
with a signal sample from simulation and a background sample consisting of 10
% of the data from the sideband region. The multivariate selection uses
geometric and kinematic variables, where the most discriminating variables are
the $\chi^{2}_{\rm IP}$ of the final state particles and the vertex quality of
the $B^{+}$ candidate. The selection with the BDT has an efficiency of 90 % on
signal surviving the initial selection while retaining 6 % of the background.
The overall efficiency for the reconstruction, trigger and selection,
normalised to the total number of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$
decays produced at the LHCb interaction point, is $2\,\%$. As the branching
fraction measurements are normalised to the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay,
only relative efficiencies are used. The yields in the $K^{+}\mu^{+}\mu^{-}$
final state from $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ and $B^{+}\\!\rightarrow\psi{(2S)}K^{+}$ decays are $9.6\times
10^{5}$ and $8\times 10^{4}$ events, respectively.
In addition to the combinatorial background, there are several small sources
of potential background that form a peak in either or both of the
$m_{K^{+}\mu^{+}\mu^{-}}$ and $m_{\mu^{+}\mu^{-}}$ distributions. The largest
of these backgrounds are the decays
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ and
$B^{+}\\!\rightarrow\psi{(2S)}K^{+}$, where the kaon and one of the muons have
been interchanged. The decays $B^{+}\\!\rightarrow K^{+}\pi^{-}\pi^{+}$ and
$B^{+}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$
followed by $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow
K^{+}\pi^{-}$, with the two pions identified as muons are also considered. To
reduce these backgrounds to a negligible level, tight particle identification
criteria and vetoes on $\mu^{-}K^{+}$ combinations compatible with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\psi{(2S)}$, or $D^{0}$ meson
decays are applied. These vetoes are 99% efficient on signal.
A kinematic fit [17] is performed for all selected candidates. In the fit the
$K^{+}\mu^{+}\mu^{-}$ mass is constrained to the nominal $B^{+}$ mass and the
candidate is required to originate from its associated PV. For
$B^{+}\\!\rightarrow\psi{(2S)}K^{+}$ decays, this improves the resolution in
$m_{\mu^{+}\mu^{-}}$ from 15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to
5${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Given the widths of the
resonances that are subsequently analysed, resolution effects are neglected.
While the $\psi{(2S)}$ state is narrow, the large branching fraction means
that its non-Gaussian tail is significant and hard to model. The $\psi{(2S)}$
contamination is reduced to a negligible level by requiring
$m_{\mu^{+}\mu^{-}}>3770{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This dimuon
mass range is defined as the low recoil region used in this analysis.
In order to estimate the amount of background present in the
$m_{\mu^{+}\mu^{-}}$ spectrum, an unbinned extended maximum likelihood fit is
performed to the $K^{+}\mu^{+}\mu^{-}$ mass distribution without the $B^{+}$
mass constraint. The signal shape is taken from a mass fit to the
$B^{+}\\!\rightarrow\psi{(2S)}K^{+}$ mode in data with the shape parameterised
as the sum of two Crystal Ball functions [18], with common tail parameters,
but different widths. The Gaussian width of the two components is increased by
$5\,\%$ for the fit to the low recoil region as determined from simulation.
The low recoil region contains 1830 candidates in the signal mass window, with
a signal to background ratio of 7.8.
The dimuon mass distribution in the low recoil region is shown in Fig. 1. Two
peaks are visible, one at the low edge corresponding to the expected decay
$\psi(3770)\\!\rightarrow\mu^{+}\mu^{-}$ and a wide peak at a higher mass. In
all fits, a vector resonance component corresponding to this decay is
included. Several fits are made to the distribution. The first introduces a
vector resonance with unknown parameters. Subsequent fits look at the
compatibility of the data with the hypothesis that the peaking structure is
due to known resonances.
Figure 1: Dimuon mass distribution of data with fit results overlaid for the
fit that includes contributions from the non-resonant vector and axial vector
components, and the $\psi(3770)$, $\psi(4040)$, and $\psi(4160)$ resonances.
Interference terms are included and the relative strong phases are left free
in the fit.
The non-resonant part of the mass fits contains a vector and axial vector
component. Of these, only the vector component will interfere with the
resonance. The probability density function (PDF) of the signal component is
given as
$\displaystyle{\cal P}_{\rm sig}$ $\displaystyle\propto P(m_{\mu^{+}\mu^{-}})\
|{\cal A}|^{2}\ f^{2}(m_{\mu^{+}\mu^{-}}^{2})\,,$ (1) $\displaystyle|{\cal
A}|^{2}$
$\displaystyle=|A^{\rm{V}}_{\text{nr}}+\sum_{k}e^{i\delta_{k}}A_{\text{r}}^{k}|^{2}+|A^{\rm{AV}}_{\text{nr}}|^{2}\,,$
(2)
where $A^{\rm{V}}_{\text{nr}}$ and $A^{\rm{AV}}_{\text{nr}}$ are the vector
and axial vector amplitudes of the non-resonant decay. The shape of the non-
resonant signal in $m_{\mu^{+}\mu^{-}}$ is driven by phase space,
$P(m_{\mu^{+}\mu^{-}})$, and the form factor, $f(m_{\mu^{+}\mu^{-}}^{2})$. The
parametrisation of Ref. [19] is used to describe the dimuon mass dependence of
the form factor. This form factor parametrisation is consistent with recent
lattice calculations [20]. In the SM at low recoil, the ratio of the vector
and axial vector contributions to the non-resonant component is expected to
have negligible dependence on the dimuon mass. The vector component accounts
for $(45\pm 6)\,\%$ of the differential branching fraction in the SM (see, for
example, Ref. [21]). This estimate of the vector component is assumed in the
fit.
The total vector amplitude is formed by summing the vector amplitude of the
non-resonant signal with a number of Breit-Wigner amplitudes,
$A_{\text{r}}^{k}$, which depend on $m_{\mu^{+}\mu^{-}}$. Each Breit-Wigner
amplitude is rotated by a phase, $\delta_{k}$, which represents the strong
phase difference between the non-resonant vector component and the resonance
with index $k$. Such phase differences are expected [19]. The $\psi(3770)$
resonance, visible at the lower edge of the dimuon mass distribution, is
included in the fit as a Breit-Wigner component whose mass and width are
constrained to the world average values [22].
The background PDF for the dimuon mass distribution is taken from a fit to
data in the $K^{+}\mu^{+}\mu^{-}$ sideband. The uncertainties on the
background amount and shape are included as Gaussian constraints to the fit in
the signal region.
The signal PDF is multiplied by the relative efficiency as a function of
dimuon mass with respect to the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decay.
As in previous analyses of the same final state [23], this efficiency is
determined from simulation after the simulation is made to match data by:
degrading by $\sim\\!20\,\%$ the impact parameter resolution of the tracks,
reweighting events to match the kinematic properties of the $B^{+}$ candidates
and the track multiplicity of the event, and adjusting the particle
identification variables based on calibration samples from data. In the region
from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass to
$4600{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ the relative efficiency drops
by around 20 %. From there to the kinematic endpoint it drops sharply,
predominantly due to the $\chi^{2}_{\rm IP}$ cut on the kaon as in this region
its direction is aligned with the $B^{+}$ candidate and therefore also with
the PV.
Initially, a fit with a single resonance in addition to the $\psi(3770)$ and
non-resonant terms is performed. This additional resonance has its phase,
mean, and width left free. The parameters of the resonance returned by the fit
are a mass of $4191^{+9}_{-8}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and a
width of $65^{+22}_{-16}{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Branching
fractions are determined by integrating the square of the Breit-Wigner
amplitude returned by the fit, normalising to the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ yield,
and multiplying with the product of branching fractions, ${\cal
B}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+})\times{\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})$ [22]. The product ${\cal
B}(B^{+}\\!\rightarrow XK^{+})\times{\cal B}(X\\!\rightarrow\mu^{+}\mu^{-})$
for the additional resonance, $X$, is determined to be
$(3.9^{+0.7}_{-0.6})\times 10^{-9}$. The uncertainty on this product is
calculated using the profile likelihood. The data are not sensitive to the
vector fraction of the non-resonant component as the branching fraction of the
resonance will vary to compensate. For example, if the vector fraction is
lowered to $30\%$, the central value of the branching fraction increases to
$4.6\times 10^{-9}$. This reflects the lower amount of interference allowed
between the resonant and non-resonant components.
The significance of the resonance is obtained by simulating pseudo-experiments
that include the non-resonant, $\psi(3770)$ and background components. The log
likelihood ratios between fits that include and exclude a resonant component
for $6\times 10^{5}$ such samples are compared to the difference observed in
fits to the data. None of the samples have a higher ratio than observed in
data and an extrapolation gives a significance of the signal above six
standard deviations.
The properties of the resonance are compatible with the mass and width of the
$\psi(4160)$ resonance as measured in Ref. [4]. To test the hypothesis that
$\psi$ resonances well above the open charm threshold are observed, another
fit including the $\psi(4040)$ and $\psi(4160)$ resonances is performed. The
mass and width of the two are constrained to the measurements from Ref. [4].
The data have no sensitivity to a $\psi(4415)$ contribution. The fit describes
the data well and the parameters of the $\psi(4160)$ meson are almost
unchanged with respect to the unconstrained fit. The fit overlaid on the data
is shown in Fig. 1 and Table 1 reports the fit parameters.
Table 1: Parameters of the dominant resonance for fits where the mass and width are unconstrained and constrained to those of the $\psi(4160)$ meson [4], respectively. The branching fractions are for the $B^{+}$ decay followed by the decay of the resonance to muons. | Unconstrained | $\psi(4160)$
---|---|---
$\cal B$$[\times 10^{-9}$] | $3.9\,^{+0.7}_{-0.6}$ | $3.5\,^{+0.9}_{-0.8}$
Mass $[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$ | $4191\,^{+9}_{-8}$ | $4190\pm 5$
Width $[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]$ | $65\,^{+22}_{-16}$ | $66\pm 12$
Phase [rad] | $-1.7\pm 0.3$ | $-1.8\pm 0.3$
The resulting profile likelihood ratio compared to the best fit as a function
of branching fraction can be seen in Fig. 2. In the fit with the three $\psi$
resonances, the $\psi(4160)$ meson is visible with ${\cal
B}(B^{+}\\!\rightarrow\psi(4160)K^{+})\times{\cal
B}(\psi(4160)\\!\rightarrow\mu^{+}\mu^{-})=(3.5^{+0.9}_{-0.8})\times 10^{-9}$
but for the $\psi(4040)$ meson, no significant signal is seen, and an upper
limit is set. The limit ${\cal
B}(B^{+}\\!\rightarrow\psi(4040)K^{+})\times{\cal
B}(\psi(4040)\\!\rightarrow\mu^{+}\mu^{-})<1.3\,(1.5)\times 10^{-9}$ at
$90\,(95)\,\%$ confidence level is obtained by integrating the likelihood
ratio compared to the best fit and assuming a flat prior for any positive
branching fraction.
Figure 2: Profile likelihood ratios for the product of branching fractions
${\cal B}(B^{+}\\!\rightarrow\psi K^{+})\times{\cal
B}(\psi\\!\rightarrow\mu^{+}\mu^{-})$ of the $\psi(4040)$ and the $\psi(4160)$
mesons. At each point all other fit parameters are reoptimised.
In Fig. 3 the likelihood scan of the fit with a single extra resonance is
shown as a function of the mass and width of the resonance. The fit is
compatible with the $\psi(4160)$ resonance, while a hypothesis where the
resonance corresponds to the decay $Y(4260)\\!\rightarrow\mu^{+}\mu^{-}$ is
disfavoured by more than four standard deviations.
Figure 3: Profile likelihood as a function of mass and width of a fit with a
single extra resonance. At each point all other fit parameters are
reoptimised. The three ellipses are (red-solid) the best fit and previous
measurements of (grey-dashed) the $\psi(4160)$ [4] and (black-dotted) the
$Y(4260)$ [22] states.
Systematic uncertainties associated with the normalisation procedure are
negligible as the decay
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ has the
same final state as the signal and similar kinematics. Uncertainties due to
the resolution and mass scale are insignificant. The systematic uncertainty
associated to the form factor parameterisation in the fit model is taken from
Ref. [21]. Finally, the uncertainty on the vector fraction of the non-resonant
amplitude is obtained using the EOS tool described in Ref. [21] and is
dominated by the uncertainty from short distance contributions. All systematic
uncertainties are included in the fit as Gaussian constraints. From comparing
the difference in the uncertainties on masses, widths and branching fractions
for fits with and without these systematic constraints, it can be seen that
the systematic uncertainties are about 20 % the size of the statistical
uncertainties and thus contribute less than 2% to the total uncertainty.
In summary, a resonance has been observed in the dimuon spectrum of
$B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays with a significance of above
six standard deviations. The resonance can be explained by the contribution of
the $\psi(4160)$, via the decays $B^{+}\\!\rightarrow\psi(4160)K^{+}$ and
$\psi(4160)\\!\rightarrow\mu^{+}\mu^{-}$. It constitutes first observations of
both decays. The $\psi(4160)$ is known to decay to electrons with a branching
fraction of $(6.9\pm 4.0)\times 10^{-6}$ [4]. Assuming lepton universality,
the branching fraction of the decay $B^{+}\\!\rightarrow\psi(4160)K^{+}$ is
measured to be $(5.1^{+1.3}_{-1.2}\pm 3.0)\times 10^{-4}$, where the second
uncertainty corresponds to the uncertainty on the $\psi(4160)\\!\rightarrow
e^{+}e^{-}$ branching fraction. The corresponding limit for
$B^{+}\\!\rightarrow\psi(4040)K^{+}$ is calculated to be $1.3\,(1.7)\times
10^{-4}$ at a 90 (95) % confidence level. The absence of the decay
$B^{+}\\!\rightarrow\psi(4040)K^{+}$ at a similar level is interesting, and
suggests future studies of $B^{+}\\!\rightarrow K^{+}\mu^{+}\mu^{-}$ decays
based on larger datasets may reveal new insights into $c\overline{}c$
spectroscopy.
The contribution of the $\psi(4160)$ resonance in the low recoil region,
taking into account interference with the non-resonant $B^{+}\\!\rightarrow
K^{+}\mu^{+}\mu^{-}$ decay, is about 20 % of the total signal. This value is
larger than theoretical estimates, where the $c\overline{}c$ contribution is
$\sim$10% of the vector amplitude, with a small correction from quark-hadron
duality violation [24]. Results presented in this Letter will play an
important role in controlling charmonium effects in future inclusive and
exclusive $b\rightarrow s\mu^{+}\mu^{-}$ measurements.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-07-29T14:24:07 |
2024-09-04T02:49:48.664569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, E. Cowie, D.C. Craik, S.\n Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P.\n Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin,\n U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez,\n P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Patrick Owen",
"url": "https://arxiv.org/abs/1307.7595"
}
|
1307.7609
|
# Search for R-parity violating Supersymmetry using the CMS detector
Fedor Ratnikov for the CMS collaboration
Karlsrihe Institute of Technology, Karlsruhe, Germany
Institute of Theoretical and Experimental Physics, Moscow, Russia
[email protected] talk presented at the LHCP 2013 Conference in
Barcelona, Spain, May 13-18th, 2013
###### Abstract
In this talk, the latest results from CMS on R-parity violating Supersymmetry
are reviewed. We present results using up to 20/fb of data from the 8 TeV LHC
run of 2012. Interpretations of the experimental results in terms of
production of squarks, gluinos, charginos, neutralinos, and sleptons within RP
violating susy models are presented.
## 1 Introduction
Supersymmetry (SUSY) [1, 2] is an attractive extension of the Standard Model.
It provides natural coupling unification, dynamic electroweak symmetry
breaking and a solution to the hierarchy problem. R-parity is assigned to
fields as $R_{p}=(-1)^{3B+L+s}$ where $B$, $L$, and $s$ are baryon and lepton
numbers, and spin of the particle respectively. In models with conserved
R-parity superpartners may only be produced in pairs, and the lightest
superpartner (LSP) is stable. However R-parity conservation is not a universal
property of SUSY models. The most general gauge-invariant and renormalizable
superpotential consists of the R-parity conserving (RPC) main part, and may
also contain extra R-parity violating (RPV) terms [3]:
$\displaystyle W_{\Delta L=1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda_{ijk}L_{i}L_{j}\bar{e}_{k}+\lambda^{\prime}_{ijk}L_{i}Q_{j}\bar{d}_{k}+\mu^{\prime}_{i}L_{i}H_{u}$
(1) $\displaystyle W_{\Delta B=1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda^{\prime\prime}_{ijk}u_{i}d_{j}\bar{d}_{k}$
(2)
The presence of non-vanishing RPV terms leads to the the LSP becoming
unstable, decaying to standard model (SM) particles. Therefore many SUSY
analyses, which are based on the expectation of high missing transverse energy
in SUSY events from non-observed stable LSPs, are not sensitive to RPV SUSY
models.
Recent CMS analyses [5, 4, 6] are focused on studying the lepton number
violating terms $\lambda_{ijk}L_{i}L_{j}\bar{e}_{k}$ and
$\lambda^{\prime}_{ijk}L_{i}Q_{j}\bar{d}_{k}$, which cause specific signatures
involving leptons in events produced in pp collisions at LHC. Section 3
discusses the search for resonant production and the following decay of
$\tilde{\mu}$ which is caused by $\lambda^{\prime}_{211}\neq 0$. Section 4
addresses a search for multi-lepton signatures caused by LSP decays due to
various $\lambda$ and $\lambda^{\prime}$ terms. Finally in Section 5 we
discuss the possibility of the generic model independent search for RPV SUSY
in 4-lepton events.
## 2 Detector, trigger, and object selection
The central feature of the CMS apparatus is a superconducting solenoid, 6 m in
internal diameter, providing a magnetic field of 3.8 T. Within the field
volume there are a silicon pixel and strip tracker, a crystal electromagnetic
calorimeter, and a brass-scintillator hadron calorimeter. Muons are measured
in gas-ionization detectors embedded in the steel return yoke. Extensive
forward calorimetry complements the coverage provided by the barrel and endcap
detectors. A more detailed description of the CMS detector can be found in
Ref. [7].
Events from pp interactions must satisfy the requirements of a two-level
trigger system. The first level performs a fast selection for physics objects
(jets, muons, electrons, and photons) above certain thresholds. The second
level performs a full event reconstruction. The principal trigger used for
these analyses requires presence of at least two light leptons, electrons or
muons. Detailed trigger conditions and off-line event selections are described
in the corresponding Ref. [5, 4, 6].
## 3 Search for resonant second generation slepton production
Figure 1: Resonant smuon (left) and sneutrino (right) production and typical
decay chain into a final state with two same-sign muons and two jets. The
R-parity violating vertices are marked by a red dot.
Figure 2: Invariant mass of the two muons and jets before (left) and after (right) applying the b-tag veto and same-sign muon requirement. Data are compared to the expectation from the simulation (left) and measured backgrounds (right). Signal distributions are shown for three different kinematic configurations for a coupling value of $\lambda^{\prime}_{211}=0.01$. Table 1: Event yields with systematic uncertainties after selection requirements, broken down in individual Standard Model background contributions, with observed 95% C.L. limits on the number of signal events $N_{sig}$ in total and for each signal region. process | totals | SR1 | SR2 | SR3
---|---|---|---|---
VVV | 0.15 $\pm$ 0.08 | 0.043 $\pm$ 0.022 | 0.054 $\pm$ 0.028 | $<$0.001
tt+V | 0.11 $\pm$ 0.06 | 0.019 $\pm$ 0.010 | 0.038 $\pm$ 0.020 | 0
rare | 0.36 $\pm$ 0.26 | 0.32 $\pm$ 0.24 | 0.042 $\pm$ 0.042 | $<$0.001
VV | 2.1 $\pm$ 1.1 | 0.69 $\pm$ 0.35 | 0.68 $\pm$ 0.34 | 0.003 $\pm$ 0.002
fakes | 8.2 $\pm$ 3.0 | 3.5 $\pm$ 1.6 | 1.9 $\pm$ 1.0 | $<$0.001
$\sum$ | 10.9 $\pm$ 3.4 | 4.6 $\pm$ 1.6 | 2.7 $\pm$ 1.1 | 0.003 $\pm$ 0.002
data | 13 | 5 | 5 | 0
95% C.L. limit on $N_{sig}$ | 11.3 | 6.9 | 8.0 | 2.8
process | | SR4 | SR5 | SR6
VVV | | 0.036 $\pm$ 0.018 | 0.010 $\pm$ 0.005 | 0.007 $\pm$ 0.004
tt+V | | 0.044 $\pm$ 0.023 | 0.006 $\pm$ 0.004 | 0.006 $\pm$ 0.004
rare | | $<$0.001 | $<$0.001 | $<$0.001
VV | | 0.49 $\pm$ 0.25 | 0.15 $\pm$ 0.08 | 0.093 $\pm$ 0.050
fakes | | 2.5 $\pm$ 1.2 | 0.22 $\pm$ 0.23 | $<$0.001
$\sum$ | | 3.1 $\pm$ 1.2 | 0.39 $\pm$ 0.25 | 0.11 $\pm$ 0.05
data | | 0 | 2 | 1
95% C.L. limit on $N_{sig}$ | | 2.9 | 6.0 | 4.6
Figure 3: Distribution of
$m_{\tilde{\mu}}=m(\text{jets},\mu_{1}^{\pm},\mu_{2}^{\pm})$ vs.
$m_{\chi}=m(\text{jets},\mu_{2}^{\pm})$ for the events selected in data
compared to the total background contribution. The crosses represent the data
points and the coloured squares show the expectation from Standard Model
backgrounds.
Figure 4: Left: observed 95% CL upper limits on $\lambda^{\prime}_{211}$ as a
function of $m_{0}$ and $m_{1/2}$ for $A_{0}=0$, sign $(\mu)=+1$ and
$\tan\beta=20$. Right: mSUGRA limits expressed in the parameter space of the
neutralino mass $m_{\tilde{\chi}^{0}_{1}}$ and smuon mass $m_{\tilde{\mu}}$.
This search which is described in details in Ref. [4], extends the results
from a previous search by the DØ collaboration [8] and is complementary to
searches for RPV SUSY performed by the LEP experiments [9]. The search
concentrates on final states with two muons and at least two jets. Fig. 1
illustrates the simplest possible Feynman diagrams leading to this final
state, which is experimentally interesting because the presence of two muons
allows to discriminate the signal from background processes. One of the muons
is expected to be produced by the resonant slepton while the other muon and
two quarks resulting in jets are expected to be produced in the subsequent
decay of the neutralino LSP. Due to the Majorana nature of the LSP, the two
muons have the same charge with about 50% probability, which allows to
discriminate further against the background. Due to the larger valence
$\mathrm{u}$-quark content of the initial state protons the configuration with
two positively charged muons is about twice as likely as the configuration
with two negatively charged muons. The kinematics of this signal is
characterized by no missing transverse energy within the detector resolution.
For the purpose of this analysis we select events with two same-sign isolated
muons with $p_{\mathrm{T}}>20$ and $p_{\mathrm{T}}>15$ GeV for the first and
second muon respectively. In addition at least two jets with
$p_{\mathrm{T}}>30$ GeV, no $\mathrm{b}$-jets, and
$E_{\mathrm{T}}^{\text{miss}}<50$ GeV are required. After this selection, two
main background components remain: low cross section backgrounds containing
two prompt same-sign leptons such as production of multiple bosons, and
backgrounds with high cross-section where leptons from semileptonic decays of
$\mathrm{c}$ or $\mathrm{b}$-hadrons or other charged particles are wrongly
identified as prompt leptons. The first contribution is estimated from the
simulation. The latter contribution is difficult to model in simulation, thus
it is estimated using data. Fig. 2 illustrates the expected backgrounds before
and after the requirement of the two same-sign muons and the $\mathrm{b}$-jets
veto.
The 13 events observed in Fig 2 (right) are further investigated using their
2D distribution in parameters
$m_{\tilde{\mu}}=m(\text{jets},\mu_{1}^{\pm},\mu_{2}^{\pm})$ vs.
$m_{\chi}=m(\text{jets},\mu_{2}^{\pm})$, where $\mu_{1}^{\pm}$ denoting the
muon with higher $p_{\mathrm{T}}$. Fig. 3 overlays the observed events with
the expected background contributions, and describes six exclusive search
regions used for the interpretation of this analysis. Table 1 presents the
observations, expected backgrounds, and respective upper limits for all search
regions. The observations are consistent with the corresponding background
estimations, therefore results are combined to put limit on
$\lambda^{\prime}_{211}$ for different mSugra models in Fig. 4.
Table 2: Observed yields for three- and four- lepton events from 19.5 fb${}^{-}1$ recorded in 2012. The channels are split by the total number of leptons (NL), the number of $\tau_{\mathrm{h}}$ candidates (Nτ), and the $S_{\mathrm{T}}$. Expected yields are the sum of simulation and estimates of backgrounds from data in each channel. SR1–SR4 require a $\mathrm{b}$-tagged jet and veto events containing $\mathrm{Z}$ bosons. SR5–SR8 contain events that either contain a $\mathrm{Z}$ boson or have no $\mathrm{b}$-tagged jet. The channels are mutually exclusive. The uncertainties include statistical and systematic uncertainties. The $S_{\mathrm{T}}$ values are given in GeV. SR | NL | Nτ | $0<S_{\mathrm{T}}<300$ | $300<S_{\mathrm{T}}<600$ | $600<S_{\mathrm{T}}<1000$ | $1000<S_{\mathrm{T}}<1500$ | $S_{\mathrm{T}}>1500$
---|---|---|---|---|---|---|---
| | | obs | exp | obs | exp | obs | exp | obs | exp | obs | exp
SR1 | 3 | 0 | 116 | 123 $\pm$ 50 | 130 | 127 $\pm$ 54 | 13 | 18.9 $\pm$ 6.7 | 1 | 1.43 $\pm$ 0.51 | 0 | 0.208 $\pm$ 0.096
SR2 | 3 | $\geq 1$ | 710 | 698 $\pm$ 287 | 746 | 837 $\pm$ 423 | 83 | 97 $\pm$ 48 | 3 | 6.9 $\pm$ 3.9 | 0 | 0.73 $\pm$ 0.49
SR3 | 4 | 0 | 0 | 0.186 $\pm$ 0.074 | 1 | 0.43 $\pm$ 0.22 | 0 | 0.19 $\pm$ 0.12 | 0 | 0.037 $\pm$ 0.039 | 0 | 0.000 $\pm$ 0.021
SR4 | 4 | $\geq 1$ | 1 | 0.89 $\pm$ 0.42 | 0 | 1.31 $\pm$ 0.48 | 0 | 0.39 $\pm$ 0.19 | 0 | 0.019 $\pm$ 0.026 | 0 | 0.000 $\pm$ 0.021
SR5 | 3 | 0 | — | — | — | — | 165 | 174 $\pm$ 53 | 16 | 21.4 $\pm$ 8.4 | 5 | 2.18 $\pm$ 0.99
SR6 | 3 | $\geq 1$ | — | — | — | — | 276 | 249 $\pm$ 80 | 17 | 19.9 $\pm$ 6.8 | 0 | 1.84 $\pm$ 0.83
SR7 | 4 | 0 | — | — | — | — | 5 | 8.2 $\pm$ 2.6 | 2 | 0.96 $\pm$ 0.37 | 0 | 0.113 $\pm$ 0.056
SR8 | 4 | $\geq 1$ | — | — | — | — | 2 | 3.8 $\pm$ 1.3 | 0 | 0.34 $\pm$ 0.16 | 0 | 0.040 $\pm$ 0.033
## 4 Search for R-parity violating SUSY in multileptons with
$\mathrm{b}$-tagged jets
Table 3: Kinematically allowed stop decay modes with RPV coupling $\lambda^{\prime}_{233}$. The allowed neutralino decay modes for $m_{\mathrm{t}}<m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}$ are $\widetilde{\chi}^{0}_{1}\to\mu\mathrm{t}\overline{\mathrm{b}}$ and $\nu\mathrm{b}\overline{\mathrm{b}}$. Label | Kinematic region | Decay mode
---|---|---
A | $m_{\mathrm{t}}<m_{\widetilde{\mathrm{t}}_{1}}<2m_{\mathrm{t}},m_{\widetilde{\chi}^{0}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{t}\nu\mathrm{b}\overline{\mathrm{b}}$
B | $2m_{\mathrm{t}}<m_{\widetilde{\mathrm{t}}_{1}}<m_{\widetilde{\chi}^{0}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{t}\mu\mathrm{t}\overline{\mathrm{b}}$ or $\mathrm{t}\nu\mathrm{b}\overline{\mathrm{b}}$
C | $m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}<m_{\mathrm{W}^{\pm}}+m_{\widetilde{\chi}^{0}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\ell\nu\mathrm{b}\widetilde{\chi}^{0}_{1}$ or $\mathrm{jj}\mathrm{b}\widetilde{\chi}^{0}_{1}$
D | $m_{\mathrm{W}^{\pm}}+m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}<m_{\mathrm{t}}+m_{\widetilde{\chi}^{0}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{b}\mathrm{W}^{\pm}\widetilde{\chi}^{0}_{1}$
E | $m_{\mathrm{t}}+m_{\widetilde{\chi}^{0}_{1}}<m_{\widetilde{\mathrm{t}}_{1}}$ | $\widetilde{\mathrm{t}}_{1}\rightarrow\mathrm{t}\widetilde{\chi}^{0}_{1}$
Figure 5: The 95% confidence level limits in the stop and bino mass plane for
models with RPV couplings $\lambda_{122}$, $\lambda_{233}$, and
$\lambda^{\prime}_{233}$. For the couplings $\lambda_{122}$ and
$\lambda_{233}$, the region to the left of the curve is excluded. For
$\lambda^{\prime}_{233}$, the region inside the curve is excluded. The
different regions, A, B, C, D, and E, for the $\lambda^{\prime}_{233}$
exclusion result from different stop decay products as explained in Table 3.
Among modern SUSY models, “natural” supersymmetry refers to those
characterized by a relatively small fine tuning to describe particle spectra.
It requires top squarks (stops), to be lighter than about 1 TeV. The
introduction of RPV does not preclude a natural hierarchy and allows the
constraints on the stop mass to be relaxed [10].
The analysis [5] searches for pair production of top squarks with RPV decays
of the lightest sparticle, using multilepton events and $\mathrm{b}$-tagged
jets. It addresses terms $\lambda_{ijk}$ and $\lambda^{\prime}_{ijk}$ in Eqn.
1.
We select events with three or more leptons (including tau leptons) that are
accepted by a trigger required two light leptons, which may be electrons or
muons. At least one electron or muon in each event is required to have
transverse momentum of $p_{\mathrm{T}}>20$ GeV. Additional electrons and muons
must have $p_{\rm T}>10$ GeV. The majority of hadronic decays of tau leptons
($\tau_{\mathrm{h}}$) yield either a single charged track (one-prong) or three
charged tracks (three-prong), occasionally with additional electromagnetic
energy from neutral pion decays. We use one- and three-prong
$\tau_{\mathrm{h}}$ candidates that have $p_{\mathrm{T}}>20$ GeV. Leptonically
decaying taus are included with other electrons and muons. The
$E_{\mathrm{T}}^{\text{miss}}$ is not a good discriminator for RPV SUSY
search. Instead we use the $S_{\mathrm{T}}$ variable, which is the scalar sum
of $E_{\mathrm{T}}^{\text{miss}}$ and the transverse energy of jets with
$p_{\mathrm{T}}>30$ GeV and charged leptons, to provide separation between
signal and the Standard Model backgrounds.
Irreducible Standard Model backgrounds are estimated from simulation.
Contributions from fakes for electrons, muons and taus are obtained using
data-driven methods.
Observed events are classified into eight topologies according to the number
of observed light leptons and the presence of hadronic tau in event. Every
topology is further split into five search regions according to the
$S_{\mathrm{T}}$ value. Table 2 summarizes observations and expected
contributions for different search regions used in this analysis.
We generate simulated samples to evaluate models with simplified mass spectra
and the only non-zero leptonic RPV couplings $\lambda_{122}$ or
$\lambda_{233}$. The stop masses in these samples range from 700–1250 GeV in
50 GeV steps, and bino masses range from 100–1300 GeV in 100 GeV steps. In a
model with only the semi-leptonic RPV coupling $\lambda^{\prime}_{233}$, we
use stop masses 300–1000 GeV in 50 GeV steps and bino masses 200–850 GeV in 50
GeV steps. In both cases, slepton and sneutrino masses are 200 GeV above the
bino mass. Other particles are irrelevant to the results for these models.
No significant excess is observed in data. The observations from Table 2 are
combined into exclusions for the corresponding models in Fig. 5.
## 5 Generalization of Unstable LSP Search
The analysis described in details in Ref. [6] presents a new approach to a
generic interpretation of experimental results. The focus of this analysis is
the lepton number violating term $\lambda_{ijk}L_{i}L_{j}\bar{e}_{k}$, which
causes the LSP in such a “Leptonic-RPV” (LRPV) SUSY model to decay into
leptons. SUSY particles are produced in pairs, thus a non-zero $\lambda$-term
would lead to events with 4 charged leptons produced in LSP decays. Recent
searches at the Tevatron [11] and LHC [12, 5] placed limits on $\lambda$. The
main challenge of RPV SUSY searches is that the RPV term exists on top of some
underlying RPC SUSY model, with properties which are currently barely
constrained. The analyses mentioned above resolve this problem by exploring
RPV on top of very specific RPC SUSY models. In this analysis we pursue a
significantly less model dependent approach. We require the presence of 4
isolated leptons in the event, as a direct signature of the LRPV SUSY. No
other restriction is applied, so the selection efficiency is not directly
affected by the underling SUSY event. Irreducible Standard Model backgrounds
are estimated from simulation, estimations of fakes are data-driven. The main
background for 4-lepton events is found to be ZZ production, so for every
event the variable $M_{1}$ is calculated as the invariant mass of same-flavor
opposite-sign lepton pair that is closest to the mass of Z-boson. $M_{2}$ is
then calculated as the invariant mass of the remaining lepton pair.
Table 4: Observed events and expected background contributions. $M_{1}$ and
$M_{2}$ intervals are in GeV.
| $M_{1}<76$ | $76<M_{1}<106$ | $M_{1}>106$
---|---|---|---
| all backgrounds | 1.4$\pm$0.5 | 18$\pm$4 | 0.47$\pm$0.10
$M_{2}>106$ | observed | 0 | 20 | 0
| all backgrounds | 0.52$\pm$0.30 | 153∗ | 0.16$\pm$0.06
$76<M_{2}<106$ | observed | 0 | 160 | 0
| all backgrounds | 10.4$\pm$2.0 | 35$\pm$8 | 1.0$\pm$0.2
$M_{2}<76$ | observed | 14 | 30 | 1
∗ ZZ prediction in “in Z”:“in Z” region is based on MC normalized to CMS ZZ
production cross section measurement, which is correlated with observation in
“in Z”:“in Z” region of this analysis.
Table 4 presents the observations and expected backgrounds in different
regions in $M_{1}:M_{2}$ space. Observations and expectations are consistent
in all regions. Based on the occupancy of different regions for typical ZZ
production events, the signal region is defined as “$M_{1}$ above Z” or
“$M_{1}$ below Z and $M_{2}$ above Z”. Then the upper limit on cross section
times integrated luminosity times efficiency
($\sigma\times\mathcal{L}\times\varepsilon$) for any physics process beyond
the SM contributing to this search region is 3.4 events. The expected upper
limit for this observation is 4.7 events. The leptonic decay of the pair of
LRPV neutralinos leads to 4 prompt leptons. The kinematics of these leptons
are in general driven by the momentum distribution of the decaying neutralinos
and their mass. In most scenarios the lepton momentum is well above threshold,
which results in high efficiency. However the following effects could reduce
the total efficiency:
* •
the presence of other leptons in the event, which affects the efficiency
through the 4-lepton requirement, as well as the calculation of the $M_{1}$
and $M_{2}$ quantities;
* •
the electron and/or muon objects reconstruction efficiency which is dependent
on $\eta$ and $p_{T}$;
* •
the isolation efficiency, which is correlated with the occupancies around the
observed prompt leptons.
The presence of an extra lepton in the SUSY event, in addition to the 4
leptons produced from neutralino decays, could veto the event. We observe no
events containing 5 isolated leptons. Thus, the potential presence of
additional leptons in fact does not significantly affect the measurement.
To evaluate the dependency of the lepton reconstruction efficiency and the
efficiency of analysis selections from details of kinematic distributions of
decaying neutralinos, we consider two extreme cases of LRPV neutralino
production:
* •
a simplified model with SUSY particles produced via a squark-anti-squark pair,
with the neutralino coming from a two-body decay $\tilde{q}\rightarrow
q\tilde{\chi}_{1}^{0}$, as presented in Fig. 6;
* •
a pair of neutralinos produced in rest in the center of the CMS detector.
Figure 6: LRPV extensions to Simplified Model [13]. The T2 RPC simplified
model is squark pair production, with $\tilde{q}\rightarrow
q\tilde{\chi}_{1}^{0}$, and $m(\tilde{g})\gg m(\tilde{q})$. The neutralinos
decay to two charged leptons and a neutrino via an LRPV term.
Figure 7: Left: efficiency for the T2+LRPV model.
$\tilde{\chi}^{0}_{1}\rightarrow\mu^{+}\mu^{-}\nu$. Right: For every
neutralino mass the efficiency value is filled corresponding to the different
squark masses.
The first approach creates the most energetic neutralinos possible,
constrained by the relevant squark and neutralino masses. Figure 7 (left)
presents the efficiency as a function of the T2 model parameters: squark mass
and neutralino mass. This distribution illustrates that the total efficiency
of this analysis is mostly driven by the neutralino mass, while the squark
mass, which drives the neutralino spectrum, affects the efficiency only
marginally. To illustrate this further Fig. 7 (right) shows the distribution
of the efficiency for different squark masses. This distribution demonstrates,
that the variations even over a wide range of squark masses, are within
$\pm$10%.
Figure 8: Left: efficiency of this analysis for neutralinos decaying in rest
(red points), overlaid with LRPV efficiency from Fig. 7. Right: efficiency
profiles for electrons and muons.
If the LSP is produced at the end of a long cascade of decays of SUSY
particles, the LSP $p_{T}$ spectra will be significantly softer than for LSPs
produced in two-body decays of the T2 scenario. To study the effect of soft
spectra we consider another extreme case: neutralino pairs produced in rest in
the detector frame. We generate the corresponding dataset by letting the
neutralino decay into ($e^{+},e^{-},\nu$) or ($\mu^{+},\mu^{-},\nu$). Figure 8
(left) shows the efficiency as a function of the neutralino mass overlaid with
the efficiency band obtained from the T2+LRPV model presented in Fig. 7
(right). It demonstrates that the difference between the T2+LRPV case and the
stopped neutralino case is below $\pm$10%.
Figure 9: Isolation efficiency for 4 leptons for the set of pMSSM models
described in the text, as a function of the neutralino mass in the model. The
green and yellow bands include 68% and 95% of the model points in the
efficiency distribution respectively.
The isolation efficiency for isolated leptons from RPV decays depends on the
occupancy of the event, which in turn depends on the content of the underlying
SUSY event. To study how strong the influence of different underlying SUSY
models and different SUSY production mechanisms is, we re-use the data samples
produced in a previous CMS analysis [14]. These are MC samples for about 7300
different RPC phenomenological MSSM (pMSSM) [15] model points, each one
containing 10000 events, selected to fulfill different pre-CMS observations.
The pMSSM model is an excellent proxy for the full MSSM with a sufficiently
small number of parameters [14]. The available datasets for this set of pMSSM
models is to date the biggest sample of varying SUSY models available to us.
To evaluate the effect of different occupancies in each event of the pMSSM, we
start by extracting the generator-level information about the neutralino. Then
we generate a neutralino RPV decay into two leptons and a neutrino and finally
calculate the reconstruction level isolation around the direction of the
obtained leptons. The event is accepted if the isolation for each of the 4
charged leptons satisfies the isolation requirements for prompt leptons used
in this analysis. Figure 9 presents the efficiencies for different SUSY models
as a function of the neutralino mass in each model. Nearly all SUSY models
have a 4-lepton isolation efficiency in the range between 0.5 and 1. The green
and yellow shaded areas in the plot contain 68% and 95% of the model points
respectively.
We use the band $[0.5,1]$ as a conservative estimate for possible variations
of the analysis signal efficiency due to different types of underlying SUSY
models. We use $30\%$ uncertainty when we combine this effect with other
uncertainties.
Combining all effects, we consider the T2+LRPV model efficiency in Fig. 8
(right) to be a representative of a “best efficiency” scenario. Large hadronic
activity in the event can reduce the isolation efficiency. In line with the
pMSSM study, we conclude that the reduction of the total efficiency for this
search may be up to 50%. Therefore, we consider an efficiency band between
these two extreme cases to cover the 4-lepton efficiency for most the SUSY
models in this analysis.
Once an upper limit on $\sigma\times\mathcal{L}\times\varepsilon$ is extracted
from the observations, and the efficiency is evaluated, the corresponding
limit on the cross section, $\sigma^{SUSY}_{total}$, may be calculated.
Figure 10: Left: 95% C.L. upper limit on total cross sections for generic
SUSY models. The band corresponds to the efficiency uncertainty as described
in the text. Right: Mass exclusions for different SUSY production mechanisms.
Left: for T2+LRPV models. Right: using a generic total RPV SUSY cross section
limit in the left plot. A $30\%$ theoretical uncertainty for NLO+NLL
calculations of SUSY production cross sections is included in the uncertainty
band.
The experimental observations together with the pMSSM based efficiency
estimation as described above drive the exclusion for the cross section of
total RPV SUSY production, which is presented in Fig. 10 (left). The bands
correspond to the 4 lepton isolation variations between 50% and 100%. Note
that this is a very generic result as this band covers RPV models with a wide
range of underlying RPC SUSY models.
To further convert the cross section limit into a mass exclusion we consider
several SUSY production mechanisms: gluino pair production, squark pair
production, and stop-quark pair production. The cross sections for these
processes as functions of the corresponding masses are NLO+NLL calculation
results of the corresponding decoupled scenarios [16]. The theoretical
uncertainties on the NLO+NLL SUSY production cross section calculations for
masses $\sim$1 TeV are about $30\%$, and are accounted for in the result.
Using these total cross sections as a function of the mass of the
corresponding SUSY particle, we convert the cross section limit bands in Fig.
10 (left) into mass exclusion bands as a function of the LSP mass. This result
is presented in Fig. 10 (right).
## 6 Conclusions
CMS developed a comprehensive program for RPV SUSY searches. In this
contribution we present the most recent results on this topic. For all
presented searches observations are consistent with expectations from the
Standard Model, thus the corresponding limits on presence of new physics are
set. We also present a new approach of generalizing physics interpretations of
experimental observations. Sampling of a big set of pMSSM models allows to
check a model dependency for obtained results, thus making more general
conclusions possible.
## References
* [1] H. P. Nilles, Phys. Rept. 110, 1 (1984).
* [2] H. E. Haber and G. L. Kane, Phys. Rept. 117 (1985) 75.
* [3] S. P. Martin, In *Kane, G.L. (ed.): Perspectives on supersymmetry II* 1-153 [hep-ph/9709356].
* [4] CMS Collaboration, Search for RPV SUSY resonant second generation slepton production in same-sign dimuon events at $\sqrt{s}=7\,$TeV, CMS-PAS-SUS-13-005 (2013).
* [5] S. Chatrchyan et al. [CMS Collaboration], arXiv:1306.6643 [hep-ex].
* [6] CMS Collaboration, Search for RPV SUSY in the four-lepton final state, CMS-PAS-SUS-13-010 (2013).
* [7] S. Chatrchyan et al. [CMS Collaboration], JINST 3 (2008) S08004.
* [8] V. M. Abazov et al. [D0 Collaboration], Phys. Rev. Lett. 97 (2006) 111801 [hep-ex/0605010].
* [9] R. Barbier, C. Berat, M. Besancon, M. Chemtob, A. Deandrea, E. Dudas, P. Fayet and S. Lavignac et al., Phys. Rept. 420 (2005) 1 [hep-ph/0406039].
* [10] J. A. Evans and Y. Kats, JHEP 1304 (2013) 028 [arXiv:1209.0764 [hep-ph]].
* [11] V. M. Abazov et al. [D0 Collaboration], Phys. Lett. B 638 (2006) 441 [hep-ex/0605005].
* [12] G. Aad et al. [ATLAS Collaboration], JHEP 1212 (2012) 124 [arXiv:1210.4457 [hep-ex]].
* [13] D. Alves et al. [LHC New Physics Working Group Collaboration], J. Phys. G 39 (2012) 105005 [arXiv:1105.2838 [hep-ph]].
* [14] CMS Collaboration, Phenomenological MSSM interpretation of the CMS 2011 5/fb results, CMS-PAS-SUS-12-030 (2013).
* [15] A. Djouadi et al. [MSSM Working Group Collaboration], hep-ph/9901246.
* [16] M. Kramer, A. Kulesza, R. van der Leeuw, M. Mangano, S. Padhi, T. Plehn and X. Portell, arXiv:1206.2892 [hep-ph].
|
arxiv-papers
| 2013-07-29T14:58:43 |
2024-09-04T02:49:48.674817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fedor Ratnikov (for the CMS collaboration)",
"submitter": "Fedor Ratnikov",
"url": "https://arxiv.org/abs/1307.7609"
}
|
1307.7628
|
Position-dependent noncommutative quantum models:
Exact solution of the harmonic oscillator
Dine Ousmane Samary
Perimeter Institute for Theoretical Physics
31 Caroline St. N. Waterloo, ON N2L 2Y5, Canada
International Chair in Mathematical Physics and Applications
(ICMPA-UNESCO Chair), University of Abomey-Calavi,
072B.P.50, Cotonou, Rep. of Benin
E-mail: [email protected]
###### Abstract
This paper is devoted to find the exact solution of the harmonic oscillator in
a position-dependent $4$-dimensional noncommutative phase space. The
noncommutative phase space that we consider is described by the commutation
relations between coordinates and momenta:
$[\hat{x}^{1},\hat{x}^{2}]=i\theta(1+\omega_{2}\hat{x}^{2})$,
$[\hat{p}^{1},\hat{p}^{2}]=i\bar{\theta}$,
$[\hat{x}^{i},\hat{p}^{j}]=i\hbar_{eff}\delta^{ij}$. We give an analytical
method to solve the eigenvalue problem of the harmonic oscillator within this
deformation algebra.
Key words: Noncommutative phase space, Moyal star product, eigenvalues
problem, harmonic oscillator.
## 1 Introduction
Noncommutative (NC) geometry plays an increasing role in the search of a
unifying theory of gravity and quantum mechanics and is a framework built for
understanding physics at short distances. Within this framework, the past two
decades have witnessed important progresses toward the solution of various
quantum models, in particular, the harmonic oscillator in NC spaces. There
exists a large number of papers which address this class of problem. We will
focus on the most recent developments discussing particular tractable models
and specific ways to realize NC spaces called Moyal spaces [1]-[22].
The Moyal type NC space is a concrete proposal for a space where the
coordinate operators $\hat{x}^{\mu}$ satisfy the commutation relation
$\displaystyle[\hat{x}^{\mu},\hat{x}^{\nu}]=i\theta^{\mu\nu}$ (1)
and where $\theta^{\mu\nu}$ is an antisymmetric tensor of space dimension
$(length)^{2}$. The noncommutativity specified by (1) can be as well realized
in terms of a star product. In this point of view, the ordinary multiplication
of functions is replaced by the Moyal star product defined for $f,g\in
C^{\infty}(\mathbb{R}^{D})$ by
$(f\star g)(x)={\bf
m}\Big{[}\exp\Big{(}\frac{i}{2}\theta^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu}\Big{)}(f\otimes
g)(x)\Big{]},\quad{\bf m}(f\otimes g)(x)=f(x)\cdot g(x).$ (2)
Then the commutation relation (1) becomes
$[x^{\mu},x^{\nu}]_{\star}=x^{\mu}\star x^{\nu}-x^{\nu}\star
x^{\mu}=i\theta^{\mu\nu}$ (3)
with now commuting coordinates $x^{\mu}$. The noncommutativity of space
coordinates can be naturally incorporated into the quantum field theory
framework. Subsequently, NC field theories and quantum mechanics have studied
extensively [2]. There is however a more general structure extending the Moyal
brackets (3). Consider that one replaces the commutation relation (1) by
following [6]-[7]
$\displaystyle[\hat{x}^{\mu},\hat{x}^{\nu}]=i\theta^{\mu\nu}e(\hat{x})$ (4)
where $e(\hat{x})$ is an arbitrary dimensionless function which depends on the
coordinates. The same space can be again realized using another star product
called the twisted Moyal product generalizing (2). Taking $e(x)$ positive, the
star product
$\displaystyle(f\star g)(x)={\bf
m}\Big{[}\exp\Big{(}\frac{i}{2}\theta^{\mu\nu}\sqrt{e(x)}\partial_{\mu}\otimes\sqrt{e(x)}\partial_{\nu}\Big{)}(f\otimes
g)(x)\Big{]}$ (5)
can be used to generate
$\displaystyle[x^{\mu},x^{\nu}]_{\star}=i\theta^{\mu\nu}e(x)$ (6)
now extending (3). The choice of the function $e(x)$ depends on the physical
considerations which may encode minimal length [4] or the integrability of
some dynamical Hamiltonians [8].
We emphasize the fact that a necessary condition for having an associative
star product from (5) is given by
$\partial_{[\mu}e(x)\partial_{\nu]}f=0,\,\,\forall f\in C^{1}(\mathbb{R}^{D})$
[6]. This is not however a sufficient condition. The associativity of the
twisted star product implies the Jacobi identity
$\displaystyle
J(\mu,\nu,\rho)=[x^{\mu},[x^{\nu},x^{\rho}]_{\star}]_{\star}+[x^{\rho},[x^{\mu},x^{\nu}]_{\star}]_{\star}+[x^{\nu},[x^{\rho},x^{\mu}]_{\star}]_{\star}=0.$
(7)
The particular case of the structure function
$\displaystyle e(x)=1+\omega_{\mu}x^{\mu}$ (8)
where $\omega_{\mu}x^{\mu}$ is dimensionless and $\omega_{\mu}\in\mathbb{R}$,
leads to
$\displaystyle
J(\mu,\nu,\rho)=-e(x)\omega_{\sigma}\Big{(}\theta^{\nu\rho}\theta^{\mu\sigma}+\theta^{\mu\nu}\theta^{\rho\sigma}+\theta^{\rho\mu}\theta^{\nu\sigma}\Big{)}.$
(9)
For such a choice of the function $e(x)$, the associativity of the star
product (5) can be shown even for the non-vanishing tensor $\omega_{\sigma}$
[8]. From this point, the authors of [8] were able to derive the equivalent of
the so-called matrix basis of the Moyal plane [10, 9].
NC spaces can be slightly more general than the above. For instance there are
several developments around the so called NC quantum mechanics [13, 18, 19,
20, 21]. NC quantum mechanics [20] can be also described by introducing
commutation relations between coordinate and momentum even also between
momentum and momentum. Thus (6) can be extended to a $2D$ NC phase space as
follows
$\displaystyle[x^{1},x^{2}]_{\star}=i\theta
e(x),\qquad[x^{1},p^{1}]_{\star}=i\hbar_{eff},\quad[x^{2},p^{2}]_{\star}=i\hbar_{eff},\qquad[p^{1},p^{2}]=i\bar{\theta}$
(10)
where $\theta$, $\bar{\theta}$ and $\hbar_{eff}$ are constant but
$e(x)=1+\omega_{1}x^{1}+\omega_{2}x^{2}$ is still a function. The present work
highlights the spectrum of the harmonic oscillator in this twisted NC phase
space defined by the commutation relations (10), with the restriction
$\omega_{1}=0$. We show, using a particular transformation of the basic
degrees of freedom that the total nonlinear harmonic oscillator Hamiltonian
factorizes. From that point, the model becomes solvable.
The paper is organized as follows. In section 2, we give some useful results
concerning the deformation of the NC phase space (10). Then, the spectrum and
states of the harmonic oscillator are solved. We give a summary of our results
in section 3.
## 2 Position-dependent NC quantum mechanics
This section addresses the construction of a position-dependent NC star
product which is induced by the deformation (10). We start with the following
definition.
###### Definition 1 (Twisted Moyal algebra).
Consider the set $E=\\{(x^{i},p^{i}),i=1,2\\}$ and
$\mathbb{C}[[x^{1},x^{2},p^{1},p^{2}]],$ the free algebra generated by $E$.
Let $\mathcal{I}$ be the ideal of $\mathbb{C}[[x^{1},x^{2},p^{1},p^{2}]],$
generated by the elements
$[x^{i},x^{j}]_{\star}-i\theta^{ij}(x),\quad[x^{i},p^{j}]_{\star}-i\hbar_{eff}\delta^{ij},\quad[p^{i},p^{j}]_{\star}-i\bar{\theta}^{ij},$
where $\theta^{ij}(x)$ is skew symmetric tensor depending on space coordinates
and $\bar{\theta}^{ij}$ a constant skew symmetric tensor. The twisted Moyal
algebra $\mathcal{M}_{\theta\bar{\theta}\hbar_{eff}}$ is the quotient
$\mathbb{C}[[x^{1},x^{2},p^{1},p^{2}]]/\mathcal{I}$. Each element in
$\mathcal{M}_{\theta\bar{\theta}\hbar_{eff}}$ is a formal power series in the
$(x^{i},p^{j})$’s for which the following relations hold:
$\displaystyle[x^{i},x^{j}]_{\star}=i\theta^{ij}(x),\quad[x^{i},p^{j}]_{\star}=i\hbar_{eff}\delta^{ij},\quad[p^{i},p^{j}]_{\star}=i\bar{\theta}^{ij}.$
(11)
The Moyal algebra can be also defined as the linear space of smooth and
rapidly decreasing functions equipped with the NC star product given in the
form $f\star g={\bf
m}\big{[}(\star_{\theta}\star_{\hbar_{eff}}\star_{\bar{\theta}})(f\otimes
g)\big{]}$, such that
$\displaystyle f\star_{\theta}g={\bf
m}\Big{[}\exp\Big{(}\frac{i}{2}\theta^{ij}(x)\partial_{x^{i}}\otimes\partial_{x^{j}}\Big{)}f\otimes
g\Big{]}$ (12) $\displaystyle f\star_{\hbar_{eff}}g={\bf
m}\Big{[}\exp\Big{(}\frac{i}{2}\hbar_{eff}\delta^{ij}(\partial_{x^{i}}\otimes\partial_{p^{j}}-\partial_{p^{i}}\otimes\partial_{x^{j}})\Big{)}f\otimes
g\Big{]}$ (13) $\displaystyle f\star_{\bar{\theta}}g={\bf
m}\Big{[}\exp\Big{(}\frac{i}{2}\bar{\theta}^{ij}\partial_{p^{i}}\otimes\partial_{p^{j}}\Big{)}f\otimes
g\Big{]}.$ (14)
For $D=2$, we set $x^{i}=(x^{1},x^{2})$, $p^{i}=(p^{1},p^{2})$ with
$(x^{i},p^{i})\in\mathbb{R}^{4}$ and we will restrict the NC structure tensors
to the following:
$\displaystyle\theta^{ij}(x)=\theta e(x)\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)=\theta
e(x)\epsilon^{ij},\quad\bar{\theta}^{ij}=\bar{\theta}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)=\bar{\theta}\epsilon^{ij}.$ (19)
For $f\in C^{\infty}(\mathbb{R}^{4})$, the following relations are satisfied
$\displaystyle x^{1}\star f=x^{1}f+\frac{i\theta
e}{2}\partial_{x^{2}}f+\frac{i\hbar_{eff}}{2}\partial_{p^{1}}f,\qquad
p^{1}\star
f=p^{1}f+\frac{i\bar{\theta}}{2}\partial_{p^{2}}f-\frac{i\hbar_{eff}}{2}\partial_{x^{1}}f,$
(20) $\displaystyle x^{2}\star f=x^{2}f-\frac{i\theta
e}{2}\partial_{x^{1}}f+\frac{i\hbar_{eff}}{2}\partial_{p^{2}}f,\qquad
p^{2}\star
f=p^{2}f-\frac{i\bar{\theta}}{2}\partial_{p^{1}}f-\frac{i\hbar_{eff}}{2}\partial_{x^{2}}f.$
(21)
The commutation relation (11) can be deduced from (20) and (21).
We can now introduce a model on that twisted NC space. Let us consider the NC
harmonic oscillator described by the Hamiltonian
$\displaystyle
H=\frac{1}{2}\Big{[}(p^{1})^{2}+(p^{2})^{2}+(x^{1})^{2}+(x^{2})^{2}\Big{]}.$
(22)
In (22), the mass parameter and the oscillator constant are taken to be $1$.
The Hamiltonian (22) is invariant under the phase space rotation. We now
address the problem we want to solve.
We will solve the eigenvalue problem associated with (22) in the NC phase
space
$\displaystyle H\star\psi=E\,\psi.$ (23)
A way to solve the eigenvalue problem of a quantum Hamiltonian is its
factorization. In the following, will introduce a particular type of
factorization. The eigenvalue problem (23) can be split into two equations
given by
$\displaystyle H_{R}\star\psi=E\,\psi,\quad\mbox{and}\quad H_{Im}\star\psi=0$
(24)
where, by expansion of the twisted star product, one should obtain the real
and imaginary part corresponding to (23) as:
$\displaystyle H_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}(p^{1})^{2}+(p^{2})^{2}+(x^{1})^{2}+(x^{2})^{2}++\frac{\bar{\theta}\hbar_{eff}}{2}(\partial_{p^{2}}\partial_{x^{1}}-\partial_{p^{1}}\partial_{x^{2}})$
(25) $\displaystyle-$
$\displaystyle\frac{\theta\hbar_{eff}e}{2}(\partial_{x^{2}}\partial_{p^{1}}-\partial_{x^{1}}\partial_{p^{2}})-\frac{\hbar_{eff}^{2}}{4}(\partial_{p^{1}}^{2}+\partial_{p^{2}}^{2}+\partial_{x^{1}}^{2}+\partial_{x^{2}}^{2})$
(26) $\displaystyle-$
$\displaystyle\frac{\bar{\theta}^{2}}{4}(\partial_{p^{1}}^{2}+\partial_{p^{2}}^{2})-\frac{\theta^{2}e}{4}(\partial_{x^{1}}^{2}+\partial_{x^{2}}^{2})-\frac{\theta
e}{4}(\omega_{2}\partial_{x^{2}}+\omega_{1}\partial_{x^{1}})\Big{]}$ (27)
and
$H_{Im}=\bar{\theta}(p^{1}\partial_{p^{2}}-p^{2}\partial_{p^{1}})+\theta
e(x^{1}\partial_{x^{2}}-x^{2}\partial_{x^{1}})+\hbar_{eff}(x^{1}\partial_{p^{1}}+x^{2}\partial_{p^{2}}-p^{1}\partial_{x^{1}}-p^{2}\partial_{x^{2}}).$
(28)
Note that the eigenvalue equation (23) can be also written as $\psi\star
H=E\psi$, due to the fact that $f\star g=\overline{g\star f}$. Equations (24)
with the Hamiltonians (25) and (28) are nonlinear. However after putting a
restriction on the type of noncommutativity that we use, we will provide
solution to these equations.
Solution - Consider $e(x)=1+\omega_{1}x^{1}+\omega_{2}x^{2}$. Let us assume
that $\omega_{1}\neq 0$ or $\omega_{2}\neq 0$. Consider the transformation
$\mathcal{T}$ mapping coordinates $(x,p)$ to the new variables
$(\widetilde{x},\widetilde{p})$ given by
$\displaystyle\mathcal{T}:\left\\{\begin{array}[]{c}x^{1}=\theta\omega_{1}e^{\widetilde{x}^{1}}-\theta\omega_{2}\widetilde{x}^{2}-\frac{\omega_{1}}{\omega_{1}^{2}+\omega_{2}^{2}}\\\
x^{2}=\theta\omega_{2}e^{\widetilde{x}^{1}}+\theta\omega_{1}\widetilde{x}^{2}-\frac{\omega_{2}}{\omega_{1}^{2}+\omega_{2}^{2}}\\\
p^{1}=\widetilde{p}^{1}\\\ p^{2}=\widetilde{p}^{2}\end{array}\right.$ (33)
For $\theta>0,$ the transformation $\mathcal{T}$ is invertible in the positive
domain of the plane $(x^{1},x^{2})$ given by relation
$e(x)=1+\omega_{1}x^{1}+\omega_{2}x^{2}>0.$ (34)
The inverse transformation $\mathcal{T}^{-1}$ is given by
$\displaystyle\mathcal{T}^{-1}:\left\\{\begin{array}[]{c}\widetilde{x}^{1}=\ln\Big{(}\frac{e(x)}{\theta(\omega_{1}^{2}+\omega_{2}^{2})}\Big{)}\\\
\widetilde{x}^{2}=\frac{-\omega_{2}x^{1}+\omega_{1}x^{2}}{\theta(\omega_{1}^{2}+\omega_{2}^{2})}\\\
\widetilde{p}^{1}=p^{1}\\\ \widetilde{p}^{2}=p^{2}\end{array}\right.$ (39)
Let us immediately remark that the transformation (39) break the ordinary
limit of the theory i.e. the limit $\theta\rightarrow 0$ cannot be taken into
account. To recover this inconvenience we can use the renormalization
procedures, which will be addressed in forthcoming work.
Under $\mathcal{T}$, the algebra $\mathcal{M}_{\theta\bar{\theta}\hbar_{eff}}$
is transformed as
$\widetilde{\mathcal{M}}_{\theta\bar{\theta}\hbar_{eff}}=\mathcal{T}[\mathcal{M}_{\theta\bar{\theta}\hbar_{eff}}]$.
The non-vanishing commutation relations satisfied by the new variables are
given by the following:
$\displaystyle[\widetilde{x}^{1},\widetilde{x}^{2}]_{\star}=i\theta\sqrt{\omega_{1}^{2}+\omega_{2}^{2}}=i\gamma,\quad[\widetilde{x}^{1},\widetilde{p}^{1}]_{\star}=i\hbar_{eff}\omega_{1}e^{-1}=i\hbar_{1}(x),$
(41)
$\displaystyle[\widetilde{x}^{2},\widetilde{p}^{2}]_{\star}=i\frac{\omega_{1}\hbar_{eff}}{\theta(\omega_{1}^{2}+\omega_{2}^{2})}=i\hbar_{2},\quad[\widetilde{p}^{1},\widetilde{p}^{2}]_{\star}=i\bar{\theta}.$
The ordinary recipes that are known for diagonalizing the algebra do not work
in this instance because of the presence of the function $e(x)$. However,
restricting to the case $\omega_{1}=0$ and $\omega_{2}\neq 0,$ we have
$\displaystyle x^{1}=-\gamma\widetilde{x}^{2},\quad x^{2}=\gamma
e^{\widetilde{x}^{1}}-\frac{1}{\omega_{2}},\quad p^{1}=\widetilde{p}^{1},\quad
p^{2}=\widetilde{p}^{2},\quad\gamma=\theta\omega_{2}.$ (42)
Therefore, for simplicity by setting $\omega_{1}=0$ and $\omega_{2}=1=\gamma$,
we understand that the transformation $\mathcal{T}$ simply induces a rotation
in the plane $(x^{1},x^{2})\to(x^{2},-x^{1})$ followed by a logarithmic scale
transformation $(x^{1},x^{2})\to(\ln[x^{1}+1],x^{2})$. Note that such a
transformation cannot be defined in the case of the Moyal plane determined by
the limiting situation $\omega_{1}=\omega_{2}=0$. Furthermore, it can be
noticed that the restriction (42) clearly breaks the symmetry between the
coordinates $x^{1}$ and $x^{2}$. In any case, the following analysis finds an
analog when we consider $\omega_{2}=0,\,\,\omega_{1}\neq 0$.
We obtain the final commutation relations
$\displaystyle[\widetilde{x}^{1},\widetilde{p}^{1}]_{\star}=0,\quad[\widetilde{x}^{2},\widetilde{p}^{2}]_{\star}=0,\quad[\widetilde{x}^{1},\widetilde{x}^{2}]_{\star}=i\gamma,\quad[\widetilde{p}^{1},\widetilde{p}^{2}]_{\star}=i\bar{\theta}.$
(43)
As a consequence, the algebra
$\widetilde{\mathcal{M}}_{\theta\bar{\theta}\hbar_{eff}}$ splits into two
sectors $\widetilde{\mathcal{M}}_{\theta}$ and
$\widetilde{\mathcal{M}}_{\bar{\theta}}$ such that
$\displaystyle\widetilde{\mathcal{M}}_{\theta}\otimes\widetilde{\mathcal{M}}_{\bar{\theta}}\equiv\widetilde{\mathcal{M}}_{\theta\bar{\theta}\hbar_{eff}},$
(44)
where the algebras $\widetilde{\mathcal{M}}_{\theta}$ and
$\widetilde{\mathcal{M}}_{\bar{\theta}}$ are each of the Moyal-type defined
such that
$\widetilde{\mathcal{M}}_{\theta}=\mathbb{C}[[\widetilde{x}^{1},\widetilde{x}^{2}]]/\mathcal{I}_{1}$
and
$\widetilde{\mathcal{M}}_{\bar{\theta}}=\mathbb{C}[[\widetilde{p}^{1},\widetilde{p}^{2}]]/\mathcal{I}_{2},$
where $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are, respectively, the ideal of
$\mathbb{C}[[\widetilde{x}^{1},\widetilde{x}^{2}]],$ generated by the elements
$[\widetilde{x}^{1},\widetilde{x}^{2}]_{\star}-i\gamma$, and the ideal of
$\mathbb{C}[[\widetilde{p}^{1},\widetilde{p}^{2}]],$ generated by the elements
$[\widetilde{p}^{1},\widetilde{p}^{2}]_{\star}-i\bar{\theta}$. In short,
$\widetilde{\mathcal{M}}_{\theta}\otimes\widetilde{\mathcal{M}}_{\bar{\theta}}$
defines a standard 4 dimensional Moyal space
$[y^{\alpha},y^{\beta}]=\Theta^{\alpha\beta}$, $\alpha,\beta=1,2,3,4$, with
tensor structure
$\Theta:=\left(\begin{array}[]{cc}\gamma J&0\\\ 0&\theta
J\end{array}\right)\qquad J:=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)$ (45)
where $y^{1,2}=\widetilde{x}^{1,2}$ and $y^{3,4}=\widetilde{p}^{1,2}$.
For simplicity, we set $\gamma=1$, $\omega_{2}=2$ and $\bar{\theta}=1$. Using
(42), the Hamiltonian (22) takes the form
$\displaystyle
H=\frac{1}{2}\Big{[}\gamma^{2}(\widetilde{x}^{2})^{2}+\gamma^{2}e^{2\widetilde{x}^{1}}-\frac{2\gamma}{\omega_{2}}e^{\widetilde{x}^{1}}+\frac{1}{\omega_{2}^{2}}+(\widetilde{p}^{1})^{2}+(\widetilde{p}^{2})^{2}\Big{]}=H_{1}(\widetilde{x}^{1},\widetilde{x}^{2})+H_{2}(\widetilde{p}^{1},\widetilde{p}^{2})$
(46)
where
$\displaystyle
H_{1}(\widetilde{x}^{1},\widetilde{x}^{2})=\frac{1}{2}\Big{[}(\widetilde{x}^{2})^{2}+e^{2\widetilde{x}^{1}}-e^{\widetilde{x}^{1}}+\frac{1}{4}\Big{]},\quad
H_{2}(\widetilde{p}^{1},\widetilde{p}^{2})=\frac{1}{2}\Big{[}(\widetilde{p}^{1})^{2}+(\widetilde{p}^{2})^{2}\Big{]}.$
(47)
Now using the commutation relations (43), we get
$\displaystyle[H_{1}(\widetilde{x}^{1},\widetilde{x}^{2}),H_{2}(\widetilde{p}^{1},\widetilde{p}^{2})]_{\star}=0.$
(48)
It appears clear that the star product
$\star=\star_{\theta}\star_{\hbar_{eff}}\star_{\bar{\theta}}$ gets mapped as
$\star\longrightarrow\mathcal{T}(\star)=\star_{1}\star_{2}$ (49)
with
$\displaystyle\star_{1}={\bf
m}\Big{[}\exp\Big{(}\frac{i}{2}(\partial_{\widetilde{x}^{1}}\otimes\partial_{\widetilde{x}^{2}}-\partial_{\widetilde{x}^{2}}\otimes\partial_{\widetilde{x}^{1}})\Big{)}\Big{]},\quad\star_{2}={\bf
m}\Big{[}\exp\Big{(}\frac{i}{2}(\partial_{\widetilde{p}^{1}}\otimes\partial_{\widetilde{p}^{2}}-\partial_{\widetilde{p}^{2}}\otimes\partial_{\widetilde{p}^{1}})\Big{)}\Big{]}.$
(50)
Then, the new coordinate and momentum operators can be described by the
following relations
$\displaystyle\widetilde{x}^{1}\star_{1}=\widetilde{x}^{1}+\frac{i}{2}\partial_{\widetilde{x}^{2}},\quad\widetilde{x}^{2}\star_{1}=\widetilde{x}^{2}-\frac{i}{2}\partial_{\widetilde{x}^{1}},\quad\widetilde{p}^{1}\star_{2}=\widetilde{p}^{1}+\frac{i}{2}\partial_{\widetilde{p}^{2}},\quad\widetilde{p}^{2}\star_{2}=\widetilde{p}^{2}-\frac{i}{2}\partial_{\widetilde{p}^{1}}.$
(51)
The initial Hamiltonian has been factorized into two commuting sectors. We can
first study the spectrum of Hamiltonian
$H_{1}(\widetilde{x}^{1},\widetilde{x}^{2})$ so called supersymmetric
Liouville Hamiltonian. Using (51), the Hamiltonian in this sector takes the
form
$\displaystyle H_{1}(\widetilde{x}^{1},\widetilde{x}^{2})\star_{1}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}(\widetilde{x}^{2})^{2}-i\widetilde{x}^{2}\partial_{\widetilde{x}^{1}}-\frac{1}{4}\partial^{2}_{\widetilde{x}^{1}}+e^{2\widetilde{x}^{1}}(\cos\partial_{\widetilde{x}^{2}}+i\sin\partial_{\widetilde{x}^{2}})$
(52) $\displaystyle-$ $\displaystyle
e^{\widetilde{x}^{1}}(\cos\frac{1}{2}\partial_{\widetilde{x}^{2}}+i\sin\frac{1}{2}\partial_{\widetilde{x}^{2}})+\frac{1}{4}\Big{]}.$
(53)
For a real $E_{1}$ and a wave function $\psi_{1,E_{1}},$ the eigenvalue
problem
$H_{1}(\widetilde{x}^{1},\widetilde{x}^{2})\star_{1}\psi_{1,E_{1}}=E_{1}\psi_{1,E_{1}}$
can be re-expressed into two parts: the real part is given by
$\displaystyle\Big{(}(\widetilde{x}^{2})^{2}-\frac{1}{4}\partial^{2}_{\widetilde{x}^{1}}+e^{2\widetilde{x}^{1}}\cos\partial_{\widetilde{x}^{2}}-e^{\widetilde{x}^{1}}\cos\frac{1}{2}\partial_{\widetilde{x}^{2}}+\frac{1}{4}-2E_{1}\Big{)}\psi_{1,E_{1}}=0$
(54)
whereas the imaginary part expresses as
$\displaystyle\Big{(}\widetilde{x}^{2}\partial_{\widetilde{x}^{1}}-e^{2\widetilde{x}^{1}}\sin\partial_{\widetilde{x}^{2}}+e^{\widetilde{x}^{1}}\sin\frac{1}{2}\partial_{\widetilde{x}^{2}}\Big{)}\psi_{1,E_{1}}=0.$
(55)
To solve consistently the equations (54) and (55), we will use a fact about
the Taylor expansion of an arbitrary function $\psi(x)$, for the small values
of parameter $\epsilon$ as
$\displaystyle\psi(x+\epsilon)=\psi(x)+\epsilon\partial_{x}\psi(x)+\frac{1}{2}\epsilon^{2}\partial^{2}_{x}\psi(x)+\cdots=e^{\epsilon\partial_{x}}\psi(x)$
(56)
$\displaystyle\psi(x-\epsilon)=\psi(x)-\epsilon\partial_{x}\psi(x)+\frac{1}{2}\epsilon^{2}\partial^{2}_{x}\psi(x)+\cdots=e^{-\epsilon\partial_{x}}\psi(x).$
(57)
Then summing (56) and (57), we get
$\displaystyle\frac{1}{2}\Big{(}\psi(x+\epsilon)+\psi(x-\epsilon)\Big{)}=\cosh\epsilon\partial_{x}\,\psi(x),$
(58)
$\displaystyle\frac{1}{2}\Big{(}\psi(x+\epsilon)-\psi(x-\epsilon)\Big{)}=\sinh\epsilon\partial_{x}\,\psi(x).$
(59)
We restrict to the case where $\epsilon=i$ and $\epsilon=\frac{i}{2}$. Then
follow from the identities
$\displaystyle\Big{(}\sin\partial_{\widetilde{x}^{2}}\Big{)}\psi(\widetilde{x}^{1},\widetilde{x}^{2})=\frac{1}{2i}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i)-\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i)\Big{)}$
(60)
$\displaystyle\Big{(}\cos\partial_{\widetilde{x}^{2}}\Big{)}\psi(\widetilde{x}^{1},\widetilde{x}^{2})=\frac{1}{2}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i)+\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i)\Big{)}$
(61)
and
$\displaystyle\Big{(}\sin\frac{1}{2}\partial_{\widetilde{x}^{2}}\Big{)}\psi(\widetilde{x}^{1},\widetilde{x}^{2})=\frac{1}{2i}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i/2)-\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i/2)\Big{)}$
(62)
$\displaystyle\Big{(}\cos\frac{1}{2}\partial_{\widetilde{x}^{2}}\Big{)}\psi(\widetilde{x}^{1},\widetilde{x}^{2})=\frac{1}{2}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i/2)+\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i/2)\Big{)}.$
(63)
The equation (55) can be simply written as
$\displaystyle\Big{(}\widetilde{x}^{2}\partial_{\widetilde{x}^{1}}\Big{)}\psi_{1,E}(\widetilde{x}^{1},\widetilde{x}^{2})$
$\displaystyle=$
$\displaystyle\frac{e^{2\widetilde{x}^{1}}}{2i}\Big{(}\psi_{1,E}(\widetilde{x}^{1},\widetilde{x}^{2}+i)-\psi_{1,E}(\widetilde{x}^{1},\widetilde{x}^{2}-i)\Big{)}$
(64) $\displaystyle-$
$\displaystyle\frac{e^{\widetilde{x}^{1}}}{2i}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i/2)-\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i/2)\Big{)}.$
(65)
Using the above relations, we then get the new equation corresponding to (54)
as
$\displaystyle\Big{(}(\widetilde{x}^{2})^{2}+\frac{1}{4}-2E_{1}\Big{)}\psi_{1,E_{1}}(\widetilde{x}^{1},\widetilde{x}^{2})-\frac{1}{4}\Big{[}\frac{e^{2\widetilde{x}^{1}}}{i\widetilde{x}^{2}}\Big{(}\psi_{1,E}(\widetilde{x}^{1},\widetilde{x}^{2}+i)-\psi_{1,E}(\widetilde{x}^{1},\widetilde{x}^{2}-i)\Big{)}$
(66)
$\displaystyle-\frac{e^{\widetilde{x}^{1}}}{2i\widetilde{x}^{2}}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i/2)-\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i/2)\Big{)}-\frac{e^{4\widetilde{x}^{1}}}{4(\widetilde{x}^{2})^{2}}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+2i)-\psi(\widetilde{x}^{1},\widetilde{x}^{2})\Big{)}$
(67)
$\displaystyle+\frac{e^{2\widetilde{x}^{1}}}{4(\widetilde{x}^{2})^{2}}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i)-\psi(\widetilde{x}^{1},\widetilde{x}^{2})+2\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i)\Big{)}\Big{]}$
(68)
$\displaystyle-\frac{e^{2\widetilde{x}^{1}}}{2}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i)+\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i)\Big{)}+\frac{e^{\widetilde{x}^{1}}}{2}\Big{(}\psi(\widetilde{x}^{1},\widetilde{x}^{2}+i/2)+\psi(\widetilde{x}^{1},\widetilde{x}^{2}-i/2)\Big{)}\Big{]}=0.$
In [14], the recursive properties of the Meijer G-function can be used to
compute the eigenvectors $\psi_{1,E_{1}}(\widetilde{x}^{1})$ as
$\displaystyle\psi_{1,E_{1}}(\widetilde{x}^{1})=\Big{(}\frac{1}{4\pi^{2}\sqrt{E_{1}}}e^{\widetilde{x}^{1}}\cosh(\pi\sqrt{E_{1}})\Big{)}^{1/2}\Big{(}K_{\frac{1}{2}-i\sqrt{E_{1}}}(e^{\widetilde{x}^{1}})+K_{\frac{1}{2}+i\sqrt{E_{1}}}(e^{\widetilde{x}^{1}})\Big{)},\,\,E_{1}\geq
0,$ (70)
where $K$ are Kelvin (modified Bessel) functions.
The second eigenvalue problem
$H_{2}(\widetilde{p}^{1},\widetilde{p}^{2})\star_{2}\psi_{2,E_{2}}=E_{2}\psi_{2,E_{2}}$
is well known as the simple quantum harmonic oscillator problem. We write
$\displaystyle\psi_{2,E_{2}}(\widetilde{p}^{1})$ $\displaystyle=$
$\displaystyle\Big{(}\frac{1}{\pi}\Big{)}^{1/4}\frac{1}{2^{n}n!}H_{n}(\widetilde{p}^{1})e^{-(\widetilde{p}^{1})^{2}/2}$
(71)
where $H_{n}$ stand for the Hermite polynomial and the oscillator energy as
$E_{2}=n+\frac{1}{2}$. Finally the solution of Hamiltonian (22) is then
$\displaystyle\psi_{E}(\widetilde{x}^{1},\widetilde{p}^{1})=\psi_{1,E_{1}}(\widetilde{x}^{1})\otimes\psi_{2,E_{2}}(\widetilde{p}^{1}),\quad
E=E_{1}+E_{2}.$ (72)
We conclude that the spectrum of the Hamiltonian $H$ is composed by two
sectors: A continuum part in the sector $H_{1}$ and a discrete one in the
sector $H_{2}$.
## 3 Conclusion
In this work, following our previous approach [7] and results based on [14],
we have found the eigenvalues and eigenvectors of the harmonic oscillator in
the twisted Moyal space with function structure $e(x)=1+\omega_{2}x^{2}$. We
have introduced a particular transformation which has allowed us to split the
total twisted Moyal algebra into two parts in which the Hamiltonian was
written in two commuting pieces.
Let us remark that the solution (72) exhibits the lack of commutative limit
$\theta\rightarrow 0$. This inconvenience is due to the form of the scale
transformation $\mathcal{T}$. Therefore the solution obtained in relation (70)
need to be renormalized. Note also that it is a more difficult problem to find
a solution for the harmonic oscillator in the more symmetric case of
$e(x)=1+\omega_{\mu}x^{\mu}.$ This two tangles deserves to be investigated.
## Acknowledgements
The author would like to thank Joseph Ben Geloun for useful comments which
have improved this work. This research was supported in part by Perimeter
Institute for Theoretical Physics. Research at Perimeter Institute is
supported by the Government of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Research and Innovation.
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|
arxiv-papers
| 2013-07-29T16:12:58 |
2024-09-04T02:49:48.683288
|
{
"license": "Public Domain",
"authors": "Dine Ousmane Samary",
"submitter": "Dine Ousmane Samary",
"url": "https://arxiv.org/abs/1307.7628"
}
|
1307.7648
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-139 LHCb-PAPER-2013-042 July 29, 2013
Study of $B_{\scriptscriptstyle(s)}^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}h^{+}h^{\prime-}$ decays with first observation of
$B_{\scriptscriptstyle s}^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}K^{\pm}\pi^{\mp}$ and $B_{\scriptscriptstyle s}^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}\pi^{+}\pi^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
A search for charmless three-body decays of $B^{0}$ and $B_{\scriptscriptstyle
s}^{0}$ mesons with a $K_{\rm\scriptscriptstyle S}^{0}$ meson in the final
state is performed using the $pp$ collision data, corresponding to an
integrated luminosity of $1.0\mbox{\,fb}^{-1}$, collected at a centre-of-mass
energy of $7\mathrm{\,Te\kern-1.00006ptV}$ recorded by the LHCb experiment.
Branching fractions of the $B_{\scriptscriptstyle(s)}^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}h^{+}h^{\prime-}$ decay modes
($h^{(\prime)}=\pi,K$), relative to the well measured $B^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}\pi^{+}\pi^{-}$ decay, are obtained. First
observation of the decay modes $B_{s}^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}K^{\pm}\pi^{\mp}$ and $B_{s}^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}\pi^{+}\pi^{-}$ and confirmation of the decay $B^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}K^{\pm}\pi^{\mp}$ are reported. The following
relative branching fraction measurements or limits are obtained
$\displaystyle\frac{{\cal B}(B^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}K^{\pm}\pi^{\mp})}{{\cal B}(B^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}\pi^{+}\pi^{-})}$ $\displaystyle=$ $\displaystyle 0.128\pm 0.017\,({\rm
stat.})\pm 0.009\,({\rm syst.})\,,$ $\displaystyle\frac{{\cal
B}(B^{0}\rightarrow K_{\rm\scriptscriptstyle S}^{0}K^{+}K^{-})}{{\cal
B}(B^{0}\rightarrow K_{\rm\scriptscriptstyle S}^{0}\pi^{+}\pi^{-})}$
$\displaystyle=$ $\displaystyle 0.385\pm 0.031\,({\rm stat.})\pm 0.023\,({\rm
syst.})\,,$ $\displaystyle\frac{{\cal B}(B_{s}^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}\pi^{+}\pi^{-})}{{\cal B}(B^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}\pi^{+}\pi^{-})}$ $\displaystyle=$
$\displaystyle 0.29\phantom{0}\pm 0.06\phantom{0}\,({\rm stat.})\pm
0.03\phantom{0}\,({\rm syst.})\pm 0.02\,(f_{s}/f_{d})\,,$
$\displaystyle\frac{{\cal B}(B_{s}^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}K^{\pm}\pi^{\mp})}{{\cal B}(B^{0}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}\pi^{+}\pi^{-})}$ $\displaystyle=$ $\displaystyle 1.48\phantom{0}\pm
0.12\phantom{0}\,({\rm stat.})\pm 0.08\phantom{0}\,({\rm syst.})\pm
0.12\,(f_{s}/f_{d})\,,$ $\displaystyle\frac{{\cal B}(B_{s}^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}K^{+}K^{-})}{{\cal B}(B^{0}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}\pi^{+}\pi^{-})}$ $\displaystyle\in$
$\displaystyle[0.004;0.068]\;{\rm at\;\;90\%\;CL}\,.$
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
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Cowie45, D.C. Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8,
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Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D.
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Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
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Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra
Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53,
T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H.
Gordon37, C. Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado
Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S.
Gregson46, P. Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu.
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Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
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P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D.
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Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44,
R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J.
Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O.
Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U.
Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
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Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T. Palczewski27, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B.
Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8,
G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N.
Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E.
Santovetti23,k, M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M.
Savrie16,e, D. Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M.
Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A.
Semennikov30, K. Senderowska26, I. Sepp52, N. Serra39, J. Serrano6, P.
Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T.
Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires9, R.
Silva Coutinho47, M. Sirendi46, N. Skidmore45, T. Skwarnicki58, N.A. Smith51,
E. Smith54,48, J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro38, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stevenson54, S.
Stoica28, S. Stone58, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K.
Subbiah37, L. Sun56, S. Swientek9, V. Syropoulos41, M. Szczekowski27, P.
Szczypka38,37, T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F.
Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin38, S. Tolk41, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E.
Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6,
P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, D. Urner53, A.
Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, M. Van
Dijk45, R. Vazquez Gomez18, P. Vazquez Regueiro36, C. Vázquez Sierra36, S.
Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B.
Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10,
D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C.
Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J.
Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wimberley57, J. Wishahi9, W.
Wislicki27, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The study of the charmless three-body decays of neutral $B$ mesons to final
states including a $K^{0}_{\rm\scriptscriptstyle S}$ meson, namely
$B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$,
$B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$
and $B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$, has
a number of theoretical applications.111Unless stated otherwise, charge
conjugated modes are implicitly included throughout the paper. The decays
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ are dominated
by $b\\!\rightarrow q\overline{}qs$ ($q=u,d,s$) loop transitions. Mixing-
induced $C\\!P$ asymmetries in such decays are predicted to be approximately
equal to those in $b\\!\rightarrow c\overline{}cs$ transitions, e.g.
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$, by the Cabibbo-Kobayashi-Maskawa
mechanism [1, 2]. However, the loop diagrams that dominate the charmless
decays can have contributions from new particles in several extensions of the
Standard Model, which could introduce additional weak phases [3, 4, 5, 6]. A
time-dependent analysis of the three-body Dalitz plot allows measurements of
the mixing-induced $C\\!P$-violating phase [7, 8, 9, 10]. The current
experimental measurements of $b\\!\rightarrow q\overline{}qs$ decays [11] show
fair agreement with the results from $b\\!\rightarrow c\overline{}cs$ decays
(measuring the weak phase $\beta$) for each of the scrutinised $C\\!P$
eigenstates. There is, however, a global trend towards lower values than the
weak phase measured from $b\\!\rightarrow c\overline{}cs$ decays. The
interpretation of this deviation is made complicated by QCD corrections, which
depend on the final state [12] and are difficult to handle. An analogous
extraction of the mixing-induced $C\\!P$-violating phase in the $B^{0}_{s}$
system will, with a sufficiently large dataset, also be possible with the
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$
decay, which can be compared with that from, e.g.
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$.
Much recent theoretical and experimental activity has focused on the
determination of the CKM angle $\gamma$ from $B\rightarrow K\pi\pi$ decays,
using and refining the methods proposed in Refs. [13, 14]. The recent
experimental results from BaBar [15] demonstrate the feasibility of the
method, albeit with large statistical uncertainties. The decay
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ is of
particular interest for this effort. Indeed, the ratio of the amplitudes of
the isospin-related mode $B^{0}_{s}\\!\rightarrow K^{-}\pi^{+}\pi^{0}$ and its
charge conjugate exhibits a direct dependence on the mixing-induced
$C\\!P$-violating phase, which would be interpreted in the Standard Model as
$(\beta_{s}+\gamma)$. Unlike the equivalent $B^{0}$ decays, the $B^{0}_{s}$
decays are dominated by tree amplitudes and the contributions from electroweak
penguin diagrams are expected to be negligible, yielding a theoretically clean
extraction of $\gamma$ [16] provided that the strong phase can be determined
from other measurements. The shared intermediate states between
$B^{0}_{s}\\!\rightarrow K^{-}\pi^{+}\pi^{0}$ and $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ (specifically $K^{*-}\pi^{+}$)
offer that possibility, requiring an analysis of the $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ Dalitz plot.
At LHCb, the first step towards this physics programme is to establish the
signals of all the decay modes. In particular, the decay modes
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{\prime-}$
($h^{(\prime)}=\pi,K$) are all unobserved and the observation of
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ by BaBar
[17] is so far unconfirmed. In this paper the results of an analysis of all
six $B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{\prime-}$ decay modes are presented. The branching fractions of the
decay modes relative to that of $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ are measured when the
significance of the signals allow it, otherwise confidence intervals are
quoted. Time-integrated branching fractions are computed, implying a non-
trivial comparison of the $B^{0}$ and $B^{0}_{s}$ decays at amplitude level
[18].
## 2 Detector and dataset
The measurements described in this paper are performed with data,
corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, from
$7\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass $pp$ collisions, collected
with the LHCb detector during 2011. Samples of simulated events are used to
estimate the efficiency of the selection requirements, to investigate possible
sources of background contributions, and to model the event distributions in
the likelihood fit. In the simulation, $pp$ collisions are generated using
Pythia 6.4 [19] with a specific LHCb configuration [20]. Decays of hadronic
particles are described by EvtGen [21], in which final state radiation is
generated using Photos [22]. The interaction of the generated particles with
the detector and its response are implemented using the Geant4 toolkit [23,
*Agostinelli:2002hh] as described in Ref. [25].
The LHCb detector [26] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream.
The combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov (RICH)
detectors [27]. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers.
## 3 Trigger and event selection
The trigger [28] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. To remove events with large occupancies, a
requirement is made at the hardware stage on the number of hits in the
scintillating-pad detector. The hadron trigger at the hardware stage also
requires that there is at least one candidate with transverse energy
$\mbox{$E_{\rm T}$}>3.5\mathrm{\,Ge\kern-1.00006ptV}$. In the offline
selection, candidates are separated into two categories based on the hardware
trigger decision. The first category are triggered by particles from candidate
signal decays that have an associated cluster in the calorimeters above the
threshold, while the second category are triggered independently of the
particles associated with the signal decay. Events that do not fall into
either of these categories are not used in the subsequent analysis.
The software trigger requires a two-, three- or four-track secondary vertex
with a high sum of the transverse momentum, $p_{\rm T}$, of the tracks and
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any primary interaction greater than 16, where $\chi^{2}_{\rm IP}$
is defined as the difference in $\chi^{2}$ of a given PV reconstructed with
and without the considered track. A multivariate algorithm [29] is used for
the identification of secondary vertices consistent with the decay of a $b$
hadron.
The events passing the trigger requirements are then filtered in two stages.
Initial requirements are applied to further reduce the size of the data
sample, before a multivariate selection is implemented. In order to minimise
the variation of the selection efficiency over the Dalitz plot it is necessary
to place only loose requirements on the momenta of the daughter particles. As
a consequence, selection requirements on topological variables such as the
flight distance of the $B$ candidate or the direction of its momentum vector
are used as the main discriminants.
The $K^{0}_{\rm\scriptscriptstyle S}$ candidates are reconstructed in the
$\pi^{+}\pi^{-}$ final state. Approximately two thirds of the reconstructed
$K^{0}_{\rm\scriptscriptstyle S}$ mesons decay downstream of the VELO. Since
those $K^{0}_{\rm\scriptscriptstyle S}$ candidates decaying within the VELO,
and those that have information only from the tracking stations, differ in
their reconstruction and selection, they are separated into two categories
labelled “Long” and “Downstream”, respectively. The pions that form the
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are required to have momentum
$\mbox{$p$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$
with respect to any PV greater than 9 (4) for Long (Downstream)
$K^{0}_{\rm\scriptscriptstyle S}$ candidates. The
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are then required to form a
vertex with $\chi^{2}_{\rm vtx}<12$ and to have invariant mass within
20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
(30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) of the nominal
$K^{0}_{\rm\scriptscriptstyle S}$ mass [30] for Long (Downstream) candidates.
The square of the separation of the $K^{0}_{\rm\scriptscriptstyle S}$ vertex
from the PV divided by the associated uncertainty ($\chi^{2}_{\rm VS}$) must
be greater than $80$ ($50$) for Long (Downstream) candidates. Downstream
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are required, in addition, to
have momentum $\mbox{$p$}>6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
The $B$ candidates are formed by combining the $K^{0}_{\rm\scriptscriptstyle
S}$ candidates with two oppositely charged tracks. Selection requirements,
common to both the Long and Downstream categories, are based on the topology
and kinematics of the $B$ candidate. The charged $B$-meson daughters are
required to have $\mbox{$p$}<100{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, a
momentum beyond which there is little pion/kaon discrimination. The scalar sum
of the three daughters’ transverse momenta must be greater than
3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and at least two of the daughters
must have $\mbox{$p_{\rm T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
impact parameter (IP) of the $B$-meson daughter with the largest $p_{\rm T}$
is required to be greater than 0.05$\rm\,mm$ relative to the PV associated to
the $B$ candidate. The $\chi^{2}$ of the distance of closest approach of any
two daughters must be less than 5. The $B$ candidates are then required to
form a vertex separated from any PV by at least 1$\rm\,mm$ and that has
$\chi^{2}_{\rm vtx}<12$ and $\chi^{2}_{\rm VS}>50$. The difference in
$\chi^{2}_{\rm vtx}$ when adding any track must be greater than 4\. The
candidates must have $\mbox{$p_{\rm
T}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and invariant mass within the
range $4779<m_{K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{\prime-}}<5866{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The cosine
of the angle between the reconstructed momentum of the $B$ meson and its
direction of flight (pointing angle) is required to be greater than 0.9999.
The candidates are further required to have a minimum $\chi^{2}_{\rm IP}$ with
respect to all PVs less than 4. Finally, the separation of the
$K^{0}_{\rm\scriptscriptstyle S}$ and $B$ vertices in the positive $z$
direction222The $z$ axis points along the beam line from the interaction
region through the LHCb detector. must be greater than 30$\rm\,mm$.
Multivariate discriminants based on a boosted decision tree (BDT) [31] with
the AdaBoost algorithm [32] have been designed in order to complete the
selection of the signal events and to further reject combinatorial
backgrounds. Simulated $B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ events and upper mass sidebands,
$5420<m_{K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}}<5866{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, in the data
are used as the signal and background training samples, respectively. The
samples of events in each of the Long and Downstream
$K^{0}_{\rm\scriptscriptstyle S}$ categories are further subdivided into two
equally-sized subsamples. Each subsample is then used to train an independent
discriminant. In the subsequent analysis the BDT trained on one subsample of a
given $K^{0}_{\rm\scriptscriptstyle S}$ category is used to select events from
the other subsample, in order to avoid bias. The input variables for the BDTs
are the $p_{\rm T}$, $\eta$, $\chi^{2}_{\rm IP}$, $\chi^{2}_{\rm VS}$,
pointing angle and $\chi^{2}_{\rm vtx}$ of the $B$ candidate; the sum
$\chi^{2}_{\rm IP}$ of the $h^{+}$ and $h^{-}$; the $\chi^{2}_{\rm IP}$,
$\chi^{2}_{\rm VS}$ and $\chi^{2}_{\rm vtx}$ of the
$K^{0}_{\rm\scriptscriptstyle S}$ candidate.
The selection requirement placed on the output of the BDTs is independently
optimised for events containing $K^{0}_{\rm\scriptscriptstyle S}$ candidates
reconstructed in either Downstream or Long categories. Two different figures
of merit are used to optimise the selection requirements, depending on whether
the decay mode in question is favoured or suppressed. If favoured, the
following is used
${\cal Q}_{1}=\frac{{\rm S}}{\sqrt{{\rm S}+{\rm B}}}\,,$ (1)
where $\rm S$ ($\rm B$) represents the number of expected signal
(combinatorial background) events for a given selection. The value of $\rm S$
is estimated based on the known branching fractions and efficiencies, while
$\rm B$ is calculated by fitting the sideband above the signal region and
extrapolating into the signal region. If the mode is suppressed, an
alternative figure of merit [33] is used
${\cal Q}_{2}=\frac{\varepsilon_{\rm sig}}{\frac{a}{2}+\sqrt{\rm B}}\,,$ (2)
where the signal efficiency ($\varepsilon_{\rm sig}$) is estimated from the
signal simulation. The value $a=5$ is used in this analysis, which corresponds
to optimising for $5\sigma$ significance to find the decay. This second figure
of merit results in a more stringent requirement than the first. Hence, the
requirements optimised with each figure of merit will from here on be referred
to as the loose and tight BDT requirements, respectively.
The fraction of selected events containing more than one candidate is at the
percent level. The candidate to be retained in each event is chosen
arbitrarily.
A number of background contributions consisting of fully reconstructed $B$
meson decays into two-body $Dh$ or $c\overline{}cK^{0}_{\rm\scriptscriptstyle
S}$ combinations, result in a $K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{\prime-}$ final state and hence are, in terms of their $B$ candidate
invariant mass distribution, indistinguishable from signal candidates. The
decays of $\mathchar 28931\relax^{0}_{b}$ baryons to $\mathchar
28931\relax^{+}_{c}h$ with $\mathchar 28931\relax^{+}_{c}\\!\rightarrow
pK^{0}_{\rm\scriptscriptstyle S}$ also peak under the signal when the proton
is misidentified. Therefore, the following $D$, $\mathchar
28931\relax^{+}_{c}$ and charmonia decays are explicitly reconstructed under
the relevant particle hypotheses and vetoed in all the spectra:
$D^{0}\rightarrow K^{-}\pi^{+}$, $D^{0}\rightarrow\pi^{+}\pi^{-}$,
$D^{0}\rightarrow K^{+}K^{-}$, $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}$, $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$,
$D^{+}_{s}\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$,
$D^{+}_{s}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$, and $\mathchar
28931\relax^{+}_{c}\rightarrow pK^{0}_{\rm\scriptscriptstyle S}$. Additional
vetoes on charmonium resonances, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\pi^{+}\pi^{-},\mu^{+}\mu^{-},K^{+}\kern-1.60004ptK^{-}$ and
$\chi_{c0}\rightarrow\pi^{+}\pi^{-},\mu^{+}\mu^{-},K^{+}\kern-1.60004ptK^{-}$,
are applied to remove the handful of fully reconstructed and well identified
peaking $B^{0}_{(s)}\\!\rightarrow\left({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu},\chi_{c0}\right)K^{0}_{\rm\scriptscriptstyle S}$ decays. The veto for
each reconstructed charm (charmonium) state $R$,
$\left|m-m_{R}\right|<30\;(48){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, is
defined around the world average mass value $m_{R}$ [30] and the range is
chosen according to the typical mass resolution obtained at LHCb.
Particle identification (PID) requirements are applied in addition to the
selection described so far. The charged pion tracks from the
$K^{0}_{\rm\scriptscriptstyle S}$ decay and the charged tracks from the $B$
decay are all required to be inconsistent with the muon track hypothesis. The
logarithm of the likelihood ratio between the kaon and pion hypotheses
($\mathrm{DLL}_{K\pi}$), mostly based on information from the RICH detectors
[27], is used to discriminate between pion and kaon candidates from the $B$
decay. Pion (kaon) candidates are required to satisfy $\mathrm{DLL}_{K\pi}<0$
($\mathrm{DLL}_{K\pi}>5$). These are also required to be inconsistent with the
proton hypothesis, in order to remove the possible contributions from
charmless $b$-baryon decays. Pion (kaon) candidates are required to satisfy
$\mathrm{DLL}_{p\pi}<10$ ($\mathrm{DLL}_{pK}<10$).
## 4 Fit model
A simultaneous unbinned extended maximum likelihood fit to the $B$-candidate
invariant mass distributions of all decay channels is performed for each of
the two BDT optimisations. In each simultaneous fit four types of components
contribute, namely signal decays, cross-feed backgrounds, partially-
reconstructed backgrounds, and combinatorial background.
Contributions from $B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{\prime-}$ decays with correct identification of the final state
particles are modelled with sums of two Crystal Ball (CB) functions [34] that
share common values for the peak position and width but have independent power
law tails on opposite sides of the peak. The $B^{0}$ and $B^{0}_{s}$ masses
(peak positions of the double-CB functions) are free in the fit. Four
parameters related to the widths of the double-CB function are also free
parameters of the fit: the common width of the $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ signals; the relative widths of
$K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ and
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ to $K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$, which are the same for $B^{0}$ and $B^{0}_{s}$ decay modes;
the ratio of Long over Downstream widths, which is the same for all decay
modes. These assumptions are made necessary by the otherwise poor
determination of the width of the suppressed mode of each spectrum. The other
parameters of the CB components are obtained by a simultaneous fit to
simulated samples, constraining the fraction of events in the two CB
components and the ratio of their tail parameters to be the same for all
double-CB contributions.
Each selected candidate belongs uniquely to one reconstructed final state, by
definition of the particle identification criteria. However, misidentified
decays yield some cross-feed in the samples and are modelled empirically by
single CB functions using simulated events. Only contributions from the decays
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ reconstructed
and selected as $K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$, or the
decays $B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}$ and $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}$ reconstructed and selected as either
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ or $K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ are considered. Other potential contributions are neglected.
The relative yield of each misidentified decay is constrained with respect to
the yield of the corresponding correctly identified decay. The constraints are
implemented using Gaussian priors included in the likelihood. The mean values
are obtained from the ratio of selection efficiencies and the resolutions
include uncertainties originating from the finite size of the simulated events
samples and the systematic uncertainties related to the determination of the
PID efficiencies.
Partially reconstructed charmed transitions such as $B^{-}\rightarrow
D^{0}\pi^{-}(K^{-})$ followed by $D^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, with a pion not reconstructed,
are expected to dominate the background contribution in the lower invariant
mass region. Charmless backgrounds such as from
$B^{0}\\!\rightarrow\eta^{\prime}(\rightarrow\rho^{0}\gamma)K^{0}_{\rm\scriptscriptstyle
S}$, $B^{0}_{s}\\!\rightarrow K^{*0}(\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{0})\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}(\rightarrow
K^{-}\pi^{+})$ and $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\pi^{+}$ decays are also expected to contribute with lower
rates. These decays are modelled by means of generalised ARGUS functions [35]
convolved with a Gaussian resolution function. Their parameters are determined
from simulated samples. In order to reduce the number of components in the
fit, only generic contributions for hadronic charmed and charmless decays are
considered in each final state, however $B^{0}$ and $B^{0}_{s}$ contributions
are explicitly included. Radiative decays and those from
$B^{0}\\!\rightarrow\eta^{\prime}(\rightarrow\rho^{0}\gamma)K^{0}_{\rm\scriptscriptstyle
S}$ are considered separately and included only in the
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ final state. The normalisation
of all such contributions is constrained with Gaussian priors using the ratio
of efficiencies from the simulation and the ratio of branching fractions from
world averages [30]. Relative uncertainties on these ratios of 100%, 20% and
10% are considered for charmless, charmed, and radiative and
$B^{0}\\!\rightarrow\eta^{\prime}(\rightarrow\rho^{0}\gamma)K^{0}_{\rm\scriptscriptstyle
S}$ decays, respectively.
The combinatorial background is modelled by an exponential function, where the
slope parameter is fitted for each of the two $K^{0}_{\rm\scriptscriptstyle
S}$ reconstruction categories. The combinatorial backgrounds to the three
final states $B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$, $B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}$ and $B^{0}_{(s)}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ are assumed to have identical
slopes. This assumption as well as the choice of the exponential model are
sources of systematic uncertainties.
The fit results for the two BDT optimisations are displayed in Figs. 1 and 2.
Table 1 summarises the fitted yields of each decay mode for the optimisation
used to determine the branching fractions. In the tight BDT optimisation the
combinatorial background is negligible in the high invariant-mass region for
the $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ final states, leading to a small
systematic uncertainty related to the assumptions used to fit this component.
An unambiguous first observation of $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ decays and a clear
confirmation of the BaBar observation [17] of $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ decays are obtained.
Significant yields for the $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays are observed above
negligible background with the tight optimisation of the selection. The
likelihood profiles are shown in Fig. 3 for Downstream and Long
$K^{0}_{\rm\scriptscriptstyle S}$ samples separately. The
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays
are observed with a combined statistical significance of $6.2\,\sigma$, which
becomes $5.9\,\sigma$ including fit model systematic uncertainties. The
statistical significance of the $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ signal is at the level of
$2.1\,\sigma$ combining Downstream and Long $K^{0}_{\rm\scriptscriptstyle S}$
reconstruction categories.
Table 1: Yields obtained from the simultaneous fit corresponding to the chosen optimisation of the selection for each mode, where the uncertainties are statistical only. The average selection efficiencies are also given for each decay mode, where the uncertainties are due to the limited simulation sample size. | | Downstream | Long
---|---|---|---
Mode | BDT | Yield | Efficiency (%) | Yield | Efficiency (%)
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ | Loose | $845$ $\,\pm\,$ | $38$ | $0.0336$ $\,\pm\,$ | $0.0010$ | $360$ $\,\pm\,$ | $21$ | $0.0117$ $\,\pm\,$ | $0.0009$
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ | Loose | $256$ $\,\pm\,$ | $20$ | $0.0278$ $\,\pm\,$ | $0.0008$ | $175$ $\,\pm\,$ | $15$ | $0.0092$ $\,\pm\,$ | $0.0016$
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ | Loose | $283$ $\,\pm\,$ | $24$ | $0.0316$ $\,\pm\,$ | $0.0007$ | $152$ $\,\pm\,$ | $15$ | $0.0103$ $\,\pm\,$ | $0.0008$
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ | Tight | $92$ $\,\pm\,$ | $15$ | $0.0283$ $\,\pm\,$ | $0.0009$ | $52$ $\,\pm\,$ | $11$ | $0.0133$ $\,\pm\,$ | $0.0005$
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ | Tight | $28$ $\,\pm\,$ | $9$ | $0.0153$ $\,\pm\,$ | $0.0013$ | $25$ $\,\pm\,$ | $6$ | $0.0109$ $\,\pm\,$ | $0.0006$
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ | Tight | $6$ $\,\pm\,$ | $4$ | $0.0150$ $\,\pm\,$ | $0.0021$ | $3$ $\,\pm\,$ | $3$ | $0.0076$ $\,\pm\,$ | $0.0016$
Figure 1: Invariant mass distributions of (top) $K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$, (middle) $K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$, and
(bottom) $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ candidate events,
with the loose selection for (left) Downstream and (right) Long
$K^{0}_{\rm\scriptscriptstyle S}$ reconstruction categories. In each plot,
data are the black points with error bars and the total fit model is overlaid
(solid black line). The $B^{0}$ ($B^{0}_{s}$) signal components are the black
short-dashed (dotted) lines, while fully reconstructed misidentified decays
are the black dashed lines close to the $B^{0}$ and $B^{0}_{s}$ peaks. The
partially reconstructed contributions from $B$ to open charm decays, charmless
hadronic decays,
$B^{0}\\!\rightarrow\eta^{\prime}(\rightarrow\rho^{0}\gamma)K^{0}_{\rm\scriptscriptstyle
S}$ and charmless radiative decays are the red dash triple-dotted, the blue
dash double-dotted, the violet dash single-dotted, and the pink short-dash
single-dotted lines, respectively. The combinatorial background contribution
is the green long-dash dotted line.
Figure 2: Invariant mass distributions of (top) $K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}$, (middle) $K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$, and
(bottom) $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ candidate events,
with the tight selection for (left) Downstream and (right) Long
$K^{0}_{\rm\scriptscriptstyle S}$ reconstruction categories. In each plot,
data are the black points with error bars and the total fit model is overlaid
(solid black line). The $B^{0}$ ($B^{0}_{s}$) signal components are the black
short-dashed (dotted) lines, while fully reconstructed misidentified decays
are the black dashed lines close to the $B^{0}$ and $B^{0}_{s}$ peaks. The
partially reconstructed contributions from $B$ to open charm decays, charmless
hadronic decays,
$B^{0}\\!\rightarrow\eta^{\prime}(\rightarrow\rho^{0}\gamma)K^{0}_{\rm\scriptscriptstyle
S}$ and charmless radiative decays are the red dash triple-dotted, the blue
dash double-dotted, the violet dash single-dotted, and the pink short-dash
single-dotted lines, respectively. The combinatorial background contribution
is the green long-dash dotted line.
Figure 3: Likelihood profiles of the $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ signal yield for the (left)
Downstream and (right) Long $K^{0}_{\rm\scriptscriptstyle S}$ samples. The
dashed red line is the statistical-only profile, while the solid blue line
also includes the fit model systematic uncertainties. The significance of the
Downstream and Long signals are $3.4\,\sigma$ and $4.8\,\sigma$, respectively,
including systematic uncertainties. Combining Downstream and Long
$K^{0}_{\rm\scriptscriptstyle S}$ samples, an observation with $5.9\,\sigma$,
including systematic uncertainties, is obtained.
## 5 Determination of the efficiencies
The measurements of the branching fractions of the $B^{0}_{(s)}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{\prime-}$ decays relative to the well
established $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ decay mode proceed according to
$\displaystyle\frac{{\cal B}(B^{0}_{(s)}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{\prime-})}{{\cal B}(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-})}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{\rm sel}_{B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}}}{\varepsilon^{\rm
sel}_{B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{\prime-}}}\frac{N_{B^{0}_{(s)}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{\prime-}}}{N_{B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}}}\frac{f_{d}}{f_{d,s}}\,,$ (3)
where $\varepsilon^{\rm sel}$ is the selection efficiency (which includes
acceptance, reconstruction, selection, trigger and particle identification
components), $N$ is the fitted signal yield, and $f_{d}$ and $f_{s}$ are the
hadronisation fractions of a $b$ quark into a $B^{0}$ and $B^{0}_{s}$ meson,
respectively. The ratio $f_{s}/f_{d}$ has been accurately determined by the
LHCb experiment from hadronic and semileptonic measurements
$f_{s}/f_{d}=0.256\pm 0.020$ [36].
Three-body decays are composed of several quasi-two-body decays and non-
resonant contributions, all of them possibly interfering. Hence, their
dynamical structure, described by the Dalitz plot [37], must be accounted for
to correct for non-flat efficiencies over the phase space. Since the dynamics
of most of the modes under study are not known prior to this analysis,
efficiencies are determined for each decay mode from simulated signal samples
in bins of the “square Dalitz plot” [38], where the usual Dalitz-plot
coordinates have been transformed into a rectangular space. The edges of the
usual Dalitz plot are spread out in the square Dalitz plot, which permits a
more precise modelling of the efficiency variations in the regions where they
are most strongly varying and where most of the signal events are expected.
Two complementary simulated samples have been produced, corresponding to
events generated uniformly in phase space or uniformly in the square Dalitz
plot. The square Dalitz-plot distribution of each signal mode is determined
from the data using the sPlot technique [39]. The binning is chosen such that
each bin is populated by approximately the same number of signal events. The
average efficiency for each decay mode is calculated as the weighted harmonic
mean over the bins. The average weighted selection efficiencies are summarised
in Table 1 and depend on the final state, the $K^{0}_{\rm\scriptscriptstyle
S}$ reconstruction category, and the choice of the BDT optimisation. Their
relative uncertainties due to the finite size of the simulated event samples
vary from 3% to 17%, reflecting the different dynamical structures of the
decay modes.
The particle identification and misidentification efficiencies are determined
from simulated signal events on an event-by-event basis by adjusting the DLL
distributions measured from calibration events to match the kinematical
properties of the tracks in the decay of interest. The reweighting is
performed in bins of $p$ and $p_{\rm T}$, accounting for kinematic
correlations between the tracks. Calibration tracks are taken from
$D^{*+}\rightarrow D^{0}\pi^{+}_{s}$ decays where the $D^{0}$ decays to the
Cabibbo-favoured $K^{-}\pi^{+}$ final state. The charge of the soft pion
$\pi^{+}_{s}$ hence provides the kaon or pion identity of the tracks. The
dependence of the PID efficiency over the Dalitz plot is included in the
procedure described above. This calibration is performed using samples from
the same data taking period, accounting for the variation in the performance
of the RICH detectors over time.
## 6 Systematic uncertainties
Most of the systematic uncertainties are eliminated or greatly reduced by
normalising the branching fraction measurements with respect to the
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ mode. The
remaining sources of systematic effects and the methods used to estimate the
corresponding uncertainties are described in this section. In addition to the
systematic effects related to the measurements performed in this analysis,
there is that associated with the measured value of $f_{s}/f_{d}$. A summary
of the contributions, expressed as relative uncertainties, is given in Table
2.
### 6.1 Fit model
The fit model relies on a number of assumptions, both in the values of
parameters being taken from simulation and in the choice of the functional
forms describing the various contributions.
The uncertainties linked to the parameters fixed to values determined from
simulated events are obtained by repeating the fit while the fixed parameters
are varied according to their uncertainties using pseudo-experiments. For
example, the five fixed parameters of the CB functions describing the signals,
as well as the ratio of resolutions with respect to $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ decays, are varied according to
their correlation matrix determined from simulated events. The nominal fit is
then performed on this sample of pseudo-experiments and the distribution of
the difference between the yield determined in each of these fits and that of
the nominal fit is fitted with a Gaussian function. The systematic uncertainty
associated with the choice of the value of each signal parameter from
simulated events is then assigned as the linear sum of the absolute value of
the mean of the Gaussian and its resolution. An identical procedure is
employed to obtain the systematic uncertainties related to the fixed
parameters of the ARGUS functions describing the partially reconstructed
backgrounds and the CB functions used for the cross-feeds.
The uncertainties related to the choice of the models used in the nominal fit
are evaluated for the signal and combinatorial background models only. Both
the partially reconstructed background and the cross-feed shapes suffer from a
large statistical uncertainty from the simulated event samples and therefore
the uncertainty related to the fixed parameters also covers any sensible
variation of the shape. The $B^{0}_{s}$ decay modes that are studied lie near
large $B^{0}$ contributions for the $K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ and $K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ spectra. The
impact of the modelling of the right hand side of the $B^{0}$ mass
distribution is addressed by removing the second CB function, used as an
alternative model.
For the combinatorial background, a unique slope parameter governs the shape
of each $K^{0}_{\rm\scriptscriptstyle S}$ reconstruction category (Long or
Downstream). Two alternative models are considered: allowing independent
slopes for each of the six spectra (testing the assumption of a universal
slope) and using a linear model in place of the exponential (testing the
functional form of the combinatorial shape). Pseudo-experiments are again used
to estimate the effect of these alternative models; in the former case, the
value and uncertainties to be considered for the six slopes are determined
from a fit to the data. The dataset is generated according to the substitute
model and the fit is performed to the generated sample using the nominal
model. The value of the uncertainty is again estimated as the linear sum of
the absolute value of the resulting bias and its resolution. The total fit
model systematic uncertainty is given by the sum in quadrature of all the
contributions and is mostly dominated by the combinatorial background model
uncertainty.
### 6.2 Selection and trigger efficiencies
The accuracy of the efficiency determination is limited in most cases by the
finite size of the samples of simulated signal events, duly propagated as a
systematic uncertainty. In addition, the effect related to the choice of
binning for the square Dalitz plot is estimated from the spread of the average
efficiencies determined from several alternative binning schemes. Good
agreement between data and the simulation is obtained, hence no further
systematic uncertainty is assigned.
Systematic uncertainties related to the hardware stage trigger have been
studied. A data control sample of $D^{*+}\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+})\pi^{+}_{s}$ decays is used to quantify differences between pions
and kaons, separated by positive and negative hadron charges, as a function of
$p_{\rm T}$ [28]. Though they show an overall good agreement for the different
types of tracks, the efficiency for pions is slightly smaller than for kaons
at high $p_{\rm T}$. Simulated events are reweighted by these data-driven
calibration curves in order to extract the hadron trigger efficiency for each
mode, propagating properly the calibration-related uncertainties. Finally, the
ageing of the calorimeters during the data taking period when the data sample
analysed was recorded induced changes in the absolute scale of the trigger
efficiencies. While this was mostly mitigated by periodic recalibration,
relative variations occurred of order $10\%$. Since the kinematics vary
marginally from one mode to the other, a systematic effect on the ratio of
efficiencies arises. It is fully absorbed by increasing the trigger efficiency
systematic uncertainty by $10\%$.
### 6.3 Particle identification efficiencies
The procedure to evaluate the efficiencies of the PID selections uses
calibration tracks that differ from the signal tracks in terms of their
kinematic distributions. While the binning procedure attempts to mitigate
these differences there could be some remaining systematic effect. To quantify
any bias due to the procedure, simulated samples of the control modes are used
in place of the data samples. The average efficiency determined from these
samples can then be compared with the efficiency determined from simply
applying the selections to the simulated signal samples. The differences are
found to be less than $1\%$, hence no correction is applied. The calibration
procedure is assigned a systematic uncertainty. The observed differences in
efficiencies are multiplied by the efficiency ratio and statistical
uncertainties from the finite sample sizes are added in quadrature.
Table 2: Systematic uncertainties on the ratio of branching fractions for
Downstream and Long $K^{0}_{\rm\scriptscriptstyle S}$ reconstruction. All
uncertainties are relative and are quoted as percentages.
Downstream | Fit | Selection | Trigger | PID | Total | $f_{s}/f_{d}$
---|---|---|---|---|---|---
${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $5$ | $6$ | $3$ | $1$ | $8$ | —
${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $1$ | $5$ | $3$ | $1$ | $6$ | —
${\cal B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $8$ | $16$ | $2$ | $1$ | $18$ | $8$
${\cal B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $2$ | $5$ | $1$ | $1$ | $6$ | $8$
${\cal B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $1$ | $18$ | $3$ | $1$ | $18$ | $8$
Long | | | | | |
${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $5$ | $10$ | $1$ | $1$ | $14$ | —
${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $3$ | $20$ | $1$ | $1$ | $20$ | —
${\cal B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $5$ | $10$ | $1$ | $1$ | $11$ | $8$
${\cal B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $3$ | $12$ | $2$ | $1$ | $13$ | $8$
${\cal B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}\right)$ / ${\cal B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)$ | $2$ | $22$ | $1$ | $1$ | $22$ | $8$
## 7 Results and conclusion
The 2011 LHCb dataset, corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$ recorded at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$, has been analysed to search for the decays
$B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{\prime-}$.
The decays $B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}$ and $B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ are observed for the first time. The former is unambiguous,
while for the latter the significance of the observation is $5.9$ standard
deviations, including statistical and systematic uncertainties. The decay mode
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$,
previously observed by the BaBar experiment [17], is confirmed. The
efficiency-corrected Dalitz-plot distributions of the three decay modes
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$,
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$, and
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ are
displayed in Fig. 4. Some structure is evident at low
$K^{0}_{\rm\scriptscriptstyle S}\pi^{\pm}$ and $K^{\pm}\pi^{\mp}$ invariant
masses in the $B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}$ decay mode, while in the $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ decay the largest structure
is seen in the low $K^{0}_{\rm\scriptscriptstyle S}K^{\pm}$ invariant mass
region. No significant evidence for $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ decays is obtained. A 90%
confidence level (CL) interval based on the CL inferences described in Ref.
[40] is hence placed on the branching fraction for this decay mode.
Figure 4: Efficiency-corrected Dalitz-plot distributions, produced using the
sPlot procedure, of (top) $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, (middle)
$B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ and
(bottom) $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$
events. Bins with negative content appear empty.
Each branching fraction is measured (or limited) relative to that of
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$. The
ratios of branching fractions are determined independently for the two
$K^{0}_{\rm\scriptscriptstyle S}$ reconstruction categories and then combined
by performing a weighted average, excluding the uncertainty due to the ratio
of hadronisation fractions, since it is fully correlated between the two
categories. The Downstream and Long results all agree within two standard
deviations, including statistical and systematic uncertainties. The results
obtained from the combination are
$\displaystyle\frac{{\cal B}\left(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}\right)}{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$ $\displaystyle 0.128\pm
0.017\;{\rm(stat.)}\;\pm 0.009\;({\rm syst.})\,,$ $\displaystyle\frac{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}K^{-}\right)}{{\cal B}\left(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$
$\displaystyle 0.385\pm 0.031\;{\rm(stat.)}\;\pm 0.023\;({\rm syst.})\,,$
$\displaystyle\frac{{\cal B}\left(B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)}{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$ $\displaystyle 0.29\phantom{0}\pm
0.06\phantom{0}\;{\rm(stat.)}\;\pm 0.03\phantom{0}\;({\rm syst.})\;\pm
0.02\phantom{0}\;(f_{s}/f_{d})\,,$ $\displaystyle\frac{{\cal
B}\left(B^{0}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{\pm}\pi^{\mp}\right)}{{\cal B}\left(B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}\right)}$ $\displaystyle=$
$\displaystyle 1.48\phantom{0}\pm 0.12\phantom{0}\;{\rm(stat.)}\;\pm
0.08\phantom{0}\;({\rm syst.})\;\pm 0.12\phantom{0}\;(f_{s}/f_{d})\,,$
$\displaystyle\frac{{\cal B}\left(B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}\right)}{{\cal
B}\left(B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}\right)}$ $\displaystyle\in$ $\displaystyle[0.004;0.068]\;{\rm
at\;\;90\%\;CL}\,.$
The measurement of the relative branching fractions of $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ and $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ are in good agreement with, and
slightly more precise than, the previous world average results [41, 42, 8, 17,
10, 11, 30]. Using the world average value, ${\cal B}(B^{0}\\!\rightarrow
K^{0}\pi^{+}\pi^{-})=(4.96\pm 0.20)\times 10^{-5}$ [11, 30], the measured
time-integrated branching fractions
$\displaystyle{\cal B}\left(B^{0}\\!\rightarrow K^{0}K^{\pm}\pi^{\mp}\right)$
$\displaystyle=$ $\displaystyle\phantom{0}(6.4\pm 0.9\pm 0.4\pm 0.3)\times
10^{-6}\,,$ $\displaystyle{\cal B}\left(B^{0}\\!\rightarrow
K^{0}K^{+}K^{-}\right)$ $\displaystyle=$ $\displaystyle(19.1\pm 1.5\pm 1.1\pm
0.8)\times 10^{-6}\,,$ $\displaystyle{\cal B}\left(B^{0}_{s}\\!\rightarrow
K^{0}\pi^{+}\pi^{-}\right)$ $\displaystyle=$ $\displaystyle(14.3\pm 2.8\pm
1.8\pm 0.6)\times 10^{-6}\,,$ $\displaystyle{\cal
B}\left(B^{0}_{s}\\!\rightarrow K^{0}K^{\pm}\pi^{\mp}\right)$ $\displaystyle=$
$\displaystyle(73.6\pm 5.7\pm 6.9\pm 3.0)\times 10^{-6}\,,$
$\displaystyle{\cal B}\left(B^{0}_{s}\\!\rightarrow K^{0}K^{+}K^{-}\right)$
$\displaystyle\in$ $\displaystyle[0.2;3.4]\times 10^{-6}\;{\rm
at\;\;90\%\;CL}\,,$
are obtained, where the first uncertainty is statistical, the second
systematic and the last due to the uncertainty on ${\cal
B}(B^{0}\\!\rightarrow K^{0}\pi^{+}\pi^{-})$.
The first observation of the decay modes $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ and $B^{0}_{s}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$ is an important step towards
extracting information on the mixing-induced $C\\!P$-violating phase in the
$B^{0}_{s}$ system and the weak phase $\gamma$ from these decays. The apparent
rich structure of the Dalitz plots, particularly for the
$B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{\pm}\pi^{\mp}$
decays, motivates future amplitude analyses of these
$B^{0}_{(s)}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{\prime-}$
modes with larger data samples.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-07-29T17:13:35 |
2024-09-04T02:49:48.691944
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, E. Cowie, D.C. Craik, S.\n Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P.\n Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin,\n U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez,\n P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Thomas Latham",
"url": "https://arxiv.org/abs/1307.7648"
}
|
1307.7759
|
The Population Genetic Signature of Polygenic Local Adaptation
Jeremy Berg1,2,3,∗, Graham Coop1,2,3,∗
1 Graduate Group in Population Biology, University of California, Davis.
2 Center for Population Biology, University of California, Davis.
3 Department of Evolution and Ecology, University of California, Davis
$\ast$ E-mail: [email protected], [email protected]
## Abstract
Adaptation in response to selection on polygenic phenotypes may occur via
subtle allele frequencies shifts at many loci. Current population genomic
techniques are not well posed to identify such signals. In the past decade,
detailed knowledge about the specific loci underlying polygenic traits has
begun to emerge from genome-wide association studies (GWAS). Here we combine
this knowledge from GWAS with robust population genetic modeling to identify
traits that may have been influenced by local adaptation. We exploit the fact
that GWAS provide an estimate of the additive effect size of many loci to
estimate the mean additive genetic value for a given phenotype across many
populations as simple weighted sums of allele frequencies. We first describe a
general model of neutral genetic value drift for an arbitrary number of
populations with an arbitrary relatedness structure. Based on this model we
develop methods for detecting unusually strong correlations between genetic
values and specific environmental variables, as well as a generalization of
$Q_{ST}/F_{ST}$ comparisons to test for over-dispersion of genetic values
among populations. Finally we lay out a framework to identify the individual
populations or groups of populations that contribute to the signal of
overdispersion. These tests have considerably greater power than their single
locus equivalents due to the fact that they look for positive covariance
between like effect alleles, and also significantly outperform methods that do
not account for population structure. We apply our tests to the Human Genome
Diversity Panel (HGDP) dataset using GWAS data for height, skin pigmentation,
type 2 diabetes, body mass index, and two inflammatory bowel disease datasets.
This analysis uncovers a number of putative signals of local adaptation, and
we discuss the biological interpretation and caveats of these results.
## Author Summary
The process of adaptation is of fundamental importance in evolutionary
biology. Within the last few decades, genotyping technologies and new
statistical methods have given evolutionary biologists the ability to identify
individual regions of the genome that are likely to have been important in
this process. When adaptation occurs in traits that are underwritten by many
genes, however, the genetic signals left behind are more diffuse, as no
individual region of the genome will show strong signatures of selection.
Identifying this signature therefore requires a detailed annotation of sites
associated with a particular phenotype. Here we develop and implement a suite
of statistical methods to integrate this sort of annotation from genome wide
association studies with allele frequency data from many populations,
providing a powerful way to identify the signal of adaptation in polygenic
traits. We apply our methods to test for the impact of selection on human
height, skin pigmentation, body mass index, type 2 diabetes risk, and
inflammatory bowel disease risk. We find relatively strong signals for height
and skin pigmentation, moderate signals for inflammatory bowel disease, and
comparatively little evidence for body mass index and type 2 diabetes risk.
## Introduction
Population and quantitative genetics were in large part seeded by Fisher’s
insight [1] that the inheritance and evolution of quantitative characters
could be explained by small contributions from many independent Mendelian loci
[2]. While still theoretically aligned [3], these two fields have often been
divergent in empirical practice. Evolutionary quantitative geneticists have
historically focused either on mapping the genetic basis of relatively simple
traits [4], or in the absence of any such knowledge, on understanding the
evolutionary dynamics of phenotypes in response to selection over relatively
short time-scales [5]. Population geneticists, on the other hand, have usually
focused on understanding the subtle signals left in genetic data by selection
over longer time scales [6, 7, 8], usually at the expense of a clear
relationship between these patterns of genetic diversity and evolution at the
phenotypic level.
Recent advances in population genetics have also allowed for the genome-wide
identification of individual recent selective events either by identifying
unusually large allele frequency differences among populations and
environments or by detecting the effects of these events on linked diversity
[9]. Such approaches are nonetheless limited because they rely on identifying
individual loci that look unusual, and thus are only capable of identifying
selection on traits where an individual allele has a large and/or sustained
effect on fitness. When selection acts on a phenotype that is underwritten by
a large number of loci, the response at any given locus is expected to be
modest, and the signal instead manifests as a coordinated shift in allele
frequency across many loci, with the phenotype increasing alleles all on
average shifting in the same direction [10, 11, 12, 13, 14]. Because this
signal is so weak at the level of the individual locus, it is impossible to
identify against the genome-wide background without a very specific annotation
of which sites are the target of selection on a given trait [15].
The advent of well-powered genome wide association studies with large sample
sizes [16] has allowed for just this sort of annotation, enabling the mapping
of many small effect alleles associated with phenotypic variation down to the
scale of linkage disequilibrium in the population. The development and
application of these methods in human populations has identified thousands of
loci associated with a wide array of traits, largely confirming the polygenic
view of phenotypic variation [17].
Although the field of human medical genetics has been the largest and most
rapid to puruse such approaches, evolutionary geneticists studying non-human
model organisms have also carried out GWAS for a wide array of fitness-
associated traits, and the development of further resources is ongoing [18,
19, 20]. In human populations, the cumulative contribution of these loci to
the additive variance so far only explain a fraction of the narrow sense
heritability for a given trait (usually less than 15%), a phenomenon known as
the missing heritability problem [21, 22]. Nonetheless, these GWAS hits
represent a rich source of information about the loci underlying phenotypic
variation.
Many investigators have begun to test whether the loci uncovered by these
studies tend to be enriched for signals of selection, in the hopes of learning
more about how adaptation has shaped phenotypic diversity and disease risk
[23, 24, 25, 26]. The tests applied are generally still predicated on the idea
of identifying individual loci that look unusual, such that a positive signal
of selection is only observed if some subset of the GWAS loci have experienced
strong enough selection to make them individually distinguishable from the
genomic background. As noted above, it is unlikely that such a signature will
exist, or at least be easy to detect, if adaptation is truly polygenic, and
thus many selective events will not be identified by this approach.
Here we develop and implement a general method based on simple quantitative
and population genetic principals, using allele frequency data at GWAS loci to
test for a signal of selection on the phenotypes they underwrite while
accounting for the hierarchical structure among populations induced by shared
history and genetic drift. Our work is most closely related to the recent work
of Turchin et al [27], Fraser [28] and Corona et al [29], who look for co-
ordinated shifts in allele frequencies of GWAS alleles for particular traits.
Our approach constitutes an improvement over the methods implemented in these
studies as it provides a high powered and theoretically grounded approach to
investigate selection in an arbitrary number of populations with an arbitrary
relatedness structure.
Using the set of GWAS effect size estimates and genome wide allele frequency
data, we estimate the mean genetic value [30, 31] for the trait of interest in
a diverse array of human populations. These genetic values may in some cases
be poor predictors of the actual phenotypes for reasons we address below and
in the Discussion. We therefore make no strong claims about their ability to
predict present day observed phenotypes. We instead focus on population
genetic modeling of the joint distribution of genetic values, which provides a
robust way of investigating how selection may have impacted the underlying
loci.
We develop a framework to describe how genetic values covary across
populations based on a flexible model of genetic drift and population history.
In Figure 1 we show a schematic diagram of our approach to aid the reader.
Using this null model, we implement simple test statistics based on
transformations of the genetic values that remove this covariance among
populations. We judge the significance of the departure from neutrality by
comparing to a null distribution of test statistics constructed from well
matched sets of control SNPs. Specifically, we test for local adaptation by
asking whether the transformed genetic values show excessive correlations with
environmental or geographic variables. We also develop and implement a less
powerful but more general test, which asks whether the genetic values are
over-dispersed among populations compared to our null model of drift. We show
that this overdispersion test, which is closely related to $Q_{ST}$ [32, 33]
and a series of approaches from the population genetics literature [34, 35,
36, 37, 38], gains considerable power to detect selection over single locus
tests by looking for unexpected covariance among loci in the deviation they
take from neutral expectations. Lastly, we develop an extension of our model
that allows us to identify individual populations or groups of populations
whose genetic values deviate from their neutral expectations given the values
observed for related populations, and thus have likely been impacted by
selection. While we develop these methods in the context of GWAS data, we also
relate them to recent methodological developments in the quantitative genetics
of observed phenotypes [39, 40], highlighting the useful connection between
these approaches.
## Results
### Estimating Genetic Values with GWAS Data
Consider a trait of interest where $L$ loci (e.g. biallelic SNPs) have been
identified from a genome-wide association study. We arbitrarily label the
phenotype increasing allele $A_{1}$ and the alternate allele $A_{2}$ at each
locus. These loci have additive effect size estimates
$\alpha_{1},\cdots\alpha_{L}$, where $\alpha_{\ell}$ is the average increase
in an individual’s phenotype from replacing an $A_{2}$ allele with an $A_{1}$
allele at locus $\ell$. We have allele frequency data for $M$ populations at
our $L$ SNPs, and denote by $p_{m\ell}$ the observed sample frequency of
allele $A_{1}$ at the $\ell^{th}$ locus in the $m^{th}$ population. From
these, we estimate the mean genetic value in the $m^{th}$ population as
$Z_{m}=2\sum_{\ell=1}^{L}\alpha_{\ell}p_{m\ell}$ (1)
and we take $\vec{Z}$ to be the vector containing the mean genetic values for
all $M$ populations.
### A Model of Genetic Value Drift
We are chiefly interested in developing a framework for testing the hypothesis
that the joint distribution of $\vec{Z}$ is driven by neutral processes alone,
with rejection of this hypothesis implying a role for selection. We first
describe a general model for the expected joint distribution of estimated
genetic values ($\vec{Z}$) across populations under neutrality, accounting for
genetic drift and shared population history.
A simple approximation to a model of genetic drift is that the current
frequency of an allele in a population is normally distributed around some
ancestral frequency ($\epsilon$). Under a Wright-Fisher model of genetic
drift, the variance of this distribution is approximately
$f\epsilon(1-\epsilon)$, where $f$ is a property of the population shared by
all loci, reflecting the compounded effect of many generations of binomially
sampling [41]. Note also that for small values, $f$ is approximately equal to
the inbreeding coefficient of the present day population relative to the
defined ancestral population, and thus has an interpretation as the
correlation between two randomly chosen alleles relative to the ancestral
population [41].
We can expand this framework to describe the joint distribution of allele
frequencies across an arbitrary number of populations for an arbitrary
demographic history by assuming that the vector of allele frequencies in $M$
populations follows a multivariate normal distribution
$\vec{p}\sim
MVN\left(\epsilon\vec{1},\epsilon\left(1-\epsilon\right)\mathbf{F}\right),$
(2)
where $\mathbf{F}$ is an $M$ by $M$ positive definite matrix describing the
correlation structure of allele frequencies across populations relative to the
mean/ancestral frequency. Note again that for small values it is also
approximately the matrix of inbreeding coefficients (on the diagonal) and
kinship coefficients (on the off-diagonals) describing relatedness among
populations [42, 37]. This flexible model was introduced, to our knowledge, by
[43] (see[44] for a review), and has subsequently been used as a
computationally tractable model for population history inference [41, 45], and
as a null model for signals of selection [46, 37, 38, 47]. So long as the
multivariate normal assumption of drift holds reasonably well, this framework
can summarize arbitrary population histories, including tree-like structures
with substantial gene flow between populations [45], or even those which lack
any coherent tree-like component, such as isolation by distance models [48,
49].
Recall that our estimated genetic values $(\vec{Z})$ are merely a sum of
sample allele frequencies weighted by effect size. If the underlying allele
frequencies are well explained by the multivariate normal model described
above, then the distribution of $\vec{Z}$ is a weighted sum of multivariate
normals, such that this distribution is itself multivariate normal
$\vec{Z}\sim MVN\left(\mu\vec{1},V_{A}\mathbf{F}\right)$ (3)
where $\mu=\frac{1}{L}\sum_{\ell=1}^{L}2\alpha_{\ell}\epsilon_{\ell}$ and
$V_{A}=4\sum_{\ell=1}^{L}\alpha_{\ell}^{2}\epsilon_{\ell}(1-\epsilon_{\ell})$
are respectively the expected genetic value and additive genetic variance of
the ancestral (global) population. The covariance matrix describing the
distribution of $\vec{Z}$ therefore differs from that describing the
distribution of frequencies at individual loci only by a scaling factor that
can be interpreted as the contribution of the associated loci to the additive
genetic variance present in a hypothetical population with allele frequencies
equal to the grand mean of the sampled populations.
The assumption that the drift of allele frequencies around their shared mean
is normally distributed (2) may be problematic if there is substantial drift.
However, even if that is the case, the estimated genetic values may still be
assumed to follow a multivariate normal distribution by appealing to the
central limit theorem, as each estimated genetic value is a sum over many
loci. We show in the Results that this assumption often holds in practice.
It is useful here to note that the relationship between the model for drift at
the individual locus level, and at the genetic value level, gives an insight
into where most of the information and statistical power for our methods will
come from. Each locus adds a contribution
$2\alpha_{\ell}(\vec{p}_{\ell}-\epsilon_{\ell}\vec{1})$ to the vector of
deviations of the genetic values from the global mean. If the allele
frequencies are unaffected by selection then the frequency deviation of an
allele at locus $\ell$ in population $m$ $(p_{m,\ell}-\epsilon_{\ell})$ will
be uncorrelated in magnitude or sign with both the effect at locus $\ell$
$(\alpha_{\ell})$ and the allele frequency deviation taken by other unlinked
loci. Thus the expected departure of a genetic value of a population from the
mean is zero, and the noise around this should be well modeled by our
multivariate normal model.
The tests described below will give positive results when these observations
are violated. The effect of selection is to induce a non-independence of
allele frequency deviation ($\vec{p}_{\ell}-\epsilon_{\ell}\vec{1}$) across
loci, determined by the sign and magnitude of the effect sizes [10, 11, 12,
14, 13] and as we demonstrate below, all of our methods rely principally on
identifying this non-independence. This observations has important
considerations for the false positive profile of our methods. Specifically,
false positives will arise only if the GWAS ascertainment procedure induces a
correlation between the estimated effect size of an allele ($\alpha_{\ell}$)
and the deviation that this allele takes across populations
$(\vec{p}_{\ell}-\epsilon_{\ell}\vec{1})$. This should not be the case if the
GWAS is performed in a single population which is well mixed compared to the
populations considered in the test. False positives can occur when a GWAS is
performed in a structured population and fails to account for the fact that
the phenotype of interest is correlated with ancestry in this population. We
address this case in greater depth in the Discussion.
These observation also allows us to exclude certain sources of statistical
error as a cause of false positives. For example, simple error in the
estimation of $\alpha_{\ell}$, or failing to include all loci affecting a
trait cannot cause false positives, because this error has no systematic
effect on $\vec{p}_{\ell}-\epsilon_{\ell}\vec{1}$ across loci. Similarly, if
the trait of interest truly is neutral, variation in the true effects of an
allele across populations or over time or space (which can arise from
epistatic interactions among loci, or from gene by environment interactions)
will not drive false positives, again because no systematic trends in
population deviations will arise. This sort of heterogeneity can, however,
reduce statistical power, as well as make straightforward interpretation of
positive results difficult, points which we address further below.
### Fitting the Model and Standardizing the Estimated Genetic Values
As described above, we obtain the vector $\vec{Z}$ by summing allele
frequencies across loci while weighting by effect size. We do not get to
observe the ancestral genetic value of the sample $(\mu)$, so we assume that
this is simply equal to the mean genetic value across populations
$(\mu=\frac{1}{M}\sum_{m}Z_{m})$. This assumption costs us a degree of
freedom, and so we must work with a vector $\vec{Z^{\prime}}$, which is the
vector of estimated genetic values for the first $M-1$ populations, centered
at the mean of the $M$ (see Methods for details). Note that this procedure
will be the norm for the rest of this paper, and thus we will always work with
vectors of length $M-1$ that are obtained by subtracting the mean of the $M$
vector and dropping the last component.
To estimate the null covariance structure of the $M-1$ populations we sample a
large number K random unlinked SNPs. In our procedure, the $K$ SNPs are
sampled so as to match certain properties of the $L$ GWAS SNPs (the specific
matching procedure is described in more depth below and in the Methods
section). Setting $\epsilon_{k}$ to be the mean sample allele frequency across
populations at the $k^{th}$ SNP, we standardize the sample allele frequency in
population $m$ as
$(p_{mk}-\epsilon_{k})/\left(\epsilon_{k}\left(1-\epsilon_{k}\right)\right)$.
We then calculate the sample covariance matrix ($\mathbf{F}$) of these
standardized frequencies, accounting for the $M-1$ rank of the matrix (see
Methods). We estimate the scaling factor of this matrix $\mathbf{F}$ as
$V_{A}=4\sum_{\ell=1}^{L}\alpha_{\ell}^{2}\epsilon_{\ell}(1-\epsilon_{\ell}).$
(4)
We now have an estimated genetic value for each population, and a simple null
model describing their expected covariance due to shared population history.
Under this multivariate normal framework, we can transform the vector of mean
centered genetic values ($\vec{Z^{\prime}}$) so as to remove this covariance.
First, we note that the Cholesky decomposition of the $\mathbf{F}$ matrix is
$\mathbf{F}=\mathbf{C}\mathbf{C^{T}}$ (5)
where $\mathbf{C}$ is a lower triangular matrix, and $\mathbf{C^{T}}$ is its
transpose. Informally, this can be thought of as taking the square root of
$\mathbf{F}$, and so $\mathbf{C}$ can loosely be thought of as analogous to
the standard deviation matrix.
Using this matrix $\mathbf{C}$ we can transform our estimated genetic values
as:
$\vec{X}=\frac{1}{\sqrt{V_{A}}}\mathbf{C}^{-1}\vec{Z^{\prime}}.$ (6)
If $\vec{Z^{\prime}}\sim\textrm{MVN}(\vec{0},V_{A}\mathbf{F})$ then
$\vec{X}\sim\textrm{MVN}(0,\mathbb{I})$, where $\mathbb{I}$ is the identity
matrix. Therefore, under the assumptions of our model, these standardized
genetic values should be independent and identically distributed $\sim N(0,1)$
random variates [38].
It is worth spending a moment to consider what this transformation has done to
the allele frequencies at the loci underlying the estimated genetic values. As
our original genetic values are written as a weighted sum of allele
frequencies, our transformed genetic values can be written as a weighted sum
of transformed allele frequencies (which have passed through the same
transform). We can write
$\vec{X}=\frac{1}{\sqrt{V_{A}}}\mathbf{C}^{-1}\vec{Z^{\prime}}=\frac{2}{\sqrt{V_{A}}}\sum_{\ell}\alpha_{\ell}\mathbf{C}^{-1}\vec{p}_{\ell}$
(7)
and so we can simply define the vector of transformed allele frequencies at
locus $\ell$ to be
$\vec{p^{\prime}}_{\ell}=\mathbf{C}^{-1}\vec{p}_{\ell}.$ (8)
This set of transformed frequencies exist within a set of transformed
populations, which by definition have zero covariance with one another under
the null, and are related by a star-like population tree with branches of
equal length.
As such, we can proceed with simple, straightforward and familiar statistical
approaches to test for the impact of spatially varying selection on the
estimated genetic values. Below we describe three simple methods for
identifying the signature of polygenic adaptation, which arise naturally from
this observation.
### Environmental Correlations
We first test if the genetic values are unusually correlated with an
environmental variable across populations compared to our null model. A
significant correlation is consistent with the hypothesis that the populations
are locally adapted, via the phenotype, to local conditions that are
correlated with the environmental variable. However, the link from correlation
to causation must be supported by alternate forms of evidence, and in the lack
of such evidence, a positive result from our environmental correlation tests
may be consistent with many explanations.
Assume we have a vector $\vec{Y}$, containing measurements of a specific
environmental variable of interest in each of the $M$ populations. We mean-
center this vector and put it through a transform identical to that which we
applied to the estimated genetic values in (7). This gives us a vector
$\vec{Y^{\prime}}$, which is in the same frame of reference as the transformed
genetic values.
There are many possible models to describe the relationship between a trait of
interest and a particular environmental variable that may act as a selective
agent. We first consider a simple linear model, where we model the
distribution of transformed genetic values ($\vec{X}$) as a linear effect of
the transformed environmental variables ($\vec{Y^{\prime}}$)
$\vec{X}\sim\beta\vec{Y^{\prime}}+\vec{e}$ (9)
where $\vec{e}$ under our null is a set of normal, independent and identically
distributed random variates (i.e. residuals), and $\beta$ can simply be
estimated as $\frac{Cov(\vec{X},\vec{Y^{\prime}})}{Var(Y)}$. We can also
calculate the associated squared Pearson correlation coefficient ($r^{2}$) as
a measure of the fraction of variance explained by our variable of choice, as
well as the non-parametric Spearman’s rank correlation
$\rho\left(\vec{X},\vec{Y^{\prime}}\right)$, which is robust to outliers that
can mislead the linear model. We note that we could equivalently pose this
linear model as a mixed effects model, with a random effect covariance matrix
$V_{A}\mathbf{F}$. However, as we know both $V_{A}$ and $\mathbf{F}$, we would
not have to estimate any of the random effect parameters, reducing it to a
fixed effect model as in (9) [50].
In the Methods (section “The Linear Model at the Individual Locus Level”) we
show that the linear environmental model applied to our transformed genetic
values has a natural interpretation in terms of the underlying individual
loci. Therefore, exploring the environmental correlations of estimated genetic
values nicely summarizes information in a sensible way at the underlying loci
identified by the GWAS.
In order to assess the significance of these measures, we implement an
empirical null hypothesis testing framework, using $\beta$, $r^{2}$, and
$\rho$ as test statistics. We sample many sets of $L$ SNPs randomly from the
genome, again applying a matching procedure discussed below and in the
Methods. With each set of $L$ SNPs we construct a vector $\vec{Z}_{null}$,
which represents a single draw from the genome-wide null distribution for a
trait with the given ascertainment profile. We then perform an identical set
of transformations and analyses on each $\vec{Z}_{null}$, thus obtaining an
empirical genome-wide null distribution for all test statistics.
### Excess Variance Test
As an alternative to testing the hypothesis of an effect by a specific
environmental variable, one might simply test whether the estimated genetic
values exhibit more variance among populations than expected due to drift.
Here we develop a simple test of this hypothesis.
As $\vec{X}$ is composed of $M-1$ independent, identically distributed
standard normal random variables, a natural choice of test statistic is given
by
$Q_{X}=\vec{X}^{T}\vec{X}=\frac{\vec{Z^{\prime}}^{T}\mathbf{F}^{-1}\vec{Z^{\prime}}}{V_{A}}.$
(10)
This $Q_{X}$ statistic represents a standardized measure of the among
population variance in estimated genetic values that is not explained by drift
and shared history. It is also worth noting that by comparing the rightmost
form in (10) to the multivariate normal likelihood function, we find that
$Q_{X}$ is proportional to the negative log likelihood of the estimated
genetic values under the neutral null model, and is thus the natural
measurement of the model’s ability to explain their distribution. Multivariate
normal theory predicts that this statistic should follow a $\chi^{2}$
distribution with $M-1$ degrees of freedom under the null hypothesis.
Nonetheless, we use a similar approach to that described for the linear model,
generating the empirical null distribution by resampling SNPs genome-wide. As
discussed below, we find that in practice the empirical null distribution
tends to be very closely matched by the theoretically predicted
$\chi^{2}_{M-1}$ distribution.
Values of this statistic that are in the upper tail correspond to an excess of
variance among populations. This excess of variance is consistent with the
differential action of natural selection on the phenotype among populations
(e.g. due to local adaptation). Values in the lower tail correspond a paucity
of variance, and thus potentially to widespread stabilizing selection, with
many populations selected for the same optimum. In this paper we mainly
concentrate on the upper tail of the distribution of $Q_{X}$, e.g. for our
power simulations, but note that either tail of the distribution is
informative about the action of selection on the phenotype.
#### The Relationship of $Q_{X}$ to Previous Tests
Our $Q_{X}$ statistic is closely related to $Q_{ST}$, the phenotypic analog of
$F_{ST}$, which measures the fraction of the genetic variance that is among
populations relative to the total genetic variance [51, 33, 32]. $Q_{ST}$ is
typically estimated in traditional local adaptation studies via careful
measurement of phenotypes from related individuals in multiple populations in
a common garden setting. If the loci underlying the trait act in a purely
additive manner and are experiencing only neutral genetic drift, then
$\mathbb{E}[Q_{ST}]=\mathbb{E}[F_{ST}]$ [52, 53].
If both quantities are well estimated, and we also assume that there is no
hierarchical structure among the populations, then
$\frac{(M-1)Q_{ST}}{F_{ST}}$ is known to have a $\chi^{2}_{M-1}$ distribution
under a wide range of models [54, 55, 56]. This statistic is thus a natural
phenotypic extension of Lewontin and Krakauer’s $F_{ST}$ based-test (LK test)
[34].
To see the close correspondence between $Q_{X}$ and $Q_{ST}$, consider the
case of a starlike population tree with branches of equal length (i.e.
$f_{mm}=F_{ST}$ and $f_{m\neq n}=0$). Under this demographic model, we have
$\displaystyle
Q_{X}=\frac{\left(\vec{Z}-\mu\right)^{T}\mathbf{F}^{-1}\left(\vec{Z}-\mu\right)}{V_{A}}$
$\displaystyle=\frac{\left(Z_{1}-\mu\right)^{2}}{V_{A}F_{ST}}+\dots+\frac{\left(Z_{M-1}-\mu\right)^{2}}{V_{A}F_{ST}}$
$\displaystyle=\frac{\left(M-1\right)\text{Var}\left(\vec{Z}\right)}{V_{A}F_{ST}}$
$\displaystyle=\frac{\left(M-1\right)\widehat{Q}_{ST}}{F_{ST}}$ (11)
where $\widehat{Q}_{ST}$ is an estimated value for $Q_{ST}$ obtained from our
estimated genetic values. This relationship between $Q_{X}$ and $Q_{ST}$
breaks down when some pairs of populations do not have zero covariance in
allele frequencies under the null, in which case the $\chi^{2}$ distribution
of the LK test also breaks down [36, 35]. Bonhomme and colleagues[37] recently
proposed an extension to the LK test that accounts for a population tree,
thereby recovering the $\chi^{2}$ distribution (see also [38], which relaxes
the tree-like assumption), and our $Q_{X}$ statistic is a natural extension of
this enhanced statistic to the problem of detecting coordinated selection at
multiple loci. This test is also nearly identical to that developed by
Ovaskainen and colleagues for application to direct phenotypic measurements
[39].
#### Writing $Q_{X}$ in Terms of Allele Frequencies
Given that our estimated genetic values are simple linear sums of allele
frequencies, it is natural to ask how $Q_{X}$ can be written in terms of these
frequencies. Again, restricting ourselves to the case where $\mathbf{F}$ is
diagonal, (i.e. $f_{mm}=F_{ST}$ and $f_{m\neq n}=0$), we can express $Q_{X}$
as
$Q_{X}=\frac{4}{V_{A}F_{ST}}\sum_{m=1}^{M-1}\sum_{\ell,\ell^{\prime}}\alpha_{\ell}\alpha_{\ell^{\prime}}(p_{m\ell}-\epsilon_{\ell})(p_{m\ell^{\prime}}-\epsilon_{\ell^{\prime}}),$
(12)
which can be rewritten as
$Q_{X}=\frac{M-1}{F_{ST}}\left(\frac{\sum_{\ell}\alpha_{\ell}^{2}Var(\vec{p}_{\ell})}{\sum_{\ell}\alpha_{\ell}^{2}\epsilon_{\ell}(1-\epsilon_{\ell})}+\frac{\sum_{\ell\neq\ell^{\prime}}\alpha_{\ell}\alpha_{\ell^{\prime}}Cov(\vec{p}_{\ell},\vec{p}_{\ell^{\prime}})}{\sum_{\ell}\alpha_{\ell}^{2}\epsilon_{\ell}(1-\epsilon_{\ell})}\right).\\\
$ (13)
The numerator of the first term inside the parentheses is the weighted sum of
the variance among populations over all GWAS loci, scaled by the contribution
of those loci to the additive genetic variance in the total population. As
such this first term is similar to $F_{ST}$ calculated for our GWAS loci, but
instead of just averaging the among population and total variances equally
across loci in the numerator and denominator, these quantities are weighted by
the squared effect size at each locus. This weighting nicely captures the
relative importance of different loci to the trait of interest.
The second term in (13) is less familiar; the numerator is the weighted sum of
the covariance of allele frequencies between all pairs of GWAS loci, and the
denominator is again the contribution of those loci to the additive genetic
variance in the total population. This term is thus a measure of the
correlation among loci in the deviation they take from the ancestral value, or
the across population component of linkage disequilibrium. For a more in depth
discussion of this relationship in the context of $Q_{ST}$, see [10, 11, 12,
14, 13].
As noted above (8), when $\mathbf{F}$ is non-diagonal, our transformed genetic
values can be written as a weighted sum of transformed allele frequencies.
Consequently, we can obtain a similar expression to (13) when population
structure exists, but now expressed in terms of the covariance of a set of
allele frequencies in transformed populations that have no covariance with
each other under the null hypothesis. Specifically, when the covariance is
non-diagonal we can write:
$Q_{X}=(M-1)\frac{\sum_{\ell}\alpha_{\ell}^{2}Var(\vec{p^{\prime}}_{\ell})}{\sum_{\ell}\alpha_{\ell}^{2}\epsilon_{\ell}(1-\epsilon_{\ell})}+(M-1)\frac{\sum_{\ell\neq\ell^{\prime}}\alpha_{\ell}\alpha_{\ell^{\prime}}Cov(\vec{p^{\prime}}_{\ell},\vec{p^{\prime}}_{\ell^{\prime}})}{\sum_{\ell}\alpha_{\ell}^{2}\epsilon_{\ell}(1-\epsilon_{\ell})}.\\\
$ (14)
We refer to the first term in this decomposition as the standardized
$F_{ST}$-like component and the second term as the standardized LD-like
component. Under the neutral null hypothesis, the expectation of the second
term is equal to zero, as drifting loci are equally likely to covary in either
direction. With differential selection among populations, however, we expect
loci underlying a trait not only to vary more than we would expect under a
neutral model, but also to covary in a consistent way across populations.
Models of local adaptation predict that it is this covariance among alleles
that is primarily responsible for differentiation at the phenotypic level [10,
11, 12, 13, 14], and we therefore expect the $Q_{X}$ statistic to offer
considerably increased power as compared to measuring average $F_{ST}$ or
identifying $F_{ST}$ outliers. We use simulations to demonstrate this fact
below, and also demonstrate the perhaps surprising result that for a broad
parameter range the standardized LD-like component exhibits almost no loss of
power when used as a test statistic.
### Identifying Outlier Populations
Having detected a putative signal of selection for a given trait, one may wish
to identify individual regions and populations which contribute to the signal.
Here we rely on our multivariate normal model of relatedness among
populations, along with well understood methods for generating conditional
multivariate normal distributions, in order to investigate specific hypotheses
about individual populations or groups of populations. Using standard results
from multivariate normal theory, we can generate the expected joint
conditional distribution of genetic values for an arbitrary set of populations
given the observed genetic values in some other set of populations. These
conditional distributions allow for a convenient way to ask whether the
estimated genetic values observed in certain populations or groups of
populations differ significantly from the values we would expect them to take
under the neutral model given the values observed in related populations.
Specifically, we exclude a population or set of populations, and then
calculate the expected mean and variance of genetic values in these excluded
populations given the values observed in the remaining populations, and the
covariance matrix relating them. Using this conditional mean and variance, we
calculate a Z-score to describe how well fit the estimated genetic values of
the excluded populations are by our model of drift, conditional on the values
in the remaining populations. In simple terms, the observation of an extreme
Z-score for a particular population or group of populations may be seen as
evidence that that group has experienced directional selection on the trait of
interest (or a correlated one) that was not experienced by the related
populations on which we condition the analyses. The approach cannot uniquely
determine the target of selection, however. For example, conditioning on
populations that have themselves been influenced by directional selection may
lead to large Z-scores for the population being tested, even if that
population has been evolving neutrally. We refer the reader to the Methods
section for a mathematical explication of these approaches.
### Datasets
We conducted power simulations and an empirical application of our methods
based on the Human Genome Diversity Panel (HGDP) population genomic dataset
[57], and a number of GWAS SNP sets. To ensure that we made the fullest
possible use of the information in the HGDP data, we took advantage of a
genome wide allele frequency dataset of $\sim$3 million SNPs imputed from the
Phase II HapMap into the 52 populations of the HGDP. These SNPs were imputed
as part of the HGDP phasing procedure in [58]; see our Methods section for a
recap of the details. We applied our method to test for signals of selection
in six human GWAS datasets identifying SNPs associated with height, skin
pigmentation, body mass index (BMI), type 2 diabetes (T2D), Crohn’s Disease
(CD) and Ulcerative Colitis (UC).
#### Choosing null SNPs
Various components of our procedure involve sampling random sets of SNPs from
across the genome. While we control for biases in our test statistics
introduced by population structure through our $\mathbf{F}$ matrix, we are
also concerned that subtle ascertainment effects of the GWAS process could
lead to biased test statistics, even under neutral conditions. We control for
this possibility by sampling null SNPs so as to match the joint distribution
of certain properties of the ascertained GWAS SNPs. Specifically, we chose our
random SNPs to match the GWAS SNPs in each study in terms of their minor
allele frequency (MAF) in the ascertainment population and the imputation
status of the allele in our population genomic dataset (i.e. whether the
allele was imputed or present in the original HGDP genotyping panel). In
addition, we were concerned that GWAS SNPs might be preferentially found close
to genes and in low recombination regions, the latter due to better tagging,
and as such may be subject to a high rate of drift due to background
selection, leading to higher levels of differentiation at these sites [59].
Therefore, in addition to MAF and imputation status, we also matched our
random SNPs to an estimate of the background selection environment experienced
by the GWAS SNPs, as measured by B value [60], which is a function of both the
density of functional sites and recombination rate calibrated to match the
reduction in genetic diversity due to background selection. We detail the
specifics of the binning scheme for matching the discretized distributions of
GWAS and random SNPs in the Methods.
### Power Simulations
To assess the power of our methods in comparison to other possible approaches,
we conducted a series of power simulations. There are two possible approaches
to simulate the effect of selection on large scale allele frequency data of
the type for which our methods are designed. The first is to simulate under
some approximate model of the evolutionary history (e.g. full forward
simulation under the Wright-Fisher model with selection). The second is to
perturb real data in such a way that approximates the effect of selection. We
choose to pursue the latter, both because it is more computationally
tractable, and because it allows us to compare the power of our different
approaches for populations with evolutionary histories of the same complexity
as the real data we analyze. Each of our simulations will thus consist of
sampling 1000 sets of SNPs matched to the height dataset (in much the same way
we sample SNPs to construct the null distributions of our test statistics),
and then adding slight shifts in frequency in various ways to mimic the effect
of selection.
Below we first describe the set of alternative statistics to which we compare
our methods. We then describe the manner in which we add perturbations to
mimic selection, and lastly describe a number of variations on this theme
which we pursued in order to better demonstrate how the power of our
statistics changes as we vary parameters of the trait of interest,
evolutionary process, or the ascertainment.
#### Statistics Tested
For our first set of simulation experiments we compared two of our statistics,
($r^{2}$ and $Q_{X}$) against their naive counterparts, which are not adjusted
for population structure (naive $r^{2}$ and $Q_{ST}$). We also include the
adjusted $F_{ST}$-like and LD-like components of $Q_{X}$ as their behavior
over certain parameter ranges is particularly illuminating. For $Q_{ST}$,
$Q_{X}$, and it’s components, we count a given simulation as producing a
positive result if the statistic lies in the upper 5% tail of the null
distribution, whereas for the environmental correlation statistics ($r^{2}$
and naive $r^{2}$) we use a two-tailed 5% test. We also compared our tests to
a single locus enrichment test, where we tested for an enrichment in the
number of SNPs that individually show a correlation with the environmental
variable. We considered this test to produce a positive result if the number
of individual loci in the 5% tail of the null distribution for individual
locus $r^{2}$ was itself in the 5% tail using a binomial test. We do not
include our alternative linear model statistics $\beta$ and $\rho$ in these
plots for the sake of figure legibility, but they generally had very similar
power to that of $r^{2}$. While slightly more powerful versions of the $r^{2}$
enrichment test that better account for sampling noise are available [46],
note that our tests could be extended similarly as well, so the comparison is
fair.
#### Simulating Selection
We base our initial power simulations on empirical data altered to have an
increasing effect of selection along a latitudinal gradient. In order to mimic
the effect of selection, we generate a new set of allele frequencies
($p_{s,m\ell}$) by taking the original frequency ($p_{m\ell}$) and adding a
small shift according to
$\displaystyle
p_{s,m\ell}=p_{m\ell}+p_{m\ell}\left(1-p_{m\ell}\right)\alpha_{\ell}\delta
Y_{m}$ (15)
where $\alpha_{\ell}$ is the effect size assigned assigned to locus $\ell$,
and $Y_{m}$ is the mean centered absolute latitude of the population. We use
1000 simulations at $\delta=0$ to form null distribution for each of our test
statistics, and from this established the $5\%$ significance level. We then
increment $\delta$ and give the power of each statistic as the fraction of
simulations whose test statistic falls beyond this cutoff. While this approach
to simulating selection is obviously naive to the way selection actually
operates, it captures many of the important effects on the loci underlying a
given trait. Namely, loci will have greater shifts if they experience extreme
environments, have large effects on the phenotype, or are at intermediate
frequencies. Because we add these shifts to allele frequencies sampled from
real, putatively neutral loci, the effect of drift on their joint distribution
is already present, and thus does not need to be simulated. The results of
these simulations are shown in Figure 2A.
Our population structure adjusted statistics clearly outperform tests that do
not account for structure, as well as the single locus outlier based test.
Particularly noteworthy is the fact that the power of a test relying on
$Q_{X}$ and that using only the LD-like component are essentially identical
over the entire range of simulation, while the $F_{ST}$-like component
achieves only about $20\%$ power by the point at which the former statistics
have reached 100%. This reinforces the observation from previous studies of
$Q_{ST}$ that for polygenic traits, nearly all of the differentiation at the
trait level arises as a consequence of across population covariance among the
underlying loci, and not as a result of substantial differentiation at the
loci themselves [14]. While our environment-genetic value correlation tests
considerably outperform $Q_{X}$, this is somewhat artificial as it assumes
that we know the environmental variable responsible for our allele frequency
shift. In reality, the power of the environmental variable test will depend on
the investigator’s ability to accurately identify the causal variable (or one
closely correlated with it) in the particular system under study, and thus in
some cases $Q_{X}$ may have have higher power in practice. Panels A and B from
Figure 2 with SNPs matched to each of the other traits we investigate can be
found in Figures S1-S5.
#### Pleiotropy and Correlated Selection
We next considered the fact that many of the loci uncovered by GWAS are may be
relatively pleiotropic, and thus may simultaneously respond to selection on
multiple different traits. To explore how our methods perform in the presence
of undetected pleiotropy, we consider the realization that from the
perspective of allele frequency change there is only one effect that matters,
and that is the effect on fitness. We therefore chose a simple and general
approach to capture a flavor of this situation. We simulate the effect of
selection as above (15), but give each locus an effect on fitness
($\alpha_{\ell}^{\prime}$) that may be only partially correlated with the
observed effect sizes for the trait of interest (with the unaccounted for
effect on fitness coming via pleiotropic relationships to any number of
unaccounted for phenotypes). For simplicity we assume that $\alpha_{\ell}$ and
$\alpha_{\ell}^{\prime}$ have a bivariate normal distribution around zero with
equal variance and correlation parameter $\phi$. We then simulate
$\alpha_{\ell}^{\prime}$ from its conditional distribution given
$\alpha_{\ell}$ (i.e. $\alpha_{\ell}^{\prime}\mid\alpha_{\ell}\sim
N(\phi\alpha_{\ell},(1-\phi^{2})Var(\alpha))$). For each SNP $\ell$ in (15) we
replaced $\alpha_{\ell}$ by its effect $\alpha_{\ell}^{\prime}$ on the
unobserved phenotype, but then perform our tests using the $\alpha_{\ell}$
measured for the trait of interest. Here $\phi$ can be thought of as the
genetic correlation between our phenotype and fitness if this simple
multivariate form held true for all of the loci contributing to the trait. The
extremes of $\phi=1$ and $\phi=0$ respectively represent the cases where
selection acts only on the focal trait and that were all the underlying loci
are affected by selection, but not due to their relationship with the focal
trait. These simulations can also informally be seen as modeling the case
where the GWAS estimated effect sizes are imperfectly correlated with the true
effect sizes that selection sees, for example due to measurement error in the
GWAS.
In Figure 2B we hold the value of $\delta$ constant at 0.14 and vary the
genetic correlation $\phi$ from one down to zero. Predictably, our GWAS
genetic value based statistics lose power as the the focal trait becomes less
correlated with fitness but do retain reasonable power out to quite low
genetic correlations (e.g. our $r^{2}$ out performs the single locus metrics
until $\phi<0.3$). In contrast, counting the number of SNPs that are
significantly correlated with a given environmental variable remains equally
powerful across all genetic correlations. This is because the single locus
environmental correlation tests treat each locus separately with no regards to
whether there is agreement across alleles with the same direction of effect
size. This may be a desirable property of the environmental outliers
enrichment approach, as it does not rely on a close relationship between the
effect sizes and the way that selection acts on the loci. On the other hand,
this is also problematic, as such tests may often be detecting selection on
only very weakly pleiotropically related phenotypes. Our approaches, however,
are more suited to determining whether the genetic basis of a trait of
interest, or a reasonably correlated trait, has been affected by
differentiating selection.
#### Ascertainment and Genetic Architecture
We next investigated the relationship between statistical power, the number of
loci associated with the trait, and the amount of variance explained by those
loci. Our simulations were motivated by the fact that the number of loci
identified by a given GWAS, and the fraction of variance explained by those
loci, will depend on both the design of the study (e.g. sample size) and the
genetic architecture of the trait. To illustrate the impact these factors have
on the power of our methods, we performed two experiments in which we again
held $\delta$ constant at 0.14. In the first, for each of the 1000 sets of 161
loci chosen above to mimic the height data ascertainment, we randomly sampled
$n$ loci, without regard to effect sizes, and recalculated the null
distribution and power for these reduced sets, allowing $n$ to range from 2 to
161\. The results of these simulations are shown in Figure 2C. This
corresponds to imagining that fewer loci had been ascertained by the initial
GWAS, and estimating the power our methods would have with this reduced set of
loci. As we down sample our loci without regard to effect sizes, the
horizontal axis of Figure 2C is proportional to the phenotypic variance
explained, e.g. the simulations in which only 80 loci are subsampled
correspond to having a dataset which explains only 50% of the variance
explained in those for which all 161 were used.
The second experiment is nearly identical to the first, except that before
adding an effect of selection to the subsampled loci, we linearly rescale the
effect sizes such that $V_{A}$ is held constant at the value calculated for
the full set of 161 loci. The results of these simulations are shown in Figure
2D. These simulations correspond to imagining that we have explained an
equivalent amount of phenotypic variance, but the number of loci over which
this variation is partitioned varies.
Our results (Figure 2C and 2D) demonstrate that even if only a small number of
loci associated with the phenotype have been identified, our tests offer
higher power than single locus-based tests. Moreover, for statistics that
appropriately deal with both covariance among loci and among populations
($r^{2}$ and $Q_{X}$), power is generally a constant function of variance
explained by the underlying loci, regardless of the number of loci over which
it is partitioned. Notably, most the power of $Q_{X}$ comes from the LD-like
component, especially when the number of loci is large. Statistics that rely
on an average of single locus metrics (the $F_{ST}$-like component of
$Q_{X}$), and those that rely on outliers ($r^{2}$ enrichment) all lose power
as the the variance explained is partitioned over more loci, as the effect of
selection at each locus is weaker. Somewhat surprisingly, the versions of our
tests that fail to adequately control for population structure (naive $r^{2}$
and $Q_{ST}$) also lose power as the phenotypic variance is spread among more
loci. We believe this reflects the fact that they are being systematically
mislead by LD among SNPs due to population structure, a problem which is
compounded as more loci are included in the test. Overall these results
suggest that accounting for population structure and using the LD between like
effect alleles is key to detecting selection on polygenic phenotypes.
#### Localizing Signatures of Selection
Lastly, we investigated the power of our conditional Z-scores to identify
signals of selection that are specific to particular populations or geographic
regions, and contrast this with the power of the global $Q_{X}$ statistic to
detect the same signal. We again perform two experiments. In the first, we
choose a single population whose allele frequencies to perturb, and leave all
other populations unchanged. In other words, an effect of selection is
mimicked according to (15), but with $Y_{m}$ set equal to one for a single
population, and zero for all others. We then increment $\delta$ to see how
power changes as the effect of selection becomes more pronounced. In Figure 2E
we display the results of these simulations for five populations chosen to
capture the range of power profiles for the populations we consider in our
empirical applications. In the last experiment, we chose a group of
populations to which to apply the allele frequency shift, again consistent
with (15), but now with $Y_{m}$ set equal to 1 for all populations in an
entire region, and zero elsewhere. In Figure 2F, we show the results of these
simulations, with each of the seven geographic/genetic clusters identified by
Rosenberg et al (2002) [61], chosen in turn as the affected region.
These simulations demonstrate that the conditional Z test can detect subtler
frequency shifts than the global $Q_{X}$ test, provided one knows which
population(s) to test a priori. They also show how unusual frequency patterns
indicative of selection are easier to detect in populations for which the
dataset contains closely related populations that are unaffected (e.g. compare
the Han and Italian to the San and Karitiana at the individual population
level, or Europe, the Middle East and Central Asia to Africa, America, and
Oceania at the regional level). Lastly, note that the horizontal axes in
Figure 2E and 2F are equivalent in the sense that for a given value of
$\delta$, alleles in (say) the Italian population have been shifted by the
same amount in the Italian specific simulations in Figure 2E as in the Europe-
wide simulation in Figure 2F, indicating that the HGDP dataset, power is
similar in efforts to detect local, population specific events, as well as
broader scale, regional level events.
### Empirical Applications
We estimated genetic values for each of six traits from the subset of GWAS
SNPs that were present in the HGDP dataset, as described above. We discuss the
analysis of each dataset in detail below, and address general points first.
For each dataset, we constructed the covariance matrix from a sample of
approximately $20,000$ appropriately matched SNPs, and the null distributions
of our test statistics from a sample of $10,000$ sets of null genetic values,
which were also constructed according to a similar matching procedure (as
described in the Methods).
In an effort to be descriptive and unbiased in our decisions about which
environmental variables to test, we tested each trait for an effect of the
major climate variables considered by Hancock et al (2008) [62] in their
analysis of adaptation to climate at the level of individual SNPs. We followed
their general procedure by running principal components (PC) analysis for both
seasons on a matrix containing six major climate variables, as well as
latitude and longitude (following Hancock et al’s rationale that these two
geographic variables may capture certain elements of the long term climatic
environment experienced by human populations). The percent of the variance
explained by these PCs and their weighting (eigenvectors) of the different
environmental variables are given in Table 1. We view these analyses largely
as a descriptive data exploration enterprise across a relatively small number
of phenotypes and distinct environmental variables, and do not impose a
multiple testing penalty against our significance measures. A multiple testing
penalization or false discovery rate approach may be needed when testing a
large number phenotypes and/or environmental variables.
We also applied our $Q_{X}$ test to identify traits whose underlying loci
showed consistent patterns of unusual differentiation across populations. In
Figure 3 we show for each GWAS set the observed value of $Q_{X}$ and its
empirical null distribution calculated using SNPs matched to the GWAS loci as
described above. We also plot the expected null distribution of the $Q_{X}$
statistic ($\sim\chi_{51}^{2}$). The expected null distribution closely
matches the empirical distribution in all cases, suggesting that our
multivariate normal framework provides a good null model for the data
(although we will use the empirical null distribution to obtain measures of
statistical significance).
For each GWAS SNP set we also separate our $Q_{X}$ statistic into its
$F_{ST}$-like and LD-like terms, as described in (14). In Figure 4 we plot the
null distributions of these two components for the height dataset as
histograms, with the observed value marked by red arrows (Figures S6-S10 give
these plots for the other five traits we examined). In accordance with the
expectation from our power simulations, the signal of selection on height is
driven entirely by covariance among loci in their deviations from neutrality,
and not by the deviations themselves being unusually large.
Lastly, we pursue a number or regionally restricted analyses. For each trait
and for each of the seven geographic/genetic clusters described by Rosenberg
et al (2002) [61], we compute a region specific $Q_{X}$ statistic to get a
sense for the extent to which global signals we detect can be explained by
variation among populations with these regions, and to highlight particular
populations and traits which may merit further examination as more association
data becomes available. The results are reported in Table 3. We also apply our
conditional Z-score approach at two levels of population structure: first at
the level of Rosenberg’s geographic/genetic clusters, testing each cluster in
turn for how differentiated it is from the rest of the world, and second at
the level of individual populations. The regional level Z-scores are useful
for identifying signals of selection acting over broad regional scale or on
deeper evolutionary timescales, while the population specific Z-scores are
useful for identifying very recent selection that has only impacted a single
population. We generally employ these regional statistics as a heuristic tool
to localize signatures of selection uncovered in global analyses, or in cases
where there is no globally interesting signal, to highlight populations or
regions which may merit further examination as more association data becomes
available. The result of these analyses are depicted in Figures 5 and 6, as
well as Tables S3-S14.
#### Height
We first analyzed the set of 180 height associated loci identified by Lango
Allen and colleagues [63], which explain about 10.5% of the total variance for
height in the mapping population, or about 15% of heritability [64]. This
dataset is an ideal first test for our methods because it contains the largest
number of associations identified for a single phenotype to date, maximizing
our power gain over single locus methods (Figure 2). In addition, Turchin and
colleagues [27] have already identified a signal of pervasive weak selection
at these same loci in European populations, and thus we should expect our
methods to replicate this observation.
Of the 180 loci identified by Lango Allen and colleagues, 161 were present in
our HGDP dataset. We used these 161 loci in conjunction with the allele
frequency data from the HGDP dataset to estimate genetic values for height in
the 52 HGDP populations. Although the genetic values are correlated with the
observed heights in these populations, they are unsurprisingly imperfect
predictions (see Figure S11 and Table S1, which compares our estimated genetic
values to observed sex-average heights for the subset of HGDP populations with
a close proxy in the dataset of Gustafsson and Lindenfors (2009)[65]).
We find a signal of excessive correlation with winter PC2 (Figure 7 and Table
2), but find no strong correlations with any other climatic variables. Our
$Q_{X}$ test also strongly rejects the neutral hypothesis, suggesting that our
estimated genetic values are overly dispersed compared to the null model of
neutral genetic drift and shared population history (Figure 3 and Table 2).
These results are consistent with with directional selection acting in concert
on alleles influencing height to drive differentiation among populations at
the level of the phenotype.
We followed up on these results by conducting regional level analyses, which
indicate that our signal of excess variance arises primarily from extreme
differentiation among populations within Europe (Table 3). Analyses using the
conditional multivariate normal model indicate that this signal is driven
largely by divergence between the French and Sardinian populations, in line
with Turchin et al’s (2012) previous observation of a North-South gradient of
height associated loci in Europe. We also find weaker signals of over-
dispersion in other regions, but the globally significant $Q_{X}$ statistic
can be erased by removing either the French or the Sardinian population from
the analysis, suggesting that the signal is primarily driven by
differentiation among those two populations.
#### Skin Pigmentation
We next analyzed data from a recent GWAS for skin pigmentation in an African-
European admixed population of Cape Verdeans [66], which identified four loci
of major effect that explain approximately 35% of the variance in skin
pigmentation in that population after controlling for admixture proportion.
Beleza et al (2013) report effect sizes in units of modified melanin (MM)
index, which is calculated as $100\times\text{log}(1/\%\text{melanin
reflectance at 650 nM})$, i.e. a higher MM index corresponds to darker skin,
and a lower value to lighter skin.
We used these four loci to calculate a genetic skin pigmentation score in each
of the HGDP populations. As expected, we identified a strong signal of excess
variance among populations, as well as a strong correlation with latitude
(Figure 7 and Table 2), again consistent with directional selection having
acted on the phenotype of skin pigmentation to drive divergence among
populations. Note, however, that this signal was driven entirely by the fact
that populations of western Eurasian descent have a lower genetic skin
pigmentation score than populations of African descent. Using only the markers
from [66], light skinned populations in East Asian and the Americas have a
genetic skin pigmentation score that is almost as high (dark) as that of most
African populations, an effect that is clearly visible when we plot the
measured skin pigmentation and skin reflectance of HGDP populations [67, 68]
against their genetic values (see Figures S12 and S13). The correlation with
latitude is thus weaker than one might expect, given the known phenotypic
distribution of skin pigmentation in human populations [67, 69]. To illustrate
this point further, we re-ran the analysis on a subsample of the HGDP
consisting of populations from Europe, the Middle East, Central Asia, and
Africa. In this subsample, the correlation with latitude, and signal of excess
variance, was notably stronger ($r^{2}=0.2$, $p=0.019$; $Q_{X}=60.1$,
$p=8\times 10^{-4}$).
This poor fit to observed skin pigmentation is due to the fact that we have
failed to capture all of the loci that contribute to variation in skin
pigmentation across the range of populations sampled, likely due to the
partial convergent evolution of light skin pigmentation in Western and Eastern
Eurasian populations [70]. Including other loci putatively involved in skin
pigmentation [71, 72] decreases the estimated genetic pigmentation score of
the other Eurasian populations (Figures S12 and S13 and Table S2), but we do
not include these in our main analyses as they differ in ascertainment (and
the role of KITLG in pigmentation variation has been contested by [66]).
Within Africa, the San population has a decidedly lower genetic skin
pigmentation score than any other HGDP African population. This is potentially
in accordance with the observation that the San are more lightly pigmented
than other African populations represented by the HGDP [67] and the
observation that other putative light skin pigmentation alleles have higher
frequency in the San than other African populations [70]. Although there is
still much work to be done on the genetic basis of skin pigment variation
within Africa, in this dataset a regional analysis of the six African
populations alone identifies a marginally significant correlation with
latitude ($r^{2}=0.62$, $p=0.0612$), and a signal of excess variance among
populations ($Q_{X}=16.19$, $p=0.01$), suggesting a possible role for
selection in the shaping of modern pigmentary variation within the continent
of Africa.
#### Body Mass Index
We next investigate two traits related to metabolic phenotypes (BMI and Type 2
diabetes), as there is a long history of adaptive hypotheses put forward to
explain phenotypic variation among populations, with little conclusive
evidence emerging thus far. We first focus on the set of 32 BMI associated
SNPs identified by Speliotes and colleagues [73] in their Table 1, which
explain approximately 1.45% of the total variance for BMI, or about 2-4% of
the additive genetic variance. Of these 32 associated SNPs, 28 were present in
the HGDP dataset, which we used to calculate a genetic BMI score for each HGDP
population. We identified no significant signal of selection at the global
level (Table 2).
Our regional level analysis indicated that the mean genetic BMI score is
significantly lower that expected in East Asia ($Z=-2.48,\ p=0.01$; see also
Figure 5 and Table S7), while marginal $Q_{X}$ statistics identify excess
intraregional variation within East Asia and the Americas (Table 3). While
these results are intriguing, given the small fraction of the additive genetic
variance explained by the ascertained SNPs and the lack of a globally
significant signal or a clear ecological pattern or explanation, it is
difficult to draw strong conclusions from them. For this reason BMI and other
related traits will warrant reexamination as more association results arise
and methods for analyzing association results from multiple correlated traits
are developed.
#### Type 2 Diabetes
We next investigated the 65 loci reported by Morris and colleagues [74] as
associated with T2D, which explain $5.7\%$ of the total variance for T2D
susceptibility, or about 8-9% of the additive genetic variance. Of these 65
SNPs, 61 were present in the HGDP dataset. We used effect sizes from the stage
1 meta-analysis, and where a range of allele frequencies are reported (due to
differing sample frequencies among cohorts), we simply used the average. Where
multiple SNPs were reported per locus we used the lead SNP from the combined
meta-analysis. Also note that Morris and colleagues report effects in terms
odds ratios (OR), which can be converted into additive effects by taking the
logarithm (the same is true of the IBD data from [25], analyzed below).
The distribution of genetic T2D risk scores showed no significant correlations
with any of the five eco-geographic axes we tested, and was in fact fairly
underdispersed worldwide relative to the null expectation due to population
structure (Table 2).
Our regional level analysis revealed that while T2D genetic risk is well
explained by drift in Africa, Central and Eastern Asia, Oceania, and the
Americas, European populations have far lower T2D genetic risk than expected
($Z=-2.79$, $p=0.005$) and Middle Eastern populations a higher genetic risk
than expected ($Z=2.37$, $p=0.018$). It’s not clear, however, that these
observations should be interpreted as evidence for selection either in Europe
or the Middle East. While the dichotomous regional labels “Europe” vs. “Middle
East” explains the majority of the variance not accounted for by population
structure ($r^{2}=0.77,p=5\times 10^{-4}$), this is essentially the same
signal detected by the regional Z scores, and our $Q_{X}$ statistic finds no
signal of excess variance ($Q_{X}=10.9$, $p=0.48$) among the twelve HGDP
populations in these two regions. Expanding to the next most closely related
region, we tested for a signal of excess differentiation between Central Asia
and either Europe or Middle East, but find no convincing signal in either case
($r^{2}=0.13,\ p=0.21;\ Q_{X}=12.0,\ p=0.75$ and $r^{2}=0.15,\ p=0.19;\
Q_{X}=9.8,\ p=0.63$ respectively). To the extent that our results are
consistent with an impact of selection on the genetic basis of T2D risk, they
appear to be consistent primarily with a scenario in which selection has
pushed the frequency of alleles that increase T2D risk up in Middle Eastern
populations and down in European populations. This is an extremely subtle
signal, which arises only after deep probing of the data, and as such we are
skeptical as to whether our results represent a meaningful signal of
selection.
A number of investigators have claimed that individual European GWAS loci for
Type 2 Diabetes show signals of selection [75, 62, 76, 77], a fact that is
seen as support for the idea that genetic variation for T2D risk has been
shaped by local adaptation, potentially consistent with a variation on the
thrifty genotype hypothesis [78]. However, our result suggest that local
adaptation has not had a large role in shaping the present day world-wide
distribution of T2D susceptibility alleles (as mapped to date in Europe). One
explanation of this discrepancy is that it is biologically unrealistic that
the phenotype of T2D susceptibility would exhibit strong adaptive
differentiation. Rather, local adaptation may have shaped some pleiotropically
related phenotype (which shares only some of the loci involved). However, as
seen in Figure 2, our methods have better power than single locus statistics
so long as there is a reasonable correlation ($\phi>0.3$) between the focal
phenotype and the one under selection. As such, the intersection of our
results with previous studies support the idea that local adaptation has had
little direct influence on the genetic basis of T2D or closely correlated
phenotypes, but that a handful of individual SNPs associated with T2D may have
experienced adaptive differentiation as a result of their function in some
other phenotype.
#### IBD
Finally, we analyzed the set of associations reported for Crohn’s Disease (CD)
and Ulcerative Colitis (UC) [25]. Because CD and UC are closely connected
phenotypes that share much of their genetic etiology, Jostins and colleagues
used a likelihood ratio test of four different models (CD only, UC only, both
CD and UC with equal effects on each, both CD and UC with independent effects)
to distinguish which SNPs where associated with either or both phenotypes, and
to assign effect sizes to SNPs (see their supplementary methods section 1d).
We take these classifications at face value, resulting in two partially
overlapping lists of 140 and 135 SNPs associated with CD and UC, which explain
$13.6\%$ and $7.5\%$ of disease susceptibility variance respectively. Of
these, there are 95 SNPs for CD and 89 SNPs for UC were present in our HGDP
dataset, and these remaining SNPs on which our analyses are based explain 9%
and 5.1% of the total variance. For now, we treat these sets of loci
independently, and leave the development of methods that appropriately deal
with correlated traits for future work.
We used these sets of SNPs to calculate genetic risk scores for CD and UC
across the 52 HGDP populations. Both CD and UC showed strong negative
correlations with summer PC2 (Figure 8), while CD also showed a significant
correlation with winter PC1, and a marginally significant correlation with
summer PC1 (Table 2).
We did not observe any significant $Q_{X}$ statistics for either trait, either
at the global or the regional level, suggesting that our environmental
correlation signals most likely arise from subtle differences between regions,
as opposed to divergence among closely related populations. Indeed, we find
moderate signals of regional level divergence in Europe (UC:
$Z=-2.08,p=0.04$), Central Asia (CD: $Z=2.21,p=0.03$), and East Asia (CD:
$Z=-1.90,p=0.06$ and UC: $Z=-2.12,p=0.03$; see also Figure 6 and Tables S13
and S14).
## Discussion
In this paper we have developed a powerful framework for identifying the
influence of local adaptation on the genetic loci underlying variation in
polygenic phenotypes. Below we discuss two major issues related to the
application of such methods, namely the effect of the GWAS ascertainment
scheme on our inference, and the interpretation of positive results.
### Ascertainment and Population Structure
Among the most significant potential pitfalls of our analysis (and the most
likely cause of a false positive) is the fact that the loci used to test for
the effect of selection on a given phenotype have been obtained through a GWAS
ascertainment procedure, which can introduce false signals of selection if
potential confounds are not properly controlled. We condition on simple
features of the ascertainment process via our allele matching procedure, but
deeper issues may arise from artifactual associations that result from the
effects of population structure in the GWAS ascertainment panel. Given the
importance of addressing this issue to the broader GWAS community, a range of
well developed methods exist for doing GWAS in structured populations, and we
refer the reader to the existing literature for a full discussion [79, 80, 81,
82, 83, 84, 85]. Here, we focus on two related issues. First, the propensity
of population structure in the GWAS ascertainment panel to generate false
positives in our selection analysis, and second, the difficulties introduced
by the sophisticated statistical approaches employed to deal with this issue
when GWAS are done in strongly structured populations.
The problem of population structure arises generally when there is a
correlation in the ascertainment panel between phenotype and ancestry such
that SNPs that are ancestry informative will appear to be associated with the
trait, even when no causal relationship exists [80]. This phenomenon can occur
regardless of whether the correlation between ancestry and phenotype is caused
by genetic or environmental effects. To make matters worse, multiple false
positive associations will tend to line up with same axis of population
structure. If the populations being tested with our methods lie at least
partially along the same axis of structure present in the GWAS ascertainment
panel, then the ascertainment process will serve to generate the very signal
of positive covariance among like effect alleles that our methods rely on to
detect the signal of selection.
The primary takeaway from this observation is that the more diverse the array
of individuals sampled for a given GWAS are with respect to ancestry, the
greater the possibility that failing to control for population structure will
generate false associations (or bias effect sizes) and hence false positives
for our method.
What bearing do these complications have on our empirical results? The GWAS
datasets we used can be divided into those conducted within populations of
European descent and the skin pigmentation dataset (which used an admixed
population). We will first discuss our analysis of the former.
The European GWAS loci we used were found in relatively homogeneous
populations, in studies with rigorous standards for replication and control
for population structure. Therefore, we are reasonably confident that these
loci are true positives. Couple this with the fact that they were ascertained
in populations that are fairly homogenous relative to the global scale of our
analyses, and it is unlikely that population structure in the ascertainment
panels is driving our positive signals. One might worry that we could still
generate false signals by including European populations in our analysis,
however many of the signals we see are driven by patterns outside of Europe
(where the influence of structure within Europe should be much lessened). For
height, where we do see a strong signal from within Europe, we use a set of
loci that have been independently verified using a family based design that is
immune to the effects of population structure [27] .
We further note that for a number of GWAS datasets, including some of those
analyzed here, studies of non-European populations have replicated many of the
loci identified in European populations [86, 87, 88, 89, 90, 91, 92], and for
many diseases, the failure of some SNPs to replicate, as well as discrepancies
in effect size estimate, are likely due to simple considerations of
statistical power and differences in patterns of LD across populations [93,
94]. This suggests that, at least for GWAS done in relatively homogenous human
populations, structure is unlikely to be a major confounding factor.
The issue of population structure may be more profound for our style of
approach when GWAS are conducted using individuals from more strongly
structured populations. In some cases it is desirable to conduct GWAS in such
populations as locally adaptive alleles will be present at intermediate
frequencies in these broader samples, whereas they may be nearly fixed in more
homogeneous samples. A range of methods have been developed to adjust for
population structure in these setting [95, 96, 97]. While generally effective
in their goal, these methods present their own issues for our selection
analysis. Consider the extreme case, such as that of Atwell et al (2010) [18],
who carried out a GWAS in Arabidopsis thaliana for 107 phenotypes across an
array of 183 inbred lines of diverse geographical and ecological origin.
Atwell and colleagues used the genome-wide mixed model program EMMA [82, 95,
96] to control for the complex structure present in their ascertainment panel.
This practice helps ensure that many of the identified associations are likely
to be real, but also means that the loci found are likely to have unusual
frequencies patterns across the species range. This follows from the fact that
the loci identified as associated with the trait must stand out as being
correlated with the trait in a way not predicted by the individual kinship
matrix (as used by EMMA and other mixed model approaches). Our approach is
predicated on the fact that we can use genome-wide patterns of kinship to
adjust for population structure, but this correction is exactly the null model
that loci significantly associated with phenotypes by mixed models have
overcome. For this reason, both the theoretical $\chi^{2}$ distribution of the
$Q_{X}$ statistic, as well as the empirical null distributions we construct
from resampling, may be inappropriate.
The Cape Verde skin pigmentation data we used may qualify as this second type
of study. The Cape Verde population is an admixed population of
African/European descent, and has substantial inter-individual variation in
admixture proportion. Due to its admixed nature, the population segregates
alleles which would not be at intermediate frequency in either parental
population, making it an ideal mapping population.
Despite the considerable population structure, the fact that intermarriage
continues to mix genotypes in this population means that much of the LD due to
the African/European population structure has been broken up (and the
remaining LD is well predicted by an individual’s genome-wide admixture
coefficient). Population structure seems to have been well controlled for in
this study, and a number of the loci have been replicated in independent
admixed populations. While we think it unlikely that the four loci we use are
false associations, they could in principle suffer from the structured
ascertainment issues described above, so it is unclear that the null
distributions we use are strictly appropriate. That said, provided that Beleza
and colleagues have appropriately controlled for population structure, under
neutrality there would be no reason to expect that the correlation among the
loci should be strongly positive with respect to the sign of their effect on
the phenotype, and thus the pattern observed is at least consistent with a
history of selection, especially in light of the multiple alternative lines of
evidence for adaptation on the basis of skin pigmentation [67, 98, 68, 99,
100, 69].
Further work is needed to determine how best to modify the tests proposed
herein to deal with GWAS performed in structured populations.
### Complications of Intepretation
Our understanding of the genetic basis of variation in complex traits remains
very incomplete, and as such the results of these analyses must be interpreted
with cautiously. That said, because our methods are based simply on the
rejection of a robust, neutral null model, an incomplete knowledge of the
genetic basis of a given trait should only lead to a loss of statistical
power, and not to a high false positive rate.
For all traits analyzed here except for skin pigmentation, the within
population variance for genetic value is considerably larger than the variance
between populations. This suggests that much of what we find is relatively
subtle adaptation even on the level of the phenotype, and emphasizes the fact
that for most genetic and phenotypic variation in humans, the majority of the
variance is within populations rather than between populations (see Figures
S14–S19). In many cases, the influence of the environment likely plays a
stronger role in the differences between populations for true phenotypes than
the subtle differences we find here (as demonstrated by the rapid change in
T2D incidence with changing diet, e.g. [101]). That said, an understanding of
how adaptation has shaped the genetic basis of a wide variety of phenotypes is
clearly of interest, even if environmental differences dominate as the cause
of present day population differences, as it informs our understanding of the
biology and evolutionary history of these traits.
The larger conceptual issues relate to the interpretation of our positive
findings, which we detail below. A number of these issues are inherent to the
conceptual interpretation of evidence for local adaptation [102].
#### Effect Size Heterogeneity and Misestimation
In all of our analyses, we have simply extrapolated GWAS effect sizes measured
in one population and one environment to the entire panel of HGDP populations.
It is therefore prudent to consider the validity of this assumption, as well
as the implications for our analyses when it is violated. Aside from simple
measurement error, there are two possible reasons that estimated effect sizes
from GWAS may not reflect the true effect sizes.
The first is that most GWAS hits likely identify tag SNPs that are in strong
LD with causal sites that are physically nearby on the chromosome, rather than
actual causal sites themselves [94, 93]. This acts to reduce the estimated
effect size in the GWAS sample. More importantly for the interpretation of our
signals, patterns of LD between tag SNPs and causal sites will change over
evolutionary time, and so a tag SNP’s allele frequencies will be an imperfect
measure of the differentiation of the causal SNP over the sampled populations.
This should lead to a reduction in our power to detect the effect of selection
in much the same way that power is reduced when selection acts on a trait that
is genetically correlated with the trait of interest (Figure 1B). This effect
will be especially pronounced when the populations under study have a shorter
scale of LD than the populations in which the effect have been mapped (e.g.
when applying effect sizes estimated in Europe to population of African
descent). In the case that selection has not affected the trait of interest,
the effect sizes have no association whatsoever with the distribution of
allele frequencies across populations unless such an association is induced by
the ascertainment process, as described above. Therefore, changes in the
patterns of LD between identified tag SNPs and causal sites will not lead to
an excess of false positives if the loci under study have not been subject to
spatially varying selection pressures.
The second is that the actual value of the additive effect at a causal site
may change across environments and genetic backgrounds due to genotype-by-
genotype (i.e. functional epistasis) and genotype-by-environment interactions.
Although the response at a given locus due to selection depends only the
additive effect of the allele in that generation, the additive effect itself
is a function of the environment and the frequencies of all interacting loci.
As all of these can change considerably during the course of evolution, the
effects estimated in one population may not apply in other populations, either
in the present day, or over history of the populations [1, 103]. We first wish
to stress that, as above, because our tests rely on rejection of a null model
of drift, differences in additive effects among populations or over time will
not lead to an excess of false positives, provided that the trait is truly
neutral. Such interactions can, however, considerably complicate the
interpretation of positive results. For example, different sets of alleles
could be locally selected to maintain a constant phenotype across populations
due to gene-by-environment interactions. Such a scenario could lead to a
signal of local adaptation on a genetic level but no change in the phenotype
across populations, a phenomenon known as countergradient variation [104].
It will be very difficult to know how reasonable it is to extrapolate effect
sizes among populations without repeating measurements in different
populations and different environments. Perhaps surprisingly, the existing
evidence suggests that for a variety of highly polygenic traits, effects sizes
and directions may be surprisingly consistent across human populations [86,
87, 88, 89, 90, 91, 92, 93, 94]. There is no particular reason to believe that
this will hold as a general rule across traits or across species, and thus
addressing this issue will require a great deal more functional genetic work
and population genetic method development, a topic which we discuss briefly
below in Future Directions.
#### Missing variants
As the majority of GWAS studies are performed in a single population they will
often miss variants contributing to phenotypic variation. This can occur due
to GxG or GxE interactions as outlined above, but also simply because those
variants are absent (or at low frequency) due to drift or selection among the
populations. Such cases will not create a false signal of selection if only
drift is involved, however, they do complicate the interpretation of positive
signals. A particularly dramatic example of this is offered by our analysis of
skin pigmentation associated loci, whose frequencies are clearly shaped by
adaptation. The alleles found by a GWAS in the Cape Verde population
completely fail to predict the skin pigmentation of East Asians and Native
Americans. This reflects the fact that a number of the alleles responsible for
light skin pigmentation in those populations are not variable in Cape Verde
due to the partially convergent adaptive evolution of light skin pigmentation
[70]. As a result, when we take the Eurasian HGDP populations we see a
significant correlation between genetic skin pigmentation score and longitude
($r^{2}=0.15,p=0.015$), despite the fact that no such phenotypic correlation
exists. While the wrong interpretation is easy to avoid here because we have a
good understanding of the true phenotypic distribution, for the majority of
GWAS studies such complications will be subtler and so care will have to be
taken in the interpretation of positive results.
#### Loss of constraint and mutational pressure
One further complication in the interpretation of our results is in how loss
of constraint may play a role in driving apparent signals of local adaptation.
Traits evolving under uniform stabilizing selection across all populations
should be less variable than predicted by our covariance model of drift, due
to negative covariances among loci, and so should be underrepresented in the
extreme tails of our environmental correlation statistics and the upper tail
of $Q_{X}$. As such, loss of constraint (i.e. weaker stabilizing selection in
some populations than others), should not on its own create a signal of local
adaptation. While the loci underpinning the phenotype can be subject to more
drift in those populations, there is no systemic bias in the direction of this
drift. Loss of constraint, therefore, will not tend to create significant
environmental correlations or systematic covariance between alleles of like
effect.
An issue may arise, however, when loss of constraint is paired with biased
mutational input (i.e. new mutations are more likely to push the phenotype in
one direction than another [105]) or asymmetric loss of constraint (selection
is relaxed on one tail of the phenotypic distribution). Under these two
scenarios, alleles that (say) increase the phenotype would tend to drift up in
frequency in the populations with loss of constraint, creating systematic
trends and positive covariance among like effect alleles at different loci,
and resulting in a positive signal under our framework. While one would be
mistaken to assume that the signal was necessarily that of recent positive
directional selection, these scenarios do still imply that selection pressures
on the genetic basis of the phenotype vary across space. Positive tests under
our methods are thus fairly robust in being signals of differential selection
among populations, but are themselves agnostic about the specific processes
involved. Further work is needed to establish whether these scenarios can be
distinguished from recent directional selection based on only allele
frequencies and effect sizes, and as always, claims of recent adaptation
should be supported by multiple lines of evidence beyond those provided by
population genomics alone.
#### Future directions
In this article we have focused on methods development and so have not fully
explored the range of populations and phenotypes to which our methods could be
applied. Of particular interest is the possibility of applying these methods
to GWAS performed in other species where the ecological determinants of local
adaptation are better understood [18, 19].
One substantial difficulty with our approach, particularly in its application
to other organisms, is that genome-wide association studies of highly
polygenic phenotypes require very large sample sizes to map even a fraction of
the total genetic variance. One promising way to partially sidestep this issue
is by applying methods recently developed in animal and plant breeding. In
these genomic prediction/selection approaches, one does not attempt to map
individual markers, but instead concentrates on predicting an individual’s
genetic value for a given phenotype using all markers simultaneously [106,
107, 108]. This is accomplished by fitting simple linear models to genome-wide
genotyping data, in principle allowing common SNPs to tag the majority of
causal sites throughout the genome without attempting to explicitly identify
them [109]. These methods have been applied to a range of species, including
humans [110, 111, 112, 113, 114, 115, 116], demonstrating that these
predictions can potentially explain a relatively high fraction of the additive
genetic variance within a population (and hence much of the total genetic
variance). As these predictions are linear functions of genotypes, and hence
allele frequencies, we might be able to predict the genetic values of sets of
closely related populations for phenotypes of interest and apply very similar
methods to those developed here. Such an approach may allow for substantial
gains in power, as it would greatly increase the fraction of the genetic
variance used in the analyses. However, if the only goal is to establish
evidence for local adaptation in a given phenotype, then because measurements
of true phenotypes inherently include all of the underlying loci, the optimal
approach is to perform a common garden experiment and employ statistical
methods such as those developed by Ovaskainen and colleagues [39, 117, 40],
assuming such experiments can be done.
As discussed in various places above, it is unlikely that all of the loci
underpinning the genetic basis of a trait will have been subject to the same
selection pressures, due to their differing roles in the trait and their
pleiotropic effects. One potential avenue of future investigation is whether,
given a large set of loci involved in a trait, we can identify sets of loci in
particular pathways or with a particular set of functional attributes that
drive the signal of selection on the additive genetic basis of a trait.
Another promising extension of our approach is to deal explicitly with
multiple correlated phenotypes. With the increasing number of GWAS efforts
both empirical and methodological work are beginning to focus on understanding
the shared genetic basis of various phenotypes [25, 118]. This raises the
possibility that we may be able to disentangle the genetic basis of which
phenotypes are more direct targets of selection, and which are responding to
correlated selection on these direct targets (for progress along these lines
using $Q_{ST}$, see [119, 120, 121, 39]). Such tools may also offer a way of
incorporating GxE interactions, as multiple GWAS for the same trait in
different environments can be treated as correlated traits [122].
As association data for a greater variety of populations, species, and traits
becomes available, we view the methods described out here as a productive way
forward in developing a quantitative framework to explore the genetic and
phenotypic basis of local adaptation.
## Materials and Methods
### Mean Centering and Covariance Matrix Estimation
Written in matrix notation, the procedure of mean centering the estimated
genetic values and dropping one population from the analysis can be expressed
as
$\vec{Z^{\prime}}=\mathbf{T}\vec{Z}$ (16)
where $\mathbf{T}$ is an $M-1$ by $M$ matrix with $\frac{M-1}{M}$ on the main
diagonal, and $-\frac{1}{M}$ elsewhere.
In order to calculate the corresponding expected neutral covariance structure
about this mean, we use the following procedure. Let $\mathbf{G}$ be an $M$ by
$K$ matrix, where each column is a vector of allele frequencies across the $M$
populations at a particular SNP, randomly sampled from the genome according to
the matching procedure described below. Let $\epsilon_{k}$ and $\epsilon_{i}$
be the mean allele frequency in columns $k$ and $i$ of $\mathbf{G}$
respectively, and let $\mathbf{S}$ be a matrix such that
$s_{ki}=\frac{1}{\sqrt{\epsilon_{k}(1-\epsilon_{k})\epsilon_{i}(1-\epsilon_{i})}}$.
With these data, we can estimate $\mathbf{F}$ as
$\mathbf{F}=\mathbf{TG}\mathbf{S}\mathbf{G^{T}T^{T}}.$ (17)
This transformation performs the operation of centering the matrix at the mean
value, and rooting the analysis with one population by dropping it from the
covariance matrix (the same one we dropped from the vector of estimated
genetic values), resulting in a covariance matrix describing the relationship
of the remaining $M-1$ populations. This procedure thus escapes the
singularity introduced by centering the matrix at the observed mean of the
sample.
As we do not get to observe the population allele frequencies, the entries of
$\mathbf{G}$ are the sample frequencies at the randomly chosen loci, and thus
the covariance matrix $\mathbf{F}$ also includes the effect of finite sample
size. Because the noise introduced by the sampling of individuals is
uncorrelated across populations (in contrast to that introduced by drift and
shared history), the primary effect is to inflate the diagonal entries of the
matrix by a factor of $\frac{1}{n_{m}}$, where $n_{m}$ is the number of
individuals sampled in population $m$ (see the supplementary material of [45]
for discussion). This means that our population structure adjusted statistics
also approximately control for differences in sample size.
#### Standardized environmental variable
Given a vector of environmental variable measurements for each population, we
apply both the $\mathbf{T}$ and Cholesky tranformation as for the estimated
genetic values
$\vec{Y^{\prime}}=\mathbf{C}^{-1}\mathbf{T}\vec{Y}.$ (18)
This provides us with a set of $M-1$ adjusted observations for the
environmental variable which can be compared to the transformed genetic values
for inference. This step is necessary as we have rotated the frame of
reference of the estimated genetic values, and so we must do the same for the
environmental variables to keep them both in a consistent reference frame.
### Identifying Outliers with Conditional MVN Distributions
As described in the Results, we can use our multivariate normal model of
relatedness to obtain the expected distribution of genetic values for an
arbitrary set of populations, conditional on the observed values in some other
arbitrary set.
We first partition our populations into two groups, those for which we want to
obtain the expected distribution of genetic values (group 1), and those on
which we condition in order to obtain this distribution (group 2). We then
re–estimate the covariance matrix such that it is centered on the mean of
group 2. This step is necessary because the amount of divergence between the
populations in group 1 and the mean of group 2 will always be greater than the
amount of divergence from the global mean, even under the neutral model, and
our covariance matrix needs to reflect this fact in order to make accurate
predictions. We can obtain this re-parameterized $\mathbf{F}$ matrix as
follows. If $M$ is the total number of populations in the sample, then let $q$
be the number of populations in group one, and let $M-q$ be the number of
populations in group 2. We then define a new $\mathbf{T_{R}}$ matrix such that
the $q$ columns corresponding the populations in group one have 1 on the
diagonal, and 0 elsewhere, while the $M-q$ columns corresponding to group two
have $\frac{M-q-1}{M-q}$ on the diagonal, and $-\frac{1}{M-q}$ elsewhere. We
can then re–estimate a covariance matrix that is centered at the mean of the
$M-q$ populations in group 2. Recalling our matrices $\mathbf{G}$ and
$\mathbf{S}$ from (17), this matrix is calculated as
$\displaystyle\mathbf{F_{R}}=\mathbf{T_{R}}\mathbf{G}\mathbf{S}\mathbf{G^{T}}\mathbf{T_{R}^{T}}$
(19)
where we write $\mathbf{F_{R}}$ to indicate that it is a covariance matrix
that has been re-centered on the mean of group two.
Once we have calculated this re–centered covariance matrix, we can use well
known results from multivariate normal theory to obtain the expected joint
distribution of the genetic values for group one, conditional on the values
observed in group two.
We partition our vector of genetic values and the re–centered covariance
matrix such that
$\displaystyle\vec{X}$ $\displaystyle=\begin{bmatrix}\vec{X}_{1}\\\
\vec{X}_{2}\end{bmatrix}$ (20) and $\displaystyle\mathbf{F}_{R}$
$\displaystyle=\begin{bmatrix}\mathbf{F}_{11}&\mathbf{F}_{12}\\\
\mathbf{F}_{21}&\mathbf{F}_{22}\end{bmatrix}$ (21)
where $\vec{X}_{1}$ and $\vec{X}_{2}$ are vectors of genetic values in group 1
and 2 respectively, and $\mathbf{F}_{11}$, $\mathbf{F}_{22}$ and
$\mathbf{F}_{12}=\mathbf{F}_{21}^{T}$ are the marginal covariance matrices of
populations within group 1, within group 2, and across the two groups,
respectively. Letting $\mu_{1}=\mu_{2}=\frac{1}{M-q}\sum_{m=M-q}^{M}X_{m}$
(i.e. the sum of the elements of $\vec{X}_{2}$), we wish to obtain the
distribution
$\displaystyle\vec{X}_{1}|\vec{X}_{2},\mu_{1},\mu_{2}\sim
MVN(\vec{\xi},\mathbf{\Omega}),$ (22)
where $\vec{\xi}$ and $\mathbf{\Omega}$ give the expected means and covariance
structure of the populations in group 1, conditional on the values observed in
group 2. These can be calculated as
$\displaystyle\vec{\xi}=\mathbb{E}[\vec{X}_{1}|\vec{X}_{2},\mu_{1},\mu_{2}]$
$\displaystyle=\mu_{1}\vec{1}+\mathbf{F}_{12}\mathbf{F}_{22}^{-1}\left(\vec{X}_{2}-\mu_{2}\vec{1}\right)$
(23) and
$\displaystyle\mathbf{\Omega}=\text{Cov}[X_{1}|X_{2},\mu_{1},\mu_{2}]$
$\displaystyle=\mathbf{F}_{11}-\mathbf{F}_{12}\mathbf{F}_{22}^{-1}\mathbf{F}_{21}.$
(24)
where the one vectors in line (23) are of length $q$ and $M-q$ respectively.
This distribution is itself multivariate normal, and as such this framework is
extremely flexible, as it allows us to obtain the expected joint distribution
for arbitrary sets of populations (e.g. geographic regions or continents), or
for each individual population. Further,
$\displaystyle\mathbb{E}\left[\frac{1}{q}\sum_{m=1}^{q}X_{m}\biggm{|}\vec{\xi},\mathbf{\Omega}\right]$
$\displaystyle=\frac{1}{q}\sum_{m=1}^{q}\xi_{m}$ (25) and
$\displaystyle\text{Var}\left[\frac{1}{q}\sum_{m=1}^{q}X_{m}\biggm{|}\vec{\xi},\mathbf{\Omega}\right]$
$\displaystyle=\frac{V_{A}}{q^{2}}\sum_{m=1}^{q}\sum_{n=1}^{q}\mathbf{\omega}_{mn}.$
(26)
where $\omega_{nm}$ denotes the elements of $\mathbf{\Omega}$. In words, the
conditional expectation of the mean estimated genetic value across group 1 is
equal to the mean of the conditional expectations, and its variance is equal
to the mean value of the elements of the conditional covariance matrix. As
such we can easily calculate a Z score (and corresponding p value) for group
one as a whole as
$\displaystyle
Z=\frac{\frac{1}{q}\sum_{m=1}^{q}X_{m}-\frac{1}{q}\sum_{m=1}^{q}\xi_{m}}{\frac{1}{q}\sqrt{V_{A}\sum_{m=1}^{q}\sum_{n=1}^{q}\omega_{m,n}}}.$
(27)
This Z score is a normal random variable with mean zero, variance one under
the null hypothesis, and thus measures the divergence of the genetic values
between the two populations relative to the null expectation under drift. Note
that the observation of a significant Z score in a given population or region
cannot necessarily be taken as evidence that selection has acted in that
population or region, as selection in the some of the populations on which we
condition (especially the closely related ones) could be responsible for such
a signal. As such, caution is warranted when interpreting the output of these
sort of analyses, and is best done in the context of more explicit information
about the demographic history, geography, and ecology of the populations.
### The Linear Model at the Individual Locus Level
As with our excess variance test, explored in the main text, it is natural to
ask how our environmental correlation tests can be written in terms of allele
frequencies at individual loci.
As noted in (8), we can obtain for each underlying locus a set of transformed
allele frequencies, which have passed through the same transformation as the
estimated genetic values. We assume that each locus $\ell$ has a regression
coefficient
$\beta_{\ell}=\gamma\alpha_{\ell}$ (28)
where $\gamma$ is shared across all loci so that
$p_{m\ell}^{\prime}\sim\gamma\alpha_{\ell}Y_{m}^{\prime}+e_{m\ell}$ (29)
where the $e_{m\ell}$ are independent and identically distributed residuals.
We can find the maximum likelihood estimate $\hat{\gamma}$ by treating
$\alpha_{\ell}Y_{m}^{\prime}$ as the linear predictor, and taking the
regression of the combined vector $\vec{p^{\prime}}$, across all populations
and loci, on the combined vector $\overrightarrow{\alpha Y^{\prime}}$. As such
$\hat{\gamma}=\frac{Cov(p^{\prime},\alpha Y^{\prime})}{Var(\alpha
Y^{\prime})}$ (30)
we can decompose this into a sum across loci such that
$\hat{\gamma}=\frac{\frac{1}{L}\sum_{\ell}Cov(p_{\ell}^{\prime},\alpha_{\ell}Y^{\prime})}{\frac{1}{L}\sum_{\ell}Var(\alpha_{\ell}Y)}=\frac{1}{\sum_{\ell}\alpha_{\ell}^{2}}\frac{\sum_{\ell}\alpha_{\ell}Cov(p_{\ell}^{\prime},Y^{\prime})}{Var(Y^{\prime})}.$
(31)
As noted in (8), our transformed genetic values can be written as
$X_{m}=2\sum_{\ell}\alpha_{\ell}p_{m\ell}^{\prime}$ (32)
and so the estimated slope ($\hat{\beta}$) of our regression
($\vec{X}=\beta\vec{Y^{\prime}}+\vec{e}$) is
$\hat{\beta}=\frac{Cov(X,Y^{\prime})}{Var(Y)}=\frac{2\sum_{\ell}\alpha_{\ell}Cov(p_{\ell}^{\prime},Y^{\prime})}{Var(Y^{\prime})}$
(33)
Comparing these equations, the mean regression coefficient at the individual
loci (31) and the regression coefficient of the estimated genetic values (33)
are proportional to each other via a constant that is given by one over two
times the sum of the effect sizes squared (i.e.
$\gamma=\frac{1}{2\sum_{\ell}\alpha_{\ell}^{2}}\beta$). Our test based on
estimating the regression of genetic values on the environmental variable is
thus mathematically equivalent to an approach in which we assume that the
regression coefficients of individual loci on the environmental variable are
proportional to one another via a constant that is a function of the effect
sizes. Such a relationship can also be demonstrated for the correlation
coefficient ($r^{2}$) calculated at the genetic value level and at the
individual locus level (this is not necessarily true for the rank correlation
$\rho$), however the algebra is more complicated, and thus we do not show it
here.
This is in contrast to the $r^{2}$ enrichment statistic we compute for the
power simulations, in which we assume that the correlations of individual loci
with the environmental variable are independent of one another, and then
perform a test for whether more loci individually show strong correlations
with the environmental variable than we would expect by chance.
### HGDP data and imputation
We used imputed allele frequency data in the HGDP, where the imputation was
performed as part of the phasing procedure of [58], as per the recommendations
of [123]. We briefly recap their procedure here:
Phasing and imputation were done using fastPHASE [124], with the settings that
allow variation in the switch rate between subpopulations. The populations
were grouped into subpopulations corresponding to the clusters identified in
[61]. Haplotypes from the HapMap YRI and CEU populations were included as
known, as they were phased in trios and are highly accurate. HapMap JPT and
CHB genotypes were also included to help with the phasing.
### Choosing null SNPs
Various components of our procedure involve sampling random sets of SNPs from
across the genome. While we control for biases in our test statistics
introduced by population structure through our $\mathbf{F}$ matrix, we are
also concerned that subtle ascertainment effects of the GWAS process could
lead to biased test statistics, even under neutral conditions. We control for
this possibility by sampling null SNPs so as to match the joint distribution
of certain properties of the ascertained GWAS SNPs. Specifically, we were
concerned that the minor allele frequency (MAF) in the ascertainment
population, the imputation status of the allele in the HGDP datasets, and the
background selection environment experienced at a given locus, as measured by
B value [60], might influence the distribution of allele frequencies across
populations in ways that we could not predict.
We partitioned SNPs into a three way contingency table, with 25 bins for MAF
(i.e. a bin size of 0.02), 2 bins for imputation (either imputed or not), and
10 bins for B value (B values range from 0 to 1, and thus our bin size was
0.1). For each set of null genetic values, we sampled one null SNP from the
same cell in the contingency table as each of the GWAS SNPs, and assigned this
null SNP the effect size associated with the GWAS SNP it was sampled to match.
While we do not assign effect sizes to sampled SNPs used to estimate the
covariance matrix $\mathbf{F}$ (instead simply scaling $\mathbf{F}$ by a
weighted sum of squared effect sizes, which is mathematically equivalent under
our assumption that all SNPs have the same covariance matrix), we follow the
same sampling procedure to ensure that $\mathbf{F}$ describes the expected
covariance structure of the GWAS SNPs.
For the skin pigmentation GWAS [66] we do not have a good proxy present in the
HGDP population, as the Cape Verdeans are an admixed population. Cape Verdeans
are admixed with $\sim 59.53\%$ African ancestry, and $41.47\%$ European
ancestry in the sample obtained by [66] (Beleza, pers. comm., April 8, 2013).
As such, we estimated genome wide allele frequencies in Cape Verde by taking a
weighted mean of the frequencies in the French and Yoruban populations of the
HGDP, such that $p_{CV}=0.5953p_{Y}+0.4147p_{F}$. We then used these estimated
frequencies to assign SNPs to frequency bins.
[66] also used an admixture mapping strategy to map the genetic basis of skin
pigmentation. However, if they had only mapped these loci in an admixture
mapping setting we would have to condition our null model on having strong
enough allele frequency differentiation between Africans and Europeans at the
functional loci for admixture mapping to have power [125]. The fact that [66]
mapped these loci in a GWAS framework allows us to simply reproduce the
strategy, and we ignore the results of the admixture mapping study (although
we note that the loci and effect sizes estimated were similar). This
highlights the need for a reasonably well defined ascertainment population for
our approach, a point which we comment further on in the Discussion.
## Acknowledgments
We would like to thank Gideon Bradburd, Yaniv Brandvain, Luke Jostins, Chuck
Langley, Joe Pickrell, Jonathan Pritchard, Peter Ralph, Jeff Ross-Ibarra,
Alisa Sedghifar, Michael Turelli and Michael Whitlock for helpful discussion
and/or comments on earlier versions of the manuscript. We thank Josh Schraiber
and Otso Ovaskainen for useful discussions via
http://haldanessieve.org/2013/07/31/the-population-genetic-signature-of-
polygenic-local-adaptation/Haldane’s Sieve.
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## Figure Legends
Figure 1: A schematic representation of the flow of our method. The boxes
colored blue are items provided by the investigator (GWAS SNP effect sizes,
the frequency of the GWAS SNPs across populations, and a environmental
variable). The boxes colored red make use of random SNPs sampled to match the
GWAS set as described in “Choosing null SNPs” in the methods section. For each
box featuring a calculated quantity a set of equation numbers are provided for
the relevant calculation. The Z score uses the untransformed genetic values,
rather than the transformed genetic values, but this relationship is not
depicted in the figure for the sake of readability. Figure 2: Power of our
statistics as compared to alternative approaches (A) across a range of
selection gradients ($\delta$) of latitude, and when we hold $\delta$ constant
at 0.14 and (B) decrease $\phi$, the genetic correlation between the trait of
interest and the selected trait, (C) vary the number of loci, and (D) vary the
number of loci while holding the fraction of variance explained constant.
Bottom panels show power of the Z-test and $Q_{X}$approaches to detect
selection affecting (E) a single population, and (F) multiple populations in a
given region. See main text for simulation details. Figure 3: Histogram of the
empirical null distribution of $Q_{X}$ for each trait, obtained from genome-
wide resampling of well matched SNPs. The mean of each distribution is marked
with a vertical black bar and the observed value is marked by a red arrow. The
expected $\chi^{2}_{M-1}$ density is shown as a black curve. Figure 4: The two
components of $Q_{X}$ for the height dataset, as described by the left and
right terms in (14). The null distribution of each statistic is shown as a
histogram. The mean value is shown as a black bar, and the observed value as a
red arrow. Figure 5: Visual representation of outlier analysis at the regional
and individual population level for (A) Height, (B) Skin Pigmentation, and (C)
Body Mass Index. For each geographic region we plot the expectation of the
regional average, given the observed values in the rest of the dataset as a
grey dashed line. The true regional average is plotted as a solid bar, with
darkness and thickness proportional to the regional Z score. For each
population we plot the observed value as a colored circle, with circle size
proportional to the population specific Z score. For example, in (A), one can
see that estimated genetic height is systematically lower than expected across
Africa. Similarly, estimated genetic height is significantly higher (lower) in
the French (Sardinian) population than expected, given the values observed for
all other populations in the dataset. Figure 6: Visual representation of
outlier analysis at the regional and individual population level for (A) Type
2 Diabetes, (B) Crohn’s Disease, and (C) Ulcerative Colitis. See Figure 5 for
explanation. Figure 7: Estimated genetic height (A) and skin pigmentation
score (B) plotted against winter PC2 and absolute latitude respectively. Both
correlations are significant against the genome wide background after
controlling for population structure (Table 2). Figure 8: Estimated genetic
risk score for Crohn’s Disease (A) and Ulcerative Colitis (B) risk plotted
against summer PC2. Both correlations are significant against the genome wide
background after controlling for population structure (Table 2). Since a large
proportion of SNPs underlying these traits are shared, we note that these
results are not independent.
## Tables
Table 1: The contribution of each geo-climatic variable to each of our four principal components, scaled such that the absolute value of the entries in each column sum to one (up to rounding error). We also show for each principal component the percent of the total variance across all eight variables that is explained by the PC. Geo-Climatic Variable | SUMPC1 | SUMPC2 | WINPC1 | WINPC2
---|---|---|---|---
Latitude | -0.16 | -0.10 | -0.17 | -0.01
Longitude | 0.02 | 0.12 | 0.04 | 0.05
Maximum Temp | 0.24 | -0.08 | 0.17 | -0.03
Minimum Temp | 0.24 | 0.07 | 0.17 | 0.08
Mean Temp | 0.25 | -0.03 | 0.17 | 0.03
Precipitation Rate | -0.01 | 0.16 | 0.07 | 0.32
Relative Humidity | -0.06 | 0.21 | -0.06 | 0.34
Short Wave Radiation Flux | -0.03 | -0.22 | 0.15 | -0.13
Percent of Variance Explained | 38% | 35% | 58% | 20%
Table 2: Climate Correlations and $Q_{X}$ statistics for all six phenotypes in
the global analysis. We report $sign(\beta)r^{2}$, for the correlation
statistics, such that they have an interpretation as the fraction of variance
explained by the environmental variable, after removing that which is
explained by the relatedness structure, with sign indicating the direction of
the correlation. P-values are two–tailed for $r^{2}$ and upper tail for
$Q_{X}$. Values for $\beta$ and $\rho$ are reported in Tables S15 and S16.
Phenotype | SUMPC1 | SUMPC2 | WINPC1 | WINPC2 | Latitude | $Q_{X}$
---|---|---|---|---|---|---
Height | $-0.03\ (0.21)$ | $10^{-5}$ (0.99) | $-0.008\ (0.52)$ | $\mathbf{0.086\ (0.035)}$ | 0.009 (0.50) | $\mathbf{86.9\ (0.002)}$
Skin Pigmentation | 0.061 (0.073) | 0.003 (0.69) | 0.048 (0.13) | $-0.008\ (0.51)$ | $\mathbf{-0.085\ (0.038)}$ | $\mathbf{79.1\ (0.006)}$
Body Mass Index | $-0.034\ (0.19)$ | 0.001 (0.82) | $-0.022\ (0.31)$ | 0.044 (0.14) | 0.031 (0.22) | $67.2\ (0.087)$
Type 2 Diabetes | 0.014 (0.40) | 0.012 (0.45) | 0.025 (0.27) | $-0.006\ (0.573)$ | $-0.05\ (0.11)$ | 39.3 (0.902)
Crohn’s Disease | 0.07 (0.062) | $\mathbf{-0.099\ (0.022)}$ | 0.0001 (0.94) | $\mathbf{-0.09\ (0.039)}$ | 0.01 (0.55) | 47.1 (0.68)
Ulcerative Colitis | 0.03 (0.21) | $\mathbf{-0.087\ (0.034)}$ | 0.004 (0.67) | $-0.049\ (0.12)$ | 0.01 (0.43) | 48.58 (0.61)
Table 3: $Q_{X}$ statistics and their empirical p-values for each of our six
traits in each of the seven geographic regions delimited by [61]. The
theoretical expected value of the statistic under neutrality for each region
is equal to $M-1$, where $M$ is the number of populations in the region. We
report the value of $M-1$ next to each region for reference.
| Europe (7) | Middle East (3) | Central Asia (8) | East Asia (16) | Americas (4) | Oceania (1) | Africa (6)
---|---|---|---|---|---|---|---
Height | $\mathbf{32.6\ (<10^{-4})}$ | 7.3 (0.07) | $\mathbf{15.5\ (0.05)}$ | 18.2 (0.33) | 4.2 (0.43) | 0.007 (0.94) | 5.4 (0.53)
Skin Pigmentation | 9.7 (0.22) | $\mathbf{9.6\ (0.026)}$ | $\mathbf{23.4\ (0.002)}$ | 13.8 (0.62) | 1.3 (0.89) | 0.38 (0.57) | $\mathbf{16.2\ (0.011)}$
Body Mass Index | 9.1 (0.24) | 1.6 (0.66) | 9.3 (0.32) | $\mathbf{28.4\ (0.03)}$ | $\mathbf{13.1\ (0.016)}$ | 1.2 (0.31) | 1.9 (0.94)
Type 2 Diabetes | 2.0 (0.96) | 0.90 (0.83) | 8.1 (0.43) | 7.5 (0.96) | 8.0 (0.13) | 2.5 (0.15) | 2.5 (0.88)
Crohn’s Disease | 6.6 (0.47) | 0.87 (0.84) | 7.56 (0.48) | 15.5 (0.52) | 1.3 (0.88) | 2.5 (0.13) | 2.6 (0.82)
Ulcerative Colitis | 8.4 (0.30) | 2.6 (0.48) | 10.9 (0.21) | 9.2 (0.907) | 0.43 (0.986) | 2.6 (0.12) | 3.5 (0.77)
## Supplementary Figure Legends
Figure S1: Power of tests described in the main text to detect a signal of
selection on the mapped genetic basis of skin pigmentation [66] as an
increasing function of the strength of selection (A), and a decreasing
function of the genetic correlation between skin pigmentation and the selected
trait with the effect of selection held constant at $\delta=0.13$ (B). Figure
S2: Power of tests described in the main text to detect a signal of selection
on the mapped genetic basis of BMI [73] as an increasing function of the
strength of selection (A), and a decreasing function of the genetic
correlation between BMI and the selected trait with the effect of selection
held constant at $\delta=0.07$ (B). Figure S3: Power of tests described in the
main text to detect a signal of selection on the mapped genetic basis of T2D
[74] as an increasing function of the strength of selection (A), and a
decreasing function of the genetic correlation between height and the selected
trait with the effect of selection held constant at $\delta=0.08$ (B). Figure
S4: Power of tests described in the main text to detect a signal of selection
on the mapped genetic basis of CD [25] as an increasing function of the
strength of selection (A), and a decreasing function of the genetic
correlation between CD and the selected trait with the effect of selection
held constant at $\delta=0.05$ (B). Figure S5: Power of tests described in the
main text to detect a signal of selection on the mapped genetic basis of UC
[25] as an increasing function of the strength of selection (A), and a
decreasing function of the genetic correlation between UC and the selected
trait with the effect of selection held constant at $\delta=0.05$ (B). Figure
S6: The two components of $Q_{X}$ for the skin pigmentation dataset, as
described by the left and right terms in (14). The null distribution of each
component is shows as a histogram. The expected value is shown as a black bar,
and the observed value as a red arrow. Figure S7: The two components of
$Q_{X}$ for the BMI dataset, as described by the left and right terms in (14).
The null distribution of each component is shows as a histogram. The expected
value is shown as a black bar, and the observed value as a red arrow. Figure
S8: The two components of $Q_{X}$ for the T2D dataset, as described by the
left and right terms in (14). The null distribution of each component is shows
as a histogram. The expected value is shown as a black bar, and the observed
value as a red arrow. Figure S9: The two components of $Q_{X}$ for the CD
dataset, as described by the left and right terms in (14). The null
distribution of each component is shows as a histogram. The expected value is
shown as a black bar, and the observed value as a red arrow. Figure S10: The
two components of $Q_{X}$ for the UC dataset, as described by the left and
right terms in (14). The null distribution of each component is shows as a
histogram. The expected value is shown as a black bar, and the observed value
as a red arrow. Figure S11: The genetic values for height in each HGDP
population plotted against the measured sex averaged height taken from [126].
Only the subset of populations with an appropriately close match in the named
population in [126]’s Appendix I are shown, values used are given in
Supplementary table S1 Figure S12: The genetic skin pigmentation score for a
each HGDP population plotted against the HGDP populations values on the skin
pigmentation index map of Biasutti 1959. Data obtained from Supplementary
table of [68]. Note that Biasutti map is interpolated, and so values are known
to be imperfect. Values used are given in Supplementary table S2 Figure S13:
The genetic skin pigmentation score for a each HGDP population plotted against
the HGDP populations values from the [67] mean skin reflectance (685nm) data
(their Table 6). Only the subset of populations with an appropriately close
match were used as in the Supplementary table of [68]. Values and populations
used are given in Table S2 Figure S14: The distribution of genetic height
score across all 52 HGDP populations. Grey bars represent the $95\%$
confidence interval for the genetic height score of an individual randomly
chosen from that population under Hardy-Weinberg assumptions Figure S15: The
distribution of genetic skin pigmentation score across all 52 HGDP
populations. Grey bars represent the $95\%$ confidence interval for the
genetic skin pigmentation score of an individual randomly chosen from that
population under Hardy-Weinberg assumptions Figure S16: The distribution of
genetic BMI score across all 52 HGDP populations. Grey bars represent the
$95\%$ confidence interval for the genetic BMI score of an individual randomly
chosen from that population under Hardy-Weinberg assumptions Figure S17: The
distribution of genetic T2D risk score across all 52 HGDP populations. Grey
bars represent the $95\%$ confidence interval for the genetic T2D risk score
of an individual randomly chosen from that population under Hardy-Weinberg
assumptions Figure S18: The distribution of genetic CD risk score across all
52 HGDP populations. Grey bars represent the $95\%$ confidence interval for
the genetic CD risk score of an individual randomly chosen from that
population under Hardy-Weinberg assumptions
Figure S19: The distribution of genetic UC risk score across all 52 HGDP
populations. Grey bars represent the $95\%$ confidence interval for the
genetic UC risk score of an individual randomly chosen from that population
under Hardy-Weinberg assumptions
## Supplementary Tables
Table S1: Genetic height scores as compared to true heights for populations
with a suitably close match in the dataset of [126]. See Figure S11 for a plot
of genetic height score against sex averaged height. Table S2: Genetic skin
pigmentation score as compared to values from Biasutti [127, 68] and [67]. We
also calculate a genetic skin pigmentation score including previously reported
associations at KITLG and OCA2 for comparisson. See also Figures S12 and S13.
Table S3: Conditional analysis at the regional level for the height dataset
Table S4: Conditional analysis at the individual population level for the
height dataset Table S5: Conditional analysis at the regional level for the
skin pigmentation dataset Table S6: Conditional analysis at the individual
population level for the skin pigmentation dataset Table S7: Condtional
analysis at the regional level for the BMI dataset Table S8: Conditional
analysis at the individual population level for the BMI dataset Table S9:
Conditional analysis at the regional level for the T2D dataset. Table S10:
Conditional analysis at the individual population level for the T2D dataset.
Table S11: Conditional analysis at the regional level for the CD dataset.
Table S12: Conditional analysis at the individual population level for the CD
dataset. Table S13: Conditional analysis at the regional level for the UC
dataset. Table S14: Conditional analysis at the individual population level
for the UC dataset Table S15: Corresponding $\beta$ statistics for all
analyses presented in Table 2. Table S16: Corresponding $\rho$ statistics for
all analyses presented in Table 2.
|
arxiv-papers
| 2013-07-29T22:29:58 |
2024-09-04T02:49:48.709226
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeremy J. Berg and Graham Coop",
"submitter": "Jeremy Berg",
"url": "https://arxiv.org/abs/1307.7759"
}
|
1307.7822
|
# Truthful Mechanisms for Secure Communication in Wireless Cooperative System
Jun Deng, Rongqing Zhang, , Lingyang Song, ,
Zhu Han, and Bingli Jiao,
Part of this work has been published in INFOCOM WKSHPS 2011 as given in
[1].Jun Deng, Rongqing Zhang, Lingyang Song and Bingli Jiao are with State Key
Laboratory of Advanced Optical Communication Systems and Networks, School of
Electronics Engineering and Computer Science, Peking University, Beijing,
China (email: {dengjun, rongqing.zhang, lingyang.song, jiaobl}@pku.edu.cn).
Zhu Han is with Electrical and Computer Engineering Department, University of
Houston, Houston, USA (email: [email protected]).
###### Abstract
To ensure security in data transmission is one of the most important issues
for wireless relay networks, and physical layer security is an attractive
alternative solution to address this issue. In this paper, we consider a
cooperative network, consisting of one source node, one destination node, one
eavesdropper node, and a number of relay nodes. Specifically, the source may
select several relays to help forward the signal to the corresponding
destination to achieve the best security performance. However, the relays may
have the incentive not to report their true private channel information in
order to get more chances to be selected and gain more payoff from the source.
We propose a Vickey-Clark-Grove (VCG) based mechanism and an
Arrow-d’Aspremont-Gerard-Varet (AGV) based mechanism into the investigated
relay network to solve this cheating problem. In these two different
mechanisms, we design different “transfer payment” functions to the payoff of
each selected relay and prove that each relay gets its maximum (expected)
payoff when it truthfully reveals its private channel information to the
source. And then, an optimal secrecy rate of the network can be achieved.
After discussing and comparing the VCG and AGV mechanisms, we prove that the
AGV mechanism can achieve all of the basic qualifications (incentive
compatibility, individual rationality and budget balance) for our system.
Moreover, we discuss the optimal quantity of relays that the source node
should select. Simulation results verify efficiency and fairness of the VCG
and AGV mechanisms, and consolidate these conclusions.
###### Index Terms:
Physical layer security, truth-telling, AGV, VCG
## I Introduction
Security is one of the most important issues in wireless communications due to
the broadcast nature of wireless radio channels. In recent years, besides the
traditional cryptographic mechanisms, information-theoretic-based physical
layer security has been developing fast. The concept of “wiretap channel” was
first introduced by Wyner [2], who showed that perfect secrecy of transmitted
data from the source to the legitimate receiver is achievable in degraded
broadcast channels. In follow-up work, Leung-Yan-Cheong and Hellman further
determined the secrecy capacity in the Gaussian wire-tap channel [3]. Later,
Csiszar and Komer extended Wyner’s work to non-degraded broadcast channels and
found an expression of secrecy capacity [4].
When considering a wireless relay network, realization of secrecy capacity is
much more complicated. In [5], the authors studied the secrecy capacity of a
relay channel with orthogonal components in the presence of a passive
eavesdropper node. In [6, 7], the authors demonstrated that cooperation among
relay nodes can dramatically improve the physical layer security in a given
wireless relay network, And in [8, 9, 10], the authors investigated the
physical layer security with friendly jammer in the relay networks. In the
related work mentioned above, the channel state information (CSI) is assumed
to be known at both the transmitter and the receiver. All these schemes are
assumed under true channel information reported by relay nodes, and the
optimal solutions in these works will not hold anymore if the fake channel
information is reported. However, in practice, the relay node always measures
its own channel gains and distributes the information to others through a
control channel. There is no guarantee that it reveals its private information
honestly. Hence, the most critical problem is how to select efficient relay
nodes to optimize the total secrecy rate in the network, while some selfish
relays may report false information to the source to increase their own
utilities. In [11], the reputation methods are designed to achieve this goal.
However, all these methods require a delicate and complex “detection scheme”
to monitor and capture the liar nodes. It needs a lot of signal consumption
like an independent entity called “Trust Manager” to runs on each node.
Besides, it also demands large amount of data like “REP_MESSAGE”,“REP_VAL” to
record the intermediate variable in the process. Moreover, this method
requires long time of observation because of low speed of convergence. It
might be impractical to use these reputation based scheme in cooperative relay
network.
In recent years, the game theory is widely applied into wireless and
communication networks to solve resource allocation problems[12, 13, 14, 15,
16, 17]. In the area of mechanism design, a field in the game theory studying
solution concepts for a class of private information games, a game designer is
interested in the game’s outcome and wants to motivate the players to disclose
their private information by designing the payoff structure [18, 19, 20]. For
example, the well-known VCG (Vickrey-Clarke-Groves) mechanism [21, 22, 23] is
a dominant-strategy mechanism, which can achieve ex-post incentive
compatibility (truth-telling is a dominant strategy for every player in the
game). However, it cannot implement the budget balance of the game [24, 25],
which costs extra payment from the players and decrease their payoffs. Thus,
it cannot be properly used in the relay network we focus on. Compared with the
VCG mechanism, the AGV (Arrow-d’Aspremont-Gerard-Varet) mechanism [26, 27] can
also solve the truth-telling problem. It is an incentive efficient mechanism
that can maximize the expected total payoff of all the players in the game.
Additionally it achieves the budget balance under a weaker participation
requirement [28, 29].
In this paper, we mainly focus on a relay network, in which all the channels
are orthogonal and each relay’s private channel information is unknown by
others. Under these conditions, we apply the ideas of the VCG and AGV
mechanism and prove that the transfer function can meet the basic requirements
of the wireless relay networks and help achieve the truth-telling target. We
find and prove that the unique Bayesian Nash Equilibrium [28] is achieved when
all the relays in the network reveal the truth. The incentive to report false
information will lead to a loss in each relay node’s own (expected) payment.
In other words, the competing relay nodes are enforced to obey the selection
criterion and cooperate with each other honestly. Furthermore, there is no
extra cost paid in the system when applying the AGV mechanism while the VCG
mechanism can not. Since the AGV mechanism is budget balanced, which means the
total transfer payment of all relay nodes equals zero. Simulation results show
that the relay nodes can maximize their utilities when they all report their
true channel information. Any cheating to the source leads to certain loss in
the total secrecy rate as well as the payoff of relays themselves. We also
observe that the optimal choice for the system is based on the relays’ channel
information, but in a majority of cases selecting only one relay node for
transmitting data can attain the largest secrecy rate of the system. In
addition, we prove with simulations that the best strategy for each relay node
under this payoff structure is to improve its own channel condition to enlarge
its secrecy rate and always report the truth to the source.
The remainder of this paper is organized as follows. In Section II, the system
model for a relay network is presented. In Section III, we elaborate on the
basic definition and qualifications of the mechanism design, and discuss the
VCG mechanism and AGV mechanism. In Section IV, we demonstrate the mechanism
solutions to enforce relays reveal the true private information, and analyze
these mechanism solutions. Simulation results are shown in Section V, and the
conclusions are drawn in Section VI.
## II System Model
Figure 1: System model for relay network with eavesdropper.
Considering a general cooperative network shown in Fig. 1. It consists of one
source node, one destination node, one eavesdropper node, and $I$ relay nodes,
which are denoted by $S$, $D$, $E$, and $R_{i}$, $i=1,2,\ldots,I$,
respectively. This cooperative network is conducted in two phases. In phase 1,
the source node broadcasts a signal $x$ to the destination node and all the
relay nodes, where only $N$ $(N\leq I)$ nodes can decode this signal correctly
due to their different geographical conditions. In phase 2, the source node
decides which nodes of those $N$ relays to forword information to the
destination node. The destination node combines messages from the source and
relays, according to the reported channel gains of both relay-destination and
relay-eavesdropper links. During the whole process, eavesdropper node wiretaps
the messages from the source node and the relay nodes. We assume the
orthogonal channel having the same bandwidth $W$. The source node hopes to
gain the highest secrecy rate by properly selecting some efficient relay nodes
based on their reported channel information. We denote the number of selected
relay nodes by $K$ $(K\leq N)$ and the set of $K$ relay nodes by
$\mathcal{K}$.
In the first phase, the received signal $y_{s,d}$, $y_{s,r_{i}}$, and
$y_{s,e}$ at destination node $D$, relays $R_{i}$, and eavesdropper $E$,
respectively, can be expressed as
$\displaystyle y_{s,d}=\sqrt{P_{s}}h_{s,d}x+n_{s,d},$ (1)
and
$\displaystyle y_{s,r_{i}}=\sqrt{P_{s}}h_{s,r_{i}}x+n_{s,r_{i}},$ (2)
and
$\displaystyle y_{s,e}=\sqrt{P_{s}}h_{s,e}x+n_{s,e},$ (3)
where $P_{s}$ represents the transmit power to the destination node from the
source node, $x$ is the unit-energy information symbol transmitted by the
source in phase 1, $h_{s,d}$, $h_{s,r_{i}}$, and $h_{s,e}$ are the channel
gains from $S$ to $D$, $R_{i}$ and $E$ respectively. $n_{s,d}$, $n_{s,r_{i}}$
and $n_{s,e}$ represent the noise at destination node, relay nodes and
eavesdropper node.
In the second phase, the received signal from the $i$-th relay node
$(R_{i}\in\mathcal{K})$ to the destination node and eavesdropper node can be
expressed as
$\displaystyle y_{r_{i},d}=\sqrt{P_{r_{i}}}h_{r_{i},d}x+n_{r_{i},d},$ (4)
and
$\displaystyle y_{r_{i},e}=\sqrt{P_{r_{i}}}h_{r_{i},e}x+n_{r_{i},e},$ (5)
respectively, where $P_{r_{i}}$ denotes the transmit power of relay node
$R_{i}$ under the power constraint $P_{r_{i}}\leq P_{max}$, $h_{r_{i},d}$ is
the channel gain between $R_{i}$ and $D$, and $h_{r_{i},e}$ is the channel
gain between $R_{i}$ and $E$. We assume that channel gain contains both the
path loss and the Rayleigh fading factor. Without loss of generality, we also
assume that all the links have the same noise power which is denoted by
${\sigma^{2}}$. The decode-and-forward (DF) protocol is used for relaying.
The direct transmission signal-to-noise-ratios (SNR) at the destination node
and eavesdropper from the source are
$\mbox{SNR}_{s,d}=\frac{P_{s}h_{s,d}^{2}}{\sigma^{2}},$ and
$\mbox{SNR}_{s,e}=\frac{P_{s}h_{s,e}^{2}}{\sigma^{2}},$ respectively. The SNR
at the destination node and eavesdropper node from relays are
$\mbox{SNR}_{r_{i},d}=\frac{{P_{r_{i}}h_{r_{i},d}^{2}}}{{\sigma^{2}}},$ and
$\mbox{SNR}_{r_{i},e}=\frac{{P_{r_{i}}h_{r_{i},e}^{2}}}{{\sigma^{2}}}.$
Therefore, the channel rate for relay $R_{i}$ to destination $D$ is
$\displaystyle C_{i,d}=W\log_{2}\left(1+\mbox{SNR}_{r_{i},d}\right).$ (6)
Similarly, the channel rate for relay $R_{i}$ to eavesdropper $E$ is
$\displaystyle C_{i,e}=W\log_{2}(1+\mbox{SNR}_{r_{i},e}).$ (7)
Then, the secrecy rate achieved by $R_{i}$ can be defined as [31]
$\displaystyle{C_{i,s}}={\left({{C_{i,d}}-{C_{i,e}}}\right)^{+}}={\left[{W{{\log}_{2}}\left({\frac{{1+\frac{{{P_{{r_{i}}}}h_{{r_{i}},d}^{2}}}{{{\sigma^{2}}}}}}{{1+\frac{{{P_{{r_{i}}}}h_{{r_{i}},e}^{2}}}{{{\sigma^{2}}}}}}}\right)}\right]^{+}},$
(8)
where $(x)^{+}=\max\\{x,0\\}$.
Besides, the secrecy rate assuming maximal ratio combining (MRC) at the
destination and the eavesdropper can be written as
$\displaystyle
C_{d,sys}=W\log_{2}\left(1+\mbox{SNR}_{s,d}+\sum\limits_{i\in\mathcal{K}}{\mbox{SNR}_{r_{i},d}}\right)$
(9)
and
$\displaystyle
C_{e,sys}=W\log_{2}\left(1+\mbox{SNR}_{s,e}+\sum\limits_{i\in\mathcal{K}}{\mbox{SNR}_{r_{i},e}}\right),$
(10)
respectively, such that the total secrecy rate attained by the system is
$\displaystyle{C_{s,sys}}$
$\displaystyle=\left({C_{d,sys}}-{C_{e,sys}}\right)^{+}$
$\displaystyle=W{\log_{2}}\left(1+\mbox{SNR}_{s,d}+\sum\limits_{i\in\mathcal{K}}{\mbox{SNR}_{r_{i},d}}\right)$
$\displaystyle-W{\log_{2}}\left(1+\mbox{SNR}_{s,e}+\sum\limits_{i\in\mathcal{K}}{\mbox{SNR}_{{r_{i}},e}}\right)$
(11)
$\displaystyle=W{\log_{2}}\left(\frac{{1+\mbox{SNR}_{s,d}+\sum\limits_{i\in\mathcal{K}}{\mbox{SNR}_{r_{i},d}}}}{1+\mbox{SNR}_{s,e}+\sum\limits_{i\in\mathcal{K}}{\mbox{SNR}_{{r_{i}},e}}}\right).$
## III Mechanism Design
In this paper, we use mechanism design as the framework to create an efficient
way to prevent relay nodes from cheating in the process of selection. This
section provides an overview of essential concepts in mechanism design, the
VCG mechanism and AGV mechanism.
### III-A Basic Definitions and Qualifications
Consider a public system consisting of $I$ agents, ${1,2,\ldots I}$. Each
agent $i\in\\{1,2\ldots I\\}$ has its private information
${\theta_{i}}\in{\Theta_{i}}$, which is known by itself only. A social choice
function $F$ is defined as
$\displaystyle
F:{\Theta_{1}}\times{\Theta_{2}}\times\ldots\times{\Theta_{I}}\to O,$
where $O$ stands for a set of possible outcomes.
A mechanism $\mathcal{M}$ is represented by the tuple
$(F,t_{1},\ldots,t_{I})$, where $t_{i}$ is the transfer payment of agent $i$
when the social choice is $F$. The utility of agent $i$:
$v_{i}\left[F\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right),\theta_{i}\right]$
depends on the outcome
$o=F\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right)$ and the true
information of agent $i$: $\theta_{i}$, where ${\hat{\theta}}_{i}$ denotes the
reported information of agent $i$, as opposed to ${\theta_{i}}$. Similarly,
${\hat{\theta}}_{-i}=\\{{\hat{\theta}_{1},\ldots,\hat{\theta}_{i-1},\hat{\theta}_{i+1},\ldots,\hat{\theta}_{I}}\\}$
is the reported information of all other agents. So the total payoff or
welfare of agent $i$ can be written as the following function:
$\displaystyle{u_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}},{\theta_{i}}\right)=v_{i}\left[F\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right),{\theta_{i}}\right]+{t_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right).$
(12)
The objective of mechanism $\mathcal{M}$ is to choose a desirable set of
transfer payments $t_{i}$. Thus, each agent in the mechanism will achieve its
maximum payoffs. In the following, we define some properties for the
mechanism.
_Definition 1_ : A mechanism is _incentive compatible (IC)_ if the truth-
telling is the best strategy for the agents: ${\hat{\theta}}_{i}=\theta_{i}$,
which means that agents have no incentives to reveal false information. The
dominant-strategy IC is defined as
$\displaystyle{u_{i}}\left({\theta_{i}},{{\hat{\theta}}_{-i}},{\theta_{i}}\right)\geq{u_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}},{\theta_{i}}\right),\;\;\;\forall{{\hat{\theta}}_{i}},{\theta_{i}}\in{\Theta_{i}},{{\hat{\theta}}_{-i}}\in{\Theta_{-i}}.$
(13)
_Definition 2_ : In an _individual rational (IR)_ mechanism, rational agents
are expected to gain a higher utility from actively participating in the
mechanism than from avoiding it. Especially, in the dominant strategy IR can
be expressed as
$\displaystyle{u_{i}}\left({\hat{\theta}_{i}},{\hat{\theta}_{-i}},{\theta_{i}}\right)\geq
0,\;\forall{\theta_{i}}\in{\Theta_{i}}.$ (14)
A mechanism that is both incentive-compatible and individual rational is said
to be _strategy-proof_.
_Definition 3_ : In a _budget balanced (BB)_ mechanism, the sum of all agents
transfer payments is zero, which implies that there is no transfer payment
paid from the mechanism designer to the agents or the other way around. The BB
is defined as
$\displaystyle\sum\limits_{i=1}^{I}{{t_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right)}=0,\;\forall{\theta_{i}}\in{\Theta_{i}}.$
(15)
### III-B VCG Mechanism
Groves introduced a group of mechanisms which satisfy IC and IR. The Groves
mechanisms are characterized by the following transfer payment function:
$\displaystyle{t_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right)=\sum\limits_{j\neq
i}^{I}{{v_{j}}\left[F\left({{\hat{\theta}}_{j}},{{\hat{\theta}}_{-j}}\right),{{\hat{\theta}}_{j}}\right]}-{\tau_{i}}\left({{\hat{\theta}}_{-i}}\right),$
(16)
where $\tau_{i}(.)$ can be any function of ${\hat{\theta}}_{i}$. The VCG
mechanism is an important special case of the Groves mechanisms for which
$\displaystyle{\tau_{i}}\left({{\hat{\theta}}_{-i}}\right)=\sum\limits_{j\neq
i}^{I}{{v_{j}}\left[{F^{*}}\left({{\hat{\theta}}_{j}},{{\hat{\theta}}_{-j}}\right),{{\hat{\theta}}_{j}}\right]},$
(17)
where ${{F^{*}}\left({{\hat{\theta}}_{j}},{{\hat{\theta}}_{-j}}\right)}$ is
the outcome of the mechanism when agent $i$ withdraws from the mechanism.
Thus, in the VCG mechanism agent $i$ could attain payoff as
$\displaystyle{u_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}},{\theta_{i}}\right)$
$\displaystyle={v_{i}}\left[F\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right),{\theta_{i}}\right]+{t_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right)$
$\displaystyle={v_{i}}\left[F\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right),{\theta_{i}}\right]+\sum\limits_{j\neq
i}^{I}{{v_{j}}\left[F\left({{\hat{\theta}}_{j}},{{\hat{\theta}}_{-j}}\right),{{\hat{\theta}}_{j}}\right]}$
$\displaystyle-\sum\limits_{j\neq
i}^{I}{{v_{j}}\left[{F^{*}}\left({{\hat{\theta}}_{j}},{{\hat{\theta}}_{-j}}\right),{{\hat{\theta}}_{j}}\right]}.$
(18)
As will be proved in the next section, the VCG mechanism can satisfy both the
incentive compatibility and individual rationality of each agent. However, the
VCG mechanism is not budget-balanced and requires a third party agent to
mediate between mechanism designer and agents.
### III-C AGV Mechanism
The AGV mechanism, an extension of the Groves mechanism, is possible to
achieve IC, IR and BB. It is an “expected form” of the Groves mechanism and
its transfer payment function is defined as
$\displaystyle{t_{i}}\left({{\hat{\theta}}_{i}},{{\hat{\theta}}_{-i}}\right)={E_{{\theta_{-i}}}}\left\\{\sum\limits_{j\neq
i}^{I}{v_{j}}\left[F\left({{\hat{\theta}}_{j}},{{\hat{\theta}}_{-j}}\right),{{\hat{\theta}}_{j}}\right]\right\\}-{\tau_{i}}\left({{\hat{\theta}}_{-i}}\right).$
(19)
The first term of $t_{i}$ is the expected total utility of agents $j\neq i$
when agent $i$ reports its information ${\hat{\theta}}_{i}$ with the
assumption that other agents report the truth. It is the function of agent
$i$’s report information only, exclusive of the actual strategies of agents
$j\neq i$, which making the AGV mechanism different from the VCG mechanism.
In the AGV mechanism it is possible to design the $\tau_{i}(.)$ to satisfy BB.
Let
$\displaystyle{\Phi_{i}}\left({{\hat{\theta}}_{i}}\right)={E_{{\theta_{-i}}}}\left\\{\sum\limits_{j\neq
i}^{I}{v_{j}}\left[F\left({{\hat{\theta}}_{j}},{{\hat{\theta}}_{-j}}\right),{{\hat{\theta}}_{j}}\right]\right\\}$
(20)
and
$\displaystyle{\tau_{i}}\left({{\hat{\theta}}_{-i}}\right)=\frac{{-\sum\limits_{j\neq
i}^{I}{{\Phi_{j}}\left({{\hat{\theta}}_{j}}\right)}}}{{I-1}},$ (21)
then budget balance can be achieved because each agent also pays an equal
share of total transfer payments distributed to the other agents, none of
which depends on its own report information. We will prove this property in
the following section.
## IV Mechanism Solutions
In this section, we first describe how mechanism design is applied in the
relay system and prove that there is no equilibrium achieved if no transfer
payment is introduced to relay nodes. Then, we show the practical mapping from
the utility, transfer payment, and payoff of the VCG mechanism and AGV
mechanism to the wireless cooperative network. Finally, we will compare and
analyze the difference between two mechanisms.
### IV-A Mechanism Implementation
In the network, each relay node reports its own channel information
$(h_{r_{i},d},h_{r_{i},e})$ to the source node which can be seen as different
agents report their own private information to mechanism designer. Assume
$\left\\{\left({{\tilde{h}}_{{r_{1}},d}},{{\tilde{h}}_{{r_{1}},e}}\right),\left({{\tilde{h}}_{{r_{2}},d}},{{\tilde{h}}_{{r_{2}},e}}\right),\ldots,\left({{\tilde{h}}_{{r_{K}},d}},{{\tilde{h}}_{{r_{K}},e}}\right)\right\\}$
is a realization of channel gains at one time slot, and relay nodes report
their information
$\left\\{\left(\hat{h}_{r_{1},d},\hat{h}_{r_{1},e}\right)\right.,$
$\left(\hat{h}_{r_{2},d},\hat{h}_{r_{2},e}\right),$ $\ldots,$
$\left.\left(\hat{h}_{r_{K},d},\hat{h}_{r_{K},e}\right)\right\\}$ to the
source node. Though the information may not be true, the source node will
still select relay nodes based on them. Define $R_{i}$’s private channel
information as
$\tilde{g}_{i}=\left\\{\tilde{h}_{r_{i},d},\tilde{h}_{r_{i},e}\right\\}$.
Thus, according to (8), the secrecy rate of relay $i$ depends on
$\tilde{g}_{i}$. The source node will choose $K$ relay nodes for transmitting
according to the relay’s reported information $\hat{g}$. The principle of
source node is to find the $K$ relays to maximize the secrecy rates. The
outcome function can be stated as
$\displaystyle
F({\bf{\hat{g}}})=\arg\max\sum\limits_{{R_{i}}\in\mathcal{K}}^{K}{{C_{i,s}}({{\hat{g}}_{i}})}.$
(22)
We define $\pi$ as the price per unit of secrecy rate achieved by the relay.
The relays in the network are assumed to be rational and fair-minded, which
means that although they are selfish, none is malicious. The object of relay
is to make itself chosen for transmitting so that it can gain payoff. Due to
the channel orthogonality, the _utility_ of $R_{i}$ can be expressed as
$\displaystyle{D_{i}}=\left\\{{\begin{array}[]{*{20}{c}}{\pi{C_{i,s}},\;\;\;\;\;\;\;\;\;\;\;\;{R_{i}}\in{\cal
K}},\\\ {0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{otherwise}.}\\\
\end{array}}\right.$ (25)
The total _payoff (utility)_ from the system can also be expressed as
$\displaystyle D=\sum\limits_{i=1}^{K}{{D_{i}}}.$ (26)
We assume that the channel information is the private information of each
relay, and thus, the source is unable to know whether the reported information
is true or not. Since only the relay nodes selected by the source for secure
data transmission can get the payoff, they will not report their true
information to the source in order to win greater opportunity to be selected.
In this situation, it can cause unfairness in selection and damage the
expected payoff of those unselected. It can also decrease the total payoff
paid by the system which can be expressed as $\hat{D}\leq\tilde{D},$ where
$\hat{D}$ represents the total the total payoff calculated according to the
information reported by the relay nodes. $\tilde{D}$ represents the total
payoff when all the relay nodes report the truth. These results can sabotage
the reliability of the system and eavesdropper can easily sniff the
transmitted messages.
Firstly, we prove that no equilibrium can be achieved under this condition.
Proposition 1: Assuming that $R_{i}$ does not know other relay’s channel
information, respectively, secrecy rate. But it knows that each relay obeys a
certain probability density function defined as
$p\left({{\tilde{C}}_{j,s}}\right)\left(0\leq{{\tilde{C}}_{j,s}}<\infty,j\neq
i\right)$. Then, $R_{i}$ has an incentive tendency to exaggerate its
${{\hat{C}}_{i,s}}$ to $\infty$ to get the maximum expected payoff.
###### Proof:
$R_{i}$’s expected payoff can be also be expressed as
$\displaystyle{D_{i}}({{\hat{g}}_{i}})=\pi{{\tilde{C}}_{i,s}}{\rm{P}}({R_{i}}\in\mathcal{K}),$
(27)
where $P\left(R_{i}\in\mathcal{K}\right)$ represents the probability of
$R_{i}$ when being chosen. Considering the principle of choosing relay,
$P\left(R_{i}\in\mathcal{K}\right)\propto{\hat{C}_{i,s}}$ and when
$\hat{C}_{i,s}\to\infty$, $P\left(R_{i}\in\mathcal{K}\right)\to 1$, so that
$R_{i}$ gets its maximum payoff at infinity. This indicates that every relay
node has the incentive to exaggerate its channel information to the source,
and thus, there is no equilibrium achieved under this kind of payoff
allocation. ∎
### IV-B VCG-based Mechanism Solution
In order to prevent relay nodes from reporting distorted channel information,
we propose an effective self-enforcing truth-telling mechanism to solve this
problem. By using the VCG-based mechanism, the honest relay nodes gain the
maximum payoff, as any cheating in the process will lead to decrease in
payoff. Like the VCG mechanism, we introduced this transfer payment of $R_{i}$
as
$\displaystyle{t_{i}}\left({{\hat{g}}_{i}},{{\hat{g}}_{-i}}\right)=\sum\limits_{j\neq
i}^{K}{{D_{j}}({{\hat{g}}_{j}})}-\sum\limits_{j\neq
i}^{K}{D_{j}^{*}({{\hat{g}}_{j}})},$ (28)
where $D_{j}^{*}(.)$ denotes the utility of $R_{j}$ when $R_{i}$ does not
participate in the system. So the total payoff of $R_{i}$ is:
$\displaystyle{U_{i}}\left({{\hat{g}}_{i}}\right)$
$\displaystyle={D_{i}}({{\hat{g}}_{i}})+{t_{i}}({{\hat{g}}_{i}},{{\hat{g}}_{-i}})$
$\displaystyle={D_{i}}({{\hat{g}}_{i}})+\sum\limits_{j\neq
i}^{N}{{D_{j}}({{\hat{g}}_{j}})}-\sum\limits_{j\neq
i}^{K}{D_{j}^{*}({{\hat{g}}_{j}})}$ (29)
$\displaystyle=\sum\limits_{j}^{K}{{D_{j}}({{\hat{g}}_{j}})}-\sum\limits_{j\neq
i}^{K}{D_{j}^{*}({{\hat{g}}_{j}}).}$
If one relay node claims a higher $\hat{h}_{r_{i},d}$ or a lower
$\hat{h}_{r_{i},e}$ than the reality to make its secrecy rate larger, it may
get more chances to be selected by the source node, but also will pay a higher
transfer payoff to those unselected. On the contrary, if one relay node
reports a lower secrecy rate than reality, it will receive the compensation
from other relay nodes at the cost of less chances to be selected. By adding
this transfer function, we will discuss some properties of this VCG-based
mechanism as follows.
Proposition 2: By using the VCG transfer function (28) to balance the payoff
allocation, relay node $R_{i}$ can gain its largest payoff when it reports the
true private channel information.
###### Proof:
We can see from (IV-B) that the payoff of each relay $R_{i}$ is the total
utility of all relays $\sum_{j}^{K}{{D_{j}}({{\hat{g}}_{j}})}$ when relay
participates in the system, minuses the total utility of all other relays
$\sum_{j\neq i}^{K}{D_{j}^{*}({{\hat{g}}_{j}})}$ when relay $i$ withdraws from
the system. It is obvious that relay $i$ cannot influence the value of
$\sum_{j\neq i}^{K}{D_{j}^{*}({{\hat{g}}_{j}})}$. Therefore, in order to
maximize its own payoff, relay $i$ seeks to maximize the total utility of the
system. According to our relay selection principle, the total utility of all
relays depends on the chosen $K$ relay’s true channel information. If and only
if each relay reports the true information $(\hat{g}_{i}=\tilde{g}_{i})$, the
total utility is maximized. Hence, the payoff of $R_{i}$ is maximized. ∎
Proposition 3: Every rational relay node in the system takes part in the VCG-
based mechanism for its own benefit.
###### Proof:
It is easy to show that $\sum_{j}^{K}{{D_{j}}({{\hat{g}}_{j}})}\geq\sum_{j\neq
i}^{K}{D_{j}^{*}({{\hat{g}}_{j}})}$ when the IC achieved
$(\hat{g}_{i}=\tilde{g}_{i})$, and the equality holds when $R_{i}$ is not
selected ($D_{i}=0$). Therefore, for each relay $R_{i}$ , $U_{i}(g_{i})\geq 0$
and participating into the system is an optimal choice for a rational relay. ∎
Proposition 4: By applying the VCG-based mechanism in our system, we cannot
achieve the BB condition: the total transfer payments
$\sum_{i=1}^{N}{{t_{i}}}<0$, which means that we need the mechanism designer
or a third party to pay parts of the payoff.
###### Proof:
There are two cases of $R_{i}$:
* •
It is not selected by the source node (${R_{i}}\notin\mathcal{K}$), then
obviously: ${t_{i}}\left({{\hat{g}}_{i}},{{\hat{g}}_{-i}}\right)=\sum_{j\neq
i}^{K}{{D_{j}}({{\hat{g}}_{j}})}-\sum_{j\neq
i}^{K}{D_{j}^{*}({{\hat{g}}_{j}})}=0$.
* •
It is selected by the source node $(R_{i}\in\mathcal{K})$, then
${t_{i}}({{\hat{g}}_{i}},{{\hat{g}}_{-i}})=\sum_{j\neq
i}^{K}{{D_{j}}({{\hat{g}}_{j}})}-\sum_{j\neq
i}^{K}{D_{j}^{*}({{\hat{g}}_{j}})}<0$. Because of the withdrawal of $R_{i}$,
another relay node with a lower secrecy rate will be selected and its utility
$D^{*}$ will get bigger.
Combine these two cases together: $K$ relay nodes will receive negative
transfer payment that makes the total transfer payments
$\sum_{i=1}^{K}{{t_{i}}}<0$. Hence, the BB is not satisfied in the VCG-based
mechanism. ∎
### IV-C AGV-based Mechanism Solution
From the discussions above we can know that the VCG-based mechanism can
enforce every relay node to tell the true private channel information, which
can effectively solve the cheating problem in our system. However, as the
mechanism designer, we need to pay some extra payments to the system because
the VCG-based mechanism fails the condition of BB. To compensate for this
loss, we improve the VCG-based mechanism to the AGV-based one.
In the AGV-based mechanism, we change the transfer payment of $R_{i}$ as
$\displaystyle{t_{i}}({{\hat{g}}_{i}},{{\hat{g}}_{-i}})={\Phi_{i}}({{\hat{g}}_{i}})-\frac{1}{{K-1}}\sum\limits_{j\neq
i}^{K}{{\Phi_{j}}({{\hat{g}}_{j}})}$ (30)
where
$\displaystyle{\Phi_{i}}\left({{\hat{g}}_{i}}\right)$
$\displaystyle={E_{{\hat{g}_{-i}}}}\left[\sum\limits_{j=1,j\neq
i}^{K}{D_{j}}({{\hat{g}}_{j}})\right]$ $\displaystyle=\sum\limits_{j=1,j\neq
i}^{K}{{E_{{\hat{g}_{-i}}}}\left[{D_{j}}({{\hat{g}}_{j}})\right]}$ (31)
represents the sum of the other relay nodes’ expected utilities given the
reported information $\hat{g}_{i}$.
Like in the VCG-based mechanism, we can prove that only if the relay nodes
reveal the true channel information, they can obtain the maximum payoff in the
AGV-based mechanism. There is only one equilibrium under this kind of payoff
allocation.
Proposition 5: By using the AGV-based mechanism, the relay node $R_{i}$ can
gain its largest expected payoff when it reports its true private channel
information to the source node.
###### Proof:
Without loss of generality, we consider the expected payoff of $R_{1}$. Since
$R_{1}$ only knows its own channel information, we can calculate the payoff
according to the transfer payment function (30) as
$\displaystyle
E[{U_{1}}({{\hat{g}}_{1}})]=E[{D_{1}}({{\hat{g}}_{1}})+{t_{1}}({{\hat{g}}_{1}},{{\hat{g}}_{-1}})]$
$\displaystyle=E[{D_{1}}({{\hat{g}}_{1}})]+{E_{{{\hat{g}}_{-1}}}}\left[\sum\limits_{j\neq
1}^{K}{D_{j}}({{\hat{g}}_{1}})\right]-\frac{1}{{K-1}}\sum\limits_{j\neq
1}^{K}{{\Phi_{j}}({{\hat{g}}_{j}})}$
$\displaystyle=E\left[\sum\limits_{j=1}^{K}{D_{j}}({{\hat{g}}_{j}})\right]-\frac{1}{{K-1}}\sum\limits_{j\neq
1}^{K}{{\Phi_{j}}({{\hat{g}}_{j}})}.$ (32)
We can see that there are two terms in the right side of (IV-C). The first one
represents the total expected payoff when $R_{1}$ reports $\hat{g}_{1}$ as its
channel information (the expectation is calculated by $R_{1}$ itself). Since
the other term being independent of $\tilde{g}_{1}$, only the first term
decides the expected payoff of $R_{1}$. As we have shown above, the total
payoff is based on the real secrecy rate. Only when the $K$ relays with top
$K$ secrecy rate are selected, the total payoff will be maximized. Any
cheating leads to a decrease in all relays’ total payoff, and therefore, the
expectation $E[U_{1}(\hat{g}_{1})]$ can get the maximum when $R_{1}$ reports
its true channel information. Similarly, each relay node in the network has an
incentive to report its true channel information
$({{\hat{g}}_{i}}={{\tilde{g}}_{i}})$. Thus, the equilibrium is achieved under
this condition. ∎
Proposition 6: Each relay node could gain a positive expected payoff in the
AGV-based mechanism, which ensures that every relay would like to take part in
this mechanism.
###### Proof:
From (30) and (IV-C) it is easy to derive $R_{i}$’s expected payoff:
$\displaystyle E[{U_{i}}({{\hat{g}}_{i}})]$
$\displaystyle=E\left[\sum\limits_{j=1}^{K}{D({{\hat{g}}_{j}})}\right]-\frac{1}{{K-1}}\sum\limits_{j\neq
i}^{K}{\sum\limits_{k\neq j}^{K}{{E_{-k}}[{D_{k}}({{\hat{g}}_{j}})]}}$
$\displaystyle=E\left[\sum\limits_{j=1}^{K}{D({{\hat{g}}_{j}})}\right]-\frac{1}{{K-1}}\left\\{(K-1)\sum\limits_{j=1}^{K}{E[{D_{j}}({{\hat{g}}_{j}})]}\right.$
$\displaystyle-\left.\sum\limits_{k\neq
i}^{K}{{E_{-k}}[{D_{k}}({{\hat{g}}_{i}})]}\right\\}$ (33)
$\displaystyle=\frac{1}{{K-1}}{E_{-j}}\left[\sum\limits_{j\neq
i}^{K}{{D_{j}}({{\hat{g}}_{i}})}\right].$
According to (27), $D_{i}>0$ if $R_{i}$ is selected and $D_{i}=0$ if not.
Since among all the relay nodes there are always some nodes being selected,
the right side of the equation above ${E_{-j}}\left[\sum\nolimits_{j\neq
i}^{N}{{D_{j}}({{\hat{g}}_{i}})}\right]>0$. Therefore, $R_{i}$ can gain a
payoff more than 0, and thus, the IR is satisfied. ∎
Proposition 7: In the AGV-based mechanism, the system can achieve budget
balance, which means that we, as the mechanism designer, will not pay any
extra payment to the system.
###### Proof:
If we calculate the total transfer payment of all relays, we could get
$\displaystyle\sum\limits_{i=1}^{N}{{t_{i}}({{\hat{g}}_{i}},{{\hat{g}}_{-i}})}$
$\displaystyle=\sum\limits_{i=1}^{N}{{\Phi_{i}}({{\hat{g}}_{i}})}-\frac{1}{{N-1}}\sum\limits_{i=1}^{N}{\sum\limits_{j=1,j\neq
i}^{N}{{\Phi_{j}}({{\hat{g}}_{j}})}}$
$\displaystyle=\sum\limits_{i=1}^{N}{{\Phi_{i}}({{\hat{g}}_{i}})}-\sum\limits_{j=1}^{N}{{\Phi_{j}}({{\hat{g}}_{j}})}=0.$
(34)
This implies that the proposed transfer function can realize a payment
reallocation among the relay nodes, and no extra payment is required to be
paid by the system or by the relay nodes. ∎
### IV-D The Value of K
In the discussion above, we assumed that the value of $K$ is fixed and the
source node always choose $K$ relays for cooperating. However, it is easy to
see that different $K$ can lead to different results in the total secrecy rate
of the network. So we did some research to figure out the optimal amount $K$
of relays the source node should select.
In our system model, the total secrecy attained of the system is (II). By
using the AGV mechanism, each relay reports the truth. Now we assume each
relay’s reported information is $\mbox{SNR}_{r_{i},d}$ and
$\mbox{SNR}_{r_{i},e}$. Let
$k_{i}=\frac{\mbox{SNR}_{r_{i},d}}{\mbox{SNR}_{r_{i},e}}$. Sort $k_{i}$ in
descending order, and get $k_{(1)}\geq k_{(2)}\geq\ldots\geq
k_{(N)},k_{(i)}\in\\{k_{1},k_{2},\ldots,k_{N}\\}$. Obviously, the relay which
has a larger $C_{i,s}$ also has a larger $k_{i}$ according to (8). We denote
that $R_{(1)}$ is the best relay which has the largest secrecy rate, $R_{(2)}$
is the second best, and so forth. Then the optimal selection strategy of the
source node is described as below:
1\. Select $R_{(1)}$ for transmitting. Let $i=1$ and calculate
$\Psi_{1}=\frac{1+\mbox{SNR}_{s,d}+\mbox{SNR}_{r_{(1)},d}}{1+\mbox{SNR}_{s,e}+\mbox{SNR}_{r_{(1)},e}}$.
2\. For $i<N$, if $\Psi_{i}<k_{(i+1)}$, proceed step 3 and if $\Psi_{i}\geq
k_{(i+1)}$, skip to step 4.
3\. Select $R_{(i+1)}$ and calculate
${\Psi_{i+1}}=\frac{{1+{\rm{SN}}{{\rm{R}}_{s,d}}+\sum\nolimits_{j=1}^{i+1}{{\rm{SN}}{{\rm{R}}_{{r_{(j)}},d}}}}}{{1+{\rm{SN}}{{\rm{R}}_{s,e}}+\sum\nolimits_{j=1}^{i+1}{{\rm{SN}}{{\rm{R}}_{{r_{(j)}},e}}}}}$.
Then let $i=i+1$ and go back to step 2.
4\. Let $K$ = i and stop.
Proposition 8: The system can attain the largest secrecy rate by selecting $K$
relays for transmitting data, where $K$ is decided by the process above.
###### Proof:
By the selection strategy of the source described above, it is easy to prove
that $\Psi_{K}$ is the maximum among
$\left\\{\Psi_{1},\Psi_{2},\ldots,\Psi_{N}\right\\}$. According to (II), the
total secrecy rate of the network when selecting $i$ relays can be expressed
as $C_{s,sys}(i)=W\log_{2}\Psi_{i}$. When $i=K$, $\Psi_{K}$ can get the
maximum, and obviously $C_{s,sys}(K)$ is the largest. Therefore, it is the
best choice for the source to select $K$ relays in the system. ∎
In many cases, because of the geographic conditions, the direct transmission
is very weak compared with relay transmission which means that the selected
relay has a
$\mbox{SNR}_{r_{i},d}>\mbox{SNR}_{r_{i},e}\gg\mbox{SNR}_{s,d}>\mbox{SNR}_{s,e}$.
So
$\mbox{SNR}_{r_{i},d}\gg(1+\mbox{SNR}_{s,d}),\mbox{SNR}_{r_{i},e}\gg(1+\mbox{SNR}_{s,e})$
and $\Psi_{1}\approx k_{(1)}>k_{(2)}$. Thus, $K=1$ is the best choice, which
means the source should only select one best relay for transmitting data. More
than one relay node would lead to a decrease in the total secrecy rate of the
system.
## V Simulation Results
In this section, we provide simulation results of the wireless relay system in
the VCG-based mechanism and AGV-based mechanism, respectively. Specifically,
to simplify the calculation and simulation, we assume that each relay node
first calculates its own secrecy rate according to its channel information,
and then reports it to the source. Without considering the process of
calculating $\pi C_{i,s}$, we assign random values $x_{i}$ to indicate $\pi
C_{i,s}(i=1,2,\ldots,N)$, which not affect the “outcome” or source’s selection
result. Furthermore, we assume that though $R_{i}$ does not know other relays’
channel information, it knows that every reported value obeys the probability
density function: $e^{-x_{i}}$
$\left(x_{i}\in[0,\infty)\;\mbox{and}\;\int_{0}^{+\infty}{e^{-x_{i}}dx_{i}}=1\right)$.
Firstly, we consider a system with $N=4$ relay nodes and from which the source
node chooses $K=2$ relays. A random sample of these relay nodes’ secrecy rates
is obtained as ${[1.0132,0.6091,0.3885,1.3210]}$ and the price per unit of
secrecy rate $\pi=1$ is assumed.
Figure 2: Payoff of $R_{i}$ when different secrecy rates are reported in the
VCG-based mechanism.
Fig. 2 shows the variation of $R_{i}$’s payoff when the reported values change
in the VCG-based mechanism. Given that the other three nodes are honest,
$R_{i}(i=1,2,3,4)$ can get its maximum payoff while reporting the truth. From
Fig. 2 we can observe that when they all tell the truth, the larger the true
value of secrecy rate of one relay node is, the more the payoff it gains. For
example, $R_{4}$ has the largest secrecy rate ($\tilde{C}_{4,s}=1.3210$) and
its payoff is the largest up to $0.5822$ when it reports the true value. It is
higher than the other three relay nodes’ payoff even though it is not as much
as $\pi C_{4,s}=1.3210$, which is paid by the destination node because of the
transfer payment.
Figure 3: Transfer payment of $R_{1}$ when different secrecy rates are
reported in the VCG-based mechanism.
In Fig. 3 we demonstrate the transfer payment of each relay they calculate
from their own angles when they report different secrecy rates. As is evident
in the figure, each relay has the same transfer payment curve. This is because
we assume each relay only knows the other relays report secrecy rates obey the
negative exponential distribution. So the difference of the utility of the
others relays whether the relay participates the mechanism or not is the same
for each relay. We can also see that they are all monotone decreasing because
the larger the reported value is, the more transfer payoff should be paid to
others. Besides, as the reported secrecy rate continuously increases, the
transfer payoff will tend to a fixed value. It is because this very large
reported value will always be larger than the others, the “outcome” or
source’s selection will be fixed whether this relay node is in this system or
not. So the transfer payment will be a fixed value when the reported value
becomes very large. The curve of $R_{i}$’s payoff in Fig. 2 is the same
reason.
Figure 4: Expected payoff of $R_{i}$ when different secrecy rates are reported
in the AGV-based mechanism. Figure 5: Expected transfer payment of $R_{i}$
when different secrecy rates are reported in the AGV-based mechanism. Figure
6: Expected payoff of $R_{1}$ with variable true secrecy rate when different
secrecy rates are reported in the AGV-based mechanism.
Fig. 4 and Fig. 5 show the results when we use the AGV-based mechanism in the
system. In Fig. 4, four curves show the expected payoff of $R_{1}$-$R_{4}$. It
is obvious to see that each relay maximizes its payoff when they report the
true secrecy rate. Compared with Fig. 2, we can see each relay’s payoff is
higher in the AGV-based mechanism than that in the VCG-based mechanism. It
means that the AGV-based mechanism can maximize all the relay nodes payoff
which is more attractive for relay node to attend. From Fig. 5 we also find
that $R_{1}$’s and $R_{4}$’s transfer payoffs are negative while the other
two’s are positive when they tell the truth. This is because $R_{1}$ and
$R_{4}$ are actually selected by the source node and need to pay the transfer
payment while $R_{2}$ and $R_{3}$ are not. By using the AGV-based mechanism,
the relay nodes with smaller secrecy rate will get compensations from those
with larger ones. It can balance the payment allocation of the system and
benefit those in worse physical conditions. Furthermore, we calculate the
expected transfer payoff of $R_{i}$ when they all report the truth:
$t_{1}=-0.1247$, $t_{2}=0.1570$, $t_{3}=0.2831$, $t_{4}=-0.3154$ and
$t_{1}+t_{2}+t_{3}+t_{4}=0$, which is in accord with the equation (27). Hence
the system is budget balanced and no extra payment is paid into or out of the
network. In conclusion, we can confirm that the AGV-based mechanism is more
compatible for our system than the VCG-based mechanism. Moreover, we show the
payoff of $R_{1}$ with changing secrecy rate when it reports different secrecy
rate in Fig. 6. It is obvious that no matter what true secrecy rate of $R_{1}$
is, $R_{1}$ always gains its maximum payoff when it reports its own true
secrecy rate.
Figure 7: Effectiveness of the reported secrecy of $R_{1}$ on the expected
payoff at different $K$ in the AGV-based mechanism. Figure 8: Effectiveness of
the reported secrecy of $R_{1}$ on the expected transfer payment at different
$K$ in the AGV-based mechanism.
In addition, we analyze the effects of the reported secrecy rate versus the
value of $K$ in the AGV-based mechanism. As an example, we set the relay node
$R_{1}$ be the interested one and its secrecy rate is $1.0132$. In Fig. 7, we
observe that the relay node $R_{1}$ achieve the maximum expected payoffs when
it reports the true secrecy rate with different $K$. When $K$ equals to $1$,
the payoff of the relay node $R_{1}$ is the lowest. And the larger the
reported value, the smaller the expected payoff shows. When $K$ equals $2$ and
$3$, the expected payoff becomes a fixed value as the reported secrecy rate
continuously increases. Because the transfer payment is a part of the total
payoff, the change of the expected payoff can be translated by the transfer
payment, which is shown in Fig. 8. When $K$ equals to $1$ and this relay
reports a larger secrecy rate, the expected transfer payment is a small and
negative value. So the expected payoff of the relay node is minor. When $K$
equals to $2$ and $3$, the slope of the curves becomes smoother as the
reported value increases. This shows the change trend of the relay node’s
payoff from the other point of view. Meanwhile, it implies that the transfer
payoff is helpful to control the payoff for fairness among relay nodes.
Figure 9: System secrecy rate at different $K$ at bad SNR in the AGV-based
mechanism.
Finally, we focus on the effect of the value of $K$ on the total system
secrecy. Here we assume $W=\ln 2$ then $C_{s,sys}(K)=\ln(\Psi(k))$. Let $N=6$,
the direct transmission SNR to destination and eavesdropper are $9.64$dB and
$5.47$dB, respectively, and given two random samples for $R_{i}$’s report
information ($\mbox{SNR}_{r_{i},d}$ and $\mbox{SNR}_{r_{i},e}$). In one sample
we assume each value has the same order of magnitude with $1$
$(1\mbox{dB}<\mbox{SNR}<10\mbox{dB}):\mbox{SNR}_{r_{i},d}=\\{6.1734,$
$7.9489,$ $9.7429,$ $7.1886,$ $6.3783,$ $7.3411\\},(\mbox{dB})$;
$\mbox{SNR}_{r_{i},e}=\\{3.7700,$ $0.9927,$ $5.6543,$ $4.3645,$ $0.6273,$
$6.1954\\},(\mbox{dB})$. In the other sample we assume each value has the same
order of magnitude with $10$
$(10\mbox{dB}<\mbox{SNR}<20\mbox{dB}):\mbox{SNR}_{r_{i},d}=\\{16.173,$
$17.948,$ $19.742,$ $17.188,$ $16.378,$ $17.341\\},(\mbox{dB});$
$\mbox{SNR}_{r_{i},e}=\\{13.770,$ $10.992,$ $15.654,$ $14.364,$ $10.627,$
$16.1954\\},(\mbox{dB})$. Similarly, we do this simulation with another group
data of high SNR and low SNR as follows: $\mbox{SNR}_{r_{i},d}=\\{8.8149,$
$5.6809,$ $9.3701,$ $8.5822,$ $3.3896,$ $10.000\\},(\mbox{dB})$;
$\mbox{SNR}_{r_{i},e}$ $=\\{3.7700,$ $0.9927,$ $5.6543,$ $4.3645,$ $0.6273,$
$6.1954\\},(\mbox{dB})$; $\mbox{SNR}_{r_{i},d}$ $=\\{$$16.173,$ $17.948,$
$19.742,$ $17.188,$ $16.378,$ $17.341\\},(\mbox{dB})$; and
$\mbox{SNR}_{r_{i},e}=\\{$$15.227,$ $12.164,$ $14.522,$ $13.278,$ $12.746,$
$13.648\\},(\mbox{dB})$. The simulation result is showed in Fig. 9, and we can
observe that in the low SNR situation, $C_{s,sys}$ is maximized when the
source select 2 and 3 relays, respectively. Thus, they attain the maximum
secrecy at $K=2$ and $K=3$. However, in the high SNR situation showed in Fig.
9, when all of the channel conditions are better, the best choice for the
system is to choose only one relay $(K=1)$ for transmitting. All these results
are based on the fact that all relays will reveal their true channel
information which is ensured by the AGV mechanism.
## VI Conclusions
In this paper, we discussed and applied the ideas of mechanism design into the
wireless relay network to guarantee the strategy-proof during the process of
relay selection when considering secure data transmission. We proved that by
using the VCG mechanism and AGV mechanism, each relay node gets its maximum
payoff only when it reveals its true channel information, and any deviation
from the truth will lead to a loss in its own (expect) payoff as well as the
total secrecy rate. We compared these two mechanisms and illustrated that the
AGV mechanism is more compatible for our system when taking the budget balance
constraint into consideration. We proved that the strategy-proof and budget
balance of the system can be achieved in the AGV mechanism, which makes our
model more practical in reality. Simulation results verified these
conclusions. Moreover, we proposed and proved the best choice for the source
node in deciding how many relays it should select to get the maximum secrecy
rate of the network. In good channel conditions with higher SNR, it is better
to select only one relay for transmitting data.
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|
arxiv-papers
| 2013-07-30T05:25:01 |
2024-09-04T02:49:48.730426
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun Deng, Rongqing Zhang, Lingyang Song, Zhu Han, and Bingli Jiao",
"submitter": "Rongqing Zhang",
"url": "https://arxiv.org/abs/1307.7822"
}
|
1307.7971
|
# Notes on a Theorem of Benci-Gluck-Ziller-Hayashi
Fengying Li111Email:[email protected] and Shiqing
Zhang222Email:[email protected]
The School of Economic and Mathematics, Southwestern University of Finance and
Economics,
Chengdu 611130, China
Mathematical School, Sichuan University, Chengdu610064 ,China
###### Abstract
We use constrained variational minimizing methods to study the existence of
periodic solutions with a prescribed energy for a class of second order
Hamiltonian systems with a $C^{2}$ potential function which may have an
unbounded potential well. Our result can be regarded as complementary to the
well-known theorem of Benci-Gluck-Ziller and Hayashi.
Key Words: $C^{2}$ second order Hamiltonian systems, periodic solutions,
constrained variational minimizing methods.
2000 Mathematical Subject Classification: 34C15, 34C25, 58F.
## 1\. Introduction
Based on the earlier works of Seifert([20]) in 1948 and Rabinowitz([18,19]) in
1978 and 1979, Benci ([4]), and Gluck-Ziller ([11]), and Hayashi([13])
published work examining the periodic solutions for second order Hamiltonian
systems
$\ddot{q}+V^{\prime}(q)=0$ (1.1) $\frac{1}{2}|\dot{q}|^{2}+V(q)=h$ (1.2)
with a fixed energy. Utilizing the Jacobi metric and very complicated geodesic
methods with algebraic topology, they proved the following general theorem:
Theorem 1.1 Suppose $V\in C^{2}(R^{n},R)$. If the potential well
$\\{x\in R^{n}|V(x)\leq h\\}$
is bounded and non-empty, then the system (1.1)-(1.2) has a periodic solution
with energy h.
Furthermore, if
$V^{\prime}(x)\not=0,\hskip 11.38092pt\forall x\in\\{x\in R^{n}|V(x)=h\\},$
then the system (1.1)-(1.2) has a nonconstant periodic solution with energy h.
For the existence of multiple periodic solutions for (1.1)-(1.2) with compact
energy surfaces, we can refer to Groessen([12]) and Long[14] and the
references therein.
In 1987, Ambrosetti-Coti Zelati[2] successfully used Clark-Ekeland’s dual
action principle and Ambrosetti-Rabinowitz’s Mountain Pass theorem to study
the existence of $T$-periodic solutions of the second-order equation
$-\ddot{x}=\nabla U(x),$
where
$U=V\in C^{2}(\Omega;\mathbb{R})$
such that
$U(x)\to\infty,x\to\Gamma=\partial\Omega;$
with $\Omega\subset\mathbb{R}^{n}$ a bounded convex domain. Their principle
result is the following:
Theorem 1.2 Suppose
1. 1.
$U(0)=0=\min U$
2. 2.
$U(x)\leq\theta(x,\nabla U(x))$ for some $\theta\in(0,\tfrac{1}{2})$ and for
all $x$ near $\Gamma$ (superquadraticity near $\Gamma$)
3. 3.
$(U^{\prime\prime}(x)y,y)\geq k|y|^{2}$ for some $k>0$ and for all
$(x,y)\in\Omega\times\mathbb{R}^{N}$.
Let $\omega_{N}$ be the greatest eigenvalue of $U^{\prime\prime}(0)$ and
$T_{0}=(2/\omega_{N})^{1/2}$.
Then $-\ddot{x}=\nabla U(x)$ has for each $T\in(0,T_{0})$ a periodic solution
with minimal period $T$.
The dual variational principle and Mountain Pass Lemma again proved the
essential ingredients for the following theorem of Coti Zelati-Ekeland-Lions
[8] concerning Hamiltonian systems in convex potential wells.
Theorem 1.3 Let $\Omega$ be a convex open subset of $R^{n}$ containing the
origin $O$. Let $V\in C^{2}(\Omega,R)$ be such that
$(V1).\ V(q)\geq V(O)=0,\forall q\in\Omega$
$(V2).\ \forall q\neq O,V^{\prime\prime}(q)>0$
$(V3).\ \exists\omega>0,$ such that
$V(q)\leq\frac{\omega}{2}\|q\|^{2},\forall\|q\|<\epsilon$
and
$(V4).V^{\prime\prime}(q)^{-1}\rightarrow 0,\|q\|\rightarrow 0$ or,
$(V4)^{\prime}.V^{\prime\prime}(q)^{-1}\rightarrow
0,q\rightarrow\partial\Omega.$
Then, for every $T<\frac{2\pi}{\sqrt{\omega}}$, the system (1.1) has a
solution with minimal period $T.$
In Theorems 1.2 and 1.3, the authors assumed the convex conditions for
potentials and potential wells in order to apply Clark-Ekeland’s dual
variational principle. We observe that Theorems 1.1-1.3 essentially make the
assumption
$V(x)\to\infty,x\to\Gamma=\partial\Omega$
so that all potential wells are bounded. We wish to generalize Theorems
1.1-1.3 from two directions: (1) We dispense with the convex assumption on
potential functions, (2) $V(x)$ can be uniformly bounded, and the potential
well can be unbounded.
In 1987, D.Offin ([16]) generalized Theorem 1.1 to some non-compact cases for
$V\in C^{3}(R^{n},R)$ under complicated geometric assumptions on the potential
wells; however, these geometric conditions appear difficult to verify for
concrete potentials. In 2009, Berg-Pasquotto-Vandervorst ([5]) studied the
closed orbits on non-compact manifolds with some complex topological
assumptions.
Using simpler constrained variational minimizing method, we obtain the
following result:
Theorem 1.4 Suppose $V\in C^{2}(R^{n},R),h\in R$ satisfies
$(V_{1}).\ V(-q)=V(q)$
$(V_{2}).\ V^{\prime}(q)q>0,\forall q\neq 0$
$(V_{3}).\ 3V^{\prime}(q)q+(V^{\prime\prime}(q)q,q)\neq 0,\forall q\neq 0$
$(V_{4}).\ \exists\mu_{1}>0,\mu_{2}\geq 0,$ such that
$V^{\prime}(q)\cdot q\geq\mu_{1}V(q)-\mu_{2}$
$(V_{5}).\ \lim_{|q|\rightarrow\infty}Sup[V(q)+\frac{1}{2}V^{\prime}(q)q]\leq
A$
$(V_{6}).\ \frac{\mu_{2}}{\mu_{1}}<h<A.$
Then the system $(1.1)-(1.2)$ has at least one non-constant periodic solution
with the given energy h.
Corollary 1.5 Suppose $V(q)=a|q|^{2n},a>0$, then the system $\forall
h>0$,$(1.1)-(1.2)$ has at least one non-constant periodic solution with the
given energy h.
Remark 1 Suppose $V(x)$ is the following well-known $C^{\infty}$ function:
$V(x)=e^{\frac{-1}{|x|}},\forall x\neq 0;$ $V(0)=0.$
Then $V(x)$ satisfies $(V_{1})-(V_{5})$ if we take $\mu_{1}=\mu_{2}>0$ and
$A=1$ in Theorem 1.4, but $(V_{6})$ does not hold .
Proof In fact,it’s easy to check $(V_{1})-(V_{5})$:
(1). It’s obvious for $(V_{1})$.
(2). For $(V_{2})$ and $(V_{3})$, we notice that
$V^{\prime}(x)x=\frac{1}{|x|}e^{\frac{-1}{|x|}}>0,\forall x\not=0,$
$(V^{\prime\prime}(x)x,x)=e^{\frac{-1}{|x|}}(\frac{-2}{|x|}+\frac{1}{|x|^{2}})$
$3V^{\prime}(x)x+(V^{\prime\prime}(x)x,x)=e^{\frac{-1}{|x|}}(\frac{1}{|x|}+\frac{1}{|x|^{2}})>0,\forall
x\neq 0.$
(3). For $(V_{4})$, we set
$w(x)=(\frac{1}{|x|}-\mu_{1})e^{\frac{-1}{|x|}};\hskip
5.69046ptx\not=0,w(0)=0.$
We will prove $w(x)>-\mu_{1}$; in fact,
$w^{\prime}(x)=[\frac{1}{|x|}-(1+\mu_{1})]\frac{x}{|x|^{3}}e^{\frac{-1}{|x|}};x\not=0,w^{\prime}(0)=0.$
From $w^{\prime}(x)=0$ ,we have $x=-\frac{1}{1+\mu_{1}}$ or $0$ or
$\frac{1}{1+\mu_{1}}$.
It’s easy to see that $w(x)$ is strictly increasing on
$(-\infty,-\frac{1}{1+\mu_{1}}]$ and $[0,\frac{1}{1+\mu_{1}}]$ but strictly
decreasing on $[\frac{-1}{1+\mu_{1}},0]$ and $[\frac{1}{1+\mu_{1}},+\infty)$.
We notice that
$\lim_{|x|\rightarrow+\infty}w(x)=-\mu_{1},$
and
$w(0)=0.$
So
$w(x)>-\mu_{1}.$
When we take $\mu_{2}=\mu_{1}>0$,$(V_{4})$ holds.
(4). For $(V_{5})$, we have
$V(x)+\frac{1}{2}V^{\prime}(x)x=e^{\frac{-1}{|x|}}(1+\frac{1}{2}\frac{1}{|x|})<1,\forall
x\neq 0;$
$V(0)+\frac{1}{2}V^{\prime}(0)0=0.$
Corollary 1.6 Given any $a>0,n\in N$, suppose
$V(x)=a|x|^{2n}+e^{\frac{-1}{|x|}},x\not=0,V(0)=0$. Then $\forall h>1$, the
system $(1.1)-(1.2)$ has at least one non-constant periodic solution with the
given energy $h$.
Remark 2 The potential $V(x)$ in Remark 1 is noteworthy since the potential
function is non-convex and bounded which satisfies neither of the conditions
of Theorems 1.1-1.3, Offin’s geometrical conditions, nor Berg-Pasquotto-
Vandervorst’s complex topological assumptions. Notice the special properties
for our potential well. It is a bounded set if $h<1$, but for $h\geq 1$ it is
$R^{n}$ \- an unbounded set. We also notice that the symmetrical condition on
the potential simplified our Theorem 1.4 and it’s proof; it seems interesting
to observe to obtain non-constant periodic solutions if the symmetrical
condition is deleted.
## 2 A Few Lemmas
Let
$H^{1}=W^{1,2}(R/Z,R^{n})=\\{u:R\rightarrow R^{n},u\in L^{2},\dot{u}\in
L^{2},u(t+1)=u(t)\\}$
Then the standard $H^{1}$ norm is equivalent to
$\|u\|=\|u\|_{H^{1}}=\left(\int^{1}_{0}|\dot{u}|^{2}dt\right)^{1/2}+|\int_{0}^{1}u(t)dt|.$
Lemma 2.1([1]) Let
$M=\\{u\in H^{1}|\int_{0}^{1}(V(u)+\frac{1}{2}V^{\prime}(u)u)dt=h\\}.$
If $(V_{3})$ holds, then $M$ is a $C^{1}$ manifold with codimension 1 in
$H^{1}.$
Let
$f(u)=\frac{1}{4}\int^{1}_{0}|\dot{u}|^{2}dt\int^{1}_{0}V^{\prime}(u)udt$
and $\widetilde{u}\in M$ be such that $f^{\prime}(\widetilde{u})=0$ and
$f(\widetilde{u})>0$. Set
$\frac{1}{T^{2}}=\frac{\int^{1}_{0}V^{\prime}(\widetilde{u})\widetilde{u}dt}{\int^{1}_{0}|\dot{\widetilde{u}}|^{2}dt}$
If $(V_{2})$ holds, then $\widetilde{q}(t)=\widetilde{u}(t/T)$ is a non-
constant $T$-periodic solution for (1.1)-(1.2).
When the potential is even, then by Palais’s symmetrical principle ([17]) and
Lemma 2.1, we have
Lemma 2.2([1]) Let
$F=\\{u\in M|u(t+1/2)=-u(t)\\}$
and suppose $(V_{1})-(V_{3})$ holds. If $\widetilde{u}\in F$ be such that
$f^{\prime}(\widetilde{u})=0$ and $f(\widetilde{u})>0$,then
$\widetilde{q}(t)=\widetilde{u}(t/T)$ is a non-constant $T$-periodic solution
for (1.1)-(1.2). In addition, we have
$\forall u\in F,\int_{0}^{1}u(t)dt=O.$
Recall the following two classic results.
Lemma 2.3(Sobolev-Rellich-Kondrachov[15],[22])
$W^{1,2}(R/Z,R^{n})\subset C(R/Z,R^{n})$
and the imbedding is compact.
Lemma 2.4(Eberlein-Smulian [21]) A Banach space $X$ is reflexive if and only
if any bounded sequence in $X$ has a weakly convergent subsequence.
Definition 2.1(Tonelli ,[15]) Let $X$ is a Banach space and $M\subset X$. If
it the case that for any sequence $\\{x_{n}\\}\subset M$ strongly convergent
to $x_{0}$ ($x_{n}\rightarrow x_{0}$), we have $x_{0}\in M$, then we call $M$
a strongly closed (closed) subset of $X$; if for any $\\{x_{n}\\}\subset M$
weakly convergent to $x_{0}$ ($x_{n}\rightharpoonup x_{0}$), we have $x_{0}\in
M$, then we call $M$ a weakly closed subset of $X$.
Let $f:M\rightarrow R$.
(i). If for any $\\{x_{n}\\}\subset M$ strongly convergent to $x_{0}$,we have
$liminff(x_{n})\geq f(x_{0}),$
then we say $f(x)$ is lower semi-continuous at $x_{0}$.
(ii). If for any $\\{x_{n}\\}\subset M$ weakly convergent to $x_{0}$, we have
$liminff(x_{n})\geq f(x_{0}),$
then we say $f(x)$ is weakly lower semi-continuous at $x_{0}$.
Using his variational principle, Ekeland proved
Lemma 2.5(Ekeland[9]) Let $X$ be a Banach space and $F\subset X$ a closed
(weakly closed) subset. Suppose that $\Phi$ defined on $X$ is Gateaux-
differentiable and lower semi-continuous (or weakly lower semi-continuous) and
that $\Phi|_{F}$ restricted on $F$ is bounded from below. Then there is a
sequence $x_{n}\subset F$ such that
$\Phi(x_{n})\rightarrow\inf_{F}\Phi\hskip 11.38092pt\hbox{ and }\hskip
11.38092pt\|\Phi|_{F}^{{}^{\prime}}(x_{n})\|\rightarrow 0.$
Definition 2.2([9,10]) Let $X$ be a Banach space and $F\subset X$ a closed
(weakly closed) subset. Suppose that $\Phi$ defined on $X$ is Gateaux-
differentiable. If it is true that whenever $\\{x_{n}\\}\subset F$ such that
$\Phi(x_{n})\rightarrow c\hskip 11.38092pt\hbox{ and }\hskip
11.38092pt\|\Phi|_{F}^{{}^{\prime}}(x_{n})\|\rightarrow 0,$
then $\\{x_{n}\\}$ has a strongly convergent (weakly convergent) subsequence,
we say $\Phi$ satisfies the $(PS)_{c,F}$ ($(WPS)_{c,F}$) condition at the
level $c$ for the closed subset $F\subset X$.
Using $\bf Lemma2.5$, it is easy to prove the following lemma.
Lemma 2.6 Let $X$ be a Banach space,
(i). Let $F\subset X$ be a closed subset. Suppose that $\Phi$ defined on $X$
is Gateaux-differentiable and lower semi-continuous and bounded from below on
$F$. If $\Phi$ satisfies $(PS)_{\inf\Phi,F}$ condition, then $\Phi$ attains
its infimum on $F$.
(ii).Let $F\subset X$ be a weakly closed subset. Suppose that $\Phi$ defined
on $F$ is Gateaux-differentiable and weakly lower semi-continuous and bounded
from below on $F$. If $\Phi$ satisfies $(WPS)_{\inf\Phi,F}$ condition, then
$\Phi$ attains its infimum on $F$.
## 3 The Proof of Theorem 1.4
We prove the Theorem as a sequence of claims.
Claim 3.1 If $(V_{1})-(V_{6})$ hold, then for any given $c>0$, $f(u)$
satisfies the $(PS)_{c,F}$ condition; that is, if $\\{u_{n}\\}\subset F$
satisfies
$\displaystyle f(u_{n})\rightarrow c>0\ \ \hbox{ and }\ \
f|_{F}^{\prime}(u_{n})\rightarrow 0,$ (3.1)
then $\\{u_{n}\\}$ has a strongly convergent subsequence.
Proof First, we prove the constrained set $F\not=\emptyset$ under our
assumptions. Using the notation of [1], for $a>0$ let
$\displaystyle
g_{u}(a)=g(au)=\int^{1}_{0}[V(au)+\frac{1}{2}V^{\prime}(au)au]dt.$ (3.2)
By the assumption $(V_{3})$, we have
$\displaystyle\frac{d}{da}g_{u}(a)\not=0$ (3.3)
and so $g_{u}$ is strictly monotone. By $(V_{5})$, we have
$\displaystyle\lim_{a\rightarrow+\infty}g_{u}(a)\leq A$ (3.4)
By $(V_{4})$, we notice that
$\displaystyle g_{u}(0)=V(O)\leq\frac{\mu_{2}}{\mu_{1}}.$ (3.5)
So for $V(O)<h<A$, the equation $g_{u}(a)=h$ has a unique solution $a(u)$ with
$a(u)u\in M.$
By $f(u_{n})\rightarrow c$, we have
$\displaystyle\frac{1}{4}\int^{1}_{0}|\dot{u_{n}}(t)|^{2}dt\cdot\int^{1}_{0}V^{\prime}(u_{n})u_{n}dt\rightarrow
c,$ (3.6)
and by $(V_{4})$ we have
$\displaystyle
h=\int^{1}_{0}(V(u_{n})+\frac{1}{2}<V^{\prime}(u_{n}),u_{n}>)dt\leq(\frac{1}{\mu_{1}}+\frac{1}{2})\int_{0}^{1}V^{\prime}(u_{n})u_{n}dt+\frac{\mu_{2}}{\mu_{1}}.$
(3.7)
By (3.6) and (3.7) we have
$\displaystyle\int_{0}^{1}V^{\prime}(u_{n})u_{n}dt\geq\frac{h-\frac{\mu_{2}}{\mu_{1}}}{\frac{1}{2}+\frac{1}{\mu_{1}}}.$
(3.8)
Condition $(V_{6})$ provides $h>\frac{\mu_{2}}{\mu_{1}}$. Then (3.6) and (3.8)
imply $\int^{1}_{0}|\dot{u_{n}}(t)|^{2}dt$ is bounded and
$\|u_{n}\|=\|\dot{u}_{n}\|_{L^{2}}$ is bounded.
We know that $H^{1}$ is a reflexive Banach space, so by the embedding theorem,
$\\{u_{n}\\}$ has a weakly convergent subsequence which uniformly strongly
converges to $u\in H^{1}$. The argument to show $\\{u_{n}\\}$ has a strongly
convergent subsequence is standard, and we can refer to Lemma 3.5 of
Ambrosetti-Coti Zelati [1].
Claim 3.2 $f(u)$ is weakly lower semi-continuous on $F$.
Proof For any $u_{n}\subset F$ with $u_{n}\rightharpoonup u$, by Sobolev’s
embedding Theorem we have the uniform convergence:
$|u_{n}(t)-u(t)|_{\infty}\rightarrow 0.$
Since $V\in C^{1}(R^{n},R)$, we have
$|V(u_{n}(t))-V(u(t))|_{\infty}\rightarrow 0.$
By the weakly lower semi-continuity of norm, we have
$\liminf(\int^{1}_{0}|\dot{u}_{n}|^{2}dt)^{\frac{1}{2}}\geq(\int^{1}_{0}|\dot{u}|^{2}dt)^{\frac{1}{2}}.$
Calculating we see
$\liminf(\int^{1}_{0}|\dot{u}_{n}|^{2}dt)\geq\int^{1}_{0}|\dot{u}|^{2}dt,$
and
$\liminf
f(u_{n})=\liminf\frac{1}{4}\int^{1}_{0}|\dot{u_{n}}|^{2}dt\int^{1}_{0}V^{\prime}(u_{n})u_{n}dt$
$\geq\frac{1}{4}\int^{1}_{0}|\dot{u}|^{2}dt\int^{1}_{0}V^{\prime}(u)udt=f(u).$
Claim 3.3 $F$ is a weakly closed subset in $H^{1}$.
Proof This follows easily from Sobolev’s embedding Theorem and $V\in
C^{1}(R^{n},R)$.
Claim 3.4 The functional $f(u)$ has positive lower bound on $F$
Proof By the definitions of $f(u)$ and $F$ and the assumption $(V_{2})$, we
have
$f(u)=\frac{1}{4}\int^{1}_{0}|\dot{u}|^{2}dt\int^{1}_{0}(V^{\prime}(u)u)dt\geq
0,\forall u\in F.$
Furthermore, we claim that
$\inf f(u)>0;$
otherwise, $u(t)=const$, and by the symmetrical property $u(t+1/2)=-u(t)$ we
have $u(t)=0,\forall t\in R$. But by assumptions $(V_{4})$ and $(V_{6})$ we
have
$V(0)\leq\frac{\mu_{2}}{\mu_{1}}<h,$
which contradicts the definition of $F$ since $V(0)=h$ if we have $0\in F$.
Now by Lemmas 3.1-3.4 and Lemma 2.6, we see that $f(u)$ attains the infimum on
$F$, and we know that the minimizer is nonconstant.
## Acknowledgements
The authors sincerely thank Professor P.Rabinowitz who brought the paper of D.
Offin ([16]) to our attention.
## References
* [1] A.Ambrosetti,V.Coti Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Rat. Mech. Anal. 112(1990), 339-362.
* [2] A.Ambrosetti,V.Coti Zelati , Solutions with minimal period for Hamiltonian systems in a potential well, Ann. Inst. H. Poincare, Analyse Non Lineare 4(1987), 235-242.
* [3] A.Ambrosetti,P.Rabinowitz,Dual variational methods in critical point theory and applications,J.of Functional Analysis,14(1973),349-381.
* [4] V.Benci ,Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems ,Ann. Inst. Henri Poincare Anal. NonLineaire 1(1984), 401-412.
* [5] J.Berg,F.Pasquotto,R.Vandervorst,Closed characteristics on non-compact hypersurfaces in $R^{2n}$ ,Math.Ann.343(2009),247-284.
* [6] K.C.Chang,Infinite dimensional Morse theory and mutiple solution problems,Birkhauser,1993.
* [7] G.Cerami ,Un criterio di esistenza per i punti critici so variete illimitate, Rend. dell academia di sc.lombardo112(1978),332-336.
* [8] V.Coti Zelati,I.Ekeland and P.L.Lions,Index estimates and critical points of functionals not satisfying Palais-Smale, Ann.Scuola Norm Sup.Pisa 17(1990),569-581.
* [9] I.Ekeland,Convexity Methods in Hamiltonian Mechanics,Springer,1990.
* [10] N.Ghoussoub,D.Preiss,A general mountain pass principle for locating and clasifying critical points,Ann. Inst. Henri Poincare Anal. NonLineaire 6(1984), 321-330.
* [11] H.Gluck and W.Ziller,Existence of periodic motions of conservative systems,in Seminar on minimal submanifolds,E.Bombieri Ed.,Princeton Univ. Press,1983.
* [12] E.W.C.Van Groesen,Analytical mini-max methods for Hamiltonian break orbits with a prescribed energy,JMAA 132(1988),1-12.
* [13] K.Hayashi,Periodic solutions of classical Hamiltonian systems,Tokyo J.Math.,1983.
* [14] Y. Long, Index Theory for Symplectic Paths with Applications ,Basel: Birkhauser,2002.
* [15] J.Mawhin , M.Willem,Critical Point Theory and Applications,Springer,1989.
* [16] D.Offin,A class of periodic orbits in classical mechanics,JDE,66(1987),90-117.
* [17] Palais R.,The principle of symmetric criticality,CMP 69(1979),19-30.
* [18] P.H.Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31(1978), 157-184.
* [19] P.H.Rabinowitz,Periodic solutions of a Hamiltonian systems on a prescribed energy surface,JDE 33(1979),336-352.
* [20] H.Seifert,Periodischer bewegungen mechanischer system,Math.Zeit51(1948),197-216.
* [21] K.Yosida,Functional Analysis,Springer,Berlin,1978.
* [22] W.P.Ziemer,Weakly differentiable functions,Springer,1989.
|
arxiv-papers
| 2013-07-30T13:33:48 |
2024-09-04T02:49:48.748088
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fengying Li and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1307.7971"
}
|
1307.8002
|
# Periodic Solutions of Non-Autonomous Second Order Hamiltonian Systems
***Supported by National Natural Science Foundation of China.
Fengying Li†††Email:[email protected]
The School of Economic and Mathematics, Southwestern University of Finance and
Economics,
Chengdu 611130, China
Shiqing Zhang and Xiaoxiao Zhao
College of Mathematics, Sichuan University, Chengdu 610064, People’s Republic
of China
> Abstract
>
> We try to generalize a result of M. Willem on forced periodic oscillations
> which required the assumption that the forced potential is periodic on
> spatial variables. In this paper, we only assume its integral on the time
> variable is periodic, and so we extend the result to cover the forced
> pendulum equation. We apply the direct variational minimizing method and
> Rabinowtz’s saddle point theorem to study the periodic solution when the
> integral of the potential on the time variable is periodic.
>
>
> Keywords
>
> Forced second order Hamiltonian systems, the forced pendulum equation,
> variational minimizers, Saddle Point Theorem.
> 2000AMS Subject Classification 34C15, 34C25.
## 1 Introduction and Main Results
In [10] and [5], M. Willem and Mawhin studied the following second order
Hamiltonian system
$\ddot{u}(t)=-\nabla F(t,u(t))=-F^{\prime}(t,u(t))$ (1.1)
where $F:[0,T]\times R^{N}\rightarrow R,\nabla F(t,u(t))=F^{\prime}(t,u(t))$
is the gradient of $F(t,u(t))$ with respect to $u$. We assume $F(t,u(t))$
satisfies the following assumption:
(A). $F(t,x)$ is measurable in $t$ for each $x\in R^{N}$, continuously
differentiable in $x$ for a.e. $t\in[0,T]$, and there exist $a\in
C(R^{+},R^{+})$ and $b\in L^{1}(0,T;R^{+})$ such that
$|F(t,x)|\leq a(|x|)b(t),$ $|\nabla F(t,x)|\leq a(|x|)b(t)$
for all $x\in R^{N}$ and a.e. $t\in[0,T]$.
M. Willem ([10]) got the following theorem :
Theorem 1.1 ([10] and [5]) Assume $F$ satisfies condition (A) and for the
canonical basis $\\{e_{i}|1\leq i\leq N\\}$ of $R^{N}$, there exist $T_{i}>0$
such that for $\forall x\in R^{N}$ and a.e. $t\in[0,T]$,
$F(t,x+T_{i}e_{i})=F(t,x),\ \ \ \ 1\leq i\leq N$ (1.2)
Then (1.1) has at least one solution which minimizes
$f(u)=\int_{0}^{T}[\frac{1}{2}|\dot{u}(t)|^{2}-F(t,u(t))]dt$
on $H_{T}^{1}=\\{u|u,\dot{u}\in L^{2}[0,T],u(t+T)=u(t)\\}$.
In order to cover the forced pendulum equation:
$\ddot{u}(t)=-a\sin u+e(t),$ (1.3)
Mawhin-Willem [5] also study the following forced equation:
$\ddot{u}(t)=-\nabla F(t,u(t))-e(t)=-F^{\prime}(t,u(t))-e(t)$ (1.4)
they got the following Theorem:
Theorem 1.2 ([10] and [5]) Assume $F$ satisfies the conditions of Theorem
1.1,and $e(t)\in L^{1}(0,T;R^{N})$ verifying
$\int_{0}^{T}e(t)dt=0,$
then (1.4) has at least one solution which minimizes on $H_{T}^{1}$ the
following functional:
$f(u)=\int_{0}^{T}[\frac{1}{2}|\dot{u}(t)|^{2}-F(t,u(t))-e(t)u(t)]dt$
We notice that the potential $F(t,x)=-(a\cos x+e(t)x)$ does not satisfy (1.2).
But if $\int_{0}^{T}e(t)dt=0$, then $F(t,x)=-(a\cos x+e(t)x)$ does satisfy
$\int_{0}^{T}F(t,x+2\pi)dt=\int_{0}^{T}F(t,x)dt.$ (1.5)
So instead of (1.2) we only assume the weaker integral condition:
$\int_{0}^{T}F(t,x+T_{i}e_{i})dt=\int_{0}^{T}F(t,x)dt\ \ \ \ i=1,2,...,N$
(1.6)
We obtain the following results:
Theorem 1.3 Assume $F:R\times R^{N}\rightarrow R$ satisfies condition (A) and
(F1). $F(t+T,x)=F(t,x)$, $\forall(t,x)\in R\times R^{N}$,
(F2). $F$ satisfies (1.6).
(F3). There exist $0<C_{1}<\frac{1}{2}(\frac{2\pi}{T})^{2}$, $C_{2}>0$ such
that
$|F(t,x)|\leq C_{1}|x|^{2}+C_{2}$
Then (1.1) has at least one $T$-periodic solution.
Corollary 1.1 (J. Mawhin, M. Willem [6]) For the pendulum equation (1.3), the
potential $F(t,x)=a\cos x+e(t)x$ satisfies all conditions in Theorem 1.3
provided $e(t+T)=e(t)$ and $\int_{0}^{T}e(t)dt=0$. In this case, (1.3) has at
least one $T$-periodic solution.
Theorem 1.4 Suppose $F:R\times R^{N}\rightarrow R$ satisfies conditions (A),
(F1), (F2) and
(F4). There are $\mu_{1}<2$, $\mu_{2}\in R$ such that
$F^{\prime}(t,x)\cdot x\leq\mu_{1}F(t,x)+\mu_{2},$
(F5). There is $\delta>0$ such that for $t\in R$, $F(t,x)>\delta$, as
$|x|\rightarrow+\infty$,
(F6). $F(t,x)\leq b|x|^{2}$.
Then if $T<\sqrt{\frac{2}{b}}\pi$, (1.1) has a $T-$periodic solution;
furthermore, if $\forall x\in R^{N}$, $\int_{0}^{T}F(t,x)dt\geq 0$, then (1.4)
has a non-constant $T-$periodic solution.
## 2 Some Important Lemmas
Lemma 2.1 (Eberlin-Smulian[11]) A Banach space X is reflexive if and only if
any bounded sequence in X has a weakly convergent subsequence.
Lemma 2.2 ([1],[5],[12]) Let $q\in W^{1,2}(R/TZ,R^{n})$ and
$\int^{T}_{0}q(t)dt=0$, then
(i). We have Poincare-Wirtinger’s inequality
$\int^{T}_{0}|\dot{q}(t)|^{2}dt\geq(\frac{2\pi}{T})^{2}\int_{0}^{T}|q(t)|^{2}dt$
(ii). We have Sobolev’s inequality
$\max_{0\leq t\leq
T}|q(t)|=\|q\|_{\infty}\leq\sqrt{\frac{T}{12}}(\int^{T}_{0}|\dot{q}(t)|^{2}dt)^{1/2}$
We define the equivalent norm in $H^{1}_{T}=H^{1}=W^{1,2}(R/TZ,R^{n}):$
$\|q\|_{H^{1}}=(\int_{0}^{T}|\dot{q}|^{2})^{1/2}+|\int_{0}^{T}q(t)dt|$
Lemma 2.3([5]) Let $X$ be a reflexive Banach space, $M\subset X$ a weakly
closed subset, and $f:M\rightarrow R\cup\\{+\infty\\}$ weakly lower semi-
continuous. If the minimizing sequence for $f$ on $M$ is bounded, then $f$
attains its infimum on $M$.
Definition 2.1([3]) Suppose $X$ is a Banach space and $f\in C^{1}(X,R)$ and
$\\{q_{n}\\}\subset X$ satisfies
$f(q_{n})\rightarrow C,\ \ \ \ (1+\|q_{n}\|)f^{\prime}(q_{n})\rightarrow 0.$
Then we say $\\{q_{n}\\}$ satisfies the $(CPS)_{C}$ condition.
Lemma 2.4(Rabinowitz’s Saddle Point Theorem[9], Mawhin-Willem[5]) Let $X$ be a
Banach space with $f\in C^{1}(X,R)$. Let $X=X_{1}\oplus X_{2}$ with
$dimX_{1}<+\infty$
and
$\sup_{S^{1}_{R}}f<\inf_{X_{2}}f,$
where $S^{1}_{R}=\\{u\in X_{1}||u|=R\\}$.
Let $B_{R}^{1}=\\{u\in X_{1},|u|\leq R\\}$, $M=\\{g\in
C(B^{1}_{R},X)|g(s)=s,s\in S^{1}_{R}\\}$
$C=\inf_{g\in M}\max_{s\in B^{1}_{R}}f(g(s)).$
Then $C>\inf_{X_{2}}f$, and if $f$ satisfies $(CPS)_{C}$ condition, then $C$
is a critical value of $f$.
## 3 The Proof of Theorem 1.3
Lemma 3.1 (Morrey [7], M-W [10]) Let $L:[0,T]\times R^{N}\times
R^{N}\rightarrow R$, $(t,x,y)\rightarrow L(t,x,y)$ be measurable in $t$ for
each $(x,y)\in R^{N}\times R^{N}$ and continuously differentiable in $(x,y)$
for a.e. $t\in[0,T]$. Suppose there exists $a\in C(R^{+},R^{+})$, $b\in
L^{1}(0,T;R^{+})$ and $c\in L^{q}(0,T;R^{+})$, $1<q<\infty$, such that for
a.e. $t\in[0,T]$ and every $(x,y)\in R^{N}\times R^{N}$ one has
$|L(t,x,y)|\leq a(|x|)(b(t)+|y|^{q}),$ $|D_{x}L(t,x,y)|\leq
a(|x|)(b(t)+|y|^{p}),$ $|D_{y}L(t,x,y)|\leq a(|x|)(c(t)+|y|^{p-1}).$
where $\frac{1}{p}+\frac{1}{q}=1$, then the functional
$\varphi(u)=\int_{0}^{T}L(t,u(t),\dot{u}(t))dt$
is continuously differentiable on the Sobolev space
$W^{1,p}=\\{u\in L^{p}(0,T),\dot{u}\in L^{p}(0,T)\\}$ (3.1)
and
$<\varphi^{\prime}(u),v>=\int_{0}^{T}[<D_{x}L(t,u,\dot{u}),v>+D_{y}L(t,u,\dot{u})\cdot\dot{v}]dt$
(3.2)
From Lemma 3.1 and the assumptions (A), we know that the variational
functional
$f(u)=\int_{0}^{T}[\frac{1}{2}|\dot{u}|^{2}-F(t,u(t))]dt$ (3.3)
is $C^{1}$ on $W_{T}^{1,2}=H^{1}_{T}$, and the critical point is just the
periodic solution for the system (1.1).
Furthermore, if (F1) and (F2) are satisfied, we will prove the functional
$f(u)$ attains its infimum on $H_{T}^{1}$; in fact,
$H_{T}^{1}=X\oplus R^{N},$ (3.4)
where
$X=\\{x\in H^{1}_{T}:\bar{x}\triangleq\frac{1}{T}\int_{0}^{T}x(t)dt=0\\}$
(3.5)
and $\forall u\in H_{T}^{1}$, we have $\widetilde{u}\in X$ and
$\overline{u}\in R^{N}$, such that $u=\widetilde{u}+\overline{u}$.
By Poincare-Wirtinger’s inequality,
$\displaystyle f(\tilde{u})$
$\displaystyle\geq\frac{1}{2}\int_{0}^{T}|\dot{\tilde{u}}|^{2}dt-
C_{1}\int_{0}^{T}|\tilde{u}|^{2}dt-C_{2}T$ (3.6)
$\displaystyle\geq[\frac{1}{2}-C_{1}(\frac{2\pi}{T})^{-2}]\int_{0}^{T}|\dot{\widetilde{u}}|^{2}dt-
C_{2}T;$
hence, $f$ is coercive on $X$.
Let $\\{u_{k}\\}$ be a minimizing sequence for $f(u)$ on $H_{T}^{1}$,
$u_{k}=\widetilde{u}_{k}+\overline{u}_{k}$, where $\widetilde{u}_{k}\in X$,
$\overline{u}_{k}\in R^{N}$, then by (3.6) we have
$\|\widetilde{u}_{k}\|_{H^{1}_{T}}\leq C.$ (3.7)
By condition (F2), we have
$f(u+T_{i}e_{i})=f(u),\ \ \ \ \forall u\in H_{T}^{1},\ \ \ \ 1\leq i\leq N.$
(3.8)
So if $\\{u_{k}\\}$ is a minimizing sequence for $f$, then
$(\widetilde{u}_{k}\cdot e_{1}+\overline{u}_{k}\cdot
e_{1}+k_{1}T_{1},...,\widetilde{u}_{k}\cdot e_{N}+\overline{u}_{k}\cdot
e_{N}+k_{N}T_{N})$
is also a minimizing sequence of $f(u)$, and so we can assume
$0\leq\overline{u}_{k}\cdot e_{i}\leq T_{i},\ \ \ \ 0\leq i\leq N.$ (3.9)
By (3.7) and (3.9), we know $\\{u_{k}\\}$ is a bounded minimizing sequence in
$H_{T}^{1}$, and it has a weakly convergent subsequence; furthermore, $f$ is
weakly lower semi-continuous since $f$ is the sum of a convex continuous
function and a weakly continuous function. We can conclude that $f$ attains
its infimum on $H_{T}^{1}$. The corresponding minimizer is a periodic solution
of (1.4).
## 4 The Proof of Theorem 1.4
Lemma 4.1: If conditions (A), (F1), (F2) and (F4) in Theorem 1.4 hold, then
$f(q)$ satisfies the $(CPS)_{C}$ condition on $H^{1}$.
Proof: For any $C$, let $\\{u_{n}\\}\subset H^{1}$ satisfy
$f(u_{n})\rightarrow C,\ \ \ \ (1+\|u_{n}\|)f^{\prime}(u_{n})\rightarrow 0.$
(4.1)
We claim $\|\dot{u}_{n}\|_{L^{2}}$ is bounded; in fact, by
$f(u_{n})\rightarrow C$, we have
$\frac{1}{2}\|\dot{u}_{n}\|^{2}_{L^{2}}-\int_{0}^{T}F(t,u_{n})dt\rightarrow
C.$ (4.2)
By (F4) we have
$\displaystyle<f^{\prime}(u_{n}),u_{n}>$ $\displaystyle=$
$\displaystyle\|\dot{u}_{n}\|^{2}_{L^{2}}-\int_{0}^{T}(<F^{\prime}(t,u_{n}),u_{n}>)dt$
(4.3) $\displaystyle\geq$
$\displaystyle\|\dot{u}_{n}\|^{2}_{L^{2}}-\int_{0}^{T}[\mu_{2}+\mu_{1}F(t,u_{n})]dt.$
By (4.2) and (4.3), we see that
$0\leftarrow<f^{\prime}(u_{n}),u_{n}>\geq
a\|\dot{u}_{n}\|^{2}_{L^{2}}+C_{1}+\delta,n\rightarrow+\infty$ (4.4)
where $C_{1}=C\mu_{1}-T\mu_{2}+\delta,\delta>0,a=1-\frac{\mu_{1}}{2}>0.$
We have shown that $\|\dot{u}_{n}\|_{L^{2}}$ is bounded.
By condition (F2), we have
$f(u+T_{i}e_{i})=f(u),\ \ \ \ \forall u\in H_{T}^{1},\ \ \ \ 1\leq i\leq N.$
(4.5)
Hence, if $\\{u_{k}\\}$ is a $(CPS)_{C}$ sequence for $f$, then
$(\widetilde{u}_{k}\cdot e_{1}+\overline{u}_{k}\cdot
e_{1}+k_{1}T_{1},...,\widetilde{u}_{k}\cdot e_{N}+\overline{u}_{k}\cdot
e_{N}+k_{N}T_{N})$
is also a $(CPS)_{C}$ sequence of $f(u)$, so we can assume
$0\leq\overline{u}_{k}\cdot e_{i}\leq T_{i},\ \ \ \ 0\leq i\leq N.$ (4.6)
By (4.6), we know $|\bar{u}_{k}|$ is bounded, and so
$\|u_{n}\|=\|\dot{u}_{n}\|_{L^{2}}+|\int_{0}^{T}u_{n}(t)dt|$ is bounded.
The rest of he lemma can be completed in a now standard fashion.
We finish the proof of Theorem 1.4. In Rabinowitz’s Saddle Point Theorem, we
take
$X_{1}=R^{N},X_{2}=\\{u\in W^{1,2}(R/TZ,R^{N}),\int_{0}^{T}udt=0\\}.$
For $u\in X_{2}$, we may use the Poincare-Wirtinger inequality, and so by
Lemma 2.2 and (F6), we have
$\displaystyle f(u)$ $\displaystyle\geq$
$\displaystyle\frac{1}{2}\int_{0}^{T}|\dot{u}|^{2}dt-b\int_{0}^{T}|u|^{2}dt$
$\displaystyle\geq$
$\displaystyle[\frac{1}{2}-b(2\pi)^{-2}T^{2}]\int_{0}^{T}|\dot{u}|^{2}dt$
$\displaystyle\geq$ $\displaystyle 0.$
On the other hand, if $u\in R^{N}$, then by $(F5)$ we have
$f(u)=-\int_{0}^{T}F(t,u)dt\leq-\delta,|u|=R\rightarrow+\infty.$
The proof of Theorem 1.4 is concluded by calling upon Rabinowitz’s Saddle
Point Theorem. In fact,there is a critical point $\bar{u}$ such that
$f(\bar{u})=C>\inf_{X_{2}}f(u)\geq 0$, which is nonconstant since otherwise
$f(\bar{u})=-\int_{0}^{T}F(\bar{u},t)dt\leq 0,$ which is a contradiction.
The authors would like to thank the referees for their valuable suggestions.
## References
* [1] Adams R.A. and Fournier J.F., Sobolev spaces, Second Edition, Academic Press, 2003.
* [2] Brezis,H.,Functional analysis,Sobolev spaces and PDE,Springer,2011.
* [3] Cerami G., Un criterio di esistenza per i punti critici so variete illimitate, Rend. dell academia di sc. lombado 112(1978) 332-336.
* [4] Chang K.C., Critical point theory and application, Shanghai Academic Press, 1986.
* [5] Mawhin J. and Willem M., Critical point theory and Hamiltonian Systems, Springer-Verlag, 1989.
* [6] Mawhin J. and Willem M., Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations 52(1984), 264-287.
* [7] Morrey C. B., Multiple integrals in the calculus of variations, Springer, Berlin, 1966.
* [8] Ortega R., The pendulum equation: from periodic to almost periodic forcings, Differential and Integral Equations 22(2009),801-814.
* [9] Rabinowitz P.H., Minimax methods in critical point theory with applications to differetial eqautions, CBMS Reg. Conf. Ser. in Math. 65, Ams, 1986.
* [10] Willem M., Oscillations forc$\acute{e}$es de syst$\grave{e}$mes Hamiltoniens, Public. S$\acute{e}$min. Analysis nonlin$\acute{e}$aire Univ. Besancon, 1981.
* [11] Yosida K., Functional analysis, 5th ed., Springer, Berlin, 1978.
* [12] Ziemer W.P., Weakly differentiable functions, Springer, 1989.
|
arxiv-papers
| 2013-07-30T14:43:22 |
2024-09-04T02:49:48.755569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fengying Li and Shiqing Zhang and Xiaoxiao Zhao",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1307.8002"
}
|
1307.8009
|
# Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-
Zehnder Interferometer
Xiaoxing Jing, Jing Liu, Wei Zhong, and Xiaoguang Wang Zhejiang Institute of
Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027,
China [email protected]
###### Abstract
We give an analytical result for the quantum Fisher information of entangled
coherent States in a lossy Mach-Zehnder Interferometer recently proposed by J.
Joo et al. [Phys. Rev. Lett. 107, 083601(2011)]. For small loss of photons, we
find that the entangled coherent state can surpass the Heisenberg limit.
Furthermore, The formalism developed here is applicable to the study of phase
sensitivity of multipartite entangled coherent states.
###### pacs:
03.67.-a, 03.65.Ta, 42.50.St
††: J. Phys. B: At. Mol. Phys.
## 1 INTRODUCTION
Precision measurements are important across all fields of science and
technology. By employing quantum features like entanglement and squeezing,
quantum metrology promises enhancing precision and has drawn a lot of
attention in the last decade [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13,
14, 15, 14, 16, 17, 18, 19]. Quantum metrology deals with the ultimate
precision limits in estimation procedures, taking into account the constraints
imposed by quantum mechanics, and allows one to gain advantages over purely
classical approaches [1, 2, 3, 4]. As a key component of the quantum metrology
theory, quantum parameter estimation has many applications in experiments,
such as the detection of gravitational radiation [12, 13], quantum frequency
standards [15, 14, 16], clock synchronization [17, 18], to name a few.
Quantum Fisher information (QFI) is another significant concept in quantum
metrology and has been studied widely [20, 21, 22, 23, 24, 25, 26, 19, 27, 28,
29]. As an extension of the classical Fisher information in statistics and
information theory, QFI plays a paramount role in quantum estimation theory.
In quantum metrology theory, these two concepts are linked by the quantum
Cramér-Rao inequality [21, 22],
${\rm var}(\hat{\varphi})\geq\frac{1}{\nu F},$ (1)
where ${\rm var}(\hat{\varphi})$ is the variance of an unbiased estimator
$\hat{\varphi}$ of a parameter $\varphi$, $\nu$ represents the number of
repeated experiments and $F$ is the QFI of the parameter. The inverse of the
QFI provides the lower bound of the error of the estimation.
In this paper, we consider a fundamental parameter estimation task in which
the parameter $\varphi$ is generated by some unitary dynamics
$U=\exp(-i\varphi H)$. This kind of parameter estimation task is common in
many experimental setups such as Mach-Zehnder interferometers and Ramsey
interferometers. Based on a recent expression of QFI [31], we show that the
QFI of $\varphi$ for a unitary parameterized dynamics is the mean variance of
$H$ over the eigenstates minus the transition terms of $H$. Next we take a two
dimension case as our interest. The eigenvalues and eigenstates of a general
$2\times 2$ density matrix have been given in terms of its determinant,
difference between diagonal elements and phase of off-diagonal elements. For
integrity we also give the eigenvalues and eigenstates for a density operator
on a nonorthogonal basis of two dimensions.
While exact results and analytical solutions are known for noiseless
situations, the determination of the ultimate precision limit in the presence
of noise is still a challenging problem in quantum mechanics. Recently, J. Joo
et al.studied the entangled coherent states in a Mach-Zehnder interferometer
under perfect and lossy conditions [5]. They found the entangled coherent
states (ECS) can reach better precision in comparison to N00N, “bat”, and
“optimal” states in both conditions. In lossy conditions, they modeled the
particle loss by fictitious beam splitters and adopted a numeric strategy to
calculate the QFI of the ECS. Utilizing our formula we give an analytic
expression of the QFI. We find that even in a lossy condition, the ECS can
still surpass the Heisenberg limit.
This paper is organized as follows. In Sec. II, we give a brief review of the
QFI and obtain an explicit formula of the QFI for a family of density matrices
parameterized through a unitary dynamics. In Sec. III, we give the eigenvalues
and eigenstates of a 2-dimensional density matrix in terms of its determinant,
difference between diagonal elements and phase of off-diagonal elements. We
also generalize the eigen problem in a nonorthogonal basis. Afterward, in Sec.
IV, we apply our result to the ECS in a lossy Mach-Zehnder interferometer and
get an analytical expression of the QFI. Finally, the conclusion is given in
Sec. V.
## 2 QFI AND PARAMETER ESTIMATION FOR UNITARY DYNAMICS
### 2.1 Brief Review of Quantum Fisher Information
In this section, we briefly review the calculation of the QFI. For a
parameterized quantum states $\rho_{\varphi}$, a widely used version of QFI
$F_{\varphi}$ is defined as [21, 22]
$F_{\varphi}:={\rm tr}(\rho_{\varphi}L^{2}),$ (2)
where the symmetric logarithmic derivative (SLD) operator $L$ is determined by
$\partial_{\varphi}\rho_{\varphi}=\frac{1}{2}[L\rho_{\varphi}+\rho_{\varphi}L].$
(3)
Consider a density operator $\rho_{\varphi}$ on a $N$-dimensional system ($N$
can be infinite). The corresponding spectrum decomposition is given by
$\rho_{\varphi}=\sum_{i=1}^{M}p_{i}|\psi_{i}\rangle\langle\psi_{i}|,$ (4)
where $p_{i}$ is the eigenvalue and $|\psi_{i}\rangle$ is the eigenstate, and
$M\leq N,$ implying that there are $N-M$ zero eigenvalues. With the
decomposition of the density matrix one can directly obtain the element of the
SLD operator from Eq. (3) as
$\langle\psi_{k}|L|\psi_{l}\rangle=\frac{2\langle\psi_{k}|\partial_{\varphi}\rho_{\varphi}|\psi_{l}\rangle}{p_{l}+p_{k}}.$
(5)
Notice that the matrix element of $L$ is not defined when $p_{l}+p_{k}=0$.
It turns out that the QFI is completely determined in the support of
$\rho_{\varphi}$, that is, the space spanned by those eigenvectors
corresponding to the nonvanishing eigenvalues. It can be expressed as [31]
$\displaystyle F_{\varphi}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\frac{1}{p_{i}}(\partial_{\varphi}p_{i})^{2}+\sum_{i=1}^{M}4p_{i}\langle\partial_{\varphi}\psi_{i}|\partial_{\varphi}\psi_{i}\rangle$
(6)
$\displaystyle-\sum_{i=1}^{M}\sum_{j=1}^{M}\frac{8p_{i}p_{j}}{(p_{i}+p_{j})}|\langle\psi_{i}|\partial_{\varphi}\psi_{j}\rangle|^{2}.$
For the special case of a pure state ($M=1$), Eq. (6) reduces to
$F(\psi_{1})=4[\langle\partial_{\varphi}\psi_{1}|\partial_{\varphi}\psi_{1}\rangle-|\langle\psi_{1}|\partial_{\varphi}\psi_{1}\rangle|^{2}].$
(7)
Using this form of the QFI for pure states, we can rewrite Eq. (6) as
$\displaystyle F_{\varphi}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\frac{1}{p_{i}}(\partial_{\varphi}p_{i})^{2}+\sum_{i=1}^{M}p_{i}F(\psi_{i})$
(8) $\displaystyle-\sum_{i\neq
j}^{M}\frac{8p_{i}p_{j}}{(p_{i}+p_{j})}|\langle\psi_{i}|\partial_{\varphi}\psi_{j}\rangle|^{2}.$
It is clear that the first term can be regarded as the classical contribution
[30, 31, 22], and the second term as the mean QFI over the eigenstates. The
third term can be regarded as a sum of harmonic mean of transition terms.
There are several similar formulas in the literature where the summation in
the last term runs over all the eigenstates, as long as $p_{i}+p_{j}\neq 0$.
Eq. (8) have some advantages over them both in analytical and numerical
calculations since $i,j$ are symmetric and one only need to find the non-
varnishing eigenstates of $\rho_{\varphi}$.
### 2.2 QFI for unitary parameterized dynamics
In quantum estimation theory, the most fundamental parameter estimation task
is to estimate a small parameter $\varphi$ generated by some unitary dynamics
$U=\exp(-i\varphi H).$ (9)
Here $H$ is a Hermitian operator and can be regarded as the generator of
parameter $\varphi$. This form of parameterization process is typical in
interferometers. For instance, in a Ramsey interferometer $H$ can be a
collective angular momentum operator $J_{n}$ [27], which can be viewed as a
generator of SU(2). In Mach-Zehnder interferometers, denoting $a_{i}$ and
$a_{i}^{\dagger}$ (i=1,2) as the annihilation and creation operators for _i_
th mode, then $H$ can be (1) the photon number difference between two modes:
$a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}$ [32], (2) the number operator in
one mode: $a_{2}^{\dagger}a_{2}$ [5, 18], (3) the number operator to the _k_
$\rm{th}$ power: $(a_{2}^{\dagger}a_{2})^{k}$, in a nonlinear interferometer
[6].
Suppose the initial state $\rho_{0}$ has already been decomposed as
$\rho_{0}=\sum_{i}^{M}p_{i}|\phi_{i}\rangle\langle\phi_{i}|.$ (10)
Here we assume $\rho_{0}$ is independent of $\varphi$. After the unitary
rotation, $\rho_{\varphi}$ can be decomposed as
$\displaystyle\rho_{\varphi}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|,$
(11)
with
$|\psi_{i}\rangle=e^{-i\varphi H}|\phi_{i}\rangle.$ (12)
Substituting Eq. (11) into Eq. (8) leads to the QFI given by
$F_{\varphi}=4\left[\sum_{i=1}^{M}p_{i}(\Delta H_{i})^{2}-\sum_{i\neq
j}^{M}\frac{2p_{i}p_{j}}{p_{i}+p_{j}}|H_{ij}|^{2}\right],$ (13)
where
$(\Delta
H_{i})^{2}=\langle\phi_{i}|H^{2}|\phi_{i}\rangle-\langle\phi_{i}|H|\phi_{i}\rangle^{2},$
and
$|H_{ij}|^{2}=|\langle\phi_{i}|H|\phi_{j}\rangle|^{2},$
are the variance and transition probability of $H$ in the eigenstates of
$\rho_{0}$. Since $p_{i}$ is independent of $\varphi$, the classical
contribution vanishes. The first term in Eq. (13) is the mean variance of $H$
over the eigenstates, while the second term is a sum of transition probability
of $H$ with a harmonic mean weight.
If $\rho_{0}$ is a pure state, we can take $p_{i}=\delta_{i1,}$ then
$F_{\varphi}=4(\Delta H_{1})^{2};$ (14)
if $\rho_{0}$ only has two nonzero components, we take $p_{1}p_{2}\neq 0$ and
$p_{i}=0$ when $i>2$, then
$F_{\varphi}=4p_{1}(\Delta H_{1})^{2}+4p_{2}(\Delta
H_{2})^{2}-16p_{1}p_{2}|H_{12}|^{2}.$ (15)
In the following, we take $M=2$ as our main interest.
## 3 EIGEN PROBLEM OF A Nonorthogonal $2\times 2$ Density Matrix
According to Eq. (8) and Eq. (13), we only need to find the non-vanishing
eigenstates of the density operator rather than all its eigenstates. However,
it is generally not feasible to get the analytical diagonalization of
$\rho_{\varphi}$. In that case, one has to resort to numeric methods or
decompose the density operator into a nonorthogonal basis and use the
convexity of QFI.
In this paper, we develop a systematic routine to find the eigenvalues and
eigenstates of a density operator of rank 2 and apply it to an interesting
scenario. Let us consider a $2\times 2$ density operator $\tilde{\rho}$ on a
nonorthogonal basis
$\tilde{\rho}=a|\Psi_{1}\rangle\langle\Psi_{1}|+b|\Psi_{1}\rangle\langle\Psi_{2}|+b^{*}|\Psi_{2}\rangle\langle\Psi_{1}|+d|\Psi_{2}\rangle\langle\Psi_{2}|,$
(16)
where $|\Psi_{1}\rangle,|\Psi_{2}\rangle$ are normalized states and $a,d$ are
real numbers due to the hermiticity of density operator. The special case when
$|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ are orthogonal is discussed in
Appendix A. In order to get the eigenvalues and eigenvectors of
$\tilde{\rho}$, we first recast it into an orthogonal basis (one can also
solve the eigen problem in the original nonorthogonal basis, see Appendix B.)
Denoting $p=\langle\Psi_{1}|\Psi_{2}\rangle$, we introduce a new set of basis
by the Gram-Schmidt procedure [33]
$\displaystyle|\Phi_{1}\rangle$ $\displaystyle=$
$\displaystyle|\Psi_{1}\rangle,$ $\displaystyle|\Phi_{2}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-|p|^{2}}}(|\Psi_{2}\rangle-p|\Psi_{1}\rangle),$
which are orthonormal. Through the inverse transformation:
$|\Psi_{1}\rangle=|\Phi_{1}\rangle$,
$|\Psi_{2}\rangle=\sqrt{1-|p|^{2}}|\Phi_{2}\rangle+p|\Phi_{1}\rangle$, the
density matrix in the new basis reads
$\tilde{\rho}=\left(\begin{array}[]{cc}a+bp^{*}+b^{*}p+d|p|^{2}&(b+dp)\sqrt{1-|p|^{2}}\\\
(b^{*}+dp^{*})\sqrt{1-|p|^{2}}&d(1-|p|^{2})\end{array}\right).$ (17)
The determinant of this density matrix, expectation value of $\sigma_{3}$ and
off-diagonal phase read
$\displaystyle{\rm det}(\tilde{\rho})$ $\displaystyle=$
$\displaystyle(1-|p|^{2})(ad-|b|^{2}),$
$\displaystyle\langle\sigma_{3}\rangle_{\tilde{\rho}}$ $\displaystyle=$
$\displaystyle 1-2d(1-|p|^{2}),$ $\displaystyle e^{i\tilde{\tau}}$
$\displaystyle=$ $\displaystyle\frac{b+dp}{|b+dp|}.$ (18)
According to appendix A, the eigenvalues and eigenstates of $\tilde{\rho}$ can
be expressed in terms of
$\rm{det}(\tilde{\rho}),\langle\sigma_{3}\rangle_{\tilde{\rho}}$ and
$\tilde{\tau}$. For clarity, we denote the eigenvalues and eigenstates as
$\tilde{\lambda}_{\pm}$ and $|\tilde{\lambda}_{\pm}\rangle$ correspondingly.
The values of $\tilde{\lambda}_{\pm}$ are
$\tilde{\lambda}_{\pm}=\frac{1\pm\sqrt{1-4{\rm det}(\tilde{\rho})}}{2},$ (19)
and the eigenstates read
$\displaystyle|\tilde{\lambda}_{+}\rangle$ $\displaystyle=$
$\displaystyle\tilde{v}_{+}e^{i\tilde{\tau}}|\Phi_{1}\rangle+\tilde{v}_{-}|\Phi_{2}\rangle,$
$\displaystyle|\tilde{\lambda}_{-}\rangle$ $\displaystyle=$
$\displaystyle-\tilde{v}_{-}e^{i\tilde{\tau}}|\Phi_{1}\rangle+\tilde{v}_{+}|\Phi_{2}\rangle,$
(20)
where
$\tilde{v}_{\pm}=\left(\frac{\sqrt{1-4{\rm
det}(\tilde{\rho})}\pm\langle\sigma_{3}\rangle_{\tilde{\rho}}}{2\sqrt{1-4{\rm
det}(\tilde{\rho})}}\right)^{\frac{1}{2}}.$ (21)
Hence the density matrix can be decomposed as
$\tilde{\rho}=\sum_{i=\pm}\tilde{\lambda}_{i}|\tilde{\lambda}_{i}\rangle\langle\tilde{\lambda}_{i}|.$
(22)
Alternatively, one can transform the eigenstates back to the nonorthogonal
basis,
$\displaystyle|\tilde{\lambda}_{+}\rangle$ $\displaystyle=$
$\displaystyle(\tilde{v}_{+}e^{i\tilde{\tau}}-\frac{p\tilde{v}_{-}}{\sqrt{1-|p|^{2}}})|\Psi_{1}\rangle+\frac{\tilde{v}_{-}}{\sqrt{1-|p|^{2}}}|\Psi_{2}\rangle,$
$\displaystyle|\tilde{\lambda}_{-}\rangle$ $\displaystyle=$
$\displaystyle(-\tilde{v}_{-}e^{i\tilde{\tau}}-\frac{p\tilde{v}_{+}}{\sqrt{1-|p|^{2}}})|\Psi_{1}\rangle+\frac{\tilde{v}_{+}}{\sqrt{1-|p|^{2}}}|\Psi_{2}\rangle.$
## 4 QFI OF ECS IN A LOSSY MACH-ZEHNDER INTERFEROMETER
### 4.1 Reformulation of the Density Matrix of ECS in A Lossy Mach-Zehnder
Interferometer
In a recent paper [5], the author analyzed the QFI of an entangled coherent
state(ECS) in the Mach-Zehnder interferometer. The main idea of their
proposition is as follows. A coherent state $|\alpha/\sqrt{2}\rangle$ and a
coherent state superposition(CSS)
$|\rm{CSS}\rangle=\mathcal{N}_{\alpha}(|\frac{\alpha}{\sqrt{2}}\rangle+|\frac{-\alpha}{\sqrt{2}}\rangle),$
(24)
are fed into the first 50:50 beam splitter of the Mach-Zehnder interferometer
and become the ECS,
$|\rm{ECS}\rangle_{1,2}=\mathcal{N}_{\alpha}[|\alpha\rangle_{1}|0\rangle_{2}+|0\rangle_{1}|\alpha\rangle_{2}],$
(25)
where
$\mathcal{N}_{\alpha}=1/\sqrt{2(1+e^{-|\alpha|^{2}}})$ (26)
is the normalized coefficient. Then a parameter is imprinted in one of the
mode by a unitary phase shift $U(\varphi)$. They modeled particle loss in the
realistic scenario by two fictitious beam splitters $B_{1,3}^{T},$
$B_{2,4}^{T}$ with the same transmission coefficient T. When $T=1$, the
interferometer has no photon loss. Here the subscript $3,4$ represent the
environment modes. After tracing out the environment modes, they got the
density matrix of the original mode $\rho_{12}$.
To calculate the QFI of $\rho_{12}$, they adopted numerical methods and
truncated the coherent state at $n=15$. Using the approach developed in Sec.
(II) and Sec. (III), we can give the analytical expression of the QFI. In the
following, we reformulate the density operator in a form as Eq. (16).
First, we denote the density operator before phase shift and particle loss as
$\rho_{\rm{in}}=|\rm{ECS}\rangle_{1,2}|0\rangle_{3}|0\rangle_{4}\langle
0|_{4}\langle 0|_{3}\langle\rm{ECS}|_{1,2}.$ (27)
In the interferometer, $\rho_{\rm{in}}$ suffers both particle loss and phase
shift before exiting the second 50:50 beam splitter. The phase accumulation
$U(\varphi)=e^{-i\varphi a_{2}^{\dagger}a_{2}}$ and the particle loss process,
indicated by the fictitious beam splitters $B_{1,3}^{T}$, $B_{2,4}^{T}$, are
interchangeable [23, 34]. Here $B_{1,3}^{T}$ and $B_{2,4}^{T}$ satisfy the
relation [35]
$B^{T}_{1,2}|\alpha_{1}\rangle_{1}|\alpha_{2}\rangle_{2}=|\alpha_{1}\sqrt{T}+\alpha_{2}\sqrt{R}\rangle_{1}|\alpha_{1}\sqrt{R}-\alpha_{2}\sqrt{T}\rangle_{2}.$
Thus the final reduced density operator can be written as
$\displaystyle\rho_{1,2}$ $\displaystyle=$ $\displaystyle{\rm
Tr_{3,4}}(B_{1,3}^{T}B_{2,4}^{T}U\rho_{\rm{in}}U^{\dagger}B_{2,4}^{T\dagger}B_{1,3}^{T\dagger})$
(28) $\displaystyle=$ $\displaystyle{\rm{\rm
Tr_{3,4}}}(UB_{1,3}^{T}B_{2,4}^{T}\rho_{\rm{in}}B_{2,4}^{T\dagger}B_{1,3}^{T\dagger}U^{\dagger}).$
(29)
The authors in Ref. [5] use the expression (28). To apply our result in Sec.
(II) and Sec. (III), we take the expression (29).
Second, the phase accumulation operator can be brought forward further, i.e.,
$\displaystyle\rho_{1,2}$ $\displaystyle=$ $\displaystyle U{\rm
Tr_{3,4}}(B_{1,3}^{T}B_{2,4}^{T}\rho_{\rm{in}}B_{2,4}^{T\dagger}B_{1,3}^{T\dagger})U^{\dagger}$
(30) $\displaystyle=$ $\displaystyle U\tilde{\rho}_{1,2}U^{\dagger},$
where
$\tilde{\rho}_{1,2}={\rm
Tr_{3,4}}(B_{1,3}^{T}B_{2,4}^{T}\rho_{\rm{in}}B_{2,4}^{T\dagger}B_{1,3}^{T\dagger}).$
(31)
That is, in such a lossy situation, the phase shift is still a unitary process
for $\tilde{\rho}_{1,2}$. Therefore we can calculate the QFI of $\rho_{1,2}$
by finding the decomposition of $\tilde{\rho}_{1,2}$. With the denotation of
$\displaystyle\alpha^{\prime}$ $\displaystyle=$ $\displaystyle\alpha\sqrt{T}$
$\displaystyle\beta^{\prime}$ $\displaystyle=$
$\displaystyle\alpha\sqrt{1-T}=\alpha\sqrt{R}$
$\tilde{\rho}_{1,2}$ can be specifically calculated as
$\displaystyle\tilde{\rho}_{1,2}$ $\displaystyle=$
$\displaystyle\mathcal{N}_{\alpha}^{2}[|\alpha^{\prime},0\rangle\langle\alpha^{\prime},0|+e^{-|\beta^{\prime}|^{2}}|\alpha^{\prime},0\rangle\langle
0,\alpha^{\prime}|$ (32)
$\displaystyle+e^{-|\beta^{\prime}|^{2}}|0,\alpha^{\prime}\rangle\langle\alpha^{\prime},0|+|0,\alpha^{\prime}\rangle\langle
0,\alpha^{\prime}|].$
We can see $\tilde{\rho}_{1,2}$ has the same form of Eq. (16). In the next
subsection we show the decomposition of $\tilde{\rho}_{1,2}$ and calculate the
QFI.
### 4.2 Calculation of the ECS’s QFI
In order to find the decomposition of $\tilde{\rho}_{1,2}$, we set
$|\Psi_{1}\rangle=|\alpha^{\prime},0\rangle,$
$|\Psi_{2}\rangle=|0,\alpha^{\prime}\rangle$ correspondingly. Comparing Eq.
(32) with Eq. (16), we can find the determinant of this density matrix,
expectation value of $\sigma_{3}$ and off-diagonal phase as
$\displaystyle{\rm det}(\tilde{\rho}_{1,2})$ $\displaystyle=$
$\displaystyle\mathcal{N}_{\alpha}^{4}(1-e^{-2|\alpha^{{}^{\prime}}|^{2}})(1-e^{-2|\beta^{{}^{\prime}}|^{2}}),$
$\displaystyle\langle\sigma_{3}\rangle_{\tilde{\rho}_{1,2}}$ $\displaystyle=$
$\displaystyle
1-2\mathcal{N}_{\alpha}^{2}+2\mathcal{N}_{\alpha}^{2}e^{-2|\alpha^{{}^{\prime}}|^{2}},$
$\displaystyle e^{i\tilde{\tau}}$ $\displaystyle=$ $\displaystyle 1.$ (33)
According to the preceding section, we can find the eigenvalues as
$\tilde{\lambda}_{\pm}=\frac{1}{2}\pm\frac{\sqrt{2e^{-|\alpha|^{2}}+e^{-2|\alpha^{\prime}|^{2}}+e^{-2|\beta^{\prime}|^{2}}}}{2+2e^{-|\alpha|^{2}}},$
(34)
and
$\tilde{v}_{\pm}=\frac{1}{2}\pm\frac{e^{-|\alpha|^{2}}+e^{-2|\alpha^{\prime}|^{2}}}{2\sqrt{2e^{-|\alpha|^{2}}+e^{-2|\alpha^{\prime}|^{2}}+e^{-2|\beta^{\prime}|^{2}}}}.$
(35)
Next we analyze the parametrization procedure. The unitary operator on
$\tilde{\rho}_{1,2}$ reads
$U(\varphi)=\exp(-i\varphi a_{2}^{\dagger}a_{2}),$ (36)
i.e., the generator of $\varphi$ is $H=a_{2}^{\dagger}a_{2}$. According to Eq.
(15), we only need to calculate the variance of $H$ in
$|\tilde{\lambda}_{\pm}\rangle$ and the transition probability of $H$ between
$|\tilde{\lambda}_{\pm}\rangle$. Since $H|\Psi_{1}\rangle=0$, we choose Eq.
(LABEL:eq:eigenstatesinPsi2) for convenience.
The variance in $|\tilde{\lambda}_{+}\rangle$ is
$\displaystyle\Delta H_{1}^{2}$ $\displaystyle=$
$\displaystyle\langle\tilde{\lambda}_{+}|(a_{2}^{\dagger}a_{2})^{2}|\tilde{\lambda}_{+}\rangle-(\langle\tilde{\lambda}_{+}|a_{2}^{\dagger}a_{2}|\tilde{\lambda}_{+}\rangle)^{2}$
(37) $\displaystyle=$
$\displaystyle\frac{\tilde{v}_{-}^{2}}{1-p^{2}}(|\alpha^{\prime
2}|^{2}+|\alpha^{\prime}|^{2}-\frac{\tilde{v}_{-}^{2}}{1-p^{2}}|\alpha^{\prime}|^{4}).$
Similarly, the variance in $|\tilde{\lambda}_{-}\rangle$ is
$\Delta H_{2}^{2}=\frac{\tilde{v}_{+}^{2}}{1-p^{2}}(|\alpha^{\prime
2}|^{2}+|\alpha^{\prime}|^{2}-\frac{\tilde{v}_{+}^{2}}{1-p^{2}}|\alpha^{\prime}|^{4}),$
(38)
and the transition term is
$|H_{12}|^{2}=(\frac{\tilde{v}_{+}\tilde{v}_{-}}{1-p^{2}}|\alpha^{\prime}|^{2})^{2}.$
(39)
Utilizing above expressions and based on Eq. (15), we can obtain the QFI of
$\rho_{1,2}$ as
$F=4\mathcal{N}_{\alpha}^{2}|\alpha|^{2}T\left[1+\mathcal{G}(T,\alpha)\right],$
(40)
where
$\mathcal{G}(T,\alpha)=\frac{(\mathcal{N}_{\alpha}^{2}-1)e^{-2|\alpha|^{2}T}+\mathcal{N}_{\alpha}^{2}e^{-2|\alpha|^{2}R}+2\mathcal{N}_{\alpha}^{2}e^{-|\alpha|^{2}}}{1-e^{-2|\alpha|^{2}T}}|\alpha|^{2}T.$
Notice that $\mathcal{N}_{\alpha}$ satisfies the relation
$2\mathcal{N}_{\alpha}^{2}e^{-|\alpha|^{2}}=1-2\mathcal{N}_{\alpha}^{2},$
then $\mathcal{G}(T,\alpha)$ can be rewritten as
$\mathcal{G}(T,\alpha)=|\alpha|^{2}T\left[1-\mathcal{N}_{\alpha}^{2}-\frac{\mathcal{N}_{\alpha}^{2}(1-e^{-2|\alpha|^{2}R})}{1-e^{-2|\alpha|^{2}T}}\right].$
Introduce the total average photon number $\bar{n}=\langle
a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}\rangle$, and it is easy to find that
in this case
$\bar{n}=2\mathcal{N}_{\alpha}^{2}|\alpha|^{2},$
then $\mathcal{G}(T,\alpha)$ can be further written into
$G(T,\alpha)=T\left[|\alpha|^{2}-\frac{\bar{n}}{2}-\frac{\bar{n}}{2}\frac{1-e^{-2|\alpha|^{2}R}}{1-e^{-2|\alpha|^{2}T}}\right],$
(41)
and the QFI (40) can be finally simplified as
$F=\bar{n}T\left[2+\left(2|\alpha|^{2}-\bar{n}-\bar{n}\frac{1-e^{-2|\alpha|^{2}R}}{1-e^{-2|\alpha|^{2}T}}\right)T\right].$
(42)
The QFI is only determined by the total average photon number $\bar{n}$ and
the transmission coefficient $T$.
When $T=R=1/2$ , the QFI reduces into
$\displaystyle F=\bar{n}+\frac{\bar{n}}{2}(|\alpha|^{2}-\bar{n})\geq\bar{n}.$
(43)
The last inequality is due to the fact that $|\alpha|^{2}\geq\bar{n}$ with the
equal sign holds in the limit of $|\alpha|^{2}\rightarrow\infty$ . Since $F$
decreases monotonically with the transmission coefficient, the ECS can surpass
the shot noise limit as long as $T>\frac{1}{2}$; when $T=1-R=1$ , i.e., there
is no particle loss in the interferometer, the QFI can be simplified as
$F=\bar{n}\left(2+2|\alpha|^{2}-\bar{n}\right),$ (44)
and due to $|\alpha|^{2}\geq\bar{n}$ , we have
$F\geq\bar{n}^{2}+2\bar{n}.$ (45)
There is a debate over the ultimate scaling of the phase sensitivity for
states with a fluctuating number of particles [36]. There are two candidates
in the literature: the so-called Hofmann limit $\delta\varphi\sim
1/\sqrt{\overline{n^{2}}}$, and the Heisenberg limit $\delta\varphi\sim
1/\overline{n}$. Here we will show that the ECS can surpass the Heisenberg
limit and Hofmann limit, even in the presence of particle loss.
From inequality (45), one can find that the QFI without particle loss is
greater than $\overline{n}^{2}$, next we will show it is also greater than
$\overline{n^{2}}$. The average of
$n^{2}=(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})^{2}$ does not change after
the first beam splitter. Then it is easy to find
$\displaystyle\overline{n^{2}}$ $\displaystyle=$
$\displaystyle\langle\mathrm{ECS}|_{1,2}(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2})^{2}|\mathrm{ECS}\rangle_{1,2}$
$\displaystyle=$ $\displaystyle
2\mathcal{N}_{\alpha}^{2}\left[|\alpha|^{2}+|\alpha|^{4}\right]$
$\displaystyle=$ $\displaystyle\left(1+|\alpha|^{2}\right)\overline{n},$
and compare with the QFI, we have
$\displaystyle
F=2\overline{n^{2}}-\overline{n}^{2}=\overline{n^{2}}+\Delta(n),$ (46)
where $\Delta(n)$ is the variance of the photon number. It is clear that $F$
is larger than both of $\overline{n^{2}}$ and $\overline{n}^{2}$.
Figure 1: The QFI of ECS with particle loss. Here $R=1-T$, $|\alpha|=2$. When
the particle loss is small, the QFI is larger than both $\overline{n^{2}}$ and
$\overline{n}^{2}$.
Figure. 1 shows the variation of QFI with the increase of $R$. Points A, B and
C represent the intersection with the Hofmann limit, Heisenberg limit and shot
noise limit repectively. The corresponding reflection coefficients read
$R_{\mathrm{A}}=0.03$, $R_{\mathrm{B}}=0.07$ and $R_{\mathrm{C}}=0.52$. From
this figure, one can find that when $R<R_{\mathrm{A}}$, the ECS can always
surpass the Hofmann limit, and for $R<R_{\mathrm{B}}$, the precision is still
better than the Heisenberg limit. This indicates that the precision is robust
and overcomes the Heisenberg limit with a small loss of photons within
$R_{\mathrm{B}}$. If the precision is only required in the range of shot noise
limit, then this interferometer can tolerate a loss of half photons.
The ECS is very useful and robust for quantum metrology [37, 38]. Our formula
gives an easy approach to the determination of the QFI of ECS and one doesn’t
have to resort to numeric methods.
## 5 Conclusion
We have derived an explicit formula for the QFI for a large class of states in
which the parameter is introduced by a unitary dynamics $U=\exp(-iH\varphi)$.
We pointed out that the QFI in this scenario is the mean variance of $H$ over
the eigenstates minus weighted cross terms. Finally, we analyzed the QFI of a
density matrix with $M=2$ and apply our result into an entangled coherent
state in a Mach-Zehnder interferometer, which was proposed in a recent paper
[5].
We have found the analytical expression of the QFI for the ECS when there is
particle loss. We find that even in the lossy condition, the ECS can still
surpass the Heisenberg limit. The formalism developed here can be applicable
to the study of more complicated states, such as the reduced two-mode mixed
state when the total multi-mode system is in a multipartite entangled coherent
states.
The authors thank Xiao-Ming Lu and Qing-Shou Tan for useful discussion. This
work was supported by the NFRPC with Grant No.2012CB921602 and NSFC with Grant
No.11025527 and No.10935010. _Note added_ : After the submission of our
manuscript, we notice that the authors in Ref. [39] do a relevant work and
have a similar conclusion.
## Appendix A Eigenvalues and Eigenstates of A $2\times 2$ Density Matrix
A general $2\times 2$ density matrix $\rho$ is given in the form
$\rho=\left(\begin{array}[]{cc}\eta&\xi e^{i\tau}\\\ \xi
e^{-i\tau}&1-\eta\end{array}\right).$ (47)
For this matrix to represent a physical state, one condition must be met: the
determinant of $\rho$ must be positive, i.e., ${\rm
det}(\rho)=\eta(1-\eta)-\xi^{2}\geq 0$ (this inequality implies $\eta\geq 0$ ,
thus fullfil the positivity requirement of density matrix). Here $\xi>0$,
$\tau\in[0,2\pi)$ are real numbers due to the Hermiticity of density matrix.
The eigenvalues of $\rho$ can be easily calculated as
$\displaystyle\lambda_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1\pm\sqrt{1-4{\rm det}(\rho)}}{2},$ (48)
and the corresponding normalized eigenvectors read
$\displaystyle|\lambda_{+}\rangle$ $\displaystyle=$
$\displaystyle\left(v_{+}e^{i\tau},v_{-}\right)^{\rm{T}},$
$\displaystyle|\lambda_{-}\rangle$ $\displaystyle=$
$\displaystyle\left(-v_{-}e^{i\tau},v_{+}\right)^{\rm{T}},$ (49)
with
$\displaystyle v_{\pm}=\left(\frac{\sqrt{1-4{\rm
det}(\rho)}\pm\langle\sigma_{3}\rangle}{2\sqrt{1-4{\rm
det}(\rho)}}\right)^{\frac{1}{2}},$ (50)
Here $\sigma_{3}$ is a Pauli matrix and $\langle\sigma_{3}\rangle={\rm
Tr}(\rho\sigma_{3})=2\eta-1$.
We can see that the eigenvalues and eigenvectors of $\rho$ are fully
determined by ${\rm det}(\rho),$ $\langle\sigma_{3}\rangle$ and $\tau.$
## Appendix B An equivalent way to solve the eigen problem of density
operator in nonorthogonal basis
In this appendix we provide an equivalent way to solve the eigen problem of
Eq. (16). Instead of recasting $\tilde{\rho}$ into an orthonormal basis, we
assume the eigenvector as
$|\phi\rangle=c_{1}|\Psi_{1}\rangle+c_{2}|\Psi_{2}\rangle.$ (51)
Then the eigen equation reads
$\tilde{\rho}|\phi\rangle=\lambda|\phi\rangle,$ (52)
specifically (in the basis of $|\Psi_{1,2}\rangle$),
$\left(\begin{array}[]{cc}a+bp^{*}&ap+b\\\
b^{*}+cp^{*}&b^{*}p+c\end{array}\right)\left(\begin{array}[]{c}c_{1}\\\
c_{2}\end{array}\right)=\lambda\left(\begin{array}[]{c}c_{1}\\\
c_{2}\end{array}\right),$ (53)
i.e., we need find the eigenvalues and eigenvectors of the left matrix. One
can easily find the trace and determinant are the same as those of Eq. (17),
thus the eigenvalues are equal according to Eq. (48).
The eigenvectors can also be easily calculated as
$\displaystyle|\phi_{1}\rangle$ $\displaystyle=$ $\displaystyle
P_{11}|\Psi_{1}\rangle+P_{21}|\Psi_{2}\rangle,$
$\displaystyle|\phi_{2}\rangle$ $\displaystyle=$ $\displaystyle
P_{12}|\Psi_{1}\rangle+P_{22}|\Psi_{2}\rangle,$ (54)
with the normalized conditions
$\displaystyle|P_{11}|^{2}+|P_{21}|^{2}+2{\rm{Re}(pP_{11}^{*}P_{21})=1},$
$\displaystyle|P_{12}|^{2}+|P_{22}|^{2}+2{\rm{Re}(pP_{12}^{*}P_{22})=1},$ (55)
where ${\rm Re}$ stands for real component. After some straightforward
calculation, we can find
$\displaystyle P_{11}$ $\displaystyle=$
$\displaystyle\tilde{v}_{+}e^{i\tilde{\tau}}-\frac{\tilde{v}_{-}p}{\sqrt{1-|p|^{2}}},$
$\displaystyle P_{21}$ $\displaystyle=$
$\displaystyle\frac{\tilde{v}_{-}}{\sqrt{1-|p|^{2}}},$ $\displaystyle P_{12}$
$\displaystyle=$
$\displaystyle-\tilde{v}_{-}e^{i\tilde{\tau}}-\frac{\tilde{v}_{+}p}{\sqrt{1-|p|^{2}}},$
$\displaystyle P_{22}$ $\displaystyle=$
$\displaystyle\frac{\tilde{v}_{+}}{\sqrt{1-|p|^{2}}},$ (56)
where $e^{i\tilde{\tau}}$ and $\tilde{v}_{\pm}$ are defined in Eq. (18) and
Eq. (21), i.e., the eigenstates in Eq. (54) are actually the same with Eq.
(LABEL:eq:eigenstatesinPsi2).
This method is a routine way to solving eigen problem. However, taking account
of the normalization condition Eq. (55), it is quite tedious in calculation.
We hope the method in the main text can offer some convenience when dealing
with similar problems.
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|
arxiv-papers
| 2013-07-30T15:03:17 |
2024-09-04T02:49:48.761585
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiaoxing Jing, Jing Liu, Wei Zhong, and Xiaoguang Wang",
"submitter": "Jing Liu",
"url": "https://arxiv.org/abs/1307.8009"
}
|
1307.8134
|
aainstitutetext: AHEP Group, Institut de Física Corpuscular –
C.S.I.C./Universitat de València
Edificio Institutos de Paterna, Apt 22085, E–46071 Valencia,
Spainbbinstitutetext: Institut für Theoretische Physik und Astrophysik,
Universität Würzburg,
97074 Würzburg, Germany.ccinstitutetext: Pontificia Universidad Católica de
Chile, Facultad de Física. Av. Vicuña Mackenna 4860. Macul. Santiago de Chile,
Chile.
# WIMP dark matter as radiative neutrino mass messenger
M. Hirsch a R. A. Lineros b S. Morisi a J. Palacio c N. Rojas a J. W. F. Valle
###### Abstract
The minimal seesaw extension of the Standard $\mathrm{SU(3)_{c}\otimes
SU(2)_{L}\otimes U(1)_{Y}}$ Model requires two electroweak singlet fermions in
order to accommodate the neutrino oscillation parameters at tree level. Here
we consider a next to minimal extension where light neutrino masses are
generated radiatively by two electroweak fermions: one singlet and one triplet
under SU(2)L. These should be odd under a parity symmetry and their mixing
gives rise to a stable weakly interactive massive particle (WIMP) dark matter
candidate. For mass in the GeV–TeV range, it reproduces the correct relic
density, and provides an observable signal in nuclear recoil direct detection
experiments. The fermion triplet component of the dark matter has gauge
interactions, making it also detectable at present and near future collider
experiments.
††arxiv: 1307.8134††dedication: IFIC/13-53
## 1 Introduction
Despite the successful discovery of the Higgs boson, so far the Large Hadron
Collider (LHC) has not discovered any new physics, so neutrino physics
remains, together with dark matter, as the main motivation to go beyond the
Standard Model (SM). Neutrino oscillation experiments indicate two different
neutrino mass squared differences Schwetz:2011zk ; Tortola:2012te . As a
result at least two of the three active neutrino must be massive, though the
oscillation interpretation is compatible with one of the neutrinos being
massless. In the Standard Model neutrinos have no mass at the renormalizable
level. However they can get a Majorana mass by means of the dimension-5
Weinberg operator,
$\frac{c}{\Lambda}\,LH\,LH\,,$ (1)
where $\Lambda$ is an effective scale, $c$ a dimensionless coefficient and $L$
and $H$ denote the lepton and Higgs isodoublets, respecively. This operator
should be understood as encoding new physics associated to heavy “messenger”
states whose fundamental renormalizable interactions should be prescribed. The
smallness of neutrino masses compared to the other fermion masses, suggests
that the messenger scale $\Lambda$ must is much higher than the electroweak
scale if the coefficient $c$ in equation 1 is of $\mathcal{O}(1)$. For
example, the scale $\Lambda$ should be close to the Grand Unification scale if
$c$ is generated at tree level. One popular mechanism to generate the
dimension-5 operator is the so–called seesaw mechanism. Its most general
$\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ realization is the so
called “1-2-3” seesaw scheme Schechter:1981cv with singlet, doublet and
triplet scalar $SU(2)_{L}$ fields with vevs respectively $v_{1}$, $v_{2}$ and
$v_{3}$. Assuming $m$ extra singlet fermions (right-handed neutrinos), the
“1-2-3” scheme is described by the $(3+m)\times(3+m)$ matrix
$M^{\nu}=\left(\begin{array}[]{cc}Y_{3}v_{3}&Y_{2}v_{2}\\\
Y_{2}^{T}v_{2}&Y_{1}v_{1}\end{array}\right).$ (2)
The vevs obey the seesaw relation
$v_{3}v_{1}\sim v_{2}^{2}\qquad\mbox{with}\qquad v_{1}\gg v_{2}\gg v_{3}\,,$
(3)
giving two contributions to the light neutrino masses
$Y_{3}v_{3}+v_{2}^{2}/v_{1}\,Y_{2}Y_{1}^{-1}Y_{2}^{T}$, called respectively
type-II and type-I seesaw. Assuming $Y_{3}=0$, namely no Higgs triplet 111Note
that in pure type-II seesaw, only one extra scalar field is required, in
contrast with type-I, where at least two fermion singlets must be assumed.,
the light neutrino masses arise only from the type-I seesaw contribution. In
this case it is well known that in order to accommodate the neutrino
oscillation parameters, at least two right-handed neutrinos are required,
namely $m\geq 2$. We call the case $m=2$ minimal. Note that in this case one
neutrino mass is zero and so the absolute neutrino mass scale is fixed.
Typically the next to minimal case is to assume three sequential right-handed
neutrinos, that is $m=3$. An alternative seesaw mechanism is the so called
type-III in which the heavy the “right-handed” neutrino “messenger” states are
replaced by SU(2)L triplet fermions Foot:1988aq . As for the type-I seesaw
case, one must assume at least two fermion triplets (if only fermion triplets
are present) in order to accommodate current neutrino oscillation data.
There is an interesting way to induce the dimension-5 operator by mimicking
the seesaw mechanism at the radiative level. This requires the fermion
messengers to be odd under an ad-hoc symmetry $Z_{2}$ in order to accommodate
a stable dark matter (DM) candidate. In this case one can have “scotogenic”
Ma:2006km neutrino masses, induced by dark matter exchange. This trick can be
realized either in type-I or type-III seesaw schemes Ma:2006km ; Ma:2008cu .
To induce Yukawa couplings between the extra fermions and the Standard Model
leptons, one must include additional scalar doublets, odd under the assumed
$Z_{2}$ symmetry, and without vacuum expectation value. In order to complete
the saga in this paper we propose a hybrid scotogenic construction which
consists in having just one singlet fermion ($m=1$) but adding one triplet
fermion as well.
Figure 1: One loop realization for the Weinberg operator.
This also gives rise to light neutrino masses, calculable at the one loop
level, as illustrated in figure 1 222Note the scalar contributions come from
the scalar and pseudoscalar pieces of the field $\eta$.. However, due to
triplet–singlet mixing, the lightest combimation of the neutral component of
the fermion triplet and the singlet will be stable and can play the role of
WIMP dark matter. We show that it provides a phenomenologically interesting
alternative to all previous “scotogenic” proposals since here the dark matter
can have sizeable gauge interactions. As a result, in addition to direct and
indirect detection signatures, it can also be kinematically accessible to
searches at present colliders such as the LHC.
Existing collider searches at LEP Ellis:1988zy ; L3:2001PhLB and LHC
CMS:2012PhLB , set a nominal lower bound of $\sim$ 100 GeV for the masses of
new charged particles. However, coannihilations present in the early universe,
between the neutral and charged components, set the dark matter mass to be of
the order of Ma:2008cu
$M_{\rm DM}\simeq 2.7\leavevmode\nobreak\ {\rm TeV}$ (4)
in order to explain the observed abundance planck:2013 :
$\Omega_{\rm DM}h^{2}=0.1196\pm 0.0031\,.$ (5)
Radiative neutrino masses generated by at least two generations of fermion
singlets or triplets have been studied in Ref. Kubo:2006yx . Here we focus on
the radiative neutrino mass generation with one singlet and one triplet
fermion which has interesting phenomenological consequences compared to the
cases aforementioned cases. In our scenario, the dark matter candidate can
indeed be observed not only in indirect but can also be kinematically
accessible to current collider searches, and need not obey Eq. (4). Moreover,
we will show that, in contrast to the proposed schemes in Refs. Ma:2006km ;
Ma:2008cu in our framework amplitudes leading naturally to direct detection
processes appear at the tree level, thanks to singlet-triplet mixing effects.
The rest of this paper is organized as follows: in section 2 we introduce the
new fields and interactions present in the model, making emphasis upon the
mixing matrices and the radiative neutrino mass generation mechanism. Section
3 is devoted to numerical results on the phenomenology of dark matter in this
model. An interesting feature of the model is the wide range of possible dark
matter masses, ranging from 1 GeV to a few TeV. We also briefly discuss some
the implications for LHC physics. In Section 4 we give our conclusions.
## 2 The model
Our model combines the ingredients employed in the models proposed in
Ma:2006km ; Ma:2008cu in such a way that it has a richer phenomenology than
either Ma:2006km or Ma:2008cu .
### 2.1 The Model and the Particle Content
The new fields with respect to the Standard Model include one Majorana fermion
triplet $\Sigma$ and a Majorana fermion singlet $N$ both with zero hypercharge
and both odd under an ad-hoc symmetry $Z_{2}$. We also include a scalar
doublet $\eta$ with same quantum numbers as the Higgs doublet, but odd under
$Z_{2}$. In addition, we require that $\eta$ not to acquire a vev. As a
result, neutrino masses are not generated at tree level by a type-I/III seesaw
mechanism. Instead they are one-loop calculable, from diagrams in Fig. 1.
Furthermore, this symmetry forbids the decays of the lightest $Z_{2}$ odd
particle into Standard Model particles, which is a mixture of the neutral
component of $\Sigma$ and $N$. As a result this becomes a viable dark matter
candidate. Note also that our proposed model does not modify quark dynamics,
since neither of the new fields couples to quarks.
The fermion triplet, can be expanded as follows ($\sigma_{i}$ are the Pauli
matrices):
$\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\Sigma_{1}\sigma_{1}+\Sigma_{2}\sigma_{2}+\Sigma_{3}\sigma_{3}\,=\,\left(\begin{array}[]{cc}\Sigma_{0}&\sqrt{2}\Sigma^{+}\\\
\sqrt{2}\Sigma^{-}&-\Sigma_{0}\\\ \end{array}\right)\,,$ (8)
where
$\displaystyle\Sigma^{+}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\Sigma_{1}+i\Sigma_{2}\right)\,,$ (9)
$\displaystyle\Sigma^{-}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\Sigma_{1}-i\Sigma_{2}\right)\,,$ (10)
$\displaystyle\Sigma^{0}$ $\displaystyle=$ $\displaystyle\Sigma_{3}\,.$ (11)
The $Z_{2}$ is exactly conserved in the Lagrangian, moreover, it allows
interactions between dark matter and leptons, in fact, this is the origin of
radiative neutrino masses. The Yukawa couplings between the triplet and
leptons play an important role in the dark matter production. Finally a
triplet scalar $\Omega$ is introduced in order to mix the neutral part of the
fermion triplet $\Sigma^{0}$ and the fermion singlet $N$. This triplet scalar
field also has zero hypercharge and is even under the $Z_{2}$ symmetry, thus,
its neutral component can acquire a nonzero vev.
| Standard Model | Fermions | Scalars
---|---|---|---
| $L$ | $e$ | $\phi$ | $\Sigma$ | N | $\eta$ | $\Omega$
$SU(2)_{L}$ | 2 | 1 | 2 | 3 | 1 | 2 | 3
$Y$ | -1 | -2 | 1 | 0 | 0 | 1 | 0
$Z_{2}$ | $+$ | $+$ | $+$ | $-$ | $-$ | $-$ | $+$
Table 1: Matter assignment of the model.
### 2.2 Yukawa Interactions and Fermion Masses
The most general $\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ and
Lorentz invariant Lagrangian is given as
$\displaystyle\mathcal{L}$ $\displaystyle\supset$ $\displaystyle-
Y_{\alpha\beta}\,\overline{L}_{\alpha}e_{\beta}\phi-
Y_{\Sigma_{\alpha}}\overline{L}_{\alpha}C\Sigma^{\dagger}\tilde{\eta}-\frac{1}{4}M_{\Sigma}\mbox{Tr}\left[\overline{\Sigma}^{c}\Sigma\right]+$
(12) $\displaystyle-
Y_{\Omega}\mbox{Tr}\left[\overline{\Sigma}\Omega\right]N-Y_{N_{\alpha}}\overline{L}_{\alpha}\tilde{\eta}N-\frac{1}{2}M_{N}\overline{N}^{c}N+h.c.\,,$
The $C$ symbol stands for the Lorentz charge conjugation matrix $i\sigma_{2}$
and $\tilde{\eta}=i\sigma_{2}\eta^{*}$.
The Yukawa term $Y_{\alpha\beta}$ is the SM Yukawa interaction for leptons,
taken as diagonal matrix in the flavor basis333We can always go to this basis
with a unitary transformation.. On the other hand the Yukawa coupling
$Y_{\Omega}$ mixes the $\Sigma$ and $N$ fields and when the neutral part of
the $\Omega$ field acquire a vev $v_{\Omega}$, the dark matter particle can be
identified to the lightest mass eigenstate of the mass matrix,
$\displaystyle M_{\chi}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}M_{\Sigma}&2Y_{\Omega}v_{\Omega}\\\
2Y_{\Omega}v_{\Omega}&M_{N}\end{array}\right)\,,$ (15)
in the basis $\psi^{T}=\left(\Sigma_{0}\,,N\right)$. As a result one gets the
following tree level fermion masses
$\displaystyle m_{\chi^{\pm}}$ $\displaystyle=$ $\displaystyle M_{\Sigma}\,,$
(16) $\displaystyle m_{\chi^{0}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(M_{\Sigma}+M_{N}-\sqrt{\displaystyle(M_{\Sigma}-M_{N})^{2}+4(2Y_{\Omega}v_{\Omega})^{2}}\right)\,,$
(17) $\displaystyle m_{\chi^{0}_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(M_{\Sigma}+M_{N}+\sqrt{\displaystyle(M_{\Sigma}-M_{N})^{2}+4(2Y_{\Omega}v_{\Omega})^{2}}\right)\,,$
(18) $\displaystyle\tan(2\,\alpha)$ $\displaystyle=$
$\displaystyle\frac{4Y_{\Omega}v_{\Omega}}{M_{\Sigma}-M_{N}}\,,$ (19)
where $\alpha$ is the mixing angle between $\Sigma_{0}$ and $N$. Here
$M_{\Sigma}$ and $M_{N}$ characterize the Majorana mass terms for the triplet
and the singlet, respectively. The $M_{\Sigma}$ term is also the mass of the
charged component of the $\Sigma$ field, this issue is important because the
mass splitting between $\Sigma^{\pm}$ and the dark matter candidate will play
a role in the calculation of its relic density. As we will see later, the
splitting induced by $v_{\Omega}$ allows us to relax the constraints on the
dark matter coming from the existence of $\Sigma^{\pm}$.
### 2.3 Scalar potential and spectrum
The most general scalar potential, even under $Z_{2}$, including the fields
$\phi$, $\eta$ and $\Omega$ and allowing for spontaneous symmetry breaking,
may be written as:
$\displaystyle V_{\rm scal}$ $\displaystyle=$ $\displaystyle-
m_{1}^{2}\phi^{\dagger}\phi+m_{2}^{2}\eta^{\dagger}\eta+\frac{\lambda_{1}}{2}\left(\phi^{\dagger}\phi\right)^{2}+\frac{\lambda_{2}}{2}\left(\eta^{\dagger}\eta\right)^{2}+\lambda_{3}\left(\phi^{\dagger}\phi\right)\left(\eta^{\dagger}\eta\right)$
(20) $\displaystyle+$
$\displaystyle\lambda_{4}\left(\phi^{\dagger}\eta\right)\left(\eta^{\dagger}\phi\right)+\frac{\lambda_{5}}{2}\left(\phi^{\dagger}\eta\right)^{2}+h.c.-\frac{M_{\Omega}^{2}}{4}Tr\left(\Omega^{\dagger}\Omega\right)+\left(\mu_{1}\phi^{\dagger}\Omega\phi+h.c.\right)$
$\displaystyle+$
$\displaystyle\lambda^{\Omega}_{1}\phi^{\dagger}\phi\,Tr\left(\Omega^{\dagger}\Omega\right)+\lambda^{\Omega}_{2}\left(Tr(\Omega^{\dagger}\Omega)\right)^{2}+\lambda^{\Omega}_{3}Tr(\left(\Omega^{\dagger}\Omega\right)^{2})+\lambda^{\Omega}_{4}\left(\phi^{\dagger}\Omega\right)\left(\Omega^{\dagger}\phi\right)$
$\displaystyle+$
$\displaystyle\left(\mu_{2}\eta^{\dagger}\Omega\eta+h.c.\right)+\lambda^{\eta}_{1}\eta^{\dagger}\eta\,Tr\left(\Omega^{\dagger}\Omega\right)+\lambda^{\eta}_{4}\left(\eta^{\dagger}\Omega\right)\left(\Omega^{\dagger}\eta\right)\,,$
where the fields $\eta$, $\phi$ and $\Omega$, can be written as follows:
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\eta^{+}\\\
(\eta^{0}+i\eta^{A})/\sqrt{2}\end{array}\right)\,,$ (23) $\displaystyle\phi$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}\varphi^{+}\\\
(h_{0}+v_{h}+i\varphi)/\sqrt{2}\end{array}\right)\,,$ (26)
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}(\Omega_{0}+v_{\Omega})&\sqrt{2}\,\Omega^{+}\\\
\sqrt{2}\,\Omega^{-}&-(\Omega_{0}+v_{\Omega})\end{array}\right)\,,$ (29)
where $v_{h}$ and $v_{\Omega}$ are the vevs of $\phi$ and $\Omega$ fields
respectively. We have three charged fields one of which is absorbed by the $W$
boson, three CP-even physical neutral fields, and two CP-odd neutral fields
one of which is absorbed by the $Z$ boson 444Remember that the neutral part of
$\Omega$ field is real, so it does not contribute to the CP-odd sector..
Let us first consider the charged scalar sector. The charged Goldstone boson
is a linear combination of the $\varphi^{+}$ and the $\Omega^{+}$, changing
the definition for the $W$ boson mass from that in the Standard Model :
$\displaystyle M_{W}=\frac{g}{2}\sqrt{v_{h}^{2}+v_{\Omega}^{2}}$. Note that
this places a constraint on the vev of $v_{\Omega}$ from electroweak precision
tests Gunion:1989ci ; Gunion:1989we , one can expect roughly this vev to be
less than 7 GeV, in order to keep the $\displaystyle
M_{Z}=\frac{\sqrt{g^{2}+{g^{\prime}}^{2}}}{2}v_{h}$ in the experimental range,
and alter the $M_{W}$ value inside the experimental error band.
Apart from the $W$ boson, the two charged scalars have mass:
$\displaystyle M_{\pm}^{2}$ $\displaystyle=$ $\displaystyle
2\mu_{1}\left(v_{h}^{2}+v_{\Omega}^{2}\right)/v_{\Omega}\,,$ (30)
$\displaystyle m_{\eta^{\pm}}^{2}$ $\displaystyle=$ $\displaystyle
m_{2}^{2}+\frac{1}{2}\lambda_{3}v_{h}^{2}+2\mu_{2}v_{\Omega}+\left(2\lambda^{\eta}_{1}+\lambda^{\eta}_{4}\right)v_{\Omega}^{2}\,.$
(31)
Notice that the nonzero vacuum expectation value $v_{\Omega}\neq 0$ will play
an important role in generating the novel phenomenological effects of interest
to us (see below). Now let us consider the neutral part: the minimization
conditions of the Higgs potential allow vevs for the neutral part of the usual
$\phi$ field as well as for the neutral part of the $\Omega$ field. The mass
matrix for neutral scalar eigenstates in the basis
$\Phi^{T}=\left(h_{0}\,,\Omega_{0}\right)$ is:
$\displaystyle\mathcal{M}_{s}^{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\lambda_{1}v_{h}^{2}+\frac{t_{h}}{v_{h}}&-2\mu_{1}v_{h}+4v_{h}v_{\Omega}\left(\lambda^{\Omega}_{1}+\frac{\lambda^{\Omega}_{4}}{2}\right)\\\
-2\mu_{1}v_{h}+4v_{h}v_{\Omega}\left(\lambda^{\Omega}_{1}+\frac{\lambda^{\Omega}_{4}}{2}\right)&\frac{\mu_{1}v_{h}^{2}}{v_{\Omega}}+16v_{\Omega}^{2}\left(2\lambda^{\Omega}_{2}+\lambda^{\Omega}_{3}\right)+\frac{t_{\Omega}}{v_{\Omega}}\\\
\end{array}\right)\,,$ (34)
where $t_{h}$ and $t_{\Omega}$ are the tadpoles for $h_{0}$ and $\Omega_{0}$
and are described in Appendix A.2. The presence of the vev $v_{\Omega}$
induces the mixing between $h_{0}$ and $\Omega_{0}$. The corresponding
eigenvalues give us the masses of the Standard Model Higgs doublet and the
second neutral scalar both labelled as $S_{i}^{0}$.
On the other hand, the $\eta$ field does not acquire vev, therefore, the mass
eigenvalues of the neutral $\eta^{0}$, charged $\eta^{\pm}$ and pseudoscalar
$\eta^{A}$ are decoupled. The spectrum for $\eta^{0}$ and $\eta^{A}$ fields
is:
$\displaystyle m_{\eta 0}^{2}$ $\displaystyle=$ $\displaystyle
m_{\eta\pm}^{2}+\frac{1}{2}\left(\lambda_{4}+\lambda_{5}\right)v_{h}^{2}-4\mu_{2}v_{\Omega}\,,$
(35) $\displaystyle m_{\eta A}^{2}$ $\displaystyle=$ $\displaystyle
m_{\eta\pm}^{2}+\frac{1}{2}\left(\lambda_{4}-\lambda_{5}\right)v_{h}^{2}-4\mu_{2}v_{\Omega}\,.$
(36)
### 2.4 Radiative Neutrino Masses
In this model, neutrino masses are generated at one loop. The dark matter
candidate particle acts as a messenger for the masses. The relevant
interactions for radiative neutrino mass generation arise from from Eqs. (12)
and (20) and can be written in terms of the tree level mass eigenstates.
Symbolically, one can rewrite the relevant terms for this purpose as:
$\displaystyle\begin{array}[]{ccccc}L\,\Sigma\,\eta&\longrightarrow&h_{ij}\nu_{i}\,{\chi}^{0}_{j}\,\eta_{0}&,&h_{ij}\nu_{i}\,{\chi}^{0}_{j}\,\eta_{A}\\\
L\,\eta\,N&\longrightarrow&h_{ij}\nu_{i}\,{\chi}^{0}_{j}\,\eta_{0}&,&h_{ij}\nu_{i}\,{\chi}^{0}_{j}\,\eta_{A}\\\
\left(\phi^{\dagger}\eta\right)^{2}&\longrightarrow&\left[\left(h+v_{h}\right)\,\eta_{0}\right]^{2}&,&\left[\left(h+v_{h}\right)\,\eta_{A}\right]^{2}\\\
\end{array}$ (40)
Here the field ${\chi}^{0}_{j}$ are the mass eigenstate of the matrix (15) and
$h$ is a $3\times 2$ matrix and is given by
$h=\left(\begin{array}[]{cc}Y_{1}^{\Sigma}&Y_{1}^{N}\\\
Y_{2}^{\Sigma}&Y_{2}^{N}\\\ Y_{3}^{\Sigma}&Y_{3}^{N}\\\
\end{array}\right)\cdot V(\alpha)\,.$ (41)
where $V(\alpha)$ is the $2\times 2$ orthogonal matrix that diagonalizes the
matrix in equation (15). There are two contributions to the neutrino masses
from the loops in figure 1, where the $\eta_{0}$ and $\eta_{A}$ fields are
involved in the loop. With the above ingredients, from the diagram in Fig. 1
one finds that the neutrino mass matrix is given by:
$\displaystyle M_{\alpha\beta}^{\nu}$ $\displaystyle=$
$\displaystyle\sum_{k=1,2}\frac{h_{\alpha\sigma}h_{\beta\sigma}}{16\pi^{2}}I_{k}\left(M_{k},m^{2}_{\eta_{0}},m^{2}_{\eta_{A}}\right)\,.$
(42)
The $I_{k}$ functions correspond essentially to a differences of the $B_{0}$
Veltman functions Passarino:1978jh , when evaluated at different scalar
masses, note they have mass dimensions. The index $k$ runs over the $\chi^{0}$
mass eigenvalues, i.e. $\sigma=1,2$. Note that these masses are independent of
the renormalization scale. In the equation below, each $M_{k}$ stands for the
mass values of the $\chi^{0}$ fields.
$\displaystyle
I_{k}\left(M_{k},m^{2}_{\eta_{0}},m^{2}_{\eta_{A}}\right)=M_{k}\frac{m^{2}_{\eta_{0}}}{m_{\eta_{0}}^{2}-M_{k}^{2}}\log\left(\frac{m_{\eta_{0}}^{2}}{M_{k}^{2}}\right)-M_{k}\frac{m^{2}_{\eta_{A}}}{m_{\eta_{A}}^{2}-M_{k}^{2}}\log\left(\frac{m_{\eta_{A}}^{2}}{M_{k}^{2}}\right)$
(43)
It is useful to rewrite the equation 42 in a compact way as follows
$\displaystyle M^{\nu}$ $\displaystyle=$ $\displaystyle
hv_{h}\cdot\left(\begin{array}[]{cc}\frac{I_{1}}{16\pi^{2}v_{h}^{2}}&0\\\
0&\frac{I_{2}}{16\pi^{2}v_{h}^{2}}\end{array}\right)\cdot h^{T}v_{h}\equiv
hv_{h}\cdot\frac{D_{I}}{v_{h}^{2}}\cdot h^{T}v_{h}\sim
m_{D}\frac{1}{M_{R}}m_{D}^{T}$ (46)
which is formally equivalent to the standard type-I seesaw relation with
$M_{R}^{-1}\to D_{I}/v_{h}^{2}$ Schechter:1980gr . This is a diagonal matrix
while $h\,v_{h}$ plays the role of the Dirac mass matrix, in our case it is a
$3\times 2$ matrix. It is not difficult to see that we can fit the required
neutrino oscillation parameters Schwetz:2011zk ; Tortola:2012te , for example,
by means of the Casas Ibarra parametrization Casas:2001sr .
In order to get an idea about the order of magnitude of the parameters
required for producing the correct neutrino masses, one can consider a special
limit in equation 42. For example, in cases where both $\chi^{0}$ are lighter
than the other fields, we have from 42:
$\displaystyle M_{\alpha\beta}^{\nu}$ $\displaystyle=$
$\displaystyle\sum_{\sigma=1,2}\frac{h_{\alpha\sigma}h_{\beta\sigma}}{8\pi^{2}}\frac{\lambda_{5}v_{h}^{2}}{m_{0}^{2}}M_{k}\,.$
(47)
Here $\lambda_{5}$ is the $\left(\phi^{\dagger}\eta\right)^{2}$ coupling
introduced in equation 20. The $M_{k}$ are the masses of the neutral $Z_{2}$
fermion fields $\chi$. The $m_{0}$ mass term comes from writing the masses of
the $\eta_{0}$, and $\eta_{A}$ in the following way:
$m^{2}_{\eta_{0},\,\eta_{A}}=m_{0}\pm\lambda_{5}v_{h}^{2}$, see appendix A.1
for more details. In particular we are interested in the magnitude of the
Yukawa couplings $h_{\alpha\beta}$ required in order to have neutrino with
masses of the order of eV. For masses of $\chi^{0}$ of order of 10 GeV and
$\eta_{0,A}$ of order of 1000 GeV, and $\lambda$ couplings not too small,
namely of order of $10^{-2}$, one finds that the values for $h_{\alpha\beta}$
are in the order of the bottom Yukawa coupling $\sim 10^{-2}$. Hence it is not
necessary to have a tiny Yukawa for obtaining the correct neutrino masses.
## 3 Fermion Dark Matter
Figure 2: $\Sigma^{0}$ and $N$ co-annihilation channels. Figures (g) and (h)
correspond to the processes involved in the $\Sigma^{\pm}$ abundace.
Figure 3: $\Sigma^{0}$ and $N$ annihilation channels.
As previously described the model contains two classes of potential dark
matter candidates. One class are the $Z_{2}$ odd scalars: $\eta^{0}$ and
$\eta^{A}$, when any of them is the lightest $Z_{2}$ odd particle. Their
phenomenology is very close to the inert doublet dark matter model
LopezHonorez:2006gr or discrete dark matter models Hirsch:2010ru ;
Boucenna:2011tj . For this reason here we focus our analysis on the other
candidates which are the fermion states $\chi^{0}_{i}$. In this case, the dark
matter candidate is a mixed state between $N$ and $\Sigma^{0}$. This interplay
brings an enriched dark matter phenomenology with respect to models with only
singlets or triplets.
For models with only fermion triplets as dark matter, equivalent in our model
to taking $M_{N}\to\infty$, the main constraints come from the observed relic
abundance (equation 5). Coannihilations between $\Sigma^{0}$ and
$\Sigma^{\pm}$ are efficient processes due to the mass degeneracy between
them, controlling the relic abundance. These processes force the dark matter
mass to be 2.7 TeV. In addition, direct detection occurs only at the one loop
level Cirelli:2005uq , see Fig. 4.
Figure 4: Direct detection in pure triplet or pure singlet models (left panel)
and in our mixed triplet-singlet case (right panel).
Most of the corresponding features have been already studied in Ma:2008cu ;
Chao:2012sz . In figure 2, we show the coannihilation channels present in our
model in terms of gauge eigenstates, except for the $Z_{2}$ even scalars. The
dark matter mass can be much smaller for singlets fulfulling the $\Omega_{\rm
DM}h^{2}$ contraint. However, processes related to direct detection are absent
at tree level Schmidt:2012yg for singlets too.
mmmmmm Parameter mmmmmm | mmmmmmRangemmmmmm
---|---
$M_{N}$ (GeV) | 1 – $10^{5}$
$M_{\Sigma}$ (GeV) | 100 – $10^{5}$
$m_{\eta^{\pm}}$ (GeV) | 100 – $10^{5}$
$M_{\pm}$ (GeV) | 100 – $10^{4}$
$|\lambda_{i}|$ | $10^{-4}$ – 1
$|\lambda_{i}^{\eta,\Omega}|$ | $10^{-4}$ – 1
$|Y_{i}|$ | $10^{-4}$ – 1
Table 2: Scanning parameter ranges. The remaing parameters are calculated from
this set.
The presence of the scalar triplet $\Omega$ and its nonzero vev induces a
mixing between $\Sigma^{0}$ and $N$, implying coannihilations that can be
important when the dark matter has a large component of $\Sigma^{0}$. This
mixing also breaks the degeneracy between the mass eigenstate fermions
$\chi_{1}^{0}$ and $\chi^{\pm}$. However, in this case, the mass degeneracy
with the charged fermion $\chi^{\pm}$ is increased and forces the dark matter
to be $\mathcal{O}({\rm TeV})$. Other coannihilation processes occur when
$M_{N}$ is also degenerate with $M_{\Sigma}$. For the opposite case, when
$\chi^{0}$ is mainly $N$, the model reproduces the phenomenology of the
fermion singlet dark matter where the main signature is the annihilation into
neutrinos and charged leptons (as in leptophilic dark matter) without any
direct detection prospective Schmidt:2012yg . The potential scenarios present
in the model have the best of singlet-only or triplets-only scenarios and
more. In addition, the dark matter phenomenology includes new annihilation and
coannihilation channels when kinematically accessible.
The presence of the scalar triplet $\Omega$ also induces an interaction
between dark matter and quarks (direct detection) via the exchange of neutral
scalar $S_{i}(h^{0},\Omega^{0})$, as illustrated in In Fig 3, we show the main
diagrams of the model related to indirect and direct searches. The model can
potentially produce the typical annihilation channels appearing in generic
weakly interactive massive particle dark matter models. Indeed, our dark
matter candidate mimicks the Lightest Supersymmetric Particle (neutralino)
present in supergravity-like versions the Minimal Supersymmetric Standard
Model with R-parity conservation. The latter would correspond here to our
assumed $Z_{2}$ symmetry.
In order to study the dark matter phenomenology, we have implemented the
lagrangian (equation 12) using the standard codes LanHEP lanhep:1996 ;
lanhep:2009 ; lanhep:2010 and Micromegas micromegas:2013 . We scan the
parameter space of the model within the ranges indicated in Tab. 2. We also
take into account the following constraints: perturbatibity and a Higgs–like
scalar at $\sim$ 125 GeV. Also we take into account the constraints from the
relic abundance planck:2013 as well as the lower bound on the masses of new
non-colored charged particles coming from LEP L3:2001PhLB and LHC
CMS:2012PhLB collider searches, roughly translated to $M_{\rm LEP}>100\,{\rm
GeV}$. We calculate the thermally averaged annihilation cross section
$\langle\sigma v\rangle$, and the spin independent cross section $\sigma_{\rm
SI}$.
Figure 5: Annihilation cross section vs dark matter mass. Color scale
represents $\log_{10}(\xi)$. Dark matter with masses larger than 1 TeV have a
larger component of $\Sigma^{0}$, cases with masses lower than 20 GeV have
larger component of $N$. The yellow line corresponds to the thermal value
$3\times 10^{-26}\textnormal{cm}^{3}/\textnormal{sec}$. Figure 6: Spin
independent cross section vs dark matter mass. Color scale is the same as in
figure 5. The yellow line is the upper bound from XENON100 experiment
XENON:2012Ph .
In figure 5, we present the results of the scan in terms of the annihilation
cross section versus the dark matter mass. Moreover, we show in color scale
the quantity:
$\xi=\frac{M_{\Sigma}-m_{\rm DM}}{m_{\rm DM}}\,,$ (48)
which estimates how degenerate is the dark matter mass with respect to
$M_{\Sigma}$. Small values of $\xi$ imply dark matter with a large component
of $\Sigma^{0}$ and large value implies a large component of $N$. This
quantity has implications for coannihilation processes discussed previously.
We notice that regions with low dark matter masses ($<20$ GeV) are less
degenerate mainly because $M_{\Sigma}>M_{\rm LEP}$. In this region the dark
matter contains a large component of $N$. As expected, the TeV region is
dominated by dark matter with large component of $\Sigma^{0}$. The mass range
100–800 GeV is particularly interesting because any of the new charged
particles are accessible at LHC. Moreover, when the $\Sigma^{0}/N$ mixing is
non-zero and $\displaystyle m_{\rm DM}\simeq\frac{m_{S_{i}}}{2}$, the
annihilation channels into quarks and leptons are naturally enhanced due to
the $s$-channel resonance in the process:
$\chi_{1}^{0}\chi_{1}^{0}\to S_{i}\to f\bar{f}\,\to\langle\sigma
v\rangle\propto\left(\frac{\sin(2\,\alpha)}{(2m_{\rm
DM})^{2}-m_{S_{i}}^{2}}\right)^{2}\,.$ (49)
This is translated into higher expected fluxes of gamma–rays and cosmic–rays
for indirect searches as well as higher spin independent cross section.
Now, turning to the direct detection perspectives, the plot of the
spin–independent cross section versus the dark matter mass is shown in figure
6. The scattering with quarks is described only with one diagram (the exchange
of scalars $S_{i}$), also shown in figure 4. The size of the interaction will
depend directly on the mixing $\Sigma^{0}/N$. For masses larger than 100 GeV,
we observe an increase of $\sigma_{SI}$ because maximal mixing can be obtained
for $M_{N}\sim M_{\Sigma}$ and for $Y_{\Omega}v_{\Omega}\neq 0$. This does not
occur for masses much lower to 100 GeV since the dark matter becomes mainly a
pure $N$. Moreover, the model produces $\sigma_{\rm SI}$ large enough to be
observed in direct detection experiments such XENON100 XENON:2012Ph (yellow
line).
Finally, we note that the new particles introduced in our model can be
kinematically accessible at the LHC. Here we briefly comment on relevant
production cross sections for the LHC. Both, ATLAS ATLAS-CONF-2013-019 and
CMS CMS:2012PhLB have searched for pair production of heavy triplet fermions:
$\Sigma^{0}+\Sigma^{+}$, deriving lower limits on $m_{\Sigma^{+}}$ of the
order of $m_{\Sigma^{+}}\gtrsim(180-210)$ GeV CMS:2012PhLB and
$m_{\Sigma^{+}}\gtrsim 245$ GeV ATLAS-CONF-2013-019 , respectively. However,
these bounds do not apply to our model, because the final state topologies
used in these searches, tri-leptons in case of CMS CMS:2012PhLB and four
charged leptons in ATLAS ATLAS-CONF-2013-019 , are based on the assumption
that $\Sigma^{0}$ decays to the final states $\Sigma^{0}\to
l^{\pm}l^{\mp}+\nu/{\bar{\nu}}$. As a result of the $Z_{2}$ symmetry present
in our model, however, the lightest fermion or scalar is stable and all
heavier $Z_{2}$-odd states will decay to this lightest state. Thus, the
intermediate states $\Sigma^{0}+\Sigma^{+}$ and $\Sigma^{-}+\Sigma^{+}$, which
have the largest production cross sections of all new particles in our model,
will not give rise to three and four charged lepton signals.
Instead, the phenomenology of $\Sigma^{0}$ and $\Sigma^{+}$ depends on the
unknown mass ordering of fermions and scalars. Since we have assumed in this
paper that the lighter of the fermions is the dark matter, we will discuss
only this case here. Then, the phenomenology depends on whether the lightest
of the neutral fermions, $\chi^{0}_{1}$, is mostly singlet or mostly triplet.
Consider first the case $\chi^{0}_{1}\simeq\Sigma^{0}$. Then, from the pair
$\chi^{0}_{1}+\Sigma^{+}$, only $\Sigma^{+}$ decays via
$\Sigma^{+}\to\chi^{0}_{1}+W^{+}$, where the $W^{+}$ can be on-shell or off-
shell. Thus, the final state consists mostly one charged lepton plus missing
energy. The other possibility is pair production of $\Sigma^{+}+\Sigma^{-}$
via photon exchange, which leads to $l^{+}+l^{-}$ plus missing energy. In both
cases, standard model backgrounds will be large and the LHC data probably does
not give any competitive limits yet. We expect that LHC data at 14 TeV with
increased statistics may constrain part of the parameter space. A quantitative
study would require a MonteCarlo analisys which is beyond the scope of this
work.
Conversely, for the case $\chi^{0}_{2}\simeq\Sigma^{0}$, the $\chi^{0}_{2}$
will decay to $\chi^{0}_{1}$ plus either one on-shell or off-shell Higgs
state, depending on kinematics. In this case the final state will be one
charged lepton plus up to four b-jets plus missing momentum. This topology is
not covered by any searches at the LHC so far, as far as we are aware.
Also, the new neutral and charged scalars can be searched for at the LHC. All
possible signals have, however, rather small production cross sections.
Neither $\eta$ nor $\Omega$ have couplings to quarks and only $\Omega$ (both
charged and neutral) can be produced at the LHC due to its mixing with the
Standard Model Higgs field $\phi$. Final states will be very much SM-Higgs
like, but the event numbers will depend quadratically on this mixing, which
supposedly is a small number, since the observed state with a mass of roughly
$(125-126)$ GeV behaves rather closely like A Standard Model Higgs. Searches
for a heavier state with Standard Model like Higgs properties
Chatrchyan:2013yoa exclude scalars with standard coupling strength now up to
roughly 700 GeV. However, upper limits on $\sin^{2}(\theta)$ in the mass range
$(130-700)$ GeV are currently only of the order $(0.2-1.0)$. The next run at
the LHC, with its projected luminosity of order ${\cal L}\simeq(100-300)$
fb-1, should allow to probe much smaller mixing angles.
## 4 Conclusions
We have presented a next-to minimal extension of the Standard Model including
new $Z_{2}$-odd majorana fermions, one singlet $N$ and one triplet $\Sigma$
under weak SU(2), as well as a $Z_{2}$-odd scalar doublet $\eta$. We also
include a $Z_{2}$-even triplet scalar $\Omega$ in order induce the mixing in
the fermionic sector $N$–$\Sigma$. The solar and atmospheric neutrino mass
scales are then generated at one-loop level, with the lightest neutrino
remaining massless. This way our model combines the ingredients present in
Refs. Ma:2006km ; Ma:2008cu with a richer phenomenology.
The unbroken $Z_{2}$ symmetry implies that the lightest $Z_{2}$-odd particle
is stable and may play the role of dark matter. We analyze the viability of
the model using state-of-art codes for dark matter phenomenology. We focus our
attention to the fermionic dark matter case. The mixing between $N$ and the
neutral component of $\Sigma$ relaxes the effects of coannihilations between
the dark matter candidate and the charged component of $\Sigma$. In the pure
triplet case, the dark matter mass is forced to be 2.7 TeV in order to
reproduce the observed dark matter abundance value. However, in the presence
of mixing the effect of coannihilations is weaker, allowing for a reduced dark
matter mass down to the GeV range. Thanks to that, the charged $\Sigma$ can be
much lighter than in the pure triplet case, openning the possibility of new
signatures at colliders such as the LHC. In addition, the dark matter
candidate can interact with quarks at tree level and then produce direct
detection signal that may be observed or constrained in current direct
searches experiments such XENON100.
## Acknowledgments
This work was supported by the Spanish MINECO under grants FPA2011-22975 and
MULTIDARK CSD2009-00064 (Consolider-Ingenio 2010 Programme), by
Prometeo/2009/091 (Generalitat Valenciana), and by the EU ITN UNILHC PITN-
GA-2009-237920. S.M. thanks to DFG grant WI 2639/4-1 for financial support.
N.R. thanks to CONICYT doctoral grant, Marco A. Díaz for useful discussions
and comments, the EPLANET grant for funding the stay in Valencia, and the
IFIC–AHEP group in Valencia for the hospitality. R.L. also thanks to V. Ţăranu
for her support.
## Appendix A Appendix
### A.1 Approximations for Neutrino Masses.
Starting from the equation 42, one can perform some approximations to examine
neutrino masses for cases of interest, for example, cases with one of the
$\chi^{0}_{1}$ masses being the lightest between $\chi^{0}_{2}$, $\eta_{0,A}$,
$\Sigma_{\pm}$ and $\Omega_{0,\pm}$.
One wants to establish the relation between neutrino masses and the other
parameters in the lagrangian in a suitable form. In principle, neutrino masses
depend on the masses of neutral $\eta$ fields and the masses of the
$\chi^{0}$, but the dependence of the parameters of the scalar sector is more
complicated, given the structure of the masses of the $\eta$ fields (see
equations 35 and 36). One can take these equations and write them in the
following way:
$\displaystyle m_{\eta_{0}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\lambda_{5}v_{h}^{2}\,,$ (50) $\displaystyle m_{\eta_{A}}^{2}$
$\displaystyle=$ $\displaystyle m_{0}^{2}-\lambda_{5}v_{h}^{2}\,.$ (51)
Where $m_{0}^{2}$ is a complicated function of the parameters of the scalar
potential. One can write the equation 43 as follows:
$\displaystyle I_{k}$ $\displaystyle=$ $\displaystyle-
M_{k}\left(\frac{m_{0}^{2}+\lambda_{5}v_{h}^{2}}{M_{k}^{2}-m_{0}^{2}-\lambda_{5}v_{h}^{2}}\right)\log\left(\frac{m_{0}^{2}+\lambda_{5}v_{h}^{2}}{M_{k}^{2}}\right)$
(52)
$\displaystyle+M_{k}\left(\frac{m_{0}^{2}-\lambda_{5}v_{h}^{2}}{M_{k}^{2}-m_{0}^{2}+\lambda_{5}v_{h}^{2}}\right)\log\left(\frac{m_{0}^{2}-\lambda_{5}v_{h}^{2}}{M_{k}^{2}}\right)\,.$
One can identify two interesting limit cases. When $\lambda_{5}v_{h}^{2}\ll
M_{k}^{2}\approx m_{0}^{2}$ then the $I_{k}$ function can be written as:
$\displaystyle I_{k}$ $\displaystyle=$
$\displaystyle\frac{2\lambda_{5}v_{h}^{2}}{M_{k}}\,.$ (53)
Therefore, the neutrino mass matrix in this approximation is given by:
$\displaystyle M_{\alpha\beta}^{\nu}$ $\displaystyle=$
$\displaystyle\sum_{\sigma=1,2}\frac{h_{\alpha\sigma}h_{\beta\sigma}}{8\pi^{2}}\frac{\lambda_{5}v_{h}^{2}}{M_{\sigma}}\,.$
(54)
The other case is given by $\lambda_{5}v_{h}^{2}\,,\,M_{k}^{2}\ll m_{0}^{2}$,
the procedure is not difficult, the result is:
$\displaystyle I_{k}$ $\displaystyle=$
$\displaystyle\frac{2\lambda_{5}v_{h}^{2}}{m_{0}^{2}}M_{k}\,.$ (55)
In this case, the neutrino mass matrix is given by:
$\displaystyle M_{\alpha\beta}^{\nu}$ $\displaystyle=$
$\displaystyle\sum_{\sigma=1,2}\frac{h_{\alpha\sigma}h_{\beta\sigma}}{8\pi^{2}}\frac{\lambda_{5}v_{h}^{2}}{m_{0}^{2}}M_{k}\,.$
(56)
### A.2 Minimization conditions
The tadpole equations were computed in order to find the minimum of the scalar
potential, thus, the linear terms of the scalar potential at tree level can be
written as:
$\displaystyle V_{(1)}$ $\displaystyle=$ $\displaystyle
t_{h}h_{0}+t_{\eta}\eta_{0}+t_{\Omega}\Omega_{0}$ (57)
Where the tadpoles are:
$\displaystyle t_{h}$ $\displaystyle=$ $\displaystyle
v_{h}\left(-m_{1}^{2}+\frac{1}{2}\lambda_{1}v_{h}^{2}+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{\eta}^{2}\right)$
(58) $\displaystyle t_{\eta}$ $\displaystyle=$ $\displaystyle
v_{\eta}\left(m_{2}^{2}+\frac{1}{2}\lambda_{2}v_{\eta}^{2}+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}+\lambda_{5}\right)v_{h}^{2}\right)$
(59) $\displaystyle t_{\Omega}$ $\displaystyle=$ $\displaystyle-
M_{\Omega}^{2}v_{\Omega}-\mu_{1}v_{h}^{2}+\left(2\lambda_{1}^{\Omega}+\lambda_{4}^{\Omega}\right)v_{h}^{2}v_{\Omega}+$
(60) $\displaystyle
8\left(2\lambda_{2}^{\Omega}+\lambda_{3}^{\Omega}\right)v_{h}^{2}v_{\Omega}^{3}+\mu_{2}v_{\eta}^{2}+\left(2\lambda_{1}^{\eta}+\lambda_{4}^{\eta}\right)v_{\Omega}^{2}v_{\eta}^{2}$
In order to have an $Z_{2}$ invariant vacuum, the vev $v_{\eta}$ has to
vanish, which is extracted from the equation 59. For the vev $v_{h}$, one can
choose the value to be nonzero solving the equation in the parenthesis, in
equal manner, one obtain the vev $v_{\Omega}$, in terms of the other
parameters of the potential.
The numerical values of the vevs $v_{h}$ and $v_{\Omega}$ are restricted to
reproduce the measured values of gauge boson masses, this allows to have the
value for $v_{h}\sim 246$ GeV, and $v_{\Omega}\leq 7$ GeV, as one can see in
the section 2.3.
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|
arxiv-papers
| 2013-07-30T20:11:08 |
2024-09-04T02:49:48.773157
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Hirsch, R. A. Lineros, S. Morisi, J. Palacio, N. Rojas, J. W. F.\n Valle",
"submitter": "Roberto Lineros",
"url": "https://arxiv.org/abs/1307.8134"
}
|
1307.8136
|
# DeBaCl: A Python Package for Interactive DEnsity-BAsed CLustering
Brian P. Kent
Carnegie Mellon University &Alessandro Rinaldo
Carnegie Mellon University &Timothy Verstynen
Carnegie Mellon University [email protected] [email protected]
[email protected]
Brian P. Kent, Alessandro Rinaldo, Timothy Verstynen DeBaCl: A Python Package
for Interactive DEnsity-BAsed CLustering The level set tree approach of
Hartigan (1975) provides a probabilistically based and highly interpretable
encoding of the clustering behavior of a dataset. By representing the
hierarchy of data modes as a dendrogram of the level sets of a density
estimator, this approach offers many advantages for exploratory analysis and
clustering, especially for complex and high-dimensional data. Several R
packages exist for level set tree estimation, but their practical usefulness
is limited by computational inefficiency, absence of interactive graphical
capabilities and, from a theoretical perspective, reliance on asymptotic
approximations. To make it easier for practitioners to capture the advantages
of level set trees, we have written the Python package DeBaCl for DEnsity-
BAsed CLustering. In this article we illustrate how DeBaCl’s level set tree
estimates can be used for difficult clustering tasks and interactive graphical
data analysis. The package is intended to promote the practical use of level
set trees through improvements in computational efficiency and a high degree
of user customization. In addition, the flexible algorithms implemented in
DeBaCl enjoy finite sample accuracy, as demonstrated in recent literature on
density clustering. Finally, we show the level set tree framework can be
easily extended to deal with functional data. density-based clustering, level
set tree, Python, interactive graphics, functional data analysis density-based
clustering, level set tree, Python, interactive graphics, functional data
analysis Brian P. Kent
Department of Statistics
Carnegie Mellon University
Baker Hall 132
Pittsburgh, PA 15213
E-mail:
URL: http://www.brianpkent.com
Alessandro Rinaldo
Department of Statistics
Carnegie Mellon University
Baker Hall 132
Pittsburgh, PA 15213
E-mail:
URL: http://www.stat.cmu.edu/~arinaldo/
Timothy Verstynen
Department of Psychology & Center for the Neural Basis of Cognition
Carnegie Mellon University
Baker Hall 340U
Pittsburgh, PA 15213
E-mail:
URL: http://www.psy.cmu.edu/~coaxlab/
## 1 Introduction
Clustering is one of the most fundamental tasks in statistics and machine
learning, and numerous algorithms are available to practitioners. Some of the
most popular methods, such as K-means (MacQueen, 1967; Lloyd, 1982) and
spectral clustering (Shi and Malik, 2000), rely on the key operational
assumption that there is one optimal partition of the data into $K$ well-
separated groups, where $K$ is assumed to be known a priori. While effective
in some cases, this flat or scale-free notion of clustering is inadequate when
the data are very noisy or corrupted, or exhibit complex multimodal behavior
and spatial heterogeneity, or simply when the value of $K$ is unknown. In
these cases, hierarchical clustering affords a more realistic and flexible
framework in which the data are assumed to have multi-scale clustering
features that can be captured by a hierarchy of nested subsets of the data.
The expression of these subsets and their order of inclusions—typically
depicted as a dendrogram—provide a great deal of information that goes beyond
the original clustering task. In particular, it frees the practitioner from
the requirement of knowing in advance the “right” number of clusters, provides
a useful global summary of the entire dataset, and allows the practitioner to
identify and focus on interesting sub-clusters at different levels of spatial
resolution.
There are, of course, myriad algorithms just for hierarchical clustering.
However, in most cases their usage is advocated on the basis of heuristic
arguments or computational ease, rather than well-founded theoretical
guarantees. The high-density hierarchical clustering paradigm put forth by
Hartigan (1975) is an exception. It is based on the simple but powerful
definition of clusters as the maximal connected components of the super-level
sets of the probability density specifying the data-generating distribution.
This formalization has numerous advantages: (1) it provides a probabilistic
notion of clustering that conforms to the intuition that clusters are the
regions with largest probability to volume ratio; (2) it establishes a direct
link between the clustering task and the fundamental problem of nonparametric
density estimation; (3) it allows for a clear definition of clustering
performance and consistency (Hartigan, 1981) that is amenable to rigorous
theoretical analysis and (4) as we show below, the dendrogram it produces is
highly interpretable, offers a compact yet informative representation of a
distribution, and can be interactively queried to extract and visualize
subsets of data at desired resolutions. Though the notion of high-density
clustering has been studied for quite some time (Polonik, 1995), recent
theoretical advances have further demonstrated the flexibility and power of
density clustering. See, for example, Rinaldo _et al._ (2012); Rinaldo and
Wasserman (2010); Kpotufe and Luxburg (2011); Chaudhuri and Dasgupta (2010);
Steinwart (2011); Sriperumbudur and Steinwart (2012); Lei _et al._ (2013);
Balakrishnan _et al._ (2013) and the refences therein.
This paper introduces the Python package DeBaCl for efficient and
statistically-principled DEnsity-BAsed CLustering. DeBaCl is not the first
implementation of level set tree estimation and clustering; the R packages
denpro (Klemelä, 2004), gslclust (Stuetzle and Nugent, 2010), and pdfCluster
(Azzalini and Menardi, 2012) also contain various level set tree estimators.
However, they tend to be too inefficient for most practical uses and rely on
methods lacking rigorous theoretical justification. The popular nonparametric
density-based clustering algorithm DBSCAN (Ester _et al._ , 1996) is
implemented in the R package fpc (Hennig, 2013) and the Python library scikit-
learn (Pedregosa _et al._ , 2011), but this method does not provide an
estimate of the level set tree.
DeBaCl handles much larger datasets than existing software, improves
computational speed, and extends the utility of level set trees in three
important ways: (1) it provides several novel visualization tools to improve
the readability and interpetability of density cluster trees; (2) it offers a
high degree of user customization; and (3) it implements several recent
methodological advances. In particular, it enables construction of level set
trees for arbitrary functions over a dataset, building on the idea that level
set trees can be used even with data that lack a bona fide probability density
fuction. DeBaCl also includes the first practical implementation of the
recent, theoretically well-supported algorithm from Chaudhuri and Dasgupta
(2010).
## 2 Level set trees
Suppose we have a collection of points
$\mathbb{X}_{n}=\\{x_{1},\ldots,x_{n}\\}$ in $\mathbb{R}^{d}$, which we model
as i.i.d. draws from an unknown probability distribution with probability
density function $f$ (with respect to Lebesgue measure). Our goal is to
identify and extract clusters of $\mathbb{X}_{n}$ without any a priori
knowledge about $f$ or the number of clusters. Following the statistically-
principled approach of Hartigan (1975), clusters can be identified as modes of
$f$. For any threshold value $\lambda\geq 0$, the $\lambda$-upper level set of
$f$ is
$L_{\lambda}(f)=\\{x\in\mathbb{R}^{d}:f(x)\geq\lambda\\}.$ (1)
The connected components of $L_{\lambda}(f)$ are called the $\lambda$-clusters
of $f$ and high-density clusters are $\lambda$-clusters for any value of
$\lambda$. It is easy to see that $\lambda$-clusters associated with larger
values of $\lambda$ are regions where the ratio of probability content to
volume is higher. Also note that for a fixed value of $\lambda$, the
corresponding set of clusters will typically not give a partition of
$\\{x:f(x)\geq 0\\}$.
The level set tree is simply the set of all high-density clusters. This
collection is a tree because it has the following property: for any two high-
density clusters $A$ and $B$, either $A$ is a subset of $B$, $B$ is a subset
of $A$, or they are disjoint. This property allows us to visualize the level
set tree with a dendrogram that shows all high-density clusters simultaneously
and can be queried quickly and directly to obtain specific cluster
assignments. Branching points of the dendrogram correspond to density levels
where two or more modes of the pdf, i.e. new clusters, emerge. Each vertical
line segment in the dendrogram represents the high-density clusters within a
single pdf mode; these clusters are all subsets of the cluster at the level
where the mode emerges. Line segments that do not branch are considered high-
density modes, which we call the leaves of the tree. For simplicity, we tend
to refer to the dendrogram as the level set tree itself.
Because $f$ is unknown, the level set tree must be estimated from the data.
Ideally we would use the high-density clusters of a suitable density estimate
$\widehat{f}$ to do this; for a well-behaved $f$ and a large sample size,
$\widehat{f}$ is close to $f$ with high probability so the level set tree for
$\widehat{f}$ would be a good estimate for the level set tree of $f$
(Chaudhuri and Dasgupta, 2010). Unfortunately, this approach is not
computationally feasible even for low-dimensional data because finding the
upper level sets of $\widehat{f}$ requires evaluating the function on a dense
mesh and identifying $\lambda$-clusters requires a combinatorial search over
all possible paths connecting any two points in the mesh.
Many methods have been proposed to overcome these computational obstacles. The
first category includes techniques that remain faithful to the idea that
clusters are regions of the sample space. Members of this family include
histogram-based partitions (Klemelä, 2004), binary tree partitions (Klemelä,
2005) (implemented in the R package denpro) and Delaunay triangulation
partitions (Azzalini and Torelli, 2007) (implemented in R package pdfCluster).
These techniques tend to work well for low-dimension data, but suffer from the
curse of dimensionality because partitioning the sample space requires an
exponentially increasing number of cells or algorithmic complexity (Azzalini
and Torelli, 2007).
In contrast, another family of estimators produces high-density clusters of
data points rather than sample space regions; this is the approach taken by
our package. Conceptually, these methods estimate the level set tree of $f$ by
intersecting the level sets of $f$ with the sample points $\mathbb{X}_{n}$ and
then evaluating the connectivity of each set by graph theoretic means. This
typically consists of three high-level steps: estimation of the probability
density $\widehat{f}(x)$ from the data; construction of a graph $G$ that
describes the similarity between each pair of data points; and a search for
connected components in a series of subgraphs of $G$ induced by removing nodes
and/or edges of insufficient weight, relative to various density levels.
The variations within the latter category are found in the definition of $G$,
the set of density levels over which to iterate, and the way in which $G$ is
restricted to a subgraph for a given density level $\lambda$. Edge iteration
methods assign a weight to the edges of $G$ based on the proximity of the
incident vertices in feature space (Chaudhuri and Dasgupta, 2010) or the value
of $\widehat{f}(x)$ at the incident vertices (Wong and Lane, 1983) or on a
line segment connecting them (Stuetzle and Nugent, 2010). For these
procedures, the relevant density levels are the edge weights of $G$.
Frequently, iteration over these levels is done by initializing $G$ with an
empty edge set and adding successively more heavily weighted edges, in the
manner of traditional single linkage clustering. In this family, the Chaudhuri
and Dasgupta algorithm (which is a generalization of Wishart (1969)) is
particularly interesting because the authors prove finite sample rates for
convergence to the true level set tree (Chaudhuri and Dasgupta, 2010). To the
best of our knowledge, however, only Stuetzle and Nugent (2010) has a publicly
available implementation, in the R package gslclust.
Point iteration methods construct $G$ so the vertex for observation $x_{i}$ is
weighted according to $\widehat{f}(x_{i})$, but the edges are unweighted. In
the simplest form, there is an edge between the vertices for observations
$x_{i}$ and $x_{j}$ if the distance between $x_{i}$ and $x_{j}$ is smaller
than some threshold value, or if $x_{i}$ and $x_{j}$ are among each other’s
$k$-closest neighbors (Kpotufe and Luxburg, 2011; Maier _et al._ , 2009). A
more complicated version places an edge $(x_{i},x_{j})$ in $G$ if the amount
of probability mass that would be needed to fill the valleys along a line
segment between $x_{i}$ and $x_{j}$ is smaller than a user-specified threshold
(Menardi and Azzalini, 2013). The latter method is available in the R package
pdfCluster.
## 3 Implementation
The default level set tree algorithm in DeBaCl is described in Algorithm 1,
based on the method proposed by Kpotufe and Luxburg (2011) and Maier _et al._
(2009). For a sample with $n$ observations in $\mathbb{R}^{d}$, the k-nearest
neighbor (kNN) density estimate is:
$\widehat{f}(x_{j})=\frac{k}{n\cdot v_{d}\cdot r^{d}_{k}(x_{j})}$ (2)
where $v_{d}$ is the volume of the Euclidean unit ball in $\mathbb{R}^{d}$ and
$r_{k}(x_{j})$ is the Euclidean distance from point $x_{j}$ to its $k$’th
closest neighbor. The process of computing subgraphs and finding connected
components of those subgraphs is implemented with the igraph package (Csardi
and Nepusz, 2006). Our package also depends on the NumPy and SciPy packages
for basic computation (Jones _et al._ , 2001) and the Matplotlib package for
plotting (Hunter, 2007).
Input: $\\{x_{1},\ldots,x_{n}\\}$, $k$, $\gamma$
Output: $\widehat{\mathcal{T}}$, a hierarchy of subsets of
$\\{x_{1},\ldots,x_{n}\\}$
$G\leftarrow$ $k$-nearest neighbor similarity graph on
$\\{x_{1},\ldots,x_{n}\\}$;
$\widehat{f}(\cdot)\leftarrow k$-nearest neighbor density estimate based on
$\\{x_{1},\ldots,x_{n}\\}$;
for _$j\leftarrow 1$ to $n$_ do
$\lambda_{j}\leftarrow\widehat{f}(x_{j})$;
$L_{\lambda_{j}}\leftarrow\\{x_{i}:\widehat{f}(x_{i})\geq\lambda_{j}\\}$;
$G_{j}\leftarrow$ subgraph of $G$ induced by $L_{j}$;
Find the connected components of $G_{\lambda_{j}}$;
$\widehat{\mathcal{T}}\leftarrow$ dendrogram of connected components of graphs
$G_{1},\ldots,G_{n}$, ordered by inclusions;
$\widehat{\mathcal{T}}\leftarrow$ remove components of size smaller than
$\gamma$;
return _$\widehat{\mathcal{T}}$_
Algorithm 1 Baseline DeBaCl level set tree estimation procedure
We use this algorithm because it is straightforward and fast; although it does
require computation of all $n\choose 2$ pairwise distances, the procedure can
be substantially shortened by estimating connected components on a sparse grid
of density levels. The implementation of this algorithm is novel in its own
right (to the best of our knowledge), and DeBaCl includes several other new
visualization and methodological tools.
### 3.1 Visualization tools
Our level set tree plots increase the amount of information contained in a
tree visualization and greatly improve interpretability relative to existing
software. Suppose a sample of 2,000 observations in $\mathbb{R}^{2}$ from a
mixture of three Gaussian distributions (Figure 1(a)). The traditional level
set tree is illustrated in Figure 1(b) and the DeBaCl version in Figure 1(c).
A plot based only on the mathematical definition of a level set tree conveys
the structure of the mode hierarchy and indicates the density levels where
each tree node begins and ends, but does not indicate how many points are in
each branch or visually associate the branches with a particular subset of
data. In the proposed software package, level set trees are plotted to
emphasize the empirical mass in each branch (i.e. the fraction of data in the
associated cluster): tree branches are sorted from left-to-right by decreasing
empirical mass, branch widths are proportional to empirical mass, and the
white space around the branches is proportional to empirical mass. For
matching tree nodes to the data, branches can be colored to correspond to
high-density data clusters (Figures 1(c) and 1(d)). Clicking on a tree branch
produces a banner that indicates the start and end levels of the associated
high-density cluster as well as its empirical mass (Figure 5(a)).
The level set tree plot is an excellent tool for interactive exploratory data
analysis because it acts as a handle for identifying and plotting spatially
coherent, high-density subsets of data. The full power of this feature can be
seen clearly with the more complex data of Section 4.
(a)
(b)
(c)
(d)
Figure 1: Level set tree plots and cluster labeling for a simple simulation. A
level set tree is constructed from a sample of 2,000 observations drawn from a
mix of three Gaussians in $\mathbb{R}^{2}$. 1(a)) The kNN density estimator
evaluated on the data. 1(b)) A plot of the tree based only on the mathematical
definition of level set trees. 1(c)) The new level set tree plot, from DeBaCl.
Tree branches emphasize empirical mass through ordering, spacing, and line
width, and they are colored to match the cluster labels in 1(d). A second
vertical axis is added that indicates that fraction of background mass at each
critical density level. 1(d)) Cluster labels from the all-mode labeling
technique, where each leaf of the level set tree is designated as a cluster.
### 3.2 Alternate scales
By construction, the nodes of a level set tree are indexed by density levels
$\lambda$, which determine the scale of the vertical axis in a plot of the
tree. While this does encode the parent-child relationships in the tree,
interpretability of the $\lambda$ scale is limited by the fact that it depends
on the height of the density estimate $\widehat{f}$. It is not clear, for
example, whether $\lambda=1$ would be a low- or a high-density threshold; this
depends on the particular distribution.
To remove the scale dependence we can instead index level set tree nodes based
on the probability content of upper level sets. Specifically, let $\alpha$ be
a number between $0$ and $1$ and define
$\lambda_{\alpha}=\sup\left\\{\lambda\colon\int_{x\in
L_{\lambda}(f)}f(x)dx\geq\alpha\right\\}$ (3)
to be the value of $\lambda$ for which the upper level set of $f$ has
probability content no smaller than $\alpha$ (Rinaldo _et al._ , 2012). The
map $\alpha\mapsto\lambda_{\alpha}$ gives a monotonically decreasing one-to-
one correspondence between values of $\alpha$ in $[0,1]$ and values of
$\lambda$ in $[0,\max_{x}f(x)]$. In particular, $\lambda_{1}=0$ and
$\lambda_{0}=\max_{x}f(x)$. For an empirical level set tree, set
$\lambda_{\alpha}$ to the $\alpha$-quantile of
$\\{\widehat{f}(x_{i})\\}_{i=1}^{n}$. Expressing the height of the tree in
terms of $\alpha$ instead of $\lambda$ does not change the topology (i.e.
number and ordering of the branches) of the tree; the re-indexed tree is a
deformation of the original tree in which some of its nodes are stretched out
and others are compressed.
$\alpha$-indexing is more interpretable and useful for several reasons. The
$\alpha$ level of the tree indexes clusters corresponding to the $1-\alpha$
fraction of “most clusterable" data points; in particular, larger $\alpha$
values yield more compact and well-separated clusters, while smaller values
can be used for de-noising and outlier removal. Because $\alpha$ is always
between $0$ and $1$, scaling by probability content also enables comparisons
of level set trees arising from data sets drawn from different pdfs, possibly
in spaces of different dimensions. Finally, the $\alpha$-index is more
effective than $\lambda$-indexing in representing regions of large probability
content but low density and is less affected by small fluctuations in density
estimates.
A common (incorrect) intuition when looking at an $\alpha$-indexed level set
tree plot is to interpret the height of the branches as the size of the
corresponding cluster, as measured by its empirical mass. However, with
$\alpha$-indexing the height of any branch depends on its empirical mass as
well as the empirical mass of all other branches that coexist with it. In
order to obtain trees that do conform to this intuition, we introduce the
$\kappa$-indexed level set tree.
Recall from Section 2 that clusters are defined as maximal connected
components of the sets $L_{\lambda}(f)$ (see equation 1) as $\lambda$ varies
from $0$ to $\max_{x}f(x)$, and that the level set tree is the dendrogram
representing the hierarchy of all clusters. Assume the tree is binary and with
tooted. Let $\\{1,2,\ldots,K\\}$ be an enumeration of the nodes of the level
set tree and let $\mathcal{C}=\\{C_{0},\ldots,C_{K}\\}$ be the corresponding
clusters. We can always choose the enumeration in a way that is consistent
with the hierarchy of inclusions of the elements of $\mathcal{C}$; that is,
$C_{0}$ is the support of $f$ (which we assume for simplicity to be a
connected set) and if $C_{i}\subset C_{j}$, then $i>j$. For a node $i>0$, we
denote with $\mathrm{parent}_{i}$ the unique node $j$ such that $C_{j}$ is the
smallest element of $\mathcal{C}$ such that $C_{j}\supset C_{i}$. Similarly,
$\mathrm{kid}_{i}$ is the pair of nodes $(j,j^{\prime})$ such that $C_{j}$ and
$C_{j^{\prime}}$ are the maximal subsets of $C_{i}$. Finally, for $i>0$,
$\mathrm{sib}_{i}$ is the node $j$ such there exists a $k$ for which
$\mathrm{kid}_{k}=(i,j)$. For a cluster $C_{i}\in\mathcal{C}$, we set
$M_{i}=\int_{C_{i}}f(x)dx,$ (4)
which we refer to as the mass of $C_{i}$.
The true $\kappa$-tree can be defined recursively by associating with each
node $i$ two numbers $\kappa^{\prime}_{i}$ and $\kappa^{\prime\prime}_{i}$
such that $\kappa^{\prime}_{i}-\kappa^{\prime\prime}_{i}$ is the salient mass
of node $i$. For leaf nodes, the salient mass is the mass of the cluster, and
for non-leaves it is the mass of the cluster boundary region.
$\kappa^{\prime}$ and $\kappa^{\prime\prime}$ are defined differently for each
node type.
1. 1.
Internal nodes, including the root node.
$\displaystyle\kappa^{\prime}_{0}=M_{0}=1,$
$\displaystyle\kappa^{\prime}_{i}=\kappa^{\prime\prime}_{\mathrm{parent}_{i}}$
$\displaystyle\kappa^{\prime\prime}_{i}=\sum_{j\in\mathrm{kid}_{i}}M_{j}+\sum_{k\in\mathrm{sib}_{i}}M_{k}$
2. 2.
Leaf nodes.
$\displaystyle\kappa^{\prime}_{i}=\kappa^{\prime\prime}_{\mathrm{parent}_{i}}$
$\displaystyle\kappa^{\prime\prime}_{i}=\kappa^{\prime}_{i}-M_{i}$
To estimate the $\kappa$-tree, we use $\widehat{f}$ instead of $f$ and let
$m_{i}$ be the fraction of data contained in the cluster for the tree node $i$
at birth. Again, define the estimated tree recursively:
$\displaystyle\widehat{\kappa}^{\prime}_{0}=1,$
$\displaystyle\widehat{\kappa}^{\prime}_{i}=\widehat{\kappa}^{\prime\prime}_{\mathrm{parent}_{i}},$
$\displaystyle\widehat{\kappa}^{\prime\prime}_{i}=\widehat{\kappa}^{\prime}_{i}-m_{i}+\sum_{j\in\mathrm{kid_{i}}}m_{j}.$
In practice we subtract the above quantities from 1 to get an increasing scale
that matches the $\lambda$ and $\kappa$ scales.
Note that switching between the $\lambda$ to $\alpha$ index does not change
the overall shape of the tree, but switching to the $\kappa$ index does. In
particular, the tallest leaf of the $\kappa$ tree corresponds to the cluster
with largest empirical mass. In both the $\lambda$ and $\alpha$ trees, on the
other hand, leaves correspond to clusters composed of points with high density
values. The difference can be substantial. Figure 3 illustrates the
differences between the three types of indexing for the “crater” example in
Figure 2. This example consists of a central Gaussian with high density and
low mass surrounded by a ring with high mass but uniformly low density. The
$\lambda$-scale tree (Figure 3(a)) correctly indicates the heights of the
modes of $\widehat{f}$, but tends to produce the incorrect intuition that the
ring (blue node and blue points in Figure 2(b)) is small. The $\alpha$-scale
plot (Figure 3(b)) ameliorates this problem by indexing node heights to the
quantiles of $\widehat{f}$. The blue node appears at $\alpha=0.35$, when 65%
of the data remains in the upper level set, and vanishes at $\alpha=0.74$,
when only 26% of the data remains in the upper level set. It is tempting to
say that this means the blue node contains $0.74-0.35=0.39$ of the mass but
this is incorrect because some of the difference in mass is due to the red
node. This interpretation is precisely the design of the $\kappa$-tree,
however, where we can say that the blue node contains $0.72-0.35=0.37$ of the
data.
(a)
(b)
Figure 2: The crater simulation. 2,000 points are sampled from a mixture of a
central Gaussian and an outer ring (Gaussian direction with uniform noise).
Roughly 70% of the points are in the outer ring. 2(a)) The kNN density
estimator evaluated on the data. 2(b)) Cluster labels from the all-mode
labeling technique, where each leaf of the level set tree is designated as a
cluster. Gray points are unlabeled low-density background observations.
(a)
(b)
(c)
Figure 3: Level set tree scales for the crater simulation. 3(a)) The $\lambda$
scale is dominant, corresponding directly to density level values. There is a
one-to-one correspondence with $\alpha$ values shown on the right y-axis. Note
the blue branch, corresponding to the outer ring in the crater simulation,
appears to be very small in this plot, despite the fact that the true group
contains about 70% of data 3(b)) The $\alpha$ scale is dominant, corresponding
to the fraction of data excluded from the upper level set at each $\lambda$
value. The blue cluster is more exaggerated but the topology of the tree
remains unchanged. 3(c)) The $\kappa$ scale. The blue cluster now appears
larger than the red, facilitating the intuitive connection between branch
height and cluster mass.
### 3.3 Cluster retrieval options
Many clustering algorithms are designed to only output a partition of the
data, whose elements are then taken to be the clusters. As we argued in the
introduction, such a paradigm is often inadequate for data exhibiting complex
and multi-scale clustering features. In contrast, hierarchical clustering in
general and level set tree clustering in particular give a more complete and
informative description of the clusters in a dataset. However, many
applications require that each data point be assigned to a single cluster
label. Much of the work on level set trees ignores this phase of a clustering
application or assumes that labels will be assigned according to the connected
components at a chosen $\lambda$ (density) or $\alpha$ (mass) level, which
DeBaCl accomodates through the upper set clustering option. Rather than
choosing a single density level, a practitioner might prefer to specify the
number of clusters $K$ (as with $K$-means). One way (of many) that this can be
done is to find the first $K-1$ splits in the level set tree and identify each
of the children from these splits as a cluster, known in DeBaCl as the first-K
clustering technique. A third, preferred, option avoids the choice of
$\lambda$, $\alpha$, or $K$ altogether and treats each leaf of the level set
tree as a separate cluster (Azzalini and Torelli, 2007). We call this the all-
mode clustering method. Use of these labeling options is illustrated in
Section 4.
Note that each of these methods assigns only a fraction of points to clusters
(the foreground points), while leaving low-density observations (background
points) unlabeled. Assigning the background points to clusters can be done
with any classification algorithm, and DeBaCl includes a handful of simple
options, including a k-nearest neighbor classifer, for the task.
### 3.4 Chaudhuri and Dasgupta algorithm
Chaudhuri and Dasgupta (2010) introduce an algorithm for estimating a level
set tree that is particularly notable because the authors prove finite-sample
convergence rates (where consistency is in the sense of Hartigan (1981)). The
algorithm is a generalization of single linkage, reproduced here for
convenience in Algorithm 2.
Input: $\\{x_{1},\ldots,x_{n}\\}$, $k$, $\alpha$
Output: $\widehat{\mathcal{T}}$, a hierarchy of subsets of
$\\{x_{1},\ldots,x_{n}\\}$
$r_{k}(x_{i})\leftarrow$ distance to the $k$’th neighbor of $x_{i}$;
for _$r\leftarrow 0$ to $\infty$_ do
$G_{r}\leftarrow$ graph with vertices $\\{x_{i}:r_{k}(x_{i})\leq r\\}$ and
edges $\\{(x_{i},x_{j}):\|x_{i}-x_{j}\|\leq\alpha r\\}$;
Find the connected components of $G_{\lambda_{r}}$;
$\widehat{\mathcal{T}}\leftarrow$ dendrogram of connected components of graphs
$G_{r}$, ordered by inclusions;
return _$\widehat{\mathcal{T}}$_
Algorithm 2 Chaudhuri and Dasgupta (2010) level set tree estimation procedure.
To translate this program into a practical implementation, we must find a
finite set of values for $r$ such that the graph $G_{r}$ can only change at
these values. When $\alpha=1$, the only values of $r$ where the graph can
change are the edge lengths in the graph $e_{ij}=\|x_{i}-x_{j}\|$ for all $i$
and $j$. Let $r$ take on each value of $e_{ij}$ in descending order; in each
iteration remove vertices and edges with larger k-neighbor radius and edge
length, respectively.
When $\alpha\neq 1$, the situation is trickier. First, note that including $r$
values where the graph does not change is not a problem, since the original
formulation of the method includes all values of $r\in\mathbb{R}^{+0}$.
Clearly, the vertex set can still change at any edge length $e_{ij}$. The edge
set can only change at values where $r=e_{ij}/\alpha$ for some $i,j$. Suppose
$e_{u,v}$ and $e_{r,s}$ are consecutive values in a descending ordered list of
edge lengths. Let $r=e/\alpha$, where $e_{u,v}<e<e_{r,s}$. Then the edge set
$E=\\{(x_{i},x_{j}):\|x_{i}-x_{j}\|\leq\alpha r=e\\}$ does not change as $r$
decreases until $r=e_{u,v}/\alpha$, where the threshold of $\alpha r$ now
excludes edge $(x_{u},x_{v})$. Thus, by letting $r$ iterate over the values in
$\bigcup_{i,j}\\{e_{ij},\frac{e_{ij}}{\alpha}\\}$, we capture all possible
changes in $G_{r}$.
In practice, starting with a complete graph and removing one edge at a time is
extremely slow because this requires $2*{n\choose 2}$ connected component
searches. The DeBaCl implementation includes an option to initialize the
algorithm at the k-nearest neighbor graph instead, which is a substantially
faster approximation to the Chaudhuri-Dasgupta method. This shortcut is still
dramatically slower than DeBaCl’s geometric tree algorithm, which is one
reason why we prefer the latter. Future development efforts will focus on
improvements in the speed of both procedures.
### 3.5 Pseudo-densities for functional data
The level set tree estimation procedure in Algorithm 1 can be extended to work
with data sampled from non-Euclidean spaces that do not admit a well-defined
pdf. The lack of a density function would seem to be an insurmountable problem
for a method defined on the levels of a pdf. In this case, however, level set
trees can be built on the levels of a pseudo-density estimate that measures
the similarity of observations and the overall connectivity of the sample
space. Pseudo-densities cannot be used to compute probabilities as in
Euclidean spaces, but are proportional to the statistical expectations of
estimates of the form $\widehat{f}$, which remain well-defined random
quantities (Ferraty and Vieu, 2006).
Random functions, for example, may have well-defined probability distributions
that cannot be represented by pdfs (Billingsley, 2012). To build level set
trees for this type of data, DeBaCl accepts very general functions for
$\widehat{f}$, including pseudo-densities, although the user must compute the
pairwise distances. The package includes a utility function for evaluating a
k-nearest neighbor pseudo-density estimator on the data based on the pairwise
distances. Specifically, equation 2 is modified by expunging the term $v^{d}$
and setting $d$ arbitrarily to 1. An application is shown in Section 4.
### 3.6 User customization
One advantage of DeBaCl over existing cluster tree software is that DeBaCl is
intended to be easily modified by the user. As described above, two major
algorithm types are offered, as well as the ability to use pseudo-densities
for functional data. In addition, the package allows a high degree of
customization in the type of similarity graph, data ordering function
(density, pseudo-density, or arbitrary function), pruning function, cluster
labeling scheme, and background point classifier. In effect, the only fixed
aspect of DeBaCl is that clusters are defined for every level to be connected
components of a geometric graph.
## 4 Usage
### 4.1 Basic Example
In this section we walk through the density-based clustering analysis of
10,000 fiber tracks mapped in a human brain with diffusion-weighted imaging.
For this analysis we use only the subcortical endpoint of each fiber track,
which is in $\mathbb{R}^{3}$. Despite this straightforward context of finite,
low-dimensional data, the clustering problem is somewhat challenging because
the data are known to have complicated striatal patterns. For this paper we
add the DeBaCl package to the Python path at run time, but this can be done in
a more persistent manner for repeated use. The NumPy library is also needed
for this example, and we assume the dataset is located in the working
directory. We use our preferred algorithm, the geometric level set tree, which
is located in the geom_tree module.
⬇
## Import DeBaCl package
import sys
sys.path.append(’/home/brian/Projects/debacl/DeBaCl/’)
from debacl import geom_tree as gtree
from debacl import utils as utl
## Import other Python libraries
import numpy as np
## Load the data
X = np.loadtxt(’0187_endpoints.csv’, delimiter=’,’)
n, p = X.shape
The next step is to define parameters for construction and pruning of the
level set tree, as well as general plot aesthetics. For this example we set
the density and connectivity smoothness parameter $k$ to $0.01n$ and the
pruning parameter $\gamma$ is set to $0.05n$. Tree branches with fewer points
than this will be merged into larger sibling branches. For the sake of speed,
we use a small subsample in this example.
⬇
## Downsample
n_samp = 5000
ix = np.random.choice(range(n), size=n_samp, replace=False)
X = X[ix, :]
n, p = X.shape
## Set level set tree parameters
p_k = 0.01
p_gamma = 0.05
k = int(p_k * n)
gamma = int(p_gamma * n)
## Set plotting parameters
utl.setPlotParams(axes_labelsize=28, xtick_labelsize=20, ytick_labelsize=20,
figsize=(8,8))
For straightforward cases like this one, we use a single convenience function
to do density estimation, similarity graph definition, level set tree
construction, and pruning. In the following example, each of these steps will
be done separately. Note the print function is overloaded to show a summary of
the tree.
⬇
## Build the level set tree with the all-in-one function
tree = gtree.geomTree(X, k, gamma, n_grid=None, verbose=False)
print tree
alpha1 alpha2 children lambda1 lambda2 parent size
key
0 0.0000 0.0040 [1, 2] 0.000000 0.000003 None 5000
1 0.0040 0.0716 [11, 12] 0.000003 0.000133 0 2030
2 0.0040 0.1278 [21, 22] 0.000003 0.000425 0 2950
11 0.0716 0.3768 [27, 28] 0.000133 0.004339 1 1437
12 0.0716 0.3124 [] 0.000133 0.002979 1 301
21 0.1278 0.9812 [] 0.000425 0.045276 2 837
22 0.1278 0.3882 [29, 30] 0.000425 0.004584 2 1410
27 0.3768 0.4244 [31, 32] 0.004339 0.005586 11 863
28 0.3768 1.0000 [] 0.004339 0.071075 11 406
29 0.3882 0.9292 [] 0.004584 0.032849 22 262
30 0.3882 0.9786 [] 0.004584 0.043969 22 668
31 0.4244 0.9896 [] 0.005586 0.048706 27 428
32 0.4244 0.9992 [] 0.005586 0.064437 27 395
The next step is to assign cluster labels to a set of foreground data points
with the function GeomTree.getClusterLabels. The desired labeling method is
specified with the method argument. When the correct number of clusters $K$ is
known, the first-k option retrieves the first $K$ disjoint clusters that
appear when $\lambda$ is increased from 0. Alternately, the upper-set option
cuts the tree at a single level, which is useful if the goal is to include or
exclude a certain fraction of the data from the upper level set. Here we use
this function with $\alpha$ set to 0.05, which removes the 5% of the
observations with the lowest estimated density (i.e. outliers) and clusters
the remainder. Finally, the all-mode option returns a foreground cluster for
each leaf of the level set tree, which avoids the need to specify either $K$,
$\lambda$, or $\alpha$.
Additional arguments for each method are specified by keyword argument; the
getClusterLabels method parses them intelligently. For all of the labeling
methods the function returns two objects. The first is an $m\times 2$ matrix,
where $m$ is the number of points in the foreground set. The first column is
the index of an observation in the full data matrix, and the second column is
the cluster label. The second object is a list of the tree nodes that are
foreground clusters. This is useful for coloring level set tree nodes to match
observations plotted in feature space.
⬇
uc_k, nodes_k = tree.getClusterLabels(method=’first-k’, k=3)
uc_lambda, nodes_lambda = tree.getClusterLabels(method=’upper-set’,
threshold=0.05, scale=’lambda’)
uc_mode, nodes_mode = tree.getClusterLabels(method=’all-mode’)
The GeomTree.plot method draws the level set tree dendrogram, with the
vertical scale controlled by the form parameter. See Section 3.2 for more
detail. The three plot forms are shown in Figure 4, foreground clusters are
derived from first-k clustering with $K$ set to 3. the plotForeground function
from the DeBaCl utils module is used to match the node colors in the
dendrogram to the clusters in feature space. Note that the plot function
returns a tuple with several objects, but only the first is useful for most
applications.
⬇
## Plot the level set tree with three different vertical scales, colored by
the first-K clustering
fig = tree.plot(form=’lambda’, width=’mass’, color_nodes=nodes_k)[0]
fig.savefig(’../figures/endpt_tree_lambda.png’)
fig = tree.plot(form=’alpha’, width=’mass’, color_nodes=nodes_k)[0]
fig.savefig(’../figures/endpt_tree_alpha.png’)
fig = tree.plot(form=’kappa’, width=’mass’, color_nodes=nodes_k)[0]
fig.savefig(’../figures/endpt_tree_kappa.png’)
## Plot the foreground points from the first-K labeling
fig, ax = utl.plotForeground(X, uc_k, fg_alpha=0.6, bg_alpha=0.4,
edge_alpha=0.3, s=22)
ax.elev = 14; ax.azim=160 # adjust the camera angle
fig.savefig(’../figures/endpt_firstK_fg.png’, bbox_inches=’tight’)
(a) Fiber endpoint data, colored by first-K foreground cluster
(b) Lambda scale
(c) Alpha scale
(d) Kappa scale
Figure 4: First-k clustering results with different vertical scales and the
clusters in feature space.
A level set tree plot is also useful as a scaffold for interactive exploration
of spatially coherent subsets of data, either by selecting individual nodes of
the tree or by retreiving high-density clusters at a selected density or mass
level. These tools are particularly useful for exploring clustering features
at multiple data resolutions. In Figure 5, for example, there are two dominant
clusters, but each one has highly salient clustering behavior at higher
resolutions. The interactive tools allow for exploration of the parent-child
relationships between these clusters.
⬇
tool1 = gtree.ComponentGUI(tree, X, form=’alpha’, output=[’scatter’], size=18,
width=’mass’)
tool1.show()
tool2 = gtree.ClusterGUI(tree, X, form=’alpha’, width=’mass’, size=18)
tool2.show()
(a)
(b)
(c)
(d)
Figure 5: The level set tree can be used as a scaffold for interactive
exploration of data subsets or upper level set clusters.
The final step of our standard data analysis is to assign background points to
a foreground cluster. DeBaCl’s utils module includes several very simple
classifiers for this task, although more sophisticated methods have been
proposed (Azzalini and Torelli, 2007). For this example we assign background
points with a k-nearest neighbor classifier. The observations are plotted a
final time, with a full data partition (Figure 6).
⬇
## Assign background points with a simple kNN classifier
segment = utl.assignBackgroundPoints(X, uc_k, method=’knn’, k=k)
## Plot all observations, colored by cluster
fig, ax = utl.plotForeground(X, segment, fg_alpha=0.6, bg_alpha=0.4,
edge_alpha=0.3, s=22)
ax.elev = 14; ax.azim=160
fig.savefig(’../figures/endpt_firstK_segment.png’, bbox_inches=’tight’)
Figure 6: Endpoint data, with background points assigned to the first-K
foreground clusters with a k-nearest neighbor classifier.
To customize the level set tree estimator, each phase can be done manually.
Here we use methods in DeBaCl’s utils module to build a k-nearest neighbor
similarity graph W, a k-nearest neighbor density estimate fhat, a grid of
density levels levels, and the background observation sets at each density
level (bg_sets). The constructTree method of the geom_tree module puts the
pieces together to make the tree and the prune function removes tree leaf
nodes that are small and likely due to random noise.
⬇
## Similarity graph and density estimate
W, k_radius = utl.knnGraph(X, k, self_edge=False)
fhat = utl.knnDensity(k_radius, n, p, k)
## Tree construction and pruning
bg_sets, levels = utl.constructDensityGrid(fhat, mode=’mass’, n_grid=None)
tree = gtree.constructTree(W, levels, bg_sets, mode=’density’, verbose=False)
tree.prune(method=’size-merge’, gamma=gamma)
print tree
alpha1 alpha2 children lambda1 lambda2 parent size
key
0 0.0000 0.0040 [1, 2] 0.000000 0.000003 None 5000
1 0.0040 0.0716 [11, 12] 0.000003 0.000133 0 2030
2 0.0040 0.1278 [21, 22] 0.000003 0.000425 0 2950
11 0.0716 0.3768 [27, 28] 0.000133 0.004339 1 1437
12 0.0716 0.3124 [] 0.000133 0.002979 1 301
21 0.1278 0.9812 [] 0.000425 0.045276 2 837
22 0.1278 0.3882 [29, 30] 0.000425 0.004584 2 1410
27 0.3768 0.4244 [31, 32] 0.004339 0.005586 11 863
28 0.3768 1.0000 [] 0.004339 0.071075 11 406
29 0.3882 0.9292 [] 0.004584 0.032849 22 262
30 0.3882 0.9786 [] 0.004584 0.043969 22 668
31 0.4244 0.9896 [] 0.005586 0.048706 27 428
32 0.4244 0.9992 [] 0.005586 0.064437 27 395
In the definition of density levels and background sets, the
constructDensityGrid allows the user to specify the n_grid parameter to speed
up the algorithm by computing the upper level set and connectivity for only a
subset of density levels. The mode parameter determines whether the grid of
density levels is based on evenly-sized blocks of observations (mode=’mass’)
or density levels (mode=’levels’); we generally prefer the ‘mass’ mode for our
own analyses.
The mode parameter of the tree construction function is usually set to be
‘density’, which treats the underlying function fhat as a density or pseudo-
density function, with a floor value of 0. This algorithm can be applied to
arbitrary functions that do not have a floor value, in which case the mode
should be set to ‘general’.
### 4.2 Extension: The Chaudhuri-Dasgupta Tree
Usage of the Chaudhuri-Dasgupta algorithm is similar to the standalone
geomTree function. First load the DeBaCl module cd_tree (labeled here for
brevity as cdt) and the utility functions in utils, as well as the data.
⬇
## Import DeBaCl package
import sys
sys.path.append(’/home/brian/Projects/debacl/DeBaCl/’)
from debacl import cd_tree as cdt
from debacl import utils as utl
## Import other Python libraries
import numpy as np
## Load the data
X = np.loadtxt(’0187_endpoints.csv’, delimiter=’,’)
n, p = X.shape
Because the straightforward implementation of the Chaudhuri-Dasgupta algorithm
is extremely slow, we use a random subset of only 200 observations (out of the
total of 10,000). The smoothing parameter is set to be 2.5% of $n$, or 5. The
pruning parameter is 5% of $n$, or 10\. The pruning parameter is slightly less
important for the Chaudhuri-Dasgupta algorithm.
⬇
## Downsample
n_samp = 200
ix = np.random.choice(range(n), size=n_samp, replace=False)
X = X[ix, :]
n, p = X.shape
## Set level set tree parameters
p_k = 0.025
p_gamma = 0.05
k = int(p_k * n)
gamma = int(p_gamma * n)
## Set plotting parameters
utl.setPlotParams(axes_labelsize=28, xtick_labelsize=20, ytick_labelsize=20,
figsize=(8,8))
The straightforward implementation of the Chaudhuri-Dasgupta algorithm starts
with a complete graph and removes one edge a time, which is extremely slow.
The start parameter of the cdTree function allows for shortcuts. These are
approximations to the method, but are necessary to make the algorithm
practical. Currently, the only implemented shortcut is to start with a
k-nearest neighbor graph.
⬇
## Construct the level set tree estimate
tree = cdt.cdTree(X, k, alpha=1.4, start=’knn’, verbose=False)
tree.prune(method=’size-merge’, gamma=gamma)
As with the geometric tree, we can print a summary of the tree, plot the tree,
retrieve foreground cluster labels, and plot the foreground clusters. This is
illustrated below for the ‘all-mode’ labeling method.
⬇
## Print/make output
print tree
fig = tree.plot()
fig.savefig(’../figures/cd_tree.png’)
uc, nodes = tree.getClusterLabels(method=’all-mode’)
fig, ax = utl.plotForeground(X, uc, fg_alpha=0.6, bg_alpha=0.4,
edge_alpha=0.3, s=60)
ax.elev = 14; ax.azim=160
fig.savefig(’../figures/cd_allmode.png’)
children parent r1 r2 size
key
0 [3, 4] None 8.134347 4.374358 200
3 [15, 16] 0 4.374358 2.220897 109
4 [23, 24] 0 4.374358 1.104121 75
15 [] 3 2.220897 0.441661 32
16 [] 3 2.220897 0.343408 55
23 [] 4 1.104121 0.445529 28
24 [] 4 1.104121 0.729226 24
(a)
(b)
Figure 7: The Chaudhuri-Dasgupta tree for the fiber track endpoint data,
downsampled from 10,000 to 200 observations to make computation feasible.
Foreground clusters based on all-mode clustering are shown on the right.
### 4.3 Extension: Functional Data
Nothing in the process of estimating a level set tree requires $\widehat{f}$
to be a bona fide probability density function, and the DeBaCl package allows
us to use this fact to use level set trees for much more complicated datasets.
To illustrate we use the phoneme dataset from Ferraty and Vieu (2006), which
contains 2000 total observations of five short speech patterns. Each
observation is recorded on a regular grid of 150 frequencies, but we treat
this as an approximation of a continuous function on an interval of
$\mathbb{R}^{1}$. Because the observations are random curves they do not have
bona fide density functions, but we can still construct a sample level set
tree by estimating a pseudo-density function that measures the proximity of
each curve to its neighbors.
To start we load the DeBaCl modules and the data, which have been pre-smoothed
for this example with cubic splines. The true class of each observation is in
the last column of the raw data object. The curves for each phoneme are shown
in Figure 8.
⬇
## Import DeBaCl package
import sys
sys.path.append(’/home/brian/Projects/debacl/DeBaCl/’)
from debacl import geom_tree as gtree
from debacl import utils as utl
## Import other Python libraries
import numpy as np
import scipy.spatial.distance as spdist
import matplotlib.pyplot as plt
## Set plotting parameters
utl.setPlotParams(axes_labelsize=28, xtick_labelsize=20, ytick_labelsize=20,
figsize=(8,8))
## Load data
speech = np.loadtxt(’smooth_phoneme.csv’, delimiter=’,’)
phoneme = speech[:, -1].astype(np.int)
speech = speech[:, :-1]
n, p = speech.shape
⬇
## Plot the curves, separated by true phoneme
fig, ax = plt.subplots(3, 2, sharex=True, sharey=True)
ax = ax.flatten()
ax[-2].set_xlabel(’frequencies’)
ax[-1].set_xlabel(’frequencies’)
for g in np.unique(phoneme):
ix = np.where(phoneme == g)[0]
for j in ix:
ax[g].plot(speech[j, :], c=’black’, alpha=0.15)
fig.savefig(’../figures/phoneme_data.png’)
Figure 8: Smoothed waveforms for spoken phonemes, separated by true phoneme.
For functional data we need to define a distance function, precluding the use
of the convenience method GeomTree.geomTree or even the utility function
utils.knnGraph. First the bandwith and tree pruning parameters are set to be
$0.01n$. In the second step all pairwise distances are computed in order to
find the $k$-nearest neighbors for each observation. For simplicity we use
Euclidean distance between a pair of curves (which happens to work well in
this example), but this is not generally optimal. Next, the adjacency matrix
for a $k$-nearest neighbor graph is constructed, which is no different than
the finite-dimensional case. Finally the pseudo-density estimator is built by
using the finite-dimenisonal $k$-nearest neighbor density estimator with the
dimension set (incorrectly) to 1. This function does not integrate to 1, but
the function induces an ordering on the observations (from smallest to largest
$k$-neighbor radius) that is invariant to the dimension. This ordering is all
that is needed for the final step of building the level set tree.
⬇
## Bandwidth and pruning parameters
p_k = 0.01
p_gamma = 0.01
k = int(p_k * n)
gamma = int(p_gamma * n)
## Find all pairwise distances and the indices of each point’s k-nearest
neighbors
D = spdist.squareform(spdist.pdist(speech, metric=’euclidean’))
rank = np.argsort(D, axis=1)
ix_nbr = rank[:, 0:k]
ix_row = np.tile(np.arange(n), (k, 1)).T
## Construct the similarity graph adjacency matrix
W = np.zeros(D.shape, dtype=np.bool)
W[ix_row, ix_nbr] = True
W = np.logical_or(W, W.T)
np.fill_diagonal(W, False)
## Compute a pseudo-density estimate and evaluate at each observation
k_nbr = ix_nbr[:, -1]
r_k = D[np.arange(n), k_nbr]
fhat = utl.knnDensity(r_k, n, p=1, k=k)
## Build the level set tree
bg_sets, levels = utl.constructDensityGrid(fhat, mode=’mass’, n_grid=None)
tree = gtree.constructTree(W, levels, bg_sets, mode=’density’, verbose=False)
tree.prune(method=’size-merge’, gamma=gamma)
print tree
alpha1 alpha2 children lambda1 lambda2 parent size
key
0 0.0000 0.2660 [1, 2] 0.000000 0.000261 None 2000
1 0.2660 0.3435 [3, 4] 0.000261 0.000275 0 1125
2 0.2660 1.0000 [] 0.000261 0.000938 0 343
3 0.3435 0.4905 [5, 6] 0.000275 0.000307 1 565
4 0.3435 0.7705 [] 0.000275 0.000426 1 413
5 0.4905 0.9920 [] 0.000307 0.000808 3 391
6 0.4905 0.7110 [] 0.000307 0.000382 3 85
Once the level set tree is constructed we can plot it and retrieve cluster
labels as with finite-dimensional data. In this case we choose the all-mode
cluster labeling which produces four clusters. The utility function
utils.plotForeground is currently designed to work only with two- or three-
dimensional data, so plotting the foreground clusters must be done manually
for functional data. The clusters from this procedure match the true groups
quite well, at least in a qualitative sense.
⬇
## Retrieve cluster labels
uc, nodes = tree.getClusterLabels(method=’all-mode’)
## Level set tree plot
fig = tree.plot(form=’alpha’, width=’mass’, color_nodes=nodes)[0]
fig.savefig(’../figures/phoneme_tree.png’)
## Plot the curves, colored by foreground cluster
palette = utl.Palette()
fig, ax = plt.subplots()
ax.set_xlabel("frequency index")
for c in np.unique(uc[:,1]):
ix = np.where(uc[:,1] == c)[0]
ix_clust = uc[ix, 0]
for i in ix_clust:
ax.plot(speech[i,:], c=np.append(palette.colorset[c], 0.25))
fig.savefig(’../figures/phoneme_allMode.png’)
Figure 9: All-mode foreground clusters for the smoothed phoneme data.
## 5 Conclusion
The Python package DeBaCl for hierarchical density-based clustering provides a
highly usable implementation of level set tree estimation and clustering. It
improves on existing software through computational efficiency and a high-
degree of modularity and customization. Namely, DeBaCl:
* •
offers the first known implementation of the theoretically well-supported
Chaudhuri-Dasgupta level set tree algorithm;
* •
allows for very general data ordering functions, which are typically
probability density estimates but could also be pseudo-density estimates for
infinite-dimensional functional data or even arbitrary functions;
* •
accepts any similarity graph, density estimator, pruning function, cluster
labeling scheme, and background point assignment classifier;
* •
includes the all-mode cluster labeling scheme, which does not require an a
priori choice of the number of clusters;
* •
incorporates the $\lambda$, $\alpha$, and $\kappa$ vertical scales for
plotting level set trees, as well as other plotting tweaks to make level set
tree plots more interpretable and usable;
* •
and finally, includes interactive GUI tools for selecting coherent data
subsets or high-density clusters based on the level set tree.
The DeBaCl package and user manual is available at
https://github.com/CoAxLab/DeBaCl. The project remains under active
development; the focus for the next version will be on improvements in
computational efficiency, particularly for the Chaudhuri-Dasgupta algorithm.
## Acknowledgments
This research was sponsored by the Army Research Laboratory and was
accomplished under Cooperative Agreement Number W911NF-10-2-0022. The views
and conclusions contained in this document are those of the authors and should
not be interpreted as representing the official policies, either expressed or
implies, of the Army Research Laboratory or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for the
Government purposes notwithstanding any copyright notation herein. This
research was also supported by NSF CAREER grant DMS 114967.
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|
arxiv-papers
| 2013-07-30T20:19:26 |
2024-09-04T02:49:48.783917
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Brian P. Kent, Alessandro Rinaldo, Timothy Verstynen",
"submitter": "Brian Kent",
"url": "https://arxiv.org/abs/1307.8136"
}
|
1307.8169
|
# Effects of the structure of charged impurities and dielectric environment on
conductivity of graphene
R. Aničić Department of Applied Mathematics, University of Waterloo,
Waterloo, Ontario, Canada N2L 3G1 Z. L. Mišković [email protected]
Department of Applied Mathematics, and Waterloo Institute for Nanotechnology,
University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
###### Abstract
We investigate the conductivity of doped single-layer graphene in the
semiclassical Boltzmann limit, as well as the conductivity minimum in neutral
graphene within the self-consistent transport theory, pointing up the effects
due to both the structure of charged impurities near graphene and the
structure of the surrounding dielectrics. Using the hard-disk model for a two-
dimensional (2D) distribution of impurities allows us to investigate
structures with large packing fractions, which are shown to give rise to both
strong increase in the slope of conductivity at low charge carrier densities
in graphene and a strongly sub-linear behavior of the conductivity at high
charge carrier densities when the correlation distance between the impurities
is large. On the other hand, we find that a super-linear dependence of the
conductivity on charge carrier density in heavily doped graphene may arise
from increasing the distance of impurities from graphene or allowing their
clustering into disk-like islands, whereas the existence of an electric dipole
polarizability of impurities may give rise to an electron-hole asymmetry in
the conductivity. Using the electrostatic Green’s function for a three-layer
structure of dielectrics, we show that finite thickness of a dielectric layer
in the top gating configuration, as well as the existence of non-zero air
gap(s) between graphene and the nearby dielectric(s) exert strong influences
on the conductivity and its minimum. While a decrease in the dielectric
thickness is shown to increase the conductivity in doped graphene and even
gives rise to finite conductivity in neutral graphene for a 2D distribution of
impurities, we find that an increase in the dielectric thickness gives rise to
a super-linear behavior of the conductivity when impurities are homogeneously
distributed throughout the dielectric. Moreover, the dependence of graphene’s
mobility on its charge carrier density is surprisingly strongly affected,
quantitatively and qualitatively, by the graphene-dielectric gap(s) when
combined with the precise position of a 2D distribution of charged impurities.
Finally, we show that the conductivity minimum in neutral graphene is
increased by increasing the correlation distance between the impurities,
reduced by increasing the graphene-dielectric gap, and increased by decreasing
the dielectric thickness in a top-gated configuration, even though the
corresponding residual charge carrier density is reduced by decreasing the
dielectric thickness.
graphene, conductivity, charged impurities, dielectric screening
###### pacs:
73.22.Pr, 72.80.Vp, 81.05.ue
## I Introduction
Graphene is a realization of a two-dimensional (2D) material made of carbon
atoms strongly bonded in a honeycomb–like lattice, exhibiting a Dirac-like
spectrum for low-energy excitations of its $\pi$ electrons, which has been
under intense scrutiny for possible applications in electronics,
photonics,Avouris_2012 and biochemical sensing.Allen_2010 Being an all-
surface material renders graphene extremely sensitive to the incident
electromagnetic fields and to the dielectric properties of the surrounding
matter,Newaz_2012 which is both a blessing and a curse from a technological
point of view. While the use of external gates and/or controlled adsorption of
atomic and molecular species present an efficient means for inducing precise
concentrations of charge carriers in graphene,Chen_2008 the presence of
indeterminate amounts of charged impurities, which may be trapped in a
substrate or directly adsorbed on graphene, render quantitative details of
many measurements of graphene’s electronic and optical properties ”sample
dependent”.Tan_2007 In addition, integrating graphene in layered structures
with different material properties may bring additional issues due to
uncertainties in the geometric structure and the chemical composition of such
structures.Fallahazad_2012 ; Hollander_2011
Possibly the most intriguing manifestation of the presence of charged
impurities is the famed minimum in the DC conductivity of single-layer
graphene in the limit of vanishing doping, i.e., when the average density of
induced charge carriers in graphene approaches zero.Tan_2007 ; Sarma_2011 It
was shown that the minimum conductivity may be explained by the manifestation
of a system of electron-hole puddles in graphene due to corrugation of the
electrostatic potential that arises from a spatial distribution of the charged
impurities in a substrate. PNAS_2007 On the other hand, the conductivity in
heavily doped graphene layers often exhibits sub-linear behavior, or
saturation with increasing charge carrier density, which is often explained by
the presence of short-range scatterers in graphene, presumably arising from
atomic-size defects in the carbon lattice. Yan_2011 However, it turned out
that spatial correlation among the nearby charged impurities may provide an
alternative and more plausible explanation of the conductivity saturation in
single-layer graphene. Li_2011 ; Yan_2011 Moreover, the atoms adsorbed on
graphene often show tendency of clustering and forming islands, which may
additionally affect the mobility of charge carriers in graphene. McCreary_2010
As far as the structure and composition of the surrounding material is
concerned, preference is usually given to insulators and metals that only
engage in weak interactions with graphene of the van der Waals type, leaving
the structure of its $\pi$ electron bands largely intact in the vicinity of
the Dirac point.Wehling_2009 Those interactions are characterized with
relatively large spatial gaps between graphene and the nearby material, on the
order of several Ångströms, which reduce the dielectric screening by that
material and often exhibit significant fluctuations in their size due to the
surface roughness of the material.Ishigami_2007 Furthermore, when graphene is
top gated with a layer of high-$\kappa$ dielectric material, the mobility of
its charge carriers may be affected by a strong image interaction with the
metallic top gate.Fallahazad_2012 ; Hollander_2011 ; Ong_2012 Finally, for
electrolytically top-gated graphene, the presence of mobile ions in the nearby
electrolyte may provide additional screening of the charged impurities in a
solid substrate.Chen_2009 ; Miskovic_2012
All of the above examples of the effects of charged impurities near graphene
and the structure of the surrounding dielectrics play important roles in its
charge carrier transport, plasmon dispersion in doped graphene, and graphene’s
capacitance, which are of interest in electronics, photonics, and sensing,
respectively. It was recently shown that those effects may be conveniently
modeled by using Green’s function (GF) for the Poisson equation for a layered
structure, Ong_2012 ; Miskovic_2012 which is easily combined in a self-
consistent manner with the polarization function of graphene within the random
phase approximation (RPA) when graphene is modeled as a zero-thickness
material. Castro_2009 In this work, we illustrate such approach to modeling
the conductivity of single-layer graphene with large area by considering a
three-layer structure of the surrounding dielectrics and using an expression
for the conductivity that results from the semiclassical Boltzmann transport
(SBT) theory for doped graphene. Sarma_2011 However, that expression is
derived here via the Energy loss method (ELM),Gerlach_1986 which explicitly
evaluates the friction force on a system of external charges with the spatial
distribution that moves rigidly parallel to graphene.Allison_2009 ;
Allison_2010 ; Radovic_2012 Hence, the ELM has an added utility as it may be
used in studying other processes, such as sliding friction of molecular layers
physisorbed on graphene,Krim_2012 or probing the streaming potential in a
flowing electrolyte by a graphene based sensor,Newaz_2012_b which will be
tackled in future work.
In this work we focus on several effects in the DC conductivity of graphene.
First, we explore the effects of long correlation distances among impurities
that give rise to large packing fractions, which cannot be described by a
simple step-like pair correlation function.Yan_2011 ; Li_2011 For that
purpose we use an analytically parameterized model of hard disks (HD) due to
Rosenfeld,Rosenfeld_1990 which gives reliable results for packing fractions
up to the freezing point of a 2D fluid. Next, whereas all the previous studies
assumed that charged impurities reside in a plane parallel to graphene, our
statistical formulation of the theory allows for a fully three-dimensional
(3D) spatial distribution of impurities that may reside at a range of
distances from graphene. In addition, we allow that individual impurities may
be characterized by atomic-like form factors, which include a finite dipole
moment and a spatial spread that accounts for the existence of disk-like
clusters near graphene. Furthermore, by taking advantage of the electrostatic
GF for a three-layer structure, we also study the effects that arise in
conductivity of graphene due to finite thickness of a nearby dielectric and a
finite gap between graphene and the nearby dielectrics. Finally, the above
effects are also studied in the context of the conductivity minimum within the
Self-consistent transport (SCT) theory.PNAS_2007
Specifically, in this paper we show via the HD model that large correlation
distances between charged impurities may give rise to significantly larger
initial slopes of the conductivity (or larger mobility) at lower charge
carrier densities, as well as to a more pronounced saturation, or sub-linear
behavior of conductivity at higher densities than in the case of small
correlation distances. Next, the effects of clustering of charge impurities,
as well as the increasing distance from graphene are confirmed to give rise to
super-linear dependence of conductivity on charge carrier density in heavily
doped graphene, in agreement with observationsMcCreary_2010 and
modeling,PNAS_2007 respectively. Impurities with finite dipolar
polarizability are shown to give rise to electron-hole asymmetry in the
conductivity as the sign of charge carrier density changes, which may be
related to experimental observations in some graphene samples.Tan_2007
Regarding the geometrical factors of a nearby dielectric layer, we find an
increase in both the conductivity and mobility of graphene when the layer
thickness decreases in the case of a 2D distribution of impurities, whereas a
homogeneous 3D distribution of impurities gives rise to a super-linear
behavior of the conductivity with increasing layer thickness. Most
intriguingly, we find a strong effect on the mobility of graphene due to the
presence of a finite gap between graphene and the nearby dielectrics in
conjunction with the varying position of impurities, which was not previously
considered in the modeling of the transport properties of graphene, but was
observed in studying the polarization forces on external charges. Allison_2009
Finally, a similarly strong effect of the finite gap between graphene and a
nearby dielectric is also demonstrated in the minimum conductivity within the
SCT theory.PNAS_2007
After outlining the theoretical model in the next section, we discuss our
numerical results, and give concluding remarks. In the Appendices we outline a
derivation of the electrostatic GF and provide details for several models of
the charged impurity structure. Note that, unless otherwise explicitly stated,
we use gaussian electrostatic units where $4\pi\epsilon_{0}\equiv 1$, with
$\epsilon_{0}$ being the dielectric permittivity of vacuum.
## II Theory
We assume that a single-layer graphene sheet of large area is embedded into a
stratified structure so that it lies parallel to layers of various dielectrics
with abrupt interfaces among them, as shown in Fig. 1. Using a 3D Cartesian
coordinate system with coordinates ${\bf R}\equiv\\{{\bf r},z\\}$, the entire
structure may be then considered translationally invariant (and is assumed to
be isotropic) in the directions of a 2D position vector ${\bf r}=\\{x,y\\}$.
Furthermore, assume that a system of charged particles is distributed
throughout the structure and is moving rigidly at a constant velocity ${\bf
v}$ parallel to graphene. If the stationary volume density of charges in the
moving frame of reference is given by $\rho_{0}({\bf R})\equiv\rho_{0}({\bf
r},z)$, then the corresponding volume density in the rest frame of graphene
(the laboratory frame of reference) is given by $\rho({\bf R},t)=\rho_{0}({\bf
r}-{\bf v}t,z)$.
Figure 1: (Color online) Diagram showing a three-layer structure of
dielectrics with the relative bulk dielectric constants $\epsilon_{j}$ for
$j=1,2,3$, which occupy the regions defined by the intervals $I_{1}=[-L,0]$,
$I_{2}=[0,H]$ and $I_{3}=[H,\infty)$ for the $z$ coordinate of a Cartesian
coordinate system, respectively.
This notion of a rigidly moving distribution of external charges may be
related to several realistic physical situations where the relative motion of
particles with respect to each other may be treated as adiabatic at the time
scale of the charge carrier dynamics in graphene. Examples include sliding of
a film of adsorbed molecular layers across graphene,Krim_2012 flow of a
molecular fluid that contains dissolved ions in thermal
equilibrium,Newaz_2012_b or propagation of ionized fragments that result from
planar Coulomb explosion of a cluster grazingly scattered from graphene.
Song_2005 In each of those examples, the movement of external charged
particles gives rise to energy dissipation due to excitations of charge
carriers in graphene.
Conversely, one my reverse the frames of reference and consider the regime of
steady-state electric conduction in graphene where its charge carriers move
with a constant drift velocity $-{\bf v}$. In this case the distribution of
external particles is static in the laboratory frame and hence may be used to
model fixed charged impurities near graphene. If the speed $v=\|{\bf v}\|$ is
sufficiently low, then the electrical resistivity of graphene may be related
to energy dissipation due to scattering of its charge carriers on external
charged impurities, giving rise to Ohmic heating of graphene. This idea of
reversing the frames of reference is a basis of the ELM that was developed for
studying the transport properties of semiconductor heterostructures by means
of the dielectric response formalism for their conducting
electrons.Gerlach_1986 This method was used successfully in studying the
scattering of conduction electrons on interface roughnessKaser_1995 and
polarizable scattering centers,Kaser_1997 as well as in discussing
vibrational damping in adsorbed layers due to surface resistivity,Persson_1991
and in studying optical properties of thin films for solar energy
materials.Jin_1988 Moreover, this same idea of the equivalence of a drag
force on a uniformly moving system of impurities and the total force on the
electron fluid in doped graphene was recently applied to evaluate the
conductivity of graphene within the semiclassical hydrodynamic model for its
charge carriers.Mendoza_2013
We note that the ELM gives an expression for the conductivity of doped
graphene, which is identical to that obtained by the SBT theory,Sarma_2011
but we chose ELM because it yields the drag force on externally moving charges
as a side result that may be more directly used in modeling other processes,
such as sliding friction of molecular layers physisorbed on graphene Krim_2012
or probing the streaming potential in a flowing electrolyte by a graphene
based sensor,Newaz_2012_b to mention a few.
### II.1 Energy loss method
To be specific, we assume that the system of external charges consists of $N$
particles, each carrying a total charge of $Z_{j}e$ (where $e>0$ is the proton
charge) that is distributed around the center of the particle according to
some function $\Delta_{j}({\bf R})$, such that $\int d^{3}{\bf
R}\,\Delta_{j}({\bf R})=Z_{j}$ with $j=1,2,\ldots,N$. If the $j$th particle is
centered at the position ${\bf R}_{j}=\\{{\bf r}_{j},z_{j}\\}$ in the moving
frame of reference, we may write for the total density of charges in that
frame
$\displaystyle\rho_{0}({\bf r},z)=e\sum_{j=1}^{N}\Delta_{j}\\!\left({\bf
r}-{\bf r}_{j},z-z_{j}\right).$ (1)
Given that the positions ${\bf R}_{j}$ of external particles, as well as their
individual charge densities $\Delta_{j}({\bf R})$ are statistically
distributed, we denote their joint ensemble average by $\langle\cdots\rangle$.
Assuming that this distribution is translationally invariant in the directions
of ${\bf r}$, we note that $\langle\rho({\bf R},t)\rangle=\langle\rho_{0}({\bf
r},z)\rangle\equiv\bar{\rho}_{0}(z)$ can only be a function of the
perpendicular coordinate $z$. Therefore, assuming that the equilibrium areal
number density of charge carriers is uniform across graphene, its value
$\bar{n}$ will be determined by both the function $\bar{\rho}_{0}(z)$ and the
potential applied through the external gates. We assume that $\bar{n}$ has a
sufficiently large value allowing us to neglect the effects of fluctuations in
the charge carrier density in graphene on its screening properties. On the
other hand, we assume $\bar{n}$ to be small enough to allow the use of a 2D
response function for graphene’s $\pi$ electrons in the approximation of Dirac
fermions. Wunsch_2006 ; Hwang_2007 Those requirements practically limit our
considerations of graphene’s DC conductivity within the SBT theory to an
approximate range of doping densities 1011 cm${}^{-2}\lesssim\bar{n}\lesssim$
1013 cm-2 (we assume $\bar{n}>0$ unless stated otherwise).
We further define the fluctuation in the charge density of external particles
by $\delta\\!\rho({\bf R},t)\equiv\rho({\bf R},t)-\langle\rho({\bf
R},t)\rangle=\rho_{0}({\bf r}-{\bf v}t,z)-\langle\rho_{0}({\bf
r},z)\rangle\equiv\delta\\!\rho_{0}({\bf r}-{\bf v}t,z)$ and use it in the
Poisson equation, allowing us to express the resulting fluctuation of the
electrostatic potential, $\delta\\!\Phi({\bf R},t)$, in terms of the
electrostatic GF for the entire system, $G({\bf R},{\bf
R}^{\prime};t-t^{\prime})\equiv G({\bf r}-{\bf
r}^{\prime};z,z^{\prime};t-t^{\prime})$, as
$\displaystyle\delta\\!\Phi({\bf R},t)=\int d^{3}{\bf
R}^{\prime}\,\int\limits_{-\infty}^{\infty}dt^{\prime}\,G({\bf R},{\bf
R}^{\prime};t-t^{\prime})\,\delta\\!\rho({\bf R}^{\prime},t^{\prime}).$ (2)
Using a tilde to denote the Fourier transform (FT) of various quantities with
respect to the 2D position (${\bf r}\rightarrow{\bf q}$) and time
($t\rightarrow\omega$), the above expression is recast in the form
$\displaystyle\delta\\!\widetilde{\Phi}({\bf
q},z,\omega)=\int\limits_{-\infty}^{\infty}dz^{\prime}\,\widetilde{G}({\bf
q};z,z^{\prime};\omega)\,\delta\\!\widetilde{\rho}({\bf
q},z^{\prime},\omega),$ (3)
where
$\displaystyle\delta\\!\widetilde{\rho}({\bf q},z,\omega)$ $\displaystyle=$
$\displaystyle\int d^{2}{\bf
r}\int\limits_{-\infty}^{\infty}dt\,\mbox{e}^{-i{\bf q}\cdot{\bf r}+i\omega
t}\,\delta\\!\rho_{0}({\bf r}-{\bf v}t,z)$ (4) $\displaystyle=$ $\displaystyle
2\pi\,\delta(\omega-{\bf q}\cdot{\bf v})\,\delta\\!\widetilde{\rho}_{0}({\bf
q},z)$
defines the relation between the FTs of the fluctuations of the external
charge densities in the two reference frames. Here,
$\delta\\!\widetilde{\rho}_{0}({\bf q},z)=\widetilde{\rho}_{0}({\bf
q},z)-(2\pi)^{2}\,\delta({\bf q})\,\bar{\rho}_{0}(z)$ is defined via the FT of
the external charge density in the moving frame of reference,
$\displaystyle\widetilde{\rho}_{0}({\bf
q},z)=e\sum_{j=1}^{N}\widetilde{\Delta}_{j}({\bf
q},z-z_{j})\,\mathrm{e}^{-i{\bf q}\cdot{\bf r}_{j}}.$ (5)
It may be shown that the ensemble average of the energy loss rate is given by
Mowbray_2010
$\displaystyle\left\langle\frac{dW}{dt}\right\rangle$ $\displaystyle=$
$\displaystyle-\int d^{3}{\bf R}\,\left\langle\delta\\!\rho({\bf
R},t)\,\frac{\partial}{\partial t}\delta\\!\Phi({\bf r},z,t)\right\rangle$ (6)
$\displaystyle=$ $\displaystyle i\int\frac{d^{2}{\bf
q}}{\left(2\pi\right)^{2}}\,({\bf q}\\!\cdot\\!{\bf v})\,\int dz\int
dz^{\prime}\,\widetilde{G}({\bf q};z,z^{\prime};{\bf q}\\!\cdot\\!{\bf v})$
$\displaystyle\times\left\langle\delta\\!\widetilde{\rho}_{0}(-{\bf
q},z)\delta\\!\widetilde{\rho}_{0}({\bf q},z^{\prime})\right\rangle.$
On using the symmetry properties of the FT of the full GF, $\widetilde{G}({\bf
q};z,z^{\prime};\omega)=\widetilde{G}(-{\bf q};z^{\prime},z;\omega)$ and
$\widetilde{G}^{\mathrm{(cc)}}({\bf
q};z,z^{\prime};\omega)=\widetilde{G}(-{\bf q};z,z^{\prime};-\omega)$, where
$\mathrm{(cc)}$ denotes complex conjugation, one notices that only the
imaginary part of the factor $\widetilde{G}({\bf q};z,z^{\prime};{\bf
q}\\!\cdot\\!{\bf v})$ in Eq. (6) contributes to the energy loss. Furthermore,
assuming that graphene has a zero thickness and is placed in the plane
$z=z_{g}$, we may express $\widetilde{G}({\bf q};z,z^{\prime};\omega)$ in
terms of the (real valued) 2D FT of the GF $\widetilde{G}^{(0)}({\bf
q};z,z^{\prime})$ for the dielectric environment _without_ graphene, as given
in Eq. (35). Thus, Eq. (6) may be rewritten as
$\displaystyle\left\langle\frac{dW}{dt}\right\rangle$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\bf q}}{\left(2\pi\right)^{2}}\,V_{C}(q)\,({\bf
q}\\!\cdot\\!{\bf v})\,\Im\\!\left[\frac{-1}{\epsilon(q,{\bf q}\\!\cdot\\!{\bf
v})}\right]$ (7)
$\displaystyle\times\left\langle\delta\\!\widetilde{\mathcal{N}}(-{\bf
q})\delta\\!\widetilde{\mathcal{N}}({\bf q})\right\rangle,$
where we have defined a dielectric function that describes the dynamic
screening of external electrostatic fields in the plane $z=z_{g}$ due to the
polarization of the entire system as
$\displaystyle\epsilon(q,\omega)=\epsilon_{\text{bg}}(q)+V_{C}(q)\,\chi(q,\omega),$
(8)
with $\epsilon_{\text{bg}}(q)\equiv
2\pi/\left[q\widetilde{G}^{(0)}(q;z_{g},z_{g})\right]$ being an effective
background dielectric function due to the polarization of the system _without_
graphene, $V_{C}(q)=2\pi e^{2}/q$ the in-plane FT of the Coulomb potential,
and $\chi(q,\omega)$ a 2D polarization function of noninteracting $\pi$
electrons in graphene. Wunsch_2006 ; Hwang_2007 Moreover, in Eq. (7) we have
introduced the fluctuation in an effective areal (or surface-projected) number
density of external particles, $\delta\\!\mathcal{N}({\bf r})$, which is
defined via its 2D FT as $\delta\\!\widetilde{\mathcal{N}}({\bf
q})=\widetilde{\mathcal{N}}({\bf q})-\langle\widetilde{\mathcal{N}}({\bf
q})\rangle$, with
$\displaystyle\widetilde{\mathcal{N}}({\bf q})\equiv\frac{1}{e}\int
dz\,\psi(q,z)\,\widetilde{\rho}_{0}({\bf
q},z)=\sum_{j=1}^{N}\mathcal{F}_{j}({\bf q})\,\mathrm{e}^{-i{\bf q}\cdot{\bf
r}_{j}},$ (9)
where
$\displaystyle\psi(q,z)=\frac{\widetilde{G}^{(0)}(q;z_{g},z)}{\widetilde{G}^{(0)}(q;z_{g},z_{g})}$
(10)
is a profile function that takes into account the decay of the Coulomb
interaction throughout the system with increasing distance from graphene, and
$\displaystyle\mathcal{F}_{j}({\bf q})=\int
dz\,\psi(q,z)\,\widetilde{\Delta}_{j}({\bf q},z-z_{j})$ (11)
may be considered to be a weighted form factor of the $j$th particle.
### II.2 Friction regime and conductivity of graphene
In order to use the ELM to obtain the DC conductivity of graphene, we require
an ensemble average of the energy loss rate to the lowest order in speed $v$,
which corresponds to the friction regime for slowly moving external charges.
This is easily accomplished by expanding the loss function
$\Im\left[-1/\epsilon(q,\omega)\right]$ in Eq. (7) to the leading order in
frequency by using the truncated expansion for the polarization function of
doped graphene,Allison_2010
$\displaystyle\chi(q,\omega)=\chi_{s}(q)+\frac{i\omega}{\pi\hbar
v_{F}^{2}}\,\mathcal{U}\\!\\!\left(2k_{F}-q\right)\sqrt{\left(\frac{2k_{F}}{q}\right)^{2}-1},$
(12)
where $\chi_{s}(q)=\chi(q,0)$ is the static polarization function and
$k_{F}=\sqrt{\pi\bar{n}}$ is an average value of the Fermi wavenumber for
Dirac electrons in graphene. We further define the auto-correlation function
of charged impurities in Eq. (7) by
$\displaystyle\mathcal{S}(q)\equiv\frac{1}{N}\left\langle\delta\\!\widetilde{\mathcal{N}}(-{\bf
q})\delta\\!\widetilde{\mathcal{N}}({\bf q})\right\rangle,$ (13)
and note that it only depends on the magnitude $q=\|{\bf q}\|$ when the
distribution of impurities is isotropic in the directions parallel to
graphene. This allows us to finally obtain from Eq. (7)
$\displaystyle\left\langle\frac{dW}{dt}\right\rangle=2\hbar
k_{F}Nv^{2}r_{s}^{2}\int\limits_{0}^{2k_{F}}\frac{dq\,\sqrt{1-\left(q/2k_{F}\right)^{2}}}{\left[\epsilon_{\text{bg}}(q)+4k_{F}r_{s}/q\right]^{2}}\,\mathcal{S}(q),$
(14)
where $r_{s}=e^{2}/\left(\hbar v_{F}\right)\approx 2$ with $v_{F}$ being the
Fermi speed of Dirac electrons. Note that the quantity
$F_{s}\equiv\left\langle dW/dt\right\rangle/v$ is an average total stopping,
or drag force that acts on the moving system of external charges, Radovic_2012
which may be used to, e.g., evaluate the friction coefficient $\eta$ for an
adsorbed layer on graphene from the expression $\eta=F_{s}/v$ in the limit
$v\rightarrow 0$.Allison_2010 ; Krim_2012
Within the ELM, by reversing the frames of reference one may express the
energy loss rate in graphene by the standard expression of classical
electrodynamics,
$\displaystyle\left\langle\frac{dW}{dt}\right\rangle=\int d^{2}{\bf
r}\,\left\langle{\bf J}\cdot{\bf E}\right\rangle,$ (15)
where ${\bf J}=\sigma{\bf E}$ is the current density of charge carriers in
graphene, induced by a constant electric field ${\bf E}$ applied across
graphene, and $\sigma$ is its DC conductivity. Assuming a uniform charge
carrier density $\bar{n}$ across graphene, we may write ${\bf J}=-e\bar{n}{\bf
v}$ in a steady-state regime, which gives $\left\langle
dW/dt\right\rangle=A\left(e\bar{n}v\right)^{2}/\sigma$, where $A$ is the
macroscopic area of graphene. We discard possible contribution to the
conductivity of graphene coming from charge carrier scattering on short-ranged
impurities, and we limit our considerations to sufficiently low temperatures
to be able to neglect the contribution from scattering on phonons. Thus, the
final expression for the DC conductivity takes a form that is familiar from
the SBT for doped graphene,Sarma_2011 ; Li_2011
$\displaystyle\sigma=\frac{e^{2}}{h}\frac{\frac{\bar{n}}{n_{\mathrm{imp}}}}{2\int\limits_{0}^{1}du\,\frac{u^{2}\sqrt{1-u^{2}}}{\left[2+\frac{u}{r_{s}}\epsilon_{\text{bg}}(2k_{F}u)\right]^{2}}\mathcal{S}(2k_{F}u)},$
(16)
where $n_{\mathrm{imp}}=N/A$ is the mean areal number density of external
charged particles.
### II.3 Variance of the potential in graphene and minimum conductivity
Equation (16) implies that the conductivity obtained within the SBT theory as
a function of the average equilibrium charge carrier density in graphene,
$\sigma(\bar{n})$, should vanish in a linear manner close to the neutrality
point, i.e., when $\bar{n}\rightarrow 0$, as long as $\epsilon_{\text{bg}}(0)$
and $\mathcal{S}(0)$ remain finite. However, experiments show that the
conductivity reaches a minimum value $\sigma_{\mathrm{min}}$ at the neutrality
point due to electron-hole puddles in the charge carrier density across
graphene, which are caused by fluctuations of the electrostatic potential in
the plane of graphene due to spatial inhomogeneity of the external charged
impurities. Chen_2008 ; Tan_2007 An estimate of $\sigma_{\mathrm{min}}$ may
be found according to the SCT theory as $\sigma_{\mathrm{min}}=\sigma(n^{*})$,
where $n^{*}$ is referred to as a residual charge carrier density that gives a
measure of the width of the plateau near the neutrality point where the
conductivity minimum is reached.PNAS_2007 It was shown that $n^{*}$ may be
found as a solution of an equation involving the square of graphene’s Fermi
energy, $\varepsilon_{F}=\hbar v_{F}k_{F}$, and the variance of the
fluctuating electrostatic potential in graphene, $\delta\\!\phi_{g}({\bf
r})\equiv\left.\delta\\!\Phi({\bf r},z)\right|_{z=z_{g}}$, that arises from a
distribution of immobile external charges,
$\displaystyle(\hbar v_{F})^{2}\pi\bar{n}=C_{0}(\bar{n}),$ (17)
where $C_{0}\equiv e^{2}\left\langle\delta\\!\phi_{g}^{2}({\bf
r})\right\rangle$. We note that the SCT theory extends the applicability of
the SBT result for the conductivity of graphene $\sigma(\bar{n})$ down to
lower charge carrier densities with typically $n^{*}\lesssim 10^{11}$
cm-2.PNAS_2007
Working in the time-independent regime, we use the 2D spatial FT to express
the fluctuating potential in graphene in terms of the 2D FT of the fluctuating
charge density $\delta\\!\widetilde{\rho}({\bf
q},z)\equiv\delta\\!\widetilde{\rho}_{0}({\bf q},z)$ as
$\displaystyle\delta\\!\widetilde{\phi}_{g}({\bf q})$ $\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}\frac{\widetilde{G}^{(0)}(q;z_{g},z)}{1+e^{2}\chi_{s}(q)\widetilde{G}^{(0)}(q;z_{g},z_{g})}\,\delta\\!\widetilde{\rho}_{0}({\bf
q},z)\,dz$ (18) $\displaystyle=$ $\displaystyle\frac{2\pi
e}{q}\frac{\delta\\!\widetilde{\mathcal{N}}({\bf q})}{\epsilon_{s}(q)},$ (19)
where $\epsilon_{s}(q)=\epsilon_{\text{bg}}(q)+V_{C}(q)\,\chi_{s}(q)$ is the
total dielectric function of the entire system in the static limit. By
invoking the translational invariance of the distribution of external charges
in the directions of ${\bf r}$, we may use a general relation,
$\displaystyle\left\langle\delta\\!\widetilde{\mathcal{N}}({\bf
q}^{\prime})\delta\\!\widetilde{\mathcal{N}}({\bf
q})\right\rangle=n_{\mathrm{imp}}\,\delta\\!({\bf q}^{\prime}+{\bf
q})\,\mathcal{S}({\bf q}),$ (20)
that allows us to write
$\displaystyle C_{0}=n_{\mathrm{imp}}\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}\,\left[\frac{V_{C}(q)}{\epsilon_{s}(q)}\right]^{2}\mathcal{S}(q).$
(21)
### II.4 Statistical description of external charges
It is important to make distinction between the geometric structure of the
external particle system and the statistical distribution of the charge
density functions $\Delta_{j}({\bf R})$ for individual particles. Assuming
that those two characteristics of the system are statistically independent,
the geometric structure may be modeled by using the one- and two–particle
distribution functions for their positions
$\displaystyle F_{1}({\bf r},z)=\frac{N}{A}f_{1}(z),$ (22)
and
$\displaystyle F_{2}({\bf r}_{1},{\bf r}_{2};z_{1},z_{2})$ $\displaystyle=$
$\displaystyle\frac{N(N-1)}{A^{2}}f_{1}(z_{1})f_{1}(z_{2})$ (23)
$\displaystyle\times g({\bf r}_{2}-{\bf r}_{1};z_{1},z_{2}),$
where $f_{1}(z)$ describes the distribution of particle positions along the
$z$ axis and is normalized to one, whereas $g({\bf r};z_{1},z_{2})$ is the
usual pair correlation function. A significant further simplification may be
achieved by assuming that the charge densities of individual particles are
identically distributed, so that $\Delta_{j}({\bf R})=\Delta({\bf R})$ for all
$j=1,2,\ldots,N$. Still, Eqs. (9) and (11) show that the corresponding
individual particle form factors generally remain entangled with the $z$
dependence of the geometric arrangement of particle positions, unless all the
particles reside in the same plane, say $z=z_{0}$.
Accordingly, we first consider a 2D geometric model with
$f_{1}(z)=\delta(z-z_{0})$, which is commonly used in all theoretical
modelings of the effects of correlated charged impurities on the conductivity
of graphene.Sarma_2011 ; PNAS_2007 ; Yan_2011 ; Li_2011 In that case, we find
that the auto-correlation function from Eq. (13) may be written as
$\displaystyle\mathcal{S}({\bf q})=\left\langle\left|\mathcal{F}_{0}({\bf
q})\right|^{2}\right\rangle-\left|\left\langle\mathcal{F}_{0}({\bf
q})\right\rangle\right|^{2}+\left|\left\langle\mathcal{F}_{0}({\bf
q})\right\rangle\right|^{2}S_{2D}(q),$ (24)
where each particle is characterized by an ”atomic” form factor
$\displaystyle\mathcal{F}_{0}({\bf q})=\int
dz\,\psi(q,z)\,\widetilde{\Delta}({\bf q},z-z_{0}),$ (25)
and
$\displaystyle S_{2D}({\bf q})=1+n_{\mathrm{imp}}\int d^{2}{\bf
r}\,\mathrm{e}^{i{\bf q}\cdot{\bf r}}\left[g_{2D}({\bf r})-1\right]$ (26)
is a ”geometric” structure factor that describes the arrangement of external
particles in the plane $z=z_{0}$. As regards the corresponding pair
correlation (or radial distribution) function $g_{2D}({\bf r})=g_{2D}(r)$, in
addition to uncorrelated particles with $g_{2D}(r)=1$, we consider two models
that contain a single parameter $r_{c}$ characterizing the inter-particle
correlation distance: a step-correlation (SC) model with
$g_{2D}(r)=\mathcal{U}(r-r_{c})$, where $\mathcal{U}$ is a Heaviside unit step
function, which was often used in the previous studies of charged impurities
in graphene,Yan_2011 ; Li_2011 and the HD model, in which particles interact
with each other as hard disks of the diameter $r_{c}$. Rosenfeld_1990
There are several advantages to using the HD model over the SC model. First,
the former model is based on a Hamiltonian equation for the thermodynamic
state of a 2D fluid with a well-defined pair potential between impurities,
whereas the latter model is an _ad hoc_ description of the impurity
distribution, made-up for simple, analytic results. That is not to say that
the SC model is poor at capturing the interesting results in the conductivity
of graphene with correlated impurities.Yan_2011 ; Li_2011 However, from Eq.
(16) it is obvious that, with $k_{F}=\sqrt{\pi\bar{n}}$, the initial slope of
$\sigma(\bar{n})$ is strongly influenced by the limiting value of the
structure factor $\mathcal{S}(q)$ as $q\rightarrow 0$, that is, by the value
of $S_{2D}(0)$ via Eq. (24). It is well known that $S_{2D}(0)$ is related to
the isothermal compressibility of a 2D fluid,Hansen_1986 which may be
expressed as a function of the packing fraction defined by $p=\pi
n_{\mathrm{imp}}r_{c}^{2}/4$. Thus, $p$ is a key measure of performance of the
two models. It was recently shown by Li _et al._Li_2011 that the SC model
gives reliable results for packing fractions $p\ll 1$ by comparing the
analytical result for the 2D structure factor in that model,
$S_{\mathrm{SC}}(q)$, with a numerically calculated structure factor of a
hexagonal lattice of impurities. However, the analytical limit
$S_{\mathrm{SC}}(0)=1-4p$ shows that the SC model already breaks down for
$p\geq 0.25$ because the corresponding compressibility becomes negative at
higher packing fractions. On the other hand, it was recently shown that the
interaction potential between two point ions near doped graphene is heavily
screened and, moreover, exhibits Friedel oscillations with inter-particle
distance, giving rise to a strongly repulsive core region of distances on the
order of $k_{F}^{-1}$ that resembles the interaction among hard disks with
diameter $r_{c}\sim k_{F}^{-1}$.Radovic_2012 Therefore, we may estimate that
the packing factor could reach values on the order $p\sim
n_{\mathrm{imp}}/\bar{n}$ that may not always be negligibly small,
necessitating the use of a model that goes well beyond the SC model, at least
for systems of adsorbed alkali-atom submonolayers on graphene.Yan_2011 In
that respect, we note that various parameterizations of the HD model extend
its applicability to include phase transitions in a 2D fluid as a function of
the packing fraction, Mak_2006 even going up about $p=0.9$, corresponding to
a crystalline closest packing where hard disks form a hexagonal structure in
2D.Guoa_2006 In this work, we use a simple analytical parametrization for the
2D structure factor in the HD model, $S_{\mathrm{HD}}(q)$, provided by
RosenfeldRosenfeld_1990 (see Appendix B) which works reasonably well for
packing fractions up to about $p=0.69$, just near the freezing point of a 2D
fluid.
Regarding the structure of individual charged particles within the 2D
geometric model, we study a few specific examples. First we consider a point
particle of charge $Ze$ that carries a dipole moment $\upmu$ with the density
function
$\displaystyle\Delta_{\mathrm{p}}({\bf R})=\left(Z-{\bf
D}\\!\cdot\\!\nabla_{{\bf R}}\right)\,\delta\\!\left({\bf R}\right),$ (27)
where ${\bf D}=\mbox{\boldmath{$\upmu$}}/e$ is an effective dipole length and
$\delta\\!({\bf R})=\delta\\!\left({\bf r}\right)\delta\\!\left(z\right)$ is a
3D delta function, which gives a form factor from Eq. (25) as
$\displaystyle\mathcal{F}_{\mathrm{p}}({\bf q})=\left(Z+i\,{\bf
q}\\!\cdot\\!{\bf
D}_{\parallel}\right)\psi(q,z_{0})+D_{\perp}\left.\frac{\partial\psi(q,z)}{\partial
z}\right|_{z=z_{0}},$ (28)
where ${\bf D}_{\parallel}=\mbox{\boldmath{$\upmu$}}_{\parallel}/e$ and
$D_{\perp}=\mu_{\perp}/e$ are the effective dipole lengths in the directions
parallel and perpendicular to graphene, respectively. We note that, having in
mind that the first two terms in the right-hand side of Eq. (24) represent the
variance of the form factor $\mathcal{F}_{0}({\bf q})$, all of the three
parameters of the point particle model, namely, $Z$, ${\bf D}_{\parallel}$ and
$D_{\perp}$ may exhibit fluctuations about their respective means (with the
mean $\langle{\bf D}_{\parallel}\rangle=0$ due to the presumed isotropy), as
well as mutual cross-correlations. In addition, assuming $n_{\mathrm{imp}}$ to
be small enough, the perpendicular dipole moment component may depend on the
local electrostatic field $E_{\perp}$ according to $\mu_{\perp}=\alpha
E_{\perp}$, where $\alpha$ is an effective dipole polarizability near
graphene.
We also consider a cluster of uniformly distributed charge $Ze$ within a disk
of radius $R_{\mathrm{cl}}$ parallel to graphene with
$\displaystyle\Delta_{\mathrm{cl}}({\bf R})=\frac{Z}{\pi
R_{\mathrm{cl}}^{2}}\,\mathcal{U}\\!\left(R_{\mathrm{cl}}-r\right)\,\delta\\!\left(z\right),$
(29)
giving
$\displaystyle\mathcal{F}_{\mathrm{cl}}({\bf
q})=\frac{2Z}{qR_{\mathrm{cl}}}\,J_{1}\\!\left(qR_{\mathrm{cl}}\right)\,\psi(q,z_{0}),$
(30)
where $J_{1}$ is a Bessel function of order one. We limit our considerations
to cases with $k_{F}R_{\mathrm{cl}}\ll 1$, validating the perturbative
treatment of charge carrier scattering on such clusters,Katsnelson_2010 and
we also assume $\pi n_{\mathrm{imp}}R_{\mathrm{cl}}^{2}\ll 1$ to avoid the
interference in scattering patterns from neighboring clusters.
On the other hand, it is of interest to explore the effects a fully
$z$-dependent geometric structure of particle positions in 3D, with arbitrary
distribution function $f_{1}(z)$ and the pair correlation function that
depends on the $z$ coordinates, $g_{3D}({\bf r}_{2}-{\bf r}_{1};z_{1},z_{2})$.
In this case, we only consider point charges with $Z=1$ and obtain the auto-
correlation function from Eq. (13) as
$\displaystyle\mathcal{S}({\bf q})$ $\displaystyle=$ $\displaystyle\int
dz\,f_{1}(z)\,\psi^{2}(q,z)+\int dz\,f_{1}(z)\,\psi(q,z)$ (31)
$\displaystyle\times\int
dz^{\prime}\,f_{1}(z^{\prime})\,\psi(q,z^{\prime})\left[S_{3D}({\bf
q};z,z^{\prime})-1\right],$
where partial structure factor in the 3D case is defined by
$\displaystyle S_{3D}({\bf q};z,z^{\prime})=1+n_{\mathrm{imp}}\int d^{2}{\bf
r}\,\mathrm{e}^{i{\bf q}\cdot{\bf r}}\left[g_{3D}({\bf
r};z,z^{\prime})-1\right].$ (32)
Any realistic modeling of the 3D pair correlation function in the presence of
charged graphene is beyond the scope of the present study, so we only consider
uncorrelated point charges with $g_{3D}({\bf r};z,z^{\prime})=1$, and focus on
the effect of their distribution over the depth $z$. In a first study of this
type, we only consider the case $f_{1}(z)=1/L$ for a homogeneous distribution
of point charges throughout a dielectric slab of finite thickness $L$. In
Appendix B we also provide a result for semi-infinite region
($L\rightarrow\infty$) based on a pair correlation function $g_{3D}(R)$ for a
bulk one-component plasma (OCP) with the volume density of charged particles
$N_{\mathrm{imp}}=N/\left(AL\right)$, which may be of interest in future work.
## III Results and discussion
In this section, we study several special configurations of graphene with the
surrounding dielectrics by using the electrostatic GF, which is derived in
Appendix A for a three-layer structure of Fig. 1, defined on the intervals
$I_{1}=[-L,0]$, $I_{2}=[0,H]$ and $I_{3}=[H,\infty)$ along the $z$ axis that
are characterized by the relative bulk dielectric constants $\epsilon_{j}$
with $j=1,2,3$, respectively.
In Fig. 2 we consider a two-layer structure that consists of a semi-infinite
SiO2 substrate ($L\rightarrow\infty$ with $\epsilon_{1}=3.9$) and a semi-
infinite layer of air ($H\rightarrow\infty$ with $\epsilon_{2}=1$, or $H=0$
with $\epsilon_{3}=1$) with graphene placed right on their boundary at
$z_{g}=0$. We show the dependence of graphene’s conductivity $\sigma$ on its
average charge carrier density $\bar{n}$ for a planar distribution of charged
impurities with fixed $Z=1$ and no dipole moment, having the areal number
density $n_{\mathrm{imp}}=10^{12}$ cm-2, which are all placed a distance $d$
away from graphene. We show the results for several values of the correlation
length $r_{c}$ among the impurities, which are obtained by using the SC and
the HD models for their 2D structure factor, and note that the SC model only
yields physical results for $r_{c}<5.6$ nm for the given value of
$n_{\mathrm{imp}}$. In addition to the case of point-like impurities being
placed directly on graphene ($d=0$ and $R_{\mathrm{cl}}=0$), we also show in
Fig. 2 the effects of point-like impurities embedded at $d=0.3$ nm inside the
SiO2 substrate, as well as disk-like impurities with fixed radius
$R_{\mathrm{cl}}=2$ nm placed on graphene ($d=0$).
Figure 2: (Color online) The dependence of conductivity (in units of
$e^{2}/h$) on the average charge carrier density $\bar{n}$ (in units of
$10^{13}$ cm-2) for a two-layer structure that consists of a semi-infinite
SiO2 substrate ($L\rightarrow\infty$, $\epsilon_{1}=3.9$) and a semi-infinite
layer of air ($H\rightarrow\infty$, $\epsilon_{2}=1$, or $H=0$,
$\epsilon_{3}=1$), with zero gap between them and graphene placed on their
boundary ($z_{g}=0$). A planar distribution of charged impurities with $Z=1$
and no dipole moment, having the areal number density
$n_{\mathrm{imp}}=10^{12}$ cm-2 and the correlation distance $r_{c}$ between
them, is placed a distance $d$ away from graphene. Results are shown for
uncorrelated impurities [thin (red) solid lines], for the SC model with
$r_{c}=$ 4 and 5 nm [thick solid and dashed gray (light blue) lines,
respectively], and for the HD model with $r_{c}=$ 4, 5, 6, and 7 nm [thick
black solid, dashed, dotted, and dash-dotted lines, respectively]. Panels (a)
and (b) show the cases of point-like impurities on graphene ($d=0$) and at
$d=0.3$ nm in the SiO2 substrate, respectively, whereas panel (c) shows disk-
like impurities with the cluster radius $R_{\mathrm{cl}}=2$ nm placed on
graphene ($d=0$). The insets show the blow-ups of the regions with
$\bar{n}\leq 10^{12}$ cm-2.
As regards the effects of finite $d$ and $R_{\mathrm{cl}}$, one notices in
Fig. 2 that they both contribute to an increase in the slope of conductivity
at higher $\bar{n}$ values, as expected, where they even give rise to a super-
linear dependence of conductivity on $\bar{n}$ for smaller values of the
correlation length $r_{c}$. (Note that the case of uncorrelated disks with
$r_{c}=0$ is somewhat unphysical as the disks are allowed to overlap.)
However, the effects of finite $d$ and $R_{\mathrm{cl}}$ are relatively weak
and only affect quantitative details of conductivity at higher $\bar{n}$,
whereas comparison among the insets in Fig. 2 shows that their effects are
barely noticeable at $\bar{n}\lesssim 10^{12}$ cm-2.
The most prominent effect in Fig. 2 is a strong increase of the initial slope
of conductivity as a function of $\bar{n}$ (and hence an increase in mobility
of graphene, $\mu=\sigma/\left(e\bar{n}\right)$) at low values of $\bar{n}$ as
the correlation length $r_{c}$ increases. One notices from the insets in Fig.
2 that the initial slopes from the SC model are higher than those from the HD
model for the same value of $r_{c}$ because
$S_{\mathrm{SC}}(0)<S_{\mathrm{HD}}(0)$, but the latter model permits the use
of much larger values of $r_{c}$ than the former model, hence giving rise to
rather large initial slopes of the conductivity at the largest packing
fractions shown. (Notice that the case with a maximum packing fraction of
$p\approx 0.38$ that is shown in Fig. 2 is still well within the interval of
confidence for the HD model used here.Rosenfeld_1990 ) As the charge carrier
density $\bar{n}$ increases, the conductivity shows a sub-linear dependence on
$\bar{n}$ that becomes more pronounced as the correlation length $r_{c}$
increases. In the case of $d=0$ and $R_{\mathrm{cl}}=0$ the sub-linear
dependence occurs for all $r_{c}>0$, whereas in the cases of finite $d$ or
$R_{\mathrm{cl}}$ values the sub-linear dependence may even overcome the
opposite effect of super-linear dependence for sufficiently large $r_{c}$s.
For the largest $r_{c}$ value shown in Fig. 2, the sub-linear behavior even
gives rise to a pronounced saturation effect in the conductivity of graphene
with increasing $\bar{n}$, which is sometimes observed in experiments.Tan_2007
; Yan_2011 Thus, high packing fractions that result from long correlation
distances among the charged impurities can give rise to both higher initial
slope of conductivity at lower $\bar{n}$ _and_ a more pronounced sub-linear
dependence of conductivity at higher $\bar{n}$ with the HD model than those
that can be achieved with the SC model. We pause to discuss those two effects
in some detail.
Various models that attempt to reproduce the experimental dependence of
graphene’s conductivity $\sigma$ on its charge carrier density $\bar{n}$ use
the areal density of charged impurities $n_{\mathrm{imp}}$ as free parameter
to fit the slope of conductivity in the range of $\bar{n}$ values where that
dependence is found to be predominantly linear. Ignoring the relatively narrow
region of $\bar{n}$ values around zero where the conductivity of a nominally
neutral graphene reaches a minimum, one sees that Eq. (16) implies a linear
dependence of conductivity in the form
$\sigma=c\,\bar{n}/\left[n_{\mathrm{imp}}S_{2D}(0)\right]$ when
$\bar{n}\rightarrow 0$, where $c$ is constant when the dielectric media are
semi-infinite. For a system of uncorrelated impurities that may be described
as a 2D gas, one simply finds $\sigma=c\,\bar{n}/n_{\mathrm{imp}}$ because
$S_{2D}(0)=1$. However, when impurities are strongly correlated, one should
consider their number $N$ to be a random variable because different samples of
graphene flakes with fixed area $A$ may cover different regions of a much
larger area of the substrate plagued by varying concentrations of impurities.
Then, the impurity density should be defined in terms of the average number of
impurities covered by the graphene flake, $n_{\mathrm{imp}}=\left\langle
N\right\rangle/A$. On the other hand, the long wavelength limit of the
structure factor may be expressed as the ratio
$S_{2D}(0)=\left\langle\delta\\!N^{2}\right\rangle/\left\langle
N\right\rangle$, where the numerator is the variance in $N$,Hansen_1986 with
$\delta\\!N=N-\left\langle N\right\rangle$ being the fluctuation in the number
of impurities that are covered by the graphene flake. Therefore, from the
statistical point of view, the $\bar{n}\rightarrow 0$ limit of the SBT
conductivity should be reinterpreted as
$\sigma=c\,\bar{n}/n_{\mathrm{imp}}^{*}$, where we define
$n_{\mathrm{imp}}^{*}=\left\langle\delta\\!N^{2}\right\rangle/A$ to be an
effective density of impurities rather than the average density. In general,
$n_{\mathrm{imp}}^{*}\neq n_{\mathrm{imp}}$ unless $N$ is Poisson distributed,
i.e., the impurities behave as an ideal 2D gas. Clearly, the distinction
between $n_{\mathrm{imp}}^{*}$ and $n_{\mathrm{imp}}$ should be borne in mind
when attempting to use $n_{\mathrm{imp}}$ as a fitting parameter in modeling
the slope of graphene’s conductivity in the presence of a liquid-like
distribution of charged impurities.
On the other hand, the sub-linear dependence of graphene’s conductivity on
$\bar{n}$ at large doping densities is often modeled by combining the
scattering processes of its charge carriers on both charged impurities and
short-ranged impurities via the Matthiessen’s rule.Yan_2011 However, the
density of atom-size defects in graphene that could give rise to short-range
scattering is extremely low due to the structural and compositional resilience
of graphene’s atomic lattice, so that ”the source of the proposed weak short-
range scattering is mysterious.”Yan_2011 Another contender for the
explanation of the sub-linear conductivity is the resonant scattering model
that invokes the existence of bound-state resonances in the $\pi$ electron
bands due to chemisorbed molecules on graphene.Ferreira_2011 However, the
fact that graphene is chemically inert also makes this mechanism unlikely in
most situations. On the other hand, it was recently shown that the charge
carrier scattering on charged impurities in a substrate may also give rise to
the sub-linear behavior of conductivity in highly doped graphene in the
presence of a strong spatial correlation among the impurities.Yan_2011 ;
Li_2011 Noting that the sub-linear behavior was demonstrated in simulations
based on the SC model with small packing fractions,Yan_2011 ; Li_2011 we
follow the same idea and suggest that, by being able to go to much larger
packing fractions in the HD model than in the SC model, one may include large
enough values of $r_{c}$ in simulations that could even give rise to
saturation of graphene’s conductivity at high enough charge carrier densities,
thus eliminating the need to invoke the existence of resonance scatterers or
atom-size defects in graphene. Namely, one may verify that, with increasing
packing fraction the structure factor $S_{\mathrm{HD}}(q)$ develops a very
pronounced peak at the wavenumber $q=q_{\mathrm{shell}}$ corresponding to the
first coordination shell due to the nearest neighbors.Rosenfeld_1990 ;
Guoa_2006 So, from Eq. (16) it follows that a relatively sudden increase in
the value of the integral over $u$ may be expected with the HD model when
$k_{F}$ surpasses the value $q_{\mathrm{shell}}/2\sim\pi/r_{c}$, causing a
slowdown in the increase of $\sigma$ when $\bar{n}\sim\pi/r_{c}^{2}$ that is
reminiscent of the saturation in conductivity. For example, in the case of the
largest correlation distance shown in Fig. 2, $r_{c}=7$ nm, one finds that a
strong saturation of the conductivity indeed occurs at about
$n_{\mathrm{imp}}=\pi/r_{c}^{2}\approx 6.4\times 10^{12}$ cm-2.
Figure 3: (Color online) The dependence of conductivity (in units of
$e^{2}/h$) on the average charge carrier density $\bar{n}$ (in units of
$10^{12}$ cm-2) for a two-layer structure that consists of a semi-infinite
SiO2 substrate ($L\rightarrow\infty$, $\epsilon_{1}=3.9$) and a semi-infinite
layer of air ($H\rightarrow\infty$, $\epsilon_{2}=1$, or $H=0$,
$\epsilon_{3}=1$), with zero gap between them and graphene placed on their
boundary ($z_{g}=0$). A planar distribution of unit ($Z=1$) point-like charged
impurities, having the areal number density $n_{\mathrm{imp}}$ and the
correlation distance $r_{c}$ between them, is placed on graphene and is
allowed to have a non-zero perpendicular dipole moment with polarizability
$\alpha$ per impurity. The results from the HD model (black solid lines) are
fitted to the experimental data from Ref. Tan_2007 (symbols), with the best
fit in panel (a) obtained for $n_{\mathrm{imp}}=3\times 10^{11}$ cm-2 with
$r_{c}=6.8$ nm (packing fraction $p=0.11$) and $\alpha=0$, and the best fit in
panel (b) obtained for $n_{\mathrm{imp}}=7.4\times 10^{11}$ cm-2 with
$r_{c}=6.3$ nm ($p=0.23$) and $\alpha=1150$ Å3. Also shown are the results for
uncorrelated impurities ($r_{c}=0$) with $\alpha=0$ on both panels [dashed
gray (red) lines], as well as for the uncorrelated impurities ($r_{c}=0$) with
$\alpha=1150$ Å3 in panel (b) [dash-dotted grey (light blue) line].
In Fig. 3 we consider the same configuration of single-layer graphene atop a
semi-infinite SiO2 substrate with a semi-infinite layer of air above it as in
Fig. 2, and attempt to model the experimental data for conductivity versus
charge carrier density $\bar{n}$ from Ref. Tan_2007 by using the HD model for
a 2D distribution of point charges with $Z=1$. We select two graphene samples
from Ref. Tan_2007 labeled K17 and K12, which both exhibit sub-linear
behavior with increasing $\bar{n}$, with K17 being symmetric and K12 showing
an electron-hole asymmetry (i.e., asymmetry with respect to the sign of
$\bar{n}$). The physical mechanism(s) that occasionally give rise to this kind
of asymmetry in graphene are still unclear, so we explore here the possibility
that the presence of the perpendicular component of dipole moment in each
impurity, $D_{\perp}$, may give rise to a sizeable asymmetry, as that seen in
Fig. 3 for the sample K12. We assume $D_{\perp}=\alpha E_{\perp}/e$, where
$\alpha$ is the effective polarizability and $E_{\perp}$ is the total
perpendicular electric field near graphene. Assuming $n_{\mathrm{imp}}$ to be
small enough, we may neglect mutual depolarization among the impurities and
simply write $E_{\perp}=4\pi e\bar{n}/\epsilon_{1}$, with $E_{\perp}$ being
positive (negative) for electron (hole) doping of graphene.Maschhoff_1994 The
two samples were fitted in Ref. Tan_2007 by assuming that the impurities
reside in graphene ($d=0$) and are uncorrelated, and the optimal linear
symmetric fits were found with $n_{\mathrm{imp}}=2.2\times 10^{11}$ cm-2 for
K17 and with $n_{\mathrm{imp}}=4\times 10^{11}$ cm-2 for K12. We also assume
the impurities to lie in graphene ($d=0$), and we use $n_{\mathrm{imp}}$,
$r_{c}$ and $\alpha$ as fitting parameters. In the case of the symmetric K17,
the best fit is found for $n_{\mathrm{imp}}=3\times 10^{11}$ cm-2 with
$r_{c}=6.8$ nm ($p=0.11$) and $\alpha=0$, whereas for the asymmetric case of
K12 the best fit is found for $n_{\mathrm{imp}}=7.4\times 10^{11}$ cm-2 with
$r_{c}=6.3$ nm ($p=0.23$) and $\alpha=1150$ Å3. Both fits obtained with the HD
model in Fig. 3 are quite satisfactory as far as the sub-linear behavior of
conductivity is concerned, and the relatively large values of packing
fractions used in both cases suggest the necessity of using the HD rather than
the SC model. On the other hand, a good fit in the asymmetric case can only be
achieved with a rather large value of $\alpha$, which indicates that the
dipole mechanism may not be the primary cause of the electron-hole asymmetry
in conductivity, at least for the experimental setting of Ref. Tan_2007
However, we note that the effective polarizability $\alpha$ of a single
impurity may be significantly increased by the presence of a nearby conducting
surface.Maschhoff_1994
In Fig. 4 we consider a structure that consists of a dielectric material of
finite thickness $L$ (we choose HfO2 with $\epsilon_{1}=22$) and a semi-
infinite layer of SiO2 (either $H\rightarrow\infty$ with $\epsilon_{2}=3.9$,
or $H=0$ with $\epsilon_{3}=3.9$) with graphene placed right on their boundary
at $z_{g}=0$. This configuration may represent the physical situation where
single-layer graphene sits on a thick SiO2 substrate (with typically $H\sim
300$ nm) and is top-gated through a thin layer of HfO2 (with $L\lesssim 10$
nm). We show the dependence of the conductivity $\sigma$ on charge carrier
density $\bar{n}$ for several model distributions of point charge impurities
in the HfO2 layer with fixed $Z=1$ and no dipole moment, having the areal
number density $n_{\mathrm{imp}}=10^{12}$ cm-2. We consider a homogeneous 3D
distribution of uncorrelated charges throughout the HfO2, which extends up to
a distance $d$ from graphene, as well as a 2D planar distribution placed in
HfO2 a distance $d$ away from graphene, with both uncorrelated ($r_{c}=0$) and
correlated ($r_{c}=6$ nm, $p\approx 0.28$) charges that are described with the
HD model.
One notices in Fig. 4 that finite thickness $L$ exhibits strong effects on
conductivity, both in quantitative and qualitative aspects, which are
dependent on the underlying structure of charged impurities. First noted is
that the overall conductivity is generally increased compared to that seen in
Figs. 1 and 2, which is expected due to the more efficient screening of
charged impurities by a high-$\kappa$ material such as HfO2. Moreover, the
conductivity is seen to increase with decreasing $L$ for all $\bar{n}$ in the
2D cases and only for lower $\bar{n}$ in the 3D case, which may be explained
by the more efficient screening of impurities due to the proximity of a metal
gate. Furthermore, the conductivity is larger in the 3D case than in the
corresponding uncorrelated 2D case because the same number of impurities is
spread over larger distances from graphene so that the resulting scattering
potential in graphene is weaker. As regards the distance $d$, one notices
similar trends as in Fig. 2, namely, a finite $d$ increases both the value of
conductivity and its slope (i.e., mobility) in both 3D and 2D models. However,
as regards the effects of finite correlation length $r_{c}$ in the 2D models
with finite $L$, one sees little evidence to the increase in the initial slope
of conductivity at lower $\bar{n}$, in contrast to the trends seen in Fig. 2,
whereas saturation of conductivity at higher $\bar{n}$ seems to get stronger
than in Fig. 2 as $L$ decreases. In fact, for the shortest thickness of $L=1$
nm for both $d=0$ and $d=0.3$ nm, this saturation turns into a broad maximum
of conductivity around $\bar{n}=10^{11}$ cm-2, followed by a still broader
minimum at higher $\bar{n}$ values.
Figure 4: The dependence of conductivity (in units of $e^{2}/h$) on the
average charge carrier density $\bar{n}$ (in units of $10^{13}$ cm-2) for a
two-layer structure that consists of a HfO2 ($\epsilon_{1}=22$) with finite
thickness $L$ and a semi-infinite layer of SiO2 ($H\rightarrow\infty$,
$\epsilon_{2}=3.9$, or $H=0$, $\epsilon_{3}=3.9$) with zero gap between them
and graphene placed on their boundary ($z_{g}=0$). The structure of the system
of unit ($Z=1$) point-like charged impurities with no dipole moment, having
the areal number density $n_{\mathrm{imp}}=10^{12}$ cm-2, is assumed to be
either (a,b) a 3D homogeneous distribution throughout the HfO2 layer extending
up to a distance $d$ from graphene, or a planar 2D distribution placed in the
HfO2 layer a distance $d$ away from graphene, with the correlation distance
being (c,d) $r_{c}=0$ or (e,f) $r_{c}$= 6 nm (giving the packing fraction
$p\approx 0.28$ within the HD model). In panels (a,c,e) we set $d=0$, while in
panels (b,d,f) we set $d=0.3$ nm. The thickness of the HfO2 layer takes values
$L$ = 1 nm (solid lines), 2 nm (dashed lines), 5 nm (dotted lines), and 10 nm
(dash-dotted lines). The insets show the blow-ups of the regions with
$\bar{n}\leq 5\times 10^{11}$ cm-2.
One remarkable feature seen in Fig. 4 is that the conductivity generally does
not vanish in the SBT limit when $\bar{n}\rightarrow 0$ for finite $L$, but
rather reaches a minimum value $\sigma(0)$. This minimum may be easily
estimated for $d=0$ by using the limiting form of the background dielectric
constant $\epsilon_{\text{bg}}(q)=\epsilon_{1}/\left(2qL\right)$ when $qL\ll
1$ in Eq. (16), which then gives
$\displaystyle\sigma(0)=\left(\frac{\epsilon_{1}}{\pi
r_{s}L}\right)^{2}\frac{e^{2}/(2h)}{n_{\mathrm{imp}}\mathcal{S}(0)}=\frac{4v_{F}}{\pi
r_{s}}\frac{C_{L}^{2}}{n_{\mathrm{imp}}\mathcal{S}(0)},$ (33)
where $\mathcal{S}(0)=1/3$ for the 3D case, $\mathcal{S}(0)=1$ for the
uncorrelated 2D case, and
$\mathcal{S}(0)=S_{\mathrm{HD}}(0)=(1-p)^{3}/(1+p)\approx 0.29$ for the
correlated 2D case in the HD model. In the second expression for $\sigma(0)$
in Eq. (33) we emphasize that the minimum conductivity in the SBT limit for
neutral graphene is governed by the geometric capacitance per unit area,
$C_{L}=\epsilon_{1}/(4\pi L)$, of the dielectric with finite thickness $L$
used in top-gating the graphene.
Finally, one notices in Fig. 4 that, as the thickness $L$ increases in the 3D
case, the conductivity gains quite strong super-linear dependence with
increasing $\bar{n}$. This dependence may be estimated by considering Eq. (16)
in the limit of large but finite $L$, such that $qL\gg 1$. In that case, the
background dielectric constant becomes
$\epsilon_{\text{bg}}\approx\left(\epsilon_{1}+\epsilon_{2}\right)/2$, whereas
the 3D structure factor, which is determined by the first term in Eq. (31),
goes as $\mathcal{S}(q)\approx 1/\left(2qL\right)$, so that Eq. (16) gives
$\sigma\propto\bar{n}^{3/2}/N_{\mathrm{imp}}$, where $N_{\mathrm{imp}}=N/(AL)$
is the volume density of charge impurities. We note that this behavior of
conductivity in graphene at large $\bar{n}$ is a consequence of the 3D nature
of a distribution of uncorrelated charges that gives rise to the special form
of structure factor, $\mathcal{S}(q)\approx 1/\left(2qL\right)$. The lack of
experimental observations of such super-linear dependence of conductivity in
graphene should not be taken as evidence to rule out the role of 3D
distributions of impurities, because both the correlation among impurities, as
that described in the Appendix B for a OCP, as well as their clustering close
to graphene seem to be capable of eliminating the super-linear dependence.
Figure 5: (Color online) The dependence of the mobility
$\mu=\sigma/\left(e\bar{n}\right)$, (in units of cm2V-1s-1) on the average
charge carrier density $\bar{n}$ (in units of $10^{13}$ cm-2) for a three-
layer structure that consists of a HfO2 ($\epsilon_{1}=22$) with thickness
$L$, a layer of air ($\epsilon_{2}=1$) with thickness $H$, and a semi-infinite
layer of SiO2 ($\epsilon_{3}=3.9$), with graphene placed at distance $z_{g}$
above the top surface of the HfO2 layer. A planar distribution of uncorrelated
unit ($Z=1$) point-like charged impurities with no dipole moment, having the
areal density $n_{\mathrm{imp}}=10^{12}$ cm-2, is placed a distance $d$
underneath graphene. The cases of graphene with equal air gaps of
$z_{g}=H-z_{g}=$ 0.3 nm towards the two dielectrics are shown with the
impurities placed on graphene ($d=0$) [thick red (dark gray) lines] or on the
top surface of the HfO2 layer ($d=0.3$ nm) (thin black lines). The case of
graphene with zero gaps ($z_{g}=H=0$) towards the two dielectrics and the
impurities placed on graphene ($d=0$) [medium green (gray) lines] corresponds
to the conductivity $\sigma$ shown Fig. 4(c). The thickness of the HfO2 layer
takes values $L$ = 1 nm (solid lines), 2 nm (dashed lines), 5 nm (dotted
lines), 10 nm (dash-dotted lines), and $\infty$ (double-dotted lines).
In Fig. 5 we consider a three-layer structure that consists of a HfO2 layer
($\epsilon_{1}=22$) with finite thickness $L$, a layer of air
($\epsilon_{2}=1$) of thickness $H=0.6$ nm, and a semi-infinite layer of SiO2
($\epsilon_{3}=3.9$), with graphene placed in the air at $z_{g}=0.3$ nm,
midway between the two dielectrics. This configuration is similar to that in
Fig. 4 with graphene sandwiched between the HfO2 and SiO2 dielectrics, but we
introduce in Fig. 5 gaps of air of equal thickness 0.3 nm on both sides of
graphene. We investigate the effects of finite thickness $L$ on the mobility
of graphene, $\mu=\sigma/\left(e\bar{n}\right)$, as a function of charge
carrier density $\bar{n}$ for a 2D planar distribution of uncorrelated point
charges with $Z=1$ and no dipole moment, having the areal density
$n_{\mathrm{imp}}=10^{12}$ cm-2. We consider three configurations, with the
impurities placed either (A) on graphene ($d=0$) or (B) on the surface of the
HfO2 layer a distance $d=0.3$ nm away from graphene, both in the presence of
the 0.3 nm gaps, as well as the case (C) from Fig. 4(c) having zero gaps
between graphene and the HfO2 and SiO2 dielectrics with the 2D distribution of
uncorrelated charges placed on graphene ($d=0$). One may see in Fig. 5 that
the mobility generally increases with decreasing $L$ within each of the three
configurations, (A), (B) and (C), but that there are remarkable differences
between them in the magnitude of the mobility and its dependence on $\bar{n}$.
In the configurations (A) and (C) with charge impurities placed on graphene,
the mobility generally decreases with increasing $\bar{n}$, whereas in the
configuration (B) with the impurities placed on the surface of the HfO2 layer
with a finite gap relative to the graphene, the mobilities with higher $L$
values pass through a minimum at a low $\bar{n}$ value and further increase as
$\bar{n}$ increases. Moreover, the magnitudes of the mobility with equal $L$
values are seen in Fig. 5 to increase in the order of configurations
(A)$\rightarrow$(C)$\rightarrow$(B), which is also the order of increasing
spread of the curves with different $L$ values within each configuration.
Finally, it is interesting to notice that differences between the magnitudes
of the mobility in the three different configurations with
$L\rightarrow\infty$ become diminished as $\bar{n}$ decreases.
One may conclude from Fig. 5 that the existence of a finite gap between
graphene and the nearby dielectric, as well as the precise location of
impurities within that gap (with the extreme positions being on graphene and
on the surface of the dielectric) both have decisive influences on the
mobility. Noting that the configuration (A) with impurities on graphene in the
presence of finite gaps was considered in Ref.Ong_2012 , it is remarkable how
closing the gaps increases the magnitude of the mobility and increases the
spread of its values for different $L$ values, whereas moving the impurities
to the surface of a HfO2 layer in the presence of finite gaps further
accentuates those two effects, and even gives rise to a non-monotonous
dependence of the mobility on $\bar{n}$ for thicker HfO2 layers. While the
role of the distance of impurities from graphene was discussed in detail for
the case of zero gaps,PNAS_2007 one may conclude from our analysis that the
size of the gap(s) between graphene and the nearby dielectric(s) plays equally
important role in modeling the conductivity of graphene in a broad range of
charge carrier densities.
We next turn to studying the conductivity minimum as $\bar{n}\rightarrow 0$
due to the presence of electron-hole puddles by using Eqs. (17) and (21) based
on the SCT theory.PNAS_2007 We only consider a 2D planar distribution of
point charges with $Z=1$ having no dipole moment and note that, unlike the
integral in Eq. (16) for conductivity, in order to render the integral in Eq.
(21) convergent one must assume that charged impurities are placed a finite
distance $d$ away from graphene.
Figure 6: (Color online) The dependence of the variance of the potential in
graphene $C_{0}$ (in units $e^{2}n_{\mathrm{imp}}$) on the average charge
carrier density $\bar{n}$ (in units of cm-2) for a two-layer structure that
consists of a semi-infinite SiO2 substrate ($L\rightarrow\infty$,
$\epsilon_{1}=3.9$) and a semi-infinite layer of air ($H\rightarrow\infty$,
$\epsilon_{2}=1$), with graphene placed either on SiO2 with zero gap
($z_{g}=0$) [thick red (grey) lines and symbols] or above SiO2 with the air
gap of $z_{g}=$ 0.3 nm (thin black lines and symbols). A planar distribution
of unit ($Z=1$) point-like charged impurities with no dipole moment, having
the areal number density $n_{\mathrm{imp}}=10^{12}$ cm-2 and the correlation
distance $r_{c}$ between them, is placed in/on SiO2 at a fixed distance
$d=0.3$ nm below graphene. The case of uncorrelated impurities ($r_{c}=0$)
(solid lines, crosses) is compared in the main panel with the cases of
correlated impurities with $r_{c}=$ 5 nm (packing fraction $p=0.2$) in the HD
model (dashed lines, circles) and in the SC model (dotted lines, squares). The
left inset shows the residual charge carrier density (in units of $10^{11}$
cm-2) and the right inset shows the conductivity minimum
$\sigma_{\mathrm{min}}$ (in units of $e^{2}/h$), as functions of the
correlation distance $r_{c}$ (in nm).
In Fig. 6 we consider a configuration similar to that in Fig. 2, with a semi-
infinite SiO2 substrate ($L\rightarrow\infty$ with $\epsilon_{1}=3.9$) and a
semi-infinite layer of air ($H\rightarrow\infty$ with $\epsilon_{2}=1$), with
graphene placed in the air at a distance $z_{g}\geq 0$ above SiO2. We show in
the main panel of Fig. 6 the $\bar{n}$ dependence of the variance of the
potential in the plane of graphene $C_{0}$ from Eq. (21) for a 2D distribution
of charged impurities with density $n_{\mathrm{imp}}=10^{12}$ cm-2 that are
placed in/on SiO2 at a fixed distance $d=0.3$ nm below graphene. Specifically,
we explore the effects of the size of the gap between graphene and the SiO2
substrate by considering both the zero gap case with $z_{g}=0$ (impurities
embedded at the depth of 0.3 nm inside SiO2) and the finite gap case with
$z_{g}=0.3$ nm (impurities placed on the surface of SiO2). In addition to
considering uncorrelated impurities, we use a finite correlation length of
$r_{c}=5$ nm ($p\approx 0.2$) allowing us to compare in the main panel the
effects of the SC and the HD models on $C_{0}$. In the insets of Fig. 6, we
show the dependence of the residual charge carrier density $n^{*}$ and the
corresponding minimum conductivity $\sigma_{\mathrm{min}}=\sigma(n^{*})$ on
$r_{c}$ for both the HD and the CS models, in the presence of both zero and
finite gaps.
Figure 7: (Color online) The dependence of the variance of the potential in
graphene $C_{0}$ (in units $e^{2}n_{\mathrm{imp}}$) on the average charge
carrier density $\bar{n}$ (in units of cm-2) for a two-layer structure that
consists of a HfO2 ($\epsilon_{1}=22$) with thickness $L$ and a semi-infinite
layer of SiO2 ($H\rightarrow\infty$, $\epsilon_{2}=3.9$, or $H=0$,
$\epsilon_{3}=3.9$) with zero gap between them and graphene placed on their
boundary ($z_{g}=0$). A planar distribution of uncorrelated unit ($Z=1$)
point-like charged impurities with no dipole moment, having the areal number
density $n_{\mathrm{imp}}=10^{12}$ cm-2 is embedded at a depth $d=0.3$ nm
inside the HfO2 layer, as in Fig. 4(d). The thickness of the HfO2 layer takes
values $L$ = 1 nm (solid lines), 2 nm (dashed lines), 5 nm (dotted lines), 10
nm (dash-dotted lines), and $\infty$ (double-dotted lines). The (red) symbols
$+$ show in the left inset the residual charge carrier density (in units of
$10^{11}$ cm-2) and in the right inset the conductivity minimum
$\sigma_{\mathrm{min}}$ (in units of $e^{2}/h$), as functions of the thickness
of the HfO2 layer $L$. The (green) symbols $\times$ in the right inset show
$\sigma(\bar{n}=0)$ as a function of the HfO2 layer thickness $L$.
One notices in Fig. 6 that the size of the gap between graphene and the SiO2
substrate exerts a very strong effect on the magnitude of $C_{0}$ for all
$\bar{n}$, echoing similar conclusion drawn from the results analyzed in Fig.
5. The gap size also strongly affects the values of $n^{*}$ for all
correlation lengths $r_{c}$, whereas the effect of the gap size on
$\sigma_{\mathrm{min}}$ is seen to diminish as $r_{c}$ decreases. The latter
result seems to justify the neglect of graphene–substrate gap, which is
implicitly invoked in all simulations of the conductivity minimum in graphene
in the presence of charged impurities with small or vanishing packing
fractions.PNAS_2007 ; Yan_2011 ; Li_2011 ; Sarma_2011 As far as the
comparison between the HD and SC models is concerned, one sees a noticeable
difference in the variance $C_{0}$ at small $\bar{n}$, which diminishes at
large $\bar{n}$ values. The differences between the two models are
surprisingly small in both $n^{*}$ and $\sigma_{\mathrm{min}}$, and only
become noticeable when the packing fraction $p$ approaches the breakdown value
of 0.25 for the SC model for sufficiently large correlation lengths $r_{c}$.
These results again lend confidence to simulations that use the SC model with
short correlation lengths among the charged impurities, which were seen to
yield robustly satisfactory interpretations for the conductivity minimum in
graphene due to electron-hole puddles.PNAS_2007 ; Yan_2011 ; Li_2011 ;
Sarma_2011
Finally, in Fig. 7 we consider a configuration that was studied in Fig. 4(d)
with graphene sandwiched between a layer of HfO2 of finite thickness $L$ and a
semi-infinite layer of SiO2, with no gaps between graphene and the two
dielectrics, and with a 2D distribution of uncorrelated charged impurities of
density $n_{\mathrm{imp}}=10^{12}$ cm-2 embedded at a depth $d=0.3$ nm inside
the HfO2 layer. In the main panel of Fig. 7 we show the dependence of the
variance $C_{0}$ on the charge carrier density in graphene $\bar{n}$, which
exhibits an overall reduction in the magnitude of $C_{0}$ in comparison to
Fig. 6 due to a larger dielectric constant of HfO2, as well as a strong
decrease of $C_{0}$ with decreasing $L$ owing to the screening of impurities
by the nearby metallic gate. As a consequence, the resulting residual density
$n^{*}$ is seen in an inset to Fog. 6 to decrease with decreasing $L$, which
indicates that fluctuations in the charge carrier density in graphene due to
electron-hole puddles would be gradually erased as the metal gate gets closer
to graphene and provides more efficient screening of the fluctuations of the
electrostatic potential. Finally, in the inset showing $\sigma_{\mathrm{min}}$
we explore the contribution of electron-hole puddles to raising the
conductivity minimum above the SBT value $\sigma(0)$ that was discussed in
Fig. 4 via Eq. (16) in the limit $\bar{n}\rightarrow 0$. It is interesting to
note that, even though the contribution $\sigma(n^{*})-\sigma(0)$ that comes
from the residual density $n^{*}$ decreases with decreasing $L$, the
dependence of $\sigma(0)\propto L^{-2}$ implied from Eq. (33) due to geometric
capacitance of the HfO2 layer appears to increase much faster with decreasing
$L$, so that the net value of the conductivity minimum
$\sigma_{\mathrm{min}}=\sigma(n^{*})$ actually increases as the thickness $L$
of the HfO2 layer decreases.
## IV Concluding remarks
We have investigated the conductivity of doped single-layer graphene in the
limit of semiclassical Boltzmann transport, as well as the conductivity
minimum of a nominally neutral graphene within the Self-consistent transport
(SCT) theory, placing emphasis on the effects due to the structure of charged
impurities near graphene and the structure of the surrounding dielectrics.
This was achieved by treating graphene as a zero-thickness layer embedded in a
stratified structure of three dielectric layers and by using the full
electrostatic Green’s function for that structure. We have used the Energy
loss method to derive the conductivity of graphene from the friction force on
a slowly moving structure of charged impurities, based on the polarization
function of graphene within the RPA for its $\pi$ electrons treated as Dirac’s
fermions. Regarding the structure of charged impurities, we have analyzed the
effects of their distance from graphene, the effects of correlation distance
between the impurities within the hard-disk (HD) model for a 2D planar
structure, and the effects of a homogeneous distribution of impurities over a
3D region. Besides point-charge impurities, we have analyzed the effects of a
finite dipole moment on each impurity, as well as the effects of clustering of
impurities into circular disks. Regarding the structure of the surrounding
dielectrics, we have analyzed the effects of finite thickness of one
dielectric layer that pertains to the top gating of graphene through a
high-$\kappa$ dielectric, as well as the effects of finite gap(s) of air
between graphene and the nearby dielectric(s).
For graphene laying on a semi-infinite substrate with zero gap, the effects of
finite distance of impurities and finite cluster size both give rise to a
slightly super-linear dependence of conductivity $\sigma$ on the average
charge carrier density $\bar{n}$ in a heavily doped graphene. Taking advantage
of the HD model that allows studying 2D structures of impurities with
relatively large packing fractions, it is shown that increasing the
correlation distance among the impurities gives rise to a strongly increasing
slope of $\sigma$ at low $\bar{n}$ values, accompanied by a pronounced sub-
linear dependence of conductivity on charge carrier density at higher
$\bar{n}$ values. Making reasonable choices of both the impurity density and
the correlation distance in the HD model gives good agreement with the
experimental data that exhibit sub-linear behavior of the conductivity in
graphene,Tan_2007 whereas inclusion of a perpendicular dipole moment with
sufficiently large polarizability also describes the electron-hole asymmetry
in soma data.
Reducing the thickness of a high-$\kappa$ dielectric gives rise to an increase
in conductivity of graphene at all charge carrier densities in the presence of
a 2D distribution of charged impurities and, in particular, causes the
conductivity at $\bar{n}=0$ to take finite values. The same conclusions are
also true for a homogeneous 3D distribution of impurities throughout the
dielectric at low charge carrier densities, but the trend is reversed at
higher charge carrier densities because of the pronounced super-linear
dependence of the conductivity on $\bar{n}$ as the thickness of the dielectric
increases. Further examination of the effects of the dielectric thickness on
graphene’s mobility, $\mu=\sigma/(e\bar{n})$, reveals that the existence of a
finite gap between graphene and the nearby dielectric and the precise location
of a 2D system of impurities both play important roles in the dependence of
$\mu$ on charge carrier density. While the role of the distance of the
impurities from graphene was discussed before, our results point to the need
of including the size of the graphene-substrate gap as another important
parameter in modeling the conductivity of graphene.
While the effects of the gap size are also important in the variance of the
electrostatic potential in graphene and in the resulting residual charge
carrier density within the SCT theory, such effects are seen to gradually
diminish in the corresponding conductance minimum as the correlation distance
among the impurities in a 2D structure is reduced. This partially justifies
the neglect of the graphene-substrate gap in previous studies of the
conductivity minimum in the presence of uncorrelated impurities. Finally,
reducing the thickness of the high-$\kappa$ dielectric in a top-gated graphene
is shown to reduce both the variance of the potential and the resulting
residual charge carrier density in graphene, showing that the effects of a
system of electron-hole puddles on conductivity in a nominally neutral
graphene are likely to be washed-out due to strong screening by a nearby
metallic top gate. However, the minimum conductivity would continue to
increase with decreasing thickness of the high-$\kappa$ dielectric due to the
effect of its geometric capacitance. These opposing roles of the electron-hole
puddles in neutral graphene and the geometric capacitance of a dielectric
layer in the minimum conductivity of top-gated graphene are worth further
exploration.
Summarizing our main findings, we have shown that the effects of finite
distance of impurities from graphene, the size of the disk-like clusters of
impurities, and the 3D distribution of impurities throughout a dielectric of
finite thickness all give rise to super-linear dependence of conductivity on
charge carrier density in heavily doped graphene. Next, the thickness of a
dielectric and its gap to graphene play important roles in both the
conductivity of doped graphene and the conductivity minimum in neutral
graphene. Those effects are conveniently taken into account using the
electrostatic Green’s function for a layered structure of dielectrics.
Finally, a strong increase in the slope of conductivity for low charge carrier
densities and its saturation at high densities are both well described by
large correlation distances among charged impurities in a 2D structure, which
may be conveniently described by means of a HD model that allows the use of
much higher packing fractions than the simple model of a step-like
correlation.
###### Acknowledgements.
This work was supported by the Natural Sciences and Engineering Research
Council of Canada.
## Appendix A Green’s function
Assume that a single layer of graphene with large area is placed in the plane
$z=z_{g}$ of a Cartesian coordinate system with coordinates ${\bf
R}\equiv\\{{\bf r},z\\}$, where ${\bf r}\equiv\\{x,y\\}$, and is embedded in a
structure that consists of several dielectric layers parallel to graphene, as
shown in Fig. 1. By invoking a translational invariance in the directions of
the 2D vector ${\bf r}$, one may obtain Green’s function (GF) $G({\bf R},{\bf
R}^{\prime};t-t^{\prime})\equiv G({\bf r}-{\bf
r}^{\prime};z,z^{\prime};t-t^{\prime})$ for the Poisson equation for the
entire structure by means of a Fourier transform (FT) with respect to position
(${\bf r}\rightarrow{\bf q}$) and time ($t\rightarrow\omega$), defined via
$\displaystyle G({\bf r}-{\bf r}^{\prime};z,z^{\prime};t-t^{\prime})$
$\displaystyle=$ $\displaystyle\int\frac{d^{2}{\bf
q}}{(2\pi)^{2}}\int\limits_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\mbox{e}^{i{\bf
q}\cdot({\bf r}-{\bf r}^{\prime})-i\omega(t-t^{\prime})}\,$ (34)
$\displaystyle\times\widetilde{G}({\bf q};z,z^{\prime};\omega).$
If one assumes that graphene has zero thickness, then the FT of the above GF
may be expressed in terms of FT of the GF (FTGF) for the dielectric structure
_without_ graphene, $G^{(0)}({\bf R},{\bf R}^{\prime};t-t^{\prime})$, as
$\displaystyle\widetilde{G}({\bf q};z,z^{\prime};\omega)$ $\displaystyle=$
$\displaystyle\widetilde{G}^{(0)}({\bf q};z,z^{\prime})$ (35) $\displaystyle-$
$\displaystyle\frac{e^{2}\chi(q,\omega)\widetilde{G}^{(0)}({\bf
q};z,z_{g})\widetilde{G}^{(0)}({\bf
q};z_{g},z^{\prime})}{1+e^{2}\chi(q,\omega)\widetilde{G}^{(0)}({\bf
q};z_{g},z_{g})},$
where $\chi(q,\omega)$ is a 2D, in-plane polarization function of graphene. We
note that this result is easily obtained from a Dyson-Schwinger equation for
the full Green s function $\widetilde{G}({\bf q};z,z^{\prime};\omega)$, which
may be generalized to solving a simple matrix algebraic problem for a system
of a finite number of graphene layers of zero-thickness that are embedded in a
stratified structure of dielectric slabs described by the FTGF
$\widetilde{G}^{(0)}({\bf q};z,z^{\prime})$.Miskovic_2012
In order to find $\widetilde{G}^{(0)}({\bf q};z,z^{\prime})$, we assume that
the dielectric structure consists of three layers that occupy the intervals
along the $z$ axis defined by $I_{1}=[-L,0]$, $I_{2}=[0,H]$ and
$I_{3}=[H,\infty)$, and are characterized by the relative bulk dielectric
constants $\epsilon_{j}$ with $j=1,2,3$, as shown in Fig. 1. To describe a
specific physical configuration, one may assume that, e.g., the interval
$I_{1}$ is occupied by a high-$\kappa$ dielectric such as HfO2
($\epsilon_{1}\approx 22$) of finite thickness $L>0$, the interval $I_{2}$
represents a layer of vacuum or air ($\epsilon_{2}=1$) of thickness $H\geq 0$
that contains graphene ($z_{g}\in I_{2}$), and $I_{3}$ is a thick (semi-
infinite) layer of SiO2 ($\epsilon_{3}\approx 3.9$). Thus, for finite
$z_{g}>0$ and $H>z_{g}$, such a configuration allows for finite vacuum gaps of
thicknesses $z_{g}$ and $H-z_{g}$ between graphene and the dielectrics
occupying the intervals $I_{1}$ and $I_{3}$, respectively.
The FTGF for the above configuration of dielectric layers,
$\widetilde{G}^{(0)}({\bf q};z,z^{\prime})$, may be obtained as a tensor
$\widetilde{G}^{(0)}_{jk}({\bf q};z,z^{\prime})$, where indices $j$ and $k$
correspond to specific locations of the observation point, $z\in I_{j}$, and
the source point, $z^{\prime}\in I_{k}$, by solving the FT of the Poisson
equation
$\displaystyle\frac{\partial^{2}}{\partial
z^{2}}\widetilde{G}^{(0)}_{jk}(z,z^{\prime})-q^{2}\widetilde{G}^{(0)}_{jk}(z,z^{\prime})=-\frac{4\pi}{\epsilon_{j}}\,\delta_{jk}\,\delta(z-z^{\prime}),$
(36)
where $\delta_{jk}$ is a Kronecker delta with $j,k=1,2,3$, and where we
dropped ${\bf q}$ in $\widetilde{G}^{(0)}({\bf q};z,z^{\prime})$ for the sake
of brevity. When the potential distribution in the system is determined by the
potentials at external, ideally conducting electrodes, solutions of Eq. (36)
need to satisfy homogeneous boundary conditions of the Dirichlet type at
$z=-L$ and $z\rightarrow\infty$ giving
$\displaystyle\widetilde{G}^{(0)}_{1k}(-L,z^{\prime})$ $\displaystyle=$
$\displaystyle 0,$ (37)
$\displaystyle\widetilde{G}^{(0)}_{3k}(\infty,z^{\prime})$ $\displaystyle=$
$\displaystyle 0$ (38)
for $k=1,2,3$. When both $z$ and $z^{\prime}$ are in the interval $I_{j}$, one
usually defines two components of the corresponding diagonal element of the
FTGF as
$\displaystyle\widetilde{G}^{(0)}_{jj}(z,z^{\prime})=\left\\{\begin{array}[]{ll}\widetilde{G}_{j}^{<}(z,z^{\prime}),&z\leq
z^{\prime},\\\ \widetilde{G}_{j}^{>}(z,z^{\prime}),&z^{\prime}\leq
z,\end{array}\right.$ (41)
which must satisfy the continuity and the jump conditions at $z=z^{\prime}$,
$\displaystyle\widetilde{G}_{j}^{<}(z^{\prime},z^{\prime})$ $\displaystyle=$
$\displaystyle\widetilde{G}_{j}^{>}(z^{\prime},z^{\prime}),$ (42)
$\displaystyle\left.\frac{\partial}{\partial
z}\widetilde{G}_{j}^{>}(z,z^{\prime})\right|_{z=z^{\prime}}-\left.\frac{\partial}{\partial
z}\widetilde{G}_{j}^{<}(z,z^{\prime})\right|_{z=z^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{\epsilon_{j}}.$ (43)
Moreover, assuming abrupt interfaces among various dielectrics, the solution
of Eq. (36) needs to satisfy the usual matching conditions at the interfaces
$z=0$ and $z=H$ between dielectric regions,
$\displaystyle\widetilde{G}^{(0)}_{1k}(0,z^{\prime})$ $\displaystyle=$
$\displaystyle\widetilde{G}^{(0)}_{2k}(0,z^{\prime}),$ (44)
$\displaystyle\epsilon_{1}\left.\frac{\partial}{\partial
z}\widetilde{G}^{(0)}_{1k}(z,z^{\prime})\right|_{z=0}$ $\displaystyle=$
$\displaystyle\epsilon_{2}\left.\frac{\partial}{\partial
z}\widetilde{G}^{(0)}_{2k}(z,z^{\prime})\right|_{z=0},$ (45)
$\displaystyle\widetilde{G}^{(0)}_{2k}(H,z^{\prime})$ $\displaystyle=$
$\displaystyle\widetilde{G}^{(0)}_{3k}(H,z^{\prime}),$ (46)
$\displaystyle\epsilon_{2}\left.\frac{\partial}{\partial
z}\widetilde{G}^{(0)}_{2k}(z,z^{\prime})\right|_{z=H}$ $\displaystyle=$
$\displaystyle\epsilon_{3}\left.\frac{\partial}{\partial
z}\widetilde{G}^{(0)}_{3k}(z,z^{\prime})\right|_{z=H},$ (47)
for $k=1,2,3$.
For the sake of definiteness, we assume that charged impurities may only
occupy the intervals $I_{1}$ and $I_{2}$, so that we only need the elements
$\widetilde{G}^{(0)}_{jk}$ of the FTGF with $k=1,2$. By solving Eq. (36)
subject to the conditions in Eqs. (37)-(38) and Eqs. (44)-(47), we obtain for
$z^{\prime}\in I_{1}$Miskovic_2012
$\displaystyle\widetilde{G}^{(0)}_{11}(z,z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\epsilon_{1}q}\,\frac{\sinh\left[q(z_{<}+L)\right]}{\sinh\left(qL)\right)}$
(48)
$\displaystyle\times\frac{\displaystyle{\frac{\epsilon_{1}}{\epsilon_{2}}}\cosh(qz_{>})-\Gamma\sinh(qz_{>})}{\Lambda+\Gamma},$
where $z_{<}=\mathrm{min}(z,z^{\prime})$, $z_{>}=\mathrm{max}(z,z^{\prime})$,
$\Lambda\equiv\left(\epsilon_{1}/\epsilon_{2}\right)\coth(qL)$, and
$\displaystyle\Gamma=\frac{\epsilon_{2}\tanh(qH)+\epsilon_{3}}{\epsilon_{2}+\epsilon_{3}\tanh(qH)},$
(49)
giving
$\displaystyle\widetilde{G}^{(0)}_{21}(z,z^{\prime})$ $\displaystyle=$
$\displaystyle\widetilde{G}^{(0)}_{11}(0,z^{\prime})\left[\cosh(qz)-\Gamma\sinh(qz)\right],$
(50)
whereas for $z^{\prime}\in I_{2}$ we findOng_2012
$\displaystyle\widetilde{G}^{(0)}_{22}(z,z^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\frac{2\pi}{\epsilon_{2}q}}{\Lambda+\Gamma}\,\left\\{\left(\Lambda+\Gamma\right)\mbox{e}^{-q|z-z^{\prime}|}+\left(\Lambda-1\right)\left(\Gamma-1\right)\cosh\left[q\left(z-z^{\prime}\right)\right]\right.$
(51)
$\displaystyle\left.-\left(\Lambda\Gamma-1\right)\cosh\left[q\left(z+z^{\prime}\right)\right]+\left(\Lambda-\Gamma\right)\sinh\left[q\left(z+z^{\prime}\right)\right]\right\\}.$
It is worthwhile mentioning that, with graphene placed at $z_{g}\in I_{2}$,
one obtains from Eq. (51) an explicit expression for the background dielectric
function $\epsilon_{\text{bg}}(q)\equiv
2\pi/\left[q\widetilde{G}^{(0)}_{22}(q;z_{g},z_{g})\right]$ as
$\displaystyle\epsilon_{\text{bg}}(q)=\frac{\epsilon_{2}}{2}\frac{\Lambda+\Gamma}{\cosh^{2}\left(qz_{g}\right)-\Lambda\Gamma\sinh^{2}\left(qz_{g}\right)+\left(\Lambda-\Gamma\right)\cosh\left(qz_{g}\right)\sinh\left(qz_{g}\right)}.$
(52)
For the sake of completeness, we briefly comment on other elements of the
FTGF. One may verify that the symmetry relation
$\widetilde{G}^{(0)}_{12}(z,z^{\prime})=\widetilde{G}^{(0)}_{21}(z^{\prime},z)$
is satisfied by defining
$\displaystyle\widetilde{G}^{(0)}_{12}(z,z^{\prime})=\widetilde{G}_{22}(0,z^{\prime})\,\frac{\sinh\left[q(z+L)\right]}{\sinh\left(qL)\right)}.$
(53)
Moreover, fluctuations of the potential in the interval $I_{3}$ may be found
from
$\displaystyle\widetilde{G}^{(0)}_{3k}(z,z^{\prime})$ $\displaystyle=$
$\displaystyle\widetilde{G}^{(0)}_{2k}(H,z^{\prime})\mathrm{e}^{-q(z-H)},$
(54)
with $k=1,2$, which may also be used to deduce components of the FTGF for the
source point $z^{\prime}\in I_{3}$ via symmetry relations
$\widetilde{G}^{(0)}_{13}(z,z^{\prime})=\widetilde{G}^{(0)}_{31}(z^{\prime},z)$
and
$\widetilde{G}^{(0)}_{23}(z,z^{\prime})=\widetilde{G}^{(0)}_{32}(z^{\prime},z)$.
Finally, it may be of interest to quote the results for the background
dielectric function $\epsilon_{\text{bg}}(q)$ and the profile function
$\psi(q,z)$ in Eq. (19) for a few cases of special interest. First, we
consider the familiar case of a semi-infinite substrate ($L\rightarrow\infty$)
with dielectric constant $\epsilon_{1}\equiv\epsilon_{s}$ that occupies the
region $z<0$, whereas we let $H\rightarrow\infty$ to represent a semi-infinite
region $z>0$ of air or vacuum with $\epsilon_{2}=1$ that contains a single
layer of graphene a distance $z_{g}\geq 0$ above the substrate. We then obtain
$\displaystyle\epsilon_{\text{bg}}(q)=\left[1-\frac{\epsilon_{s}-1}{\epsilon_{s}+1}\exp\\!\left(-2qz_{g}\right)\right]^{-1},$
(55)
and
$\displaystyle\psi(q,z)=\left\\{\begin{array}[]{lll}\displaystyle{\frac{\exp(qz)}{\cosh(qz_{g})+\epsilon_{s}\sinh(qz_{g})}},&z\leq
0,\\\ \\\
\displaystyle{\frac{\cosh(qz)+\epsilon_{s}\sinh(qz)}{\cosh(qz_{g})+\epsilon_{s}\sinh(qz_{g})}},&0\leq
z\leq z_{g},\\\ \\\ \exp\\!\left[-q(z-z_{g})\right],&z\geq
z_{g}.\end{array}\right.$ (61)
As a second example, we consider a semi-infinite substrate
($L\rightarrow\infty$) with dielectric constant $\epsilon_{1}$ that occupies
the region $z<0$, but we retain $H$ finite and allow for three different
dielectric constants as in the original model, and we place graphene at
$z_{g}=H$, i.e., at the boundary between the regions with dielectric constants
$\epsilon_{2}$ and $\epsilon_{3}$. Assuming that the impurities may only
reside in the region $z<0$, this configuration describes a case with a
dielectric spacer of thickness $H$ between graphene and the region with
impurities, giving
$\displaystyle\epsilon_{\text{bg}}(q)=\frac{\epsilon_{3}-\epsilon_{2}}{2}+\epsilon_{2}\left[1+\frac{\epsilon_{2}-\epsilon_{1}}{\epsilon_{2}+\epsilon_{1}}\exp\\!\left(-2qH\right)\right]^{-1},$
(62)
and $\psi(q,z)=\psi_{0}(q)\,\mbox{e}^{qz}$ for $z<0$, where
$\displaystyle\psi_{0}(q)=\displaystyle{\frac{\epsilon_{2}}{\epsilon_{2}\cosh(qH)+\epsilon_{1}\sinh(qH)}}.$
(63)
## Appendix B Geometric structure models
We summarize expressions that define the structure factor for the Hard disk
(HD) model due to RosenfeldRosenfeld_1990 for a 2D planar distribution of
charged impurities with the packing fraction $p=\pi
n_{\mathrm{imp}}r_{c}^{2}/4$, where $n_{\mathrm{imp}}=N/A$ is their areal
number density and $r_{c}$ is the disk diameter,
$\displaystyle S_{\mathrm{HD}}(q)$ $\displaystyle=$
$\displaystyle\left\\{1+16a\left[\frac{J_{1}(qr_{c}/2)}{qr_{c}}\right]^{2}\right.$
(64) $\displaystyle+$
$\displaystyle\left.8b\frac{J_{0}(qr_{c}/2)J_{1}(qr_{c}/2)}{qr_{c}}+\frac{8p}{1-p}\frac{J_{1}(qr_{c})}{qr_{c}}\right\\}^{-1}$
with
$\displaystyle a$ $\displaystyle=$ $\displaystyle 1+x(2p-1)+\frac{2p}{1-p},$
$\displaystyle b$ $\displaystyle=$ $\displaystyle x(1-p)-1-\frac{3p}{1-p},$
$\displaystyle x$ $\displaystyle=$ $\displaystyle\frac{1+p}{(1-p)^{3}}.$
Note that the important long wavelength limit is given by
$S_{\mathrm{HD}}(0)=1/x=(1-p)^{3}/(1+p)$. The expression in Eq. (64) should be
compared with the structure factor for a model with the step-like pair
correlation function,Yan_2011 ; Li_2011
$\displaystyle S_{\mathrm{SC}}(q)=1-\frac{8p}{qr_{c}}J_{1}(qr_{c}),$ (65)
which gives $S_{\mathrm{SC}}(0)=1-4p$.
Next consider a 3D distribution of $N$ point charges $Ze$ occupying the region
$-L\leq z\leq 0$ with a large but finite thickness $L$ and the dielectric
constant $\epsilon_{1}$, while graphene sits in a region with the dielectric
constant $\epsilon_{2}$ at the distance $z_{g}=H\geq 0$. If one disregards the
effects of the proximity of graphene and uses the pair correlation (or radial
distribution) function for the bulk of a homogeneous charge distribution,
$g_{3D}({\bf r}_{2}-{\bf r}_{1};z_{2}-z_{1})=g_{3D}(R)$ with $R=\sqrt{({\bf
r}_{2}-{\bf r}_{1})^{2}+(z_{2}-z_{1})^{2}}$, Eqs. (31) and (63) give
$\displaystyle\mathcal{S}(q)=\frac{Z^{2}}{\pi
L}\psi_{0}^{2}(q)\int\limits_{q}^{\infty}\frac{dQ}{Q}\,\frac{S_{3D}(Q)}{\sqrt{Q^{2}-q^{2}}},$
(66)
where
$\displaystyle S_{3D}(Q)=1+N_{\mathrm{imp}}\int d^{3}{\bf
R}\,\mathrm{e}^{i{\bf Q}\cdot{\bf R}}\left[g_{3D}(R)-1\right],$ (67)
with $N_{\mathrm{imp}}=N/\left(AL\right)$ being the volume density of
particles and ${\bf Q}=\left({\bf q},q_{z}\right)$ a 3D wavevector. For
example, we may consider a model for electrostatic correlations among mobile
charges in a one-component plasma (OCP)Ichimaru_1982 at temperature $T$ with
the square of the inverse Debye length defined by $Q_{D}^{2}=3\pi
N_{\mathrm{imp}}Z^{2}e^{2}/\left(\epsilon_{1}k_{B}T\right)$, and use the long
wavelength result for this system
$S_{3D}(Q)=Q^{2}/\left(Q^{2}+Q_{D}^{2}\right)$ in Eq. (66) to obtain
$\displaystyle\mathcal{S}(q)=\frac{Z^{2}\,\psi_{0}^{2}(q)}{2L\sqrt{q^{2}+Q_{D}^{2}}}.$
(68)
This result is not used in this work, but it may be found useful in future
modeling of the interaction of graphene with an OCP with a spacer layer of
thickness $H$ and dielectric constant $\epsilon_{2}$ between graphene and the
OCP.
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|
arxiv-papers
| 2013-07-30T23:13:48 |
2024-09-04T02:49:48.797652
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rastko Ani\\v{c}i\\'c and Zoran L. Mi\\v{s}kovi\\'c",
"submitter": "Rastko Anicic",
"url": "https://arxiv.org/abs/1307.8169"
}
|
1307.8227
|
# Pointed Hopf algebras with classical Weyl groups (II)
Weicai Wu, Shouchuan Zhang, Zhengtang Tan
Department of Mathematics, Hunan University
Changsha 410082, P.R. China, Emails: [email protected]
###### Abstract
We prove that except in several cases Nichols algebras of irreducible Yetter-
Drinfeld modules over classical Weyl groups
$\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ are infinite dimensional. We also
prove that except in several cases conjugacy classes of classical Weyl groups
are of type $D$; hence they collapse. We give the relationship between
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ and ${\mathcal{E}}_{n}$.
2000 Mathematics Subject Classification: 16W30, 16G10
keywords: Rack, Hopf algebra, Weyl group.
## 0 Introduction
This work is a contribution to the classification of finite-dimensional Hopf
algebras over an algebraically closed field of characteristic 0, problem posed
by I. Kaplansky in 1975. N. Andruskiewitsch and H.-J. Schneider classify
finite-dimensional complex Hopf algebras by Lifting method [AS10]. N.
Andruskiewitsch and M. Grana study Nichols algebra of the most important class
of braided vector spaces $(CX,cq)$, where X is a rack and q is a 2-cocycle on
X with values in C in [AG03]. N. Andruskiewitsch, F. Fantino, M. Graña,
L.Vendramin and S. Zhang obtain that Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ over symmetry groups have infinite
dimension, except for a small list of examples and remarkable cases
corresponding to ${\mathcal{O}}_{\sigma}$ in [AFGV08, AFZ, AZ07]. Shouchuan
Zhang and Yao-Zhong Zhang show that except in three cases Nichols algebras of
irreducible Yetter-Drinfeld (YD in short) modules over classical Weyl groups
$A\rtimes\mathbb{S}_{n}$ supported by $\mathbb{S}_{n}$ are infinite
dimensional in [ZZ12], but this has not completed for general elements.
This paper follows from [ZZ12] to keep on classifying finite dimensional
complex pointed Hopf algebras with classic Weyl groups.
In this paper we prove that except in several cases Nichols algebras of
irreducible Yetter-Drinfeld modules over classical Weyl groups
$\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ are infinite dimensional. We also
prove that except in several cases conjugacy classes of classical Weyl groups
are of type $D$; hence they collapse.
The main results in this paper are summarized in the following statements.
###### Theorem 0.1.
Let $G=\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ with $n>4$. Let
$\tau\in\mathbb{S}_{n}$ be of type
$(1^{\lambda_{1}},2^{\lambda_{2}},\dots,n^{\lambda_{n}})$ and
$a\in\mathbb{Z}_{2}^{n}$ with $\sigma=(a,\tau)\in G$ with $\tau\not=1$. If
$\dim\mathfrak{B}(\mathcal{O}_{\sigma}^{G},\rho)<\infty$, then some of the
following hold.
1. (i)
$(2,3)$; $(2^{3});$
2. (ii)
$(2^{4});$ $(1,2^{2}),$
3. (iii)
$(1^{2},2^{2})$, $(1^{n-2},2)$ and $(1^{n-3},3)$ with $a_{i}=a_{j}$ when
$\tau(i)=i$ and $\tau(j)=j$.
Indeed, It follows from Theorem 2.6 and Theorem 3.2.
## Preliminaries and Conventions
Let ${k}$ be the complex field, A quiver $Q=(Q_{0},Q_{1},s,t)$ is an oriented
graph, where $Q_{0}$ and $Q_{1}$ are the sets of vertices and arrows,
respectively; $s$ and $t$ are two maps from $Q_{1}$ to $Q_{0}$. For any arrow
$a\in Q_{1}$, $s(a)$ and $t(a)$ are called its start vertex and end vertex,
respectively, and $a$ is called an arrow from $s(a)$ to $t(a)$. For any $n\geq
0$, an $n$-path or a path of length $n$ in the quiver $Q$ is an ordered
sequence of arrows $p=a_{n}a_{n-1}\cdots a_{1}$ with $t(a_{i})=s(a_{i+1})$ for
all $1\leq i\leq n-1$. Note that a 0-path is exactly a vertex and a 1-path is
exactly an arrow. In this case, we define $s(p)=s(a_{1})$, the start vertex of
$p$, and $t(p)=t(a_{n})$, the end vertex of $p$. For a 0-path $x$, we have
$s(x)=t(x)=x$. Let $Q_{n}$ be the set of $n$-paths. Let ${}^{y}Q_{n}^{x}$
denote the set of all $n$-paths from $x$ to $y$, $x,y\in Q_{0}$. That is,
${}^{y}Q_{n}^{x}=\\{p\in Q_{n}\mid s(p)=x,t(p)=y\\}$.
A quiver $Q$ is finite if $Q_{0}$ and $Q_{1}$ are finite sets. A quiver $Q$ is
locally finite if ${}^{y}Q_{1}^{x}$ is a finite set for any $x,y\in Q_{0}$.
Let ${\mathcal{K}}(G)$ denote the set of conjugacy classes in $G$. A formal
sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugacy classes of $G$ with
cardinal number coefficients is called a ramification (or ramification data )
of $G$, i.e. for any $C\in{\mathcal{K}}(G)$, $r_{C}$ is a cardinal number. In
particular, a formal sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugacy
classes of $G$ with non-negative integer coefficients is a ramification of
$G$.
For any ramification $r$ and $C\in{\mathcal{K}}(G)$, since $r_{C}$ is a
cardinal number, we can choose a set $I_{C}(r)$ such that its cardinal number
is $r_{C}$ without loss of generality. Let
${\mathcal{K}}_{r}(G):=\\{C\in{\mathcal{K}}(G)\mid
r_{C}\not=0\\}=\\{C\in{\mathcal{K}}(G)\mid I_{C}(r)\not=\emptyset\\}$. If
there exists a ramification $r$ of $G$ such that the cardinal number of
${}^{y}Q_{1}^{x}$ is equal to $r_{C}$ for any $x,y\in G$ with $x^{-1}y\in
C\in{\mathcal{K}}(G)$, then $Q$ is called a Hopf quiver with respect to the
ramification data $r$. In this case, there is a bijection from $I_{C}(r)$ to
${}^{y}Q_{1}^{x}$, and hence we write ${\ }^{y}Q_{1}^{x}=\\{a_{y,x}^{(i)}\mid
i\in I_{C}(r)\\}$ for any $x,y\in G$ with $x^{-1}y\in C\in{\mathcal{K}}(G)$.
$(G,r,\overrightarrow{\rho},u)$ is called a ramification system with
irreducible representations (or RSR in short ), if $r$ is a ramification of
$G$; $u$ is a map from ${\mathcal{K}}(G)$ to $G$ with $u(C)\in C$ for any
$C\in{\mathcal{K}}(G)$; $I_{C}(r,u)$ and $J_{C}(i)$ are sets with
$\mid\\!J_{C}(i)\\!\mid$ = ${\rm deg}(\rho_{C}^{(i)})$ and
$I_{C}(r)=\\{(i,j)\mid i\in I_{C}(r,u),j\in J_{C}(i)\\}$ for any
$C\in{\mathcal{K}}_{r}(G)$, $i\in I_{C}(r,u)$;
$\overrightarrow{\rho}=\\{\rho_{C}^{(i)}\\}_{i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)}\
\in\prod_{C\in{\mathcal{K}}_{r}(G)}(\widehat{{G^{u(C)}}})^{\mid
I_{C}(r,u)\mid}$ with $\rho_{C}^{(i)}\in\widehat{{G^{u(C)}}}$ for any $i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)$. In this paper we always assume that
$I_{C}(r,u)$ is a finite set for any $C\in{\mathcal{K}}_{r}(G).$ Furthermore,
if $\rho_{C}^{(i)}$ is a one dimensional representation for any
$C\in{\mathcal{K}}_{r}(G)$, then $(G,r,\overrightarrow{\rho},u)$ is called a
ramification system with characters (or RSC $(G,r,\overrightarrow{\rho},u)$
in short ) (see [ZZC04, Definition 1.8]). In this case, $a_{y,x}^{(i,j)}$ is
written as $a_{y,x}^{(i)}$ in short since $J_{C}(i)$ has only one element.
For ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, let $\chi_{C}^{(i)}$ denote the
character of $\rho_{C}^{(i)}$ for any $i\in I_{C}(r,u)$,
$C\in{\mathcal{K}}_{r}(C)$. If ramification $r=r_{C}C$ and
$I_{C}(r,u)=\\{i\\}$ then we say that ${\rm RSR}(G,r,\overrightarrow{\rho},u)$
is bi-one, written as ${\rm RSR}(G,{\mathcal{O}}_{s},\rho)$ with $s=u(C)$ and
$\rho=\rho_{C}^{(i)}$ in short, since $r$ only has one conjugacy class $C$ and
$\mid\\!I_{C}(r,u)\\!\mid=1$. Quiver Hopf algebras, Nichols algebras and
Yetter-Drinfeld modules, corresponding to a bi-one ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$, are said to be bi-one.
If $(G,r,\overrightarrow{\rho},u)$ is an ${\rm RSR}$, then it is clear that
${\rm RSR}(G,{\mathcal{O}}_{u(C)},\rho_{C}^{(i)})$ is bi-one for any
$C\in{\mathcal{K}}$ and $i\in I_{C}(r,u)$, which is called a bi-one sub-${\rm
RSR}$ of ${\rm RSR}(G,r,\overrightarrow{\rho},u)$,
For $s\in G$ and $(\rho,V)\in\widehat{G^{s}}$, here is a precise description
of the YD module $M({\mathcal{O}}_{s},\rho)$, introduced in [Gr00, AZ07]. Let
$t_{1}=s$, …, $t_{m}$ be a numeration of ${\mathcal{O}}_{s}$, which is a
conjugacy class containing $s$, and let $g_{i}\in G$ such that $g_{i}\rhd
s:=g_{i}sg_{i}^{-1}=t_{i}$ for all $1\leq i\leq m$. Then
$M({\mathcal{O}}_{s},\rho)=\oplus_{1\leq i\leq m}g_{i}\otimes V$. Let
$g_{i}v:=g_{i}\otimes v\in M({\mathcal{O}}_{s},\rho)$, $1\leq i\leq m$, $v\in
V$. If $v\in V$ and $1\leq i\leq m$, then the action of $h\in G$ and the
coaction are given by
$\displaystyle\delta(g_{i}v)=t_{i}\otimes g_{i}v,\qquad
h\cdot(g_{i}v)=g_{j}(\gamma\cdot v),$ (0.1)
where $hg_{i}=g_{j}\gamma$, for some $1\leq j\leq m$ and $\gamma\in G^{s}$.
The explicit formula for the braiding is then given by
$c(g_{i}v\otimes g_{j}w)=t_{i}\cdot(g_{j}w)\otimes
g_{i}v=g_{j^{\prime}}(\gamma\cdot w)\otimes g_{i}v$ (0.2)
for any $1\leq i,j\leq m$, $v,w\in V$, where $t_{i}g_{j}=g_{j^{\prime}}\gamma$
for unique $j^{\prime}$, $1\leq j^{\prime}\leq m$ and $\gamma\in G^{s}$. Let
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ denote
$\mathfrak{B}(M({\mathcal{O}}_{s},\rho))$. $M({\mathcal{O}}_{s},\rho)$ is a
simple YD module (see [AZ07, Section 1.2 ]).
Set $sq(x,y):=x\rhd(y\rhd(x\rhd y)).$ $G$ is called collapse if every finite
dimensional pointed Hopf algebra with group $G$ is a group algebra. It follows
from [AFGV10] that $G$ is called collapse if and only if
$\dim\mathfrak{B}(\mathcal{O}_{\sigma}^{G},\rho)=\infty$ for any
$\rho\in\widehat{G^{\sigma}}$. Let
$\mathbb{B}_{n}:=\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$. Obviously,
$(a,\tau)=a\tau$ for any $(a,\tau)\in\mathbb{B}_{n}.$
Assume that $X$ is a rack. $R\cup S$ is called a subrack decomposition of $X$
if $R$ and $S$ are two disjoint subracks satisfying $y\rhd x\in R$ and $x\rhd
y\in S$, for any $x\in R,y\in S$. $X$ is said to be of type $D$ if there
exists a subrack decomposition $R\cup S$ and two element $r\in R,s\in S$ such
that $sq(r,s)\not=s.$ We say that a finite rack $X$ collapses if for any
finite faithful cocycle $q$ ( associated to any decomposition of $X$ and of
any degree $n$ ) $\dim\mathfrak{B}(X,q)=\infty$ (see [AFGV08, Section 2.4] and
[AG03]).
## 1 Extension
In this section we apply juxtapositions to decide if Nichols algebras
associated to the irreducible Yetter-Drinfeld modules over classic Weyl groups
are finite dimensional or not.
###### Lemma 1.1.
Let $G=\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ and $(a,\tau),(b,\mu)\in G$.
Let $(c,\lambda)$ denote $(a,\tau)\rhd((b,\mu)\rhd((a,\tau)\rhd(b,\mu)))$.
(i) Then
$\displaystyle c$ $\displaystyle=$
$\displaystyle(a+\tau\cdot[b+\mu\cdot(a+\tau\cdot b+(\tau\rhd\mu)\cdot
a)+(\mu\rhd(\tau\rhd\mu))\cdot b]$ (1.1) $\displaystyle+$
$\displaystyle(\tau\rhd(\mu\rhd(\tau\rhd\mu)))\cdot a.$
(ii) If $\tau$ and $\mu$ are commutative, then
$\displaystyle c=a+\tau\mu\cdot a+\tau\mu^{2}\cdot a+\mu\cdot a+\tau\cdot
b+\tau^{2}\mu\cdot b+\tau\mu\cdot b.$ (1.2)
(iii) If $\tau$ and $\mu$ are commutative, then $sq(a\tau,b\mu)=b\mu$ if and
only if
$\displaystyle a+\tau\mu\cdot a+\tau\mu^{2}\cdot a+\mu\cdot a=b+\tau\cdot
b+\tau^{2}\mu\cdot b+\tau\mu\cdot b.$ (1.3)
(iv) If $\tau$ and $\mu$ are commutative with $(b,\mu)=\xi\rhd(a,\tau)$ and
$\xi\cdot a=a$, then $c=a+\tau\mu^{2}\cdot a+\mu\cdot a+\tau\cdot
a+\tau^{2}\mu\cdot a.$
(v) If $\tau$ and $\xi$ are commutative with $\tau^{2}=1$ and
$(b,\mu)=\xi\rhd(a,\tau)$ and $\xi\cdot a=a$, then $c=a$.
Proof. It is clear. $\Box$
If lengths of independent sign cycles of $(a,\pi)$ and $(b,\tau)$ are
different each other, then they are called orthogonal each other, written as
$a\pi\bot b\tau.$
For any $a\pi\in\mathbb{B}_{n}$ and $b\tau\in\mathbb{B}_{m}$, define
$a\pi\\#b\tau\in\mathbb{B}_{m+n}$ as follows:
$(a\\#b)_{i}:=\left\\{\begin{array}[]{ll}a_{i}&\hbox{when }i\leq n\\\
b_{i-n}&\hbox{when }i>n\end{array}\right.,$
$(\pi\\#\tau)(i):=\left\\{\begin{array}[]{ll}\pi(i)&\hbox{when }i\leq n\\\
\tau(i-n)+n&\hbox{when }i>n\end{array}\right.$ and
$(a\pi)\\#(b\tau):=(a\\#b,\pi\\#\tau),$ which is called a juxtaposition of
$a\pi$ and $b\tau$. Obviously, $(a\pi)\\#(b\tau)\in\mathbb{B}_{m+n}$. Let
$\overrightarrow{\nu_{n,m}}$ be a map from $\mathbb{B}_{n}$ to
$\mathbb{B}_{m+n}$ by sending $a\pi$ to
$\overrightarrow{\nu_{n,m}}(a\pi):=a\pi\\#1_{\mathbb{B}_{m}}$; let
$\overleftarrow{\nu_{n,m}}$ be a map from $\mathbb{B}_{m}$ to
$\mathbb{B}_{m+n}$ by sending $b\tau$ to
$\overleftarrow{\nu_{n,m}}(b\tau):=1_{\mathbb{B}_{n}}\\#b\tau$.
###### Lemma 1.2.
Assume $a\pi\bot b\tau$ with $a\pi,a^{\prime}\pi^{\prime}\in\mathbb{B}_{n}$
and $b\tau,b^{\prime}\tau^{\prime}\in\mathbb{B}_{m}$. Then
(i) $(a\pi\\#b\tau)(a^{\prime}\pi^{\prime}\\#b^{\prime}\tau^{\prime})=(a\pi
a^{\prime}\pi^{\prime}\\#b\tau b^{\prime}\tau^{\prime})$ .
(ii)
$a\pi\\#b\tau=\overrightarrow{\nu_{n,m}}(a\pi)\overleftarrow{\nu_{n,m}}(b\tau)=\overleftarrow{\nu_{n,m}}(b\tau)\overrightarrow{\nu_{n,m}}(a\pi)$.
(iii)
$\mathbb{B}_{m+n}^{a\pi\\#b\tau}=\mathbb{B}_{n}^{a\pi}\\#\mathbb{B}_{m}^{b\tau}=\overrightarrow{\nu_{n,m}}(\mathbb{B}_{n}^{a\pi})\overleftarrow{\nu_{n,m}}(\mathbb{B}_{m}^{b\tau})$
as directed products.
(iv) For any $\rho\in\widehat{\mathbb{B}_{m+n}^{a\pi\\#b\tau}}$, there exist
$\mu\in\widehat{\mathbb{B}_{n}^{a\pi}}$,
$\lambda\in\widehat{\mathbb{B}_{m}^{b\tau}}$ such that
$\rho=\mu\otimes\lambda$.
(v)
$(a\pi\\#b\tau)\rhd(a^{\prime}\pi^{\prime}\\#b^{\prime}\tau^{\prime})=(a\pi\rhd
a^{\prime}\pi^{\prime})\\#(b\tau\rhd b^{\prime}\tau^{\prime})$
(vi)
$\mathcal{O}_{a\pi\\#b\tau}^{\mathbb{B}_{m+n}}=\mathcal{O}_{a\pi}^{\mathbb{B}_{n}}\\#\mathcal{O}_{b\tau}^{\mathbb{B}_{m}}$.
Proof. (i) , (ii) and (v) are clear.
(iii) By [AZ07, Section 2.2],
$\mathbb{S}_{m+n}^{\pi\\#\tau}=\mathbb{S}_{n}^{\pi}\\#\mathbb{S}_{m}^{\tau}.$
Obviously,
$\mathbb{B}_{n}^{a\pi}\\#\mathbb{B}_{m}^{b\tau}\subseteq\mathbb{B}_{m+n}^{a\pi\\#b\tau}$.
For any $c\xi\in\mathbb{B}_{m+n}^{a\pi\\#b\tau},$ then
$\xi\in\mathbb{S}_{m+n}^{\pi\\#\tau}$ and there exist
$\mu\in\mathbb{S}_{n}^{\pi}$ and $\lambda\in\mathbb{S}_{m}^{\tau}$ such that
$\xi=\mu\\#\lambda$. Consequently, $c\xi=d\mu\\#f\lambda.$ Considering
$c\xi(a\pi\\#b\tau)=(a\pi\\#b\tau)c\xi$, we have
$d\mu\in\mathbb{B}_{n}^{a\pi}$ and $f\lambda\in\mathbb{B}_{m}^{b\tau}$. This
completes the proof.
(iv) It follows from (iii).
(vi) By (v),
$\mathcal{O}_{a\pi}^{\mathbb{B}_{n}}\\#\mathcal{O}_{b\tau}^{\mathbb{B}_{m}}\subseteq\mathcal{O}_{a\pi\\#b\tau}^{\mathbb{B}_{m+n}}.$
Consequently, (vi) follows from (iii). $\Box$
Remark: (i), (ii) and (v) above still hold when $a\pi$ and $b\tau$ are not
orthogonal each other.
###### Theorem 1.3.
If $\mathcal{O}_{a\tau}$ is of type $D$, then $\mathcal{O}_{a\tau\\#b\mu}$ is
of type $D$, too.
Proof. Let $X=R\cup S$ be a subrack decomposition of $\mathcal{O}_{a\tau}$ and
of type $D$. It is clear that $X\\#b\mu=R\\#b\mu\cup S\\#b\mu$ is a subrack
decomposition of $\mathcal{O}_{a\tau\\#b\mu}$ and of type $D$.$\Box$
###### Lemma 1.4.
Assume $a\pi\bot b\tau$. Let
$\rho=\mu\otimes\lambda\in\widehat{\mathbb{B}_{m+n}^{a\pi\\#b\tau}}$ ,
$\mu\in\widehat{\mathbb{B}_{n}^{a\pi}}$,
$\lambda\in\widehat{\mathbb{B}_{m}^{b\tau}}$, $a\pi\in\mathbb{B}_{n}$ and
$b\tau\in\mathbb{B}_{m}$ with $q_{a\pi}id=\mu(a\pi)$ and
$q_{b\tau}id=\lambda(b\tau)$. If
$\dim\mathfrak{B}({\mathcal{O}}_{a\pi\\#b\tau}^{\mathbb{B}_{n+m}},\rho)<\infty$,
then
(i) $M({\mathcal{O}}_{a\pi}^{\mathbb{B}_{n}},\mu)$ is isomorphic to a YD
submodule of $M({\mathcal{O}}_{a\pi\\#b\tau}^{\mathbb{B}_{n+m}},\rho)$ over
$\mathbb{B}_{n}$ when $q_{b\tau}=1$; hence
$\dim\mathfrak{B}({\mathcal{O}}_{a\pi}^{\mathbb{B}_{n}},\mu)<\infty$.
(ii) $q_{a\pi}q_{b\tau}=-1$.
(iii) $q_{b\tau}=1$ and $q_{a\pi}=-1$ when ${\rm ord}(b\tau)\leq 2$ and ${\rm
ord}(q_{a\pi})\not=1$.
(iv) $q_{b\tau}=1$ and $q_{a\pi}=-1$ when ${\rm ord}(a\pi)$ and ${\rm
ord}(b\tau)$ are coprime and ${\rm ord}(b\tau)$ is odd.
Proof. (i) We can show this as the proof in [AZ07, Section 2.2].
(ii) and (iii) are clear.
(iv) Obviously, $ordq_{a\pi}\mid ord(a\pi)$ and $ordq_{b\tau}\mid ord(b\tau)$.
Therefore, $ordq_{b\tau}$ is odd. $(ord(q_{a\pi}),ord(q_{b\tau}))=1$ since
$(ord{(a\pi}),ord({b\tau}))=1$. By Part (ii), $ordq_{a\pi}ordq_{b\tau}=2$.
Consequently, $ordq_{a\pi}=2$ and $ordq_{b\tau}=1$. $\Box$
## 2 Rack of $\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$
In this section we prove that except in several cases conjugacy classes of
classical Weyl groups are of type $D$. Let
$\alpha:=(1,1,\cdots,1)\in\mathbb{Z}_{2}^{n}.$
###### Lemma 2.1.
Let $n$ be odd with $n\geq 5$ and $a\tau\in\mathbb{B}_{n}$ with $\tau=(1\
2\cdots\ n)$. Then $\mathcal{O}_{a\tau}^{\mathbb{B}_{n}}$ is of type $D$.
Proof. (i) Assume that $a\tau$ is a negative cycle and $b=(1,0,\cdots,0)$.
Thus $\alpha\tau$, $a\tau$ and $b\tau$ are conjugate. We assume $a=\alpha$
without lost generality. Obviously, the side of right hand (1.3) $\not=0$ for
$\mu=\tau^{2}$, i.e. $sq(\alpha\tau,b\tau^{2})\not=b\tau^{2}.$ Let
$R:=\mathbb{Z}_{2}^{n}\rtimes\tau\cap\mathcal{O}_{\alpha\tau}^{\mathbb{B}_{n}}$
and
$S:=\mathbb{Z}_{2}^{n}\rtimes\tau^{2}\cap\mathcal{O}_{\alpha\tau}^{\mathbb{B}_{n}}$.
It is clear that $R\cup S$ is a subrack decomposition and of type $D$.
(ii) Assume that $a\tau$ is a positive cycle. Let $b=(1,0,0,1,0)$ when $n=5;$
$b=(1,1,0,\cdots,0)$ when $n>5.$ Thus $0\tau$, $a\tau$ and $b\tau$ are
conjugate. We assume $a=0$ without lost generality. It is clear that the right
side of (1.3) $\not=0$. Consequently, $R\cup S$ is a subrack decomposition and
of type $D$ as Part (i). $\Box$
###### Lemma 2.2.
If $\sigma$ is of type ($3^{2}$), then
$\mathcal{O}_{a\sigma}^{\mathbb{B}_{6}}$ is of type $D.$
Proof. Let $\tau=(1\ 2\ 3)(4\ 5\ 6)$, $\mu=(1\ 2\ 3)^{2}(4\ 5\ 6)$, $\pi=(1\
2\ 3)$ and $\xi=(4\ 5\ 6)$. It is clear that we have
$\displaystyle(a_{6}+a_{5},a_{4}+a_{6},a_{5}+a_{4})=(b_{6}+b_{5},b_{4}+b_{6},b_{5}+b_{4}).$
(2.1)
by (1.3).
(i) If $a=\alpha$ and $b=(1,0,0,1,0,0)$, then (2.1) does not hold.
(ii) If $a=0$ and $b=(0,0,0,1,1,0)$, then (2.1) does not hold.
(iii) If $a=(1,0,0,0,0,0)$ and $b=(1,0,0,1,1,0)$, then (2.1) does not hold.
(iv) If $a=(0,0,0,1,0,0)$ and $b=(0,0,0,0,1,0)$, then (2.1) does not hold.
Let $R:=\mathbb{Z}_{2}^{6}\rtimes\tau$ and $S:=\mathbb{Z}_{2}^{6}\rtimes\mu$.
It is clear that $R\cup S$ is a subrack decomposition of
$\mathcal{O}_{a\sigma}^{\mathbb{B}_{6}}$. Consequently
$\mathcal{O}_{a\sigma}^{\mathbb{B}_{6}}$ is of type $D.$ $\Box$
###### Lemma 2.3.
If $\sigma$ is of type ($2^{2}\ 3^{1}$), then
$\mathcal{O}_{a\sigma}^{\mathbb{B}_{7}}$ is of type $D.$
Proof. Let $\pi=(5\ 6\ 7)$, $\xi=(1\ 2)(3\ 4)$, $\lambda=(1\ 3)(2\ 4)$,
$\tau=\pi\xi$ and $\mu=\pi\lambda$.
If $a=(a_{1},a_{2},a_{3},a_{4},0,0,0)$, let
$b=(b_{1},b_{2},b_{3},b_{4},1,1,0)$; If $a=(a_{1},a_{2},a_{3},a_{4},,1,1,1)$,
let $b=(b_{1},b_{2},b_{3},b_{4},1,0,0)$. The 5th, 6th, 7th component of (1.3)
are
$(a_{5}+a_{7},a_{6}+a_{5},a_{7}+a_{6})=(b_{5}+b_{7},b_{6}+b_{5},b_{7}+b_{6})$.
Consequently, (1.3) does not hold.
Let $R:=\mathbb{Z}_{2}^{7}\rtimes\tau$ and $S:=\mathbb{Z}_{2}^{7}\rtimes\mu$.
It is clear that $R\cup S$ is a subrack decomposition of
$\mathcal{O}_{a\sigma}^{\mathbb{B}_{7}}$. Consequently
$\mathcal{O}_{a\sigma}^{\mathbb{B}_{7}}$ is of type $D.$ $\Box$
###### Lemma 2.4.
(i) Assume that $\tau$ and $\mu$ are conjugate with $sq(\tau,\mu)\not=\mu$ in
$\mathbb{S}_{n}$ and $\tau(n)=\mu(n)=n$. If $a\in\mathbb{Z}_{2}^{n}$ and there
exist $i,j$ such that $a_{i}\not=a_{j}$ with $\tau(i)=i$ and $\tau(j)=j,$ then
$\mathbb{O}_{a\tau}$ is of type $D.$
(ii) Assume that $n>3$ and $a\tau\in\mathbb{B}_{n}$ with type $(1^{n-2},2)$ of
$\tau.$ If there exist $i,j$ such that $\tau(i)=i$ and $\tau(j)=j$ with
$a_{i}\not=a_{j}$, then $\mathcal{O}_{a\tau}$ is of type $D$.
(iii) Assume that $n>4$ and $a\tau\in\mathbb{B}_{n}$ with type $(1^{n-3},3)$
of $\tau.$ If there exist $i,j$ such that $\tau(i)=i$ and $\tau(j)=j$ with
$a_{i}\not=a_{j}$, then $\mathcal{O}_{a\tau}$ is of type $D$.
(iv) Assume that $n=6$ and $a\tau\in\mathbb{B}_{n}$ with type $(1^{2},2^{2})$
of $\tau.$ If there exist $i,j$ such that $\tau(i)=i$ and $\tau(j)=j$ with
$a_{i}\not=a_{j}$, then $\mathcal{O}_{a\tau}$ is of type $D$.
Proof. (i) Let
$R:=\\{d\xi\in\mathcal{O}_{a\tau}^{\mathbb{B}_{n}}\mid\xi\in\mathbb{S}_{n-1};d_{n}=0\\}$
and
$S:=\\{d\xi\in\mathcal{O}_{a\tau}^{\mathbb{B}_{n}}\mid\xi\in\mathbb{S}_{n-1};d_{n}=1\\}$.
Obviously, $R\cup S$ is a subrack decomposition. It is clear that there exists
$b,c\in\mathbb{Z}_{2}^{n}$, $\xi,\lambda\in\mathbb{S}_{n-1}$ such that
$b\xi\in R$ and $c\lambda\in S$. Since $sq(\tau,\mu)\not=\mu$, we have that
$sq(b\tau,c\mu)\not=c\mu$. That implies that $\mathbb{O}_{a\tau}$ is of type
$D.$
(ii) It is clear that $sq(\tau,\mu)\not=\mu$ with $\tau:=(1,2)$ and
$\mu:=(2,3)$. Applying Part (i) we complete the proof.
(iii)It is clear that $sq(\tau,\mu)\not=\mu.$ with $\tau=(1,2,3)$ and
$\mu=(2,4,3)$. Applying Part (i) we complete the proof.
(iv) It is clear that $sq(\tau,\mu)\not=\mu.$ with $\tau=(1\ 2)(3\ 4)$ and
$\mu=(2\ 3)(4\ 5)$. Applying Part (i) we complete the proof. $\Box$
###### Lemma 2.5.
Let $G=\mathbb{Z}_{2}^{n}\rtimes H$ and $\sigma=(a,\tau)\in G$ with
$a\in\mathbb{Z}_{2}^{n},\tau\in H.$ If $\mathcal{O}_{\tau}^{H}$ is of type
$D$, then so is $\mathcal{O}_{(a,\tau)}^{G}$.
Proof. Let $X=S\cup T$ is a subrack decomposition of $\mathcal{O}_{\tau}^{H}$
and there exist $s\in S,t\in T$ such that
$\displaystyle s\rhd(t\rhd(s\rhd t))\not=t.$ (2.2)
Let $h\rhd\tau=s$ and $g\rhd\tau=t$ with $h,g\in H$. It is clear $sq((h\cdot
a,s),(g\cdot a,t))\not=(g\cdot a,t)$ since (2.2); $(h\cdot a,s)=h\rhd(a,\tau)$
and $(g\cdot a,t)=g\rhd(a,\tau)$.
$(<H\cdot a>,X)$ is a subrack, where $<H\cdot a>$ is the subgroup generated by
set $H\cdot a$ of $\mathbb{Z}_{2}^{n}$. In fact, for any $h,g\in H,\xi,\mu\in
X,$ we have that
$\displaystyle(h\cdot a,\xi)\rhd(g\cdot a,\mu)=((h+\xi g+\xi\mu\xi^{-1}h)\cdot
a,\xi\rhd\mu)\in(<H\cdot a>,X).$
Thus $(<H\cdot a>,X)$ is a subrack. Consequently $(<H\cdot
a>,X)\cap\mathcal{O}_{(a,\tau)}^{G}=(<H\cdot
a>,S)\cap\mathcal{O}_{(a,\tau)}^{G}\cup(<H\cdot
a>,T)\cap\mathcal{O}_{(a,\tau)}^{G}$ is a subrack decomposition of
$\mathcal{O}_{(a,\tau)}^{G}$ of type $D$. $\Box$
###### Theorem 2.6.
Let $G=\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ with $n>4$. Let
$\tau\in\mathbb{S}_{n}$ be of type
$(1^{\lambda_{1}},2^{\lambda_{2}},\dots,n^{\lambda_{n}})$ and
$a\in\mathbb{Z}_{2}^{n}$ with $\sigma=(a,\tau)\in G$ and $\tau\not=1$. If
$\mathcal{O}_{\sigma}^{G}$ is not of type $D$, or rack $\mathcal{O}_{a\tau}$
does not collapse, then some of the following hold.
1. (i)
$(2,3)$; $(2^{3});$
2. (ii)
$(2^{4});$ $(1,2^{2}),$
3. (iii)
$(1^{2},2^{2})$, $(1^{n-2},2)$ and $(1^{n-3},3)$ with $a_{i}=a_{j}$ when
$\tau(i)=i$ and $\tau(j)=j$.
Proof. It follows from Lemma 2.3, Lemma 2.4, Lemma 2.5, Lemma 2.1, Lemma 2.2
and [AFGV08, Th. 4.1]. $\Box$
## 3 Nichols algebras over $\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$
In this section we show that except in three cases Nichols algebras of
irreducible YD modules over classical Weyl groups are infinite dimensional.
Let ${\rm supp}M=:\\{g\in G\mid M_{g}\not=0\\}$ for $G$-comodule $M$.
###### Theorem 3.1.
([AFGV08]) If $G$ is a finite group and $\mathcal{O}_{\sigma}^{G}$ is of type
$D$, then dim $\mathfrak{B}(\mathcal{O}_{\sigma}^{G},\rho)=\infty$ for any
$\rho\in\widehat{G^{\sigma}}$.
Proof. It is clear that $\mathcal{O}_{\sigma}^{G}$ is a subrack of ${\rm
supp}M(\mathcal{O}_{\sigma}^{G},\rho)$. If $G$ is a finite group and
$\mathcal{O}_{\sigma}^{G}$ is of type $D$, then (B) in [AFGV08, Rem 2.3] holds
according to the proof of [AFGV08, Th. 3.6]. Consequently, dim
$\mathfrak{B}(\mathcal{O}_{\sigma}^{G},\rho)=\infty$. $\Box$
###### Theorem 3.2.
Let $G=\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ with $n>4$. Let
$\tau\in\mathbb{S}_{n}$ be of type
$(1^{\lambda_{1}},2^{\lambda_{2}},\dots,n^{\lambda_{n}})$ and
$a\in\mathbb{Z}_{2}^{n}$ with $\sigma=(a,\tau)\in G$ and $\tau\not=1$. If dim
$\mathfrak{B}(\mathcal{O}_{\sigma}^{G},\rho)<\infty$, then some of the
following hold.
1. (i)
$(2,3)$; $(2^{3});$
2. (ii)
$(2^{4});$$(1,2^{2}),$
3. (iii)
$(1^{2},2^{2}),$ $(1^{n-2},2)$ and $(1^{n-3},3)$ with $a_{i}=a_{j}$ when
$\tau(i)=i$ and $\tau(j)=j$.
Proof. It follows from Theorem 2.6 and [AFGV08, Th. 4.1]. $\Box$
Assume that $\tau\in\mathbb{S}_{m}$ and $a\mu=c\tau\\#d\xi\in\mathbb{B}_{n}$
with $c\tau\in\mathbb{B}_{m}$, $d\xi\in\mathbb{B}_{n-m}$, $c\tau\bot d\xi$,
$d_{1}=d_{2}=\cdots=d_{n-m}$, $\xi=id$, $m<n$. Obviously,
$\rho=\rho_{1}\otimes\rho_{2}\in\widehat{\mathbb{B}_{n}^{a\mu}}=\widehat{\mathbb{B}_{m}^{c\tau}}\times\widehat{\mathbb{B}_{n-m}^{d\xi}}.$
$\rho_{2}=\chi_{2}\otimes\mu_{2}$ with $\chi_{2}\in\widehat{Z_{2}^{n-m}}$,
$\mu_{2}\in\widehat{\mathbb{S}_{n-m}}.$
$\rho_{1}=(\chi_{1}\otimes\mu_{1})\uparrow_{G^{a\mu}_{\chi_{1}}}^{G^{a\mu}}$
with $\chi_{1}\in\widehat{(Z_{2}^{m})^{\tau}}$,
$\mu_{1}\in\widehat{(\mathbb{S}_{m}^{\tau})_{\chi_{1}}}$ when
$c=(1,1,\cdots,1)$ (see [ZZ12, Section 2.5] and [Se]). Case $a=0$ and
$a=(1,1,\cdots,1)$ were studied in paper [ZZ12, Theorem 1.1 and Table 1]. The
other case are listed as follows:
###### Corollary 3.3.
Under notation above assume
$\dim\mathfrak{B}({\mathcal{O}}_{a\mu}^{\mathbb{B}_{n}},\rho)<\infty$.
(i) Then $\rho_{1}(c\tau)=-id$ when $\rho_{2}(d\xi)=id$ and
$\rho_{1}(c\tau)=id$ when $\rho_{2}(d\xi)=-id$.
(ii) Case $\tau=(1\ 2)$ , $c=0$ and $d=(1,1,\cdots,1)$. Then
$\rho_{1}(c\tau)=\pm id$, $\rho_{2}(d\xi)=\mp id$, $\chi_{1}(c)=1$ and
$\mu_{1}(\tau)=\pm 1.$
(iii) Case $\tau=(1\ 2)$ , $c=(1,1)$ and $d=0$. Then $\rho_{1}(c\tau)=-id$,
$\rho_{2}(d\xi)=id$.
(iv) Case $\tau=(1\ 2\ 3)$ , $c=0$ and $d=(1,1,\cdots,1)$. Then
$\rho_{1}(c\tau)=id$, $\rho_{2}(d\xi)=-id$, $\chi_{1}(c)=1$ and
$\mu_{1}(\tau)=1.$
(v) Case $\tau=(1\ 2\ 3)$ , $c=(1,1,1)$ and $d=0$. Then $\rho_{1}(c\tau)=-id$,
$\rho_{2}(d\xi)=id$, $\chi_{1}(c)=-1$ and $\mu_{1}(\tau)=1.$
(vi) Case $\tau=(1\ 2)(3\ 4)$, $c=0$ and $d=(1,1)$. Then $\rho_{1}(c\tau)=id$,
$\rho_{2}(d\xi)=-id$.
(vii) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,1,0)$ and $d=(1,1)$. Then
$\rho_{1}(c\tau)=\pm id$, $\rho_{2}(d\xi)=\mp id$.
(viii) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,1,0)$ and $d=0$ . Then
$\rho_{1}(c\tau)=-id$, $\rho_{2}(d\xi)=id$.
(ix) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,0,0)$ and $d=(1,1)$. Then
$\rho_{1}(c\tau)=\pm id$, $\rho_{2}(d\xi)=\mp id$.
(x) Case $\tau=(1\ 2)(3\ 4)$, $c=(1,0,0,0)$ and $d=(0,0)$ . Then
$\rho_{1}(c\tau)=-id$, $\rho_{2}(d\xi)=id$.
Proof. (iv) If $\rho_{2}(d\xi)=id$, then
$\dim\mathfrak{B}({\mathcal{O}}_{c\tau}^{\mathbb{B}_{3}},\rho_{1})<\infty$ by
Lemma 1.4 , which constracts to [ZZ12, Th. 1.1].
(v) If $\chi_{1}(c)=1$, then there exists a contradiction by [ZZ12, Pro. 2.4,
Th. 1.1].
The others follow from Lemma 1.4 and [ZZ12, Theorem 1.1 ]. $\Box$
Let $\bar{a}$ denote $(a_{1},\cdots,a_{n},0)$ for
$a=(a_{1},a_{2},\cdots,a_{n})\in\mathbb{Z}_{2}^{n}.$
###### Lemma 3.4.
Let $\varphi$ be a map from $\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$ to
$\mathbb{Z}_{2}^{n+1}\rtimes\mathbb{S}_{n+1}$ sending $(a,\tau)$ to
$(\bar{a},\tau)$ for any $(a,\tau)\in\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$.
Then $\varphi$ is monomorphic as groups.
Proof.
$\displaystyle\varphi((a,\tau)(b,\mu))$ $\displaystyle=$
$\displaystyle\varphi(a+\tau\cdot b,\tau\mu)$ $\displaystyle=$
$\displaystyle(\bar{a}+\overline{(\tau\cdot b)},\tau\mu)$ $\displaystyle=$
$\displaystyle\varphi((a,\tau))\varphi((b,\mu)),$
for any $(a,\tau),(b,\mu)\in\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}$. $\Box$
###### Lemma 3.5.
If
$\dim\mathfrak{B}({\mathcal{O}}_{(a,\tau)}^{\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n}},\rho)=\infty$
for any
$\rho\in\widehat{(\mathbb{Z}_{2}^{n}\rtimes\mathbb{S}_{n})^{(a,\tau)}}$ , then
$\dim\mathfrak{B}({\mathcal{O}}_{\bar{a}\tau}^{\mathbb{Z}_{2}^{n+1}\rtimes\mathbb{S}_{n+1}},\mu)=\infty$,
for any
$\mu\in\widehat{(\mathbb{Z}_{2}^{n+1}\rtimes\mathbb{S}_{n+1})^{(\bar{a},\tau)}}$.
Proof. It follows from Lemma 3.4 and [ZZ12]. $\Box$
## 4 Relation between bi-one arrow Nichols algebras and
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$
In this section it is shown that bi-one arrow Nichols algebras and
$\mathfrak{B}({\mathcal{O}}_{s},\rho)$ introduced in [Gr00, AZ07, AFZ] are the
same up to isomorphisms.
For any ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, we can construct an arrow
Nichols algebra $\mathfrak{B}(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},$ $u))$
( see [ZCZ, Pro. 2.4]), written as $\mathfrak{B}(G,r,\overrightarrow{\rho},$
$u)$ in short. Let us recall the precise description of arrow YD module. For
an ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and a $kG$-Hopf bimodule
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ with the module operations
$\alpha^{-}$ and $\alpha^{+}$, define a new left $kG$-action on $kQ_{1}$ by
$g\rhd x:=g\cdot x\cdot g^{-1},\ g\in G,x\in kQ_{1},$
where $g\cdot x=\alpha^{-}(g\otimes x)$ and $x\cdot g=\alpha^{+}(x\otimes g)$
for any $g\in G$ and $x\in kQ_{1}$. With this left $kG$-action and the
original left (arrow) $kG$-coaction $\delta^{-}$, $kQ_{1}$ is a Yetter-
Drinfeld $kG$-module. Let $Q_{1}^{1}:=\\{a\in Q_{1}\mid s(a)=1\\}$, the set of
all arrows with starting vertex $1$. It is clear that $kQ_{1}^{1}$ is a
Yetter-Drinfeld $kG$-submodule of $kQ_{1}$, denoted by
$(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$, called the arrow YD module.
###### Lemma 4.1.
For any $s\in G$ and $\rho\in\widehat{G^{s}}$, there exists a bi-one arrow
Nichols algebra $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ such that
$\mathfrak{B}({\mathcal{O}}_{s},\rho)\cong\mathfrak{B}(G,r,\overrightarrow{\rho},u)$
as graded braided Hopf algebras in ${}^{kG}_{kG}\\!{\mathcal{Y}D}$.
Proof. Assume that $V$ is the representation space of $\rho$ with
$\rho(g)(v)=g\cdot v$ for any $g\in G,v\in V$. Let $C={\mathcal{O}_{s}}$,
$r=r_{C}C$, $r_{C}={\rm deg}\rho$, $u(C)=s$, $I_{C}(r,u)=\\{1\\}$ and
$(v)\rho_{C}^{(1)}(h)=\rho(h^{-1})(v)$ for any $h\in G$, $v\in V$. We get a
bi-one arrow Nichols algebra $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$.
We now only need to show that
$M({\mathcal{O}}_{s},\rho)\cong(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$
in ${}^{kG}_{kG}\\!{\mathcal{Y}D}$. We recall the notation in [ZCZ,
Proposition 1.2]. Assume $J_{C}(1)=\\{1,2,\cdots,n\\}$ and $X_{C}^{(1)}=V$
with basis $\\{x_{C}^{(1,j)}\mid j=1,2,\cdots,n\\}$ without loss of
generality. Let $v_{j}$ denote $x_{C}^{(1,j)}$ for convenience. In fact, the
left and right coset decompositions of $G^{s}$ in $G$ are
$\displaystyle G=\bigcup_{i=1}^{m}g_{i}G^{s}\ \ \hbox{and }\ \ G$
$\displaystyle=$ $\displaystyle\bigcup_{i=1}^{m}G^{s}g_{i}^{-1}\ \ ,$ (4.1)
respectively.
Let $\psi$ be a map from $M({\mathcal{O}}_{s},\rho)$ to $(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))$ by sending $g_{i}v_{j}$ to
$a_{t_{i},1}^{(1,j)}$ for any $1\leq i\leq m,1\leq j\leq n$. Since the
dimension is $mn$, $\psi$ is a bijective. See
$\displaystyle\delta^{-}(\psi(g_{i}v_{j}))$ $\displaystyle=$
$\displaystyle\delta^{-}(a_{t_{i},1}^{(1,j)})$ $\displaystyle=$ $\displaystyle
t_{i}\otimes a_{t_{i},1}^{(1,j)}=(id\otimes\psi)\delta^{-}(g_{i}v_{j}).$
Thus $\psi$ is a $kG$-comodule homomorphism. For any $h\in G$, assume
$hg_{i}=g_{i^{\prime}}\gamma$ with $\gamma\in G^{s}$. Thus
$g_{i}^{-1}h^{-1}=\gamma^{-1}g_{i^{\prime}}^{-1}$, i.e.
$\zeta_{i}(h^{-1})=\gamma^{-1}$, where $\zeta_{i}$ was defined in [ZZC04,
(0.3)]. Since $\gamma\cdot x^{(1,j)}\in V$, there exist
$k_{C,h^{-1}}^{(1,j,p)}\in k$, $1\leq p\leq n$, such that $\gamma\cdot
x^{(1,j)}=\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}x^{(1,p)}$. Therefore
$\displaystyle x^{(1,j)}\cdot\zeta_{i}(h^{-1})$ $\displaystyle=$
$\displaystyle\gamma\cdot x^{(1,j)}\ \ (\hbox{by definition of
}\rho_{C}^{(1)})$ (4.2) $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}x^{(1,p)}.$
See
$\displaystyle\psi(h\cdot g_{i}v_{j})$ $\displaystyle=$
$\displaystyle\psi(g_{i^{\prime}}(\gamma v_{j}))$ $\displaystyle=$
$\displaystyle\psi(g_{i^{\prime}}(\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}v_{p}))$
$\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}a_{t_{i^{\prime}},1}^{(1,p)}$
and
$\displaystyle h\rhd(\psi(g_{i}v_{j}))$ $\displaystyle=$ $\displaystyle
h\rhd(a_{t_{i},1}^{(1,j)})$ $\displaystyle=$ $\displaystyle
a_{ht_{i},h}^{(1,j)}\cdot h^{-1}$ $\displaystyle=$
$\displaystyle\sum_{p=1}^{n}k_{C,h^{-1}}^{(1,j,p)}a_{t_{i^{\prime}},1}^{(1,p)}\
\ (\hbox{by \cite[cite]{[\@@bibref{}{ZCZ08}{}{}, Pro.1.2]} and
}(\ref{e1.11})).$
Therefore $\psi$ is a $kG$-module homomorphism. $\Box$
Therefore $\mathfrak{B}({\mathcal{O}}_{s},\rho)$ is viewed as
$\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ sometimes.
###### Remark 4.2.
The representation $\rho$ in $\mathfrak{B}({\mathcal{O}}_{s},\rho)$ introduced
in [Gr00, AZ07] and $\rho_{C}^{(i)}$ in RSR are different. $\rho(g)$ acts on
its representation space from the left and $\rho_{C}^{(i)}(g)$ acts on its
representation space from the right.
Otherwise, when $\rho=\chi$ is a one dimensional representation, then
$(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$ is PM (see [ZZC04, Def. 1.1]).
Thus the formulae are available in [ZZC04, Lemma 1.9]. That is, $g\cdot
a_{t}=a_{gt_{i},g}$, $a_{t_{i}}\cdot g=\chi(\zeta_{i}(g))a_{t_{i}g,g}$.
## 5 Transposition
In this section we consider Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ of transposition $\sigma$ over
symmetry groups, where $\rho=sgn\otimes sgn$ or $\rho=\epsilon\otimes sgn.$
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ is finite dimensional when $n<5$
according to [MS, FK97, AZ07]. However, it is open whether
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ is finite dimensional when $n>4.$
We give the relation between $\mathfrak{B}({\mathcal{O}}_{\sigma},\rho)$ and
${\mathcal{E}}_{n}$ defined in [FK97, Def. 2.1].
Let $\sigma=(12)\in S_{n}:=G$, $\mathcal{O}_{\sigma}=\\{(ij)|1\leq i,j\leq
n\\}$, $G^{\sigma}=\\{g\in G\mid g\sigma=\sigma g\\}$.
$G=\bigcup\limits_{1\leq i<j\leq n}G^{\sigma}g_{ij}$.
Let $g_{kj}:=\left\\{\begin{array}[]{lll}id\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
k=1,j=2\\\ (2j)\ \ \ \ \ \ \ \ \ \ \ \ \ \ k=1,j>2\\\ (1j)\ \ \ \ \ \ \ \ \ \
\ \ \ \ k=2,j>2\\\ (1k)(2j)\ \ \ \ \ \ \ \ \ k>2,j>k\\\ \end{array}\right.$
and $t_{ij}=(ij).$
###### Lemma 5.1.
In The following equations in $\mathbb{S}_{m}$ hold.
$(12)id=id(12)$
$(12)(2j)=(2j1)=(1j)(12)$
$(12)(1j)=(1j2)=(2j)(12)$
$(12)(1k)(2j)=(1k2)(2j)=(1k)(2j)(kj)$
$(1j)id=(1j)id$
$(1j)(2j)=(j21)=(2j)(12)$
$(1j)(2j_{1})=(1j)(2j_{1})id\ \ j<j_{1}$
$(1j)(2j_{1})=(1j_{1})(2j)(jj_{1})(12)\ \ j>j_{1}$
$(1j)(1j)=id$
$(1j)(1j_{1})=(1j_{1}j)=(1j_{1})(jj_{1})$
$(1j)(1k)(2j)=(1kj)(2j)=(2k)(12)(kj)$
$(1j)(1k)(2j_{1})=(2j_{1})id\ \ \ j=k$
$(1j)(1k)(2j_{1})=(1k)(kj)(2j_{1})=(1k)(2j_{1})(kj)\ \ j\neq j_{1}$
$(2j)id=(2j)id$
$(2j)(2j)=id$
$(2j)(2j_{1})=(2j_{1}j)=(2j_{1})(jj_{1})$
$(2j)(1j)=(j12)=(1j)(12)$
$(2j)(1j_{1})=(1j)(2j_{1})(jj_{1})(12)\ \ j<j_{1}$
$(2j)(1j_{1})=(1j_{1})(2j)id\ \ j>j_{1}$
$(2j)(1k)(2j)=(1k)=(1k)id$
$(2j)(1k)(2j_{1})=(1k)(12)(2j_{1})=(1k)(1j_{1})(12)=(1j_{1})(12)(kj_{1})\ \
j=k$
$(2j)(1k)(2j_{1})=(2j)(2j_{1})(1k)=(2j_{1})(j_{1}j)(1k)=(2j_{1})(1k)(j_{1}j)\
\ j\neq j_{1},j\neq k$
$(kj)id=id(kj)$
$(kj)(2j)=(2k)(kj)$
$(kj)(2k)=(2j)(kj)$
$(kj)(2j_{1})=(2j_{1})(kj)\ \ k\neq j_{1},j\neq j_{1}$
$(kj)(1j)=(1k)(kj)$
$(kj)(1k)=(1j)(kj)$
$(kj)(1j_{1})=(1j_{1})(kj)\ \ k\neq j_{1},j\neq j_{1}$
$(kj)(1k)(2j)=(1k)(2j)(12)$
$(kj)(1k_{1})(2j_{1})=(1k)(2j_{1})(kj)\ \ k_{1}=j$
$(kj)(1k_{1})(2j_{1})=(1k_{1})(2k)(kj)\ \ k_{1}<k,j_{1}=j$
$(kj)(1k_{1})(2j_{1})=(1k)(2k_{1})(12)(kjk_{1})\ \ k_{1}>k,j_{1}=j$
$(kj)(1k_{1})(2j_{1})=(1j)(2j_{1})(kj)\ \ j_{1}>j,k_{1}=k$
$(kj)(1k_{1})(2j_{1})=(1j_{1})(2j)(12)(jkj_{1})\ \ j_{1}<j,k_{1}=k$
$(kj)(1k_{1})(2j_{1})=(1k_{1})(2j)(kj)\ \ k_{1}\neq j,j_{1}=k$
$(kj)(1k_{1})(2j_{1})=(1k_{1})(2j_{1})(kj)\ \ k_{1}\neq k,k_{1}\neq
j,j_{1}\neq j,j_{1}\neq k$.
Remark: By Lemma above, we can obtain $\zeta_{st}(t_{ij})\in G^{\sigma}$ such
that $g_{st}t_{ij}=\zeta_{st}(t_{ij})g_{s^{\prime}t^{\prime}}$ for any $1\leq
i,j,s,t\leq n.$
Let $a_{ij}$ denote the arrow $a_{t_{ij},1}$ from $1$ to $t_{ij}$. By Lemma
4.1, $\\{a_{ij}\mid i\not=j,1\leq i,j\leq n\\}$ generates algebra in its co-
path Hopf algebra is isomorphic to Nichols algebra
$\mathfrak{B}({\mathcal{O}}_{\sigma},\rho).$
###### Lemma 5.2.
In $\mathfrak{B}({\mathcal{O}}_{(12)},\chi)$ with $\chi=sgn\otimes sgn$ or
$\chi=\epsilon\otimes sgn$,
(i) If $i,j$ and $k$ are different each other, then there exists
$\alpha_{i,j,k}$, $\beta_{i,j,k}\in\\{1,-1\\}$ such that
$\displaystyle
a_{ij}a_{jk}+\alpha_{ijk}a_{jk}a_{ki}+\beta_{ijk}a_{ki}a_{ij}=0.$ (5.1)
(ii)
The left hand side of (5.1) $\displaystyle=$
$\displaystyle(\chi(\zeta_{ij}(t_{jk}))a_{t_{ij}t_{jk},t_{jk}}a_{t_{jk},1}+a_{t_{ij}t_{jk},t_{ij}}a_{t_{ij},1})$
(5.2) $\displaystyle+$
$\displaystyle\alpha_{ijk}(\chi(\zeta_{jk}(t_{ik}))a_{t_{jk}t_{ik},t_{ik}}a_{t_{ik},1}+a_{t_{jk}t_{ik},t_{jk}}a_{t_{jk},1})$
$\displaystyle+$
$\displaystyle\beta_{ijk}(\chi(\zeta_{ik}(t_{ij}))a_{t_{ik}t_{ij},t_{ij}}a_{t_{ij},1}+a_{t_{ik}t_{ij},t_{ik}}a_{t_{ik},1})$
(iii) If
$\chi(\zeta_{ij}(t_{jk}))\chi(\zeta_{jk}(t_{ik}))\chi(\zeta_{ik}(t_{ij}))=-1,$
then Part (i) holds.
(iv) If $i,j$ and $k$ are different each other, then Part (i) holds if and
only if
$\chi(\zeta_{ij}(t_{jk}))\chi(\zeta_{jk}(t_{ik}))\chi(\zeta_{ik}(t_{ij}))=-1.$
(v) If $i,j,k$ and $l$ are different each other, then there exist
$\lambda_{ijkl}\in\\{1,-1\\}$ such that
$a_{ij}a_{kl}=\lambda_{ijkl}a_{kl}a_{ij}.$
Proof. Let $\chi^{\prime}:=sgn\otimes sgn$ and
$\chi^{\prime\prime}:=\epsilon\otimes sgn.$ It is clear that
$M({\mathcal{O}}_{(12)},\chi)$ is a PM $\mathbb{S}_{n}$\- YD module (see
[ZZC04, Def. 1.1]) and $g\cdot a_{ij}=a_{gt_{ij},g}$, $a_{t_{ij}}\cdot
g=\chi(\zeta_{ij}(g))a_{t_{ij}g,g}$ (see [ZZC04, Lemma 1.9]). By [CR02],
$\displaystyle
a_{{ij}}a_{{kl}}=\chi(\zeta_{ij}({t_{kl}}))a_{{t_{ij}}{t_{kl}},{t_{kl}}}a_{{t_{kl}},1}+a_{{t_{ij}}{t_{kl}},{t_{ij}}}a_{{t_{ij}},1}.$
(5.3)
(ii) It follows from (5.3).
(iii) and (iv) follow from Part (ii).
(i) Let $a,b$ and $c$ denote $\zeta_{ij}(t_{jk})$, $\zeta_{jk}(t_{ik})$ and
$\zeta_{ik}(t_{ij})$ in the following Table 1 in short, respectively.
case $a$ $b$ $c$ $\chi^{\prime}(a)$ $\chi^{\prime}(b)$ $\chi^{\prime}(c)$
$\chi^{\prime\prime}(a)$ $\chi^{\prime\prime}(b)$ $\chi^{\prime\prime}(c)$
$2<i<j<k$ $(kj)$ $(12)(ijk)$ $(ij)$ $-1$ $-1$ $-1$ $1$ $-1$ $1$ $i=1,j=2<k$
$(1)$ $(1)$ $(12)$ $1$ $1$ $-1$ $1$ $1$ $-1$ $i=1,2<j<k$ $(kj)$ $(12)(kj)$
$(1)$ $-1$ $1$ $1$ $1$ $-1$ $1$ $i=2<j<k$ $(kj)$ $(1)$ $(12)(jk)$ $-1$ $1$ $1$
$1$ $1$ $-1$ $2<i<k<j$ $(kj)$ $(ik)$ $(12)(ijk)$ $-1$ $-1$ $-1$ $1$ $1$ $-1$
$i=1,k=2<j$ $(1)$ $(12)$ $(1)$ $1$ $-1$ $1$ $1$ $-1$ $1$ $i=1,2<k<j$ $(kj)$
$(1)$ $(12)(jk)$ $-1$ $1$ $1$ $1$ $1$ $-1$ $i=2<k<j$ $(kj)$ $(kj)(12)$ $(1)$
$-1$ $1$ $1$ $1$ $-1$ $1$
$\hbox{Table }1$
By Table 1,
$\chi(\zeta_{ij}(t_{jk}))\chi(\zeta_{jk}(t_{ik}))\chi(\zeta_{ik}(t_{ij}))=-1$.
Consequently, Part (i) holds by Part (iii).
(v) By (5.3), $a_{ij}a_{kl}=\lambda_{ijkl}a_{kl}a_{ij}$ if and only if
$(\chi(\zeta_{ij}(t_{kl}))-\lambda_{ijkl})=(1-\chi(\zeta_{kl}(t_{ij}))\lambda_{ijkl})=0$.
Since $(ij)(kl)=(kl)(ij)$, we have $g_{ij}(kl)=g_{ij}(kl)g_{ij}g_{ij}$ with
$g_{ij}(kl)g_{ij}\in\mathbb{S}_{n}^{(12)}$ and
$\zeta_{ij}((kl))=g_{ij}(kl)g_{ij}.$ See
$\zeta_{ij}(t_{kl})=g_{ij}t_{kl}g_{ij}=$
$\left\\{\begin{array}[]{lll l}(kl)&\hbox{ if }k,l>2\\\ (kl)&\hbox{ if
}i=1,j=2\\\ =(2j)(kl)(2j)=(lj)\hbox{ or }(kj)\hbox{ or }(kl)&\hbox{ if
}i=1,j>2,k=2\hbox{ or }i=1,j>2,l=2\\\ &\hbox{ or }i=1,j>2,k\not=2,l\not=2\\\
=(1j)(kl)(1j)=(lj)\hbox{ or }(kj)\hbox{ or }(kl)&\hbox{ if }i=2,j>2,k=1\hbox{
or }i=1,j>2,l=1\\\ &\hbox{ or }i=1,j>2,k\not=1,l\not=1\\\ \hbox{ is a
transposition of }\\\ \hbox{ two greater numbers than }$2$&\hbox{ if
}i,j>2\end{array}\right..$ Consequently, $\zeta_{ij}(t_{kl})$ is a
transposition of two greater numbers than $2$ and
$\chi^{\prime}(\zeta_{ij}(t_{kl}))=-1$ and
$\chi^{\prime\prime}(\zeta_{ij}(t_{kl}))=1$. Similarly,
$\chi^{\prime}(\zeta_{kl}(t_{ij}))=-1$ and
$\chi^{\prime\prime}(\zeta_{kl}(t_{ij}))=1$. Consequently, it is enough to set
$\lambda_{ijkl}=:\chi(\zeta_{ij}(t_{kl}))$. $\Box$
###### Definition 5.3.
(See [FK97, Def. 2.1]) algebra ${\mathcal{E}}_{n}$ is generated by
$\\{x_{ij}\mid 1\leq i<j\leq n\\}$ with definition relations:
(i) $x_{ij}^{2}=0$ for $i<j.$
(ii) $x_{ij}x_{jk}=x_{jk}x_{ik}+x_{ik}x_{ij}$ and
$x_{jk}x_{ij}=x_{ik}x_{jk}+x_{ij}x_{ik},$ for $i<j<k.$
(iii) $x_{ij}x_{kl}+x_{kl}x_{ij}=0$ for any distinct $i,j,k,l$ and $l,$
$i<j,k<l.$
Equivalently, algebra ${\mathcal{E}}_{n}$ is generated by $\\{x_{ij}\mid
i\not=j,1\leq i,j\leq n\\}$ with definition relations:
(i) $x_{ij}^{2}=0$, $x_{ij}=-x_{ji}$, for $1\leq i,j\leq n.$
(ii) $x_{ij}x_{jk}+x_{jk}x_{ki}+x_{ki}x_{ij}=0,$ for $1\leq i,j,k\leq n.$
(iii) $x_{ij}x_{kl}=x_{kl}x_{ij}$ for any distinct $i,j,k$ and $l.$
By [FK97, Th. 7.1], a subring of $\mathcal{E}_{n}$ is isomorphic to the
cohomology ring of the flag manifold.
By [AFGV08] and [HS08], most of Nichols algebras over $\mathbb{S}_{n}$ are
infinite dimensional when $n>5$. Consequently, we have
###### Conjecture 5.4.
Let $\alpha_{ijk},\beta_{ijk},\gamma_{ij},\lambda_{ijkl}\in\\{1,-1\\}$. Assume
that algebra $A(\alpha,\beta,\gamma,\lambda)$ is generated by $\\{x_{ij}\mid
i\not=j,1\leq i,j\leq n\\}$ with definition relations:
(i) $x_{ij}^{2}=0$, $x_{ij}=\gamma_{ij}x_{ji}$, for $1\leq i,j\leq n.$
(ii) $x_{ij}x_{jk}+\alpha_{ijk}x_{jk}x_{ki}+\beta_{ijk}x_{ki}x_{ij}=0,$ for
$1\leq i,j,k\leq n.$
(iii) $x_{ij}x_{kl}=\lambda_{ijkl}x_{kl}x_{ij}$ for any distinct $i,j,k$ and
$l.$
Then $A(\alpha,\beta,\gamma,\lambda)$ is infinite dimensional when $n>4$.
Furthermore, $\mathfrak{B}({\mathcal{O}}_{(12)},\rho)$ is infinite dimensional
with $\rho=sgn\otimes sgn$ or $\rho=\epsilon\otimes sgn$ when $n>6$.
Obviously, $\mathcal{E}_{n}=A(1,1,-1,1).$
## 6 Appendix
In this section we give another proof of Lemma 2.5.
###### Lemma 6.1.
(i) If $\varphi$ is epimorphic from group $G$ onto group $\bar{G}$, then
$\varphi\mid_{\mathcal{O}_{a}^{G}}$ is epimorphic from $\mathcal{O}_{a}^{G}$
onto $\mathcal{O}_{\varphi(a)}^{\bar{G}}$ as racks.
(ii) If $\pi$ is a map from $\mathbb{Z}_{2}^{n}\rtimes H$ to $H$ by sending
$a\tau$ to $\tau$ for any $a\tau\in A\rtimes H,$ then $\pi$ is epimorphic as
groups.
Proof. (i) For any $\bar{x},\bar{y}\in\mathcal{O}_{\varphi(a)}^{\bar{G}}$ with
$x,y\in G$ and $\bar{x}=\varphi(x),\bar{y}=\varphi(y)$, It is clear
$\varphi(x)\rhd\varphi(y)=\varphi(x\rhd y)$. i.e. $\varphi$ is a homomorphism
of racks. Since $\bar{x}\in\mathcal{O}_{\varphi(a)}^{\bar{G}}$, there exists
$h\in G$ such that $\varphi(h)\rhd\varphi(a)=\bar{x}$. Consequently,
$\bar{x}\in\varphi(\mathcal{O}_{a}^{G})$.
(ii) It is clear. $\Box$
###### Lemma 6.2.
(i) If $\varphi$ is epimorphic from $X$ to $Y$ as racks and $Y$ is of type
$D$, then $X$ is of type $D$.
(ii) Let $G:=\mathbb{Z}_{2}^{n}\rtimes H$ and $\sigma=(a,\tau)\in G$ with
$a\in\mathbb{Z}_{2}^{n},\tau\in H.$ If $\mathcal{O}_{\tau}^{H}$ is of type
$D$, then so is $\mathcal{O}_{(a,\tau)}^{G}$.
Proof. (i) Let $Y=R\cup S$ be a decomposition of subracks with type $D$ and
let $r\in R$, $s\in S$ such that $sq(r,s)\not=s.$ Set
$\bar{R}:=\varphi^{-1}(R)$, $\bar{S}:=\varphi^{-1}(S)$;
$\varphi(r^{\prime})=r$ and $\varphi(s^{\prime})=s$ with
$r^{\prime}\in\bar{R}$ and $s^{\prime}\in\bar{S}$. It is clear that
$\bar{R}\cup\bar{S}$ be a decomposition of subracks and
$sq(r^{\prime},s^{\prime})\not=s^{\prime}.$
(ii) It follows from Part (i) and Lemma 6.1. $\Box$
Remark: Lemma 6.2 (i) appeared in [AFGV10, Section 2.4], but they had not
proof.
## References
* [AFZ] N. Andruskiewitsch, F. Fantino, S. Zhang, On pointed Hopf algebras associated with the symmetric groups, Manuscripta Math., 128(2009) 3, 359-371.
* [AFGV08] N. Andruskiewitsch, F. Fantino, M. Graña and L.Vendramin, Finite-dimensional pointed Hopf algebras with alternating groups are trivial, preprint arXiv:0812.4628, to appear Aparecer en Ann. Mat. Pura Appl..
* [AFGV10] N. Andruskiewitsch, F. Fantino, M. Graña and L.Vendramin, On Nichols algebras associated to simple racks, preprint arXiv:1006.5727.
* [AZ07] N. Andruskiewitsch and S. Zhang, On pointed Hopf algebras associated to some conjugacy classes in $\mathbb{S}_{n}$, Proc. Amer. Math. Soc. 135 (2007), 2723-2731.
* [AG03] N. Andruskiewitsch, M. Grana, From racks to pointed Hopf algebras, Adv. in Math. 178 (2003)2, 177–243.
* [AS10] N. Andruskiewitsch, H.-J. Schneider, On the classification of finite-dimensional pointed Hopf algebras, Ann. Math., 171 (2010) 1, 375-417.
* [CR02] C. Cibils and M. Rosso, _Hopf quivers_ , J. Alg., 254 (2002), 241-251.
* [FK97] S. Fomin and A.N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Progress in Geometry, ed. J.-L. Brylinski and R. Brylinski, 1997.
* [Gr00] M. Graña, On Nichols algebras of low dimension, Contemp. Math. 267 (2000), 111–134.
* [HS08] I. Heckenberger and H.-J. Schneider, Root systems and Weyl groupoids for Nichols algebras, preprint arXiv:0807.0691, to appear Proc. London Math. Soc..
* [MS] A. Milinski and H-J. Schneider, _Pointed Indecomposable Hopf Algebras over Coxeter Groups_ , Contemp. Math. 267 (2000), 215–236.
* [Se] J.-P. Serre, Linear representations of finite groups, Springer-Verlag, New York 1977.
* [ZZC04] Shouchuan Zhang, Y-Z Zhang and H. X. Chen, Classification of PM Quiver Hopf Algebras, J. Alg. and Its Appl. 6 (2007)(6), 919-950. Also see in math. QA/0410150.
* [ZCZ] S. Zhang, H. X. Chen and Y.-Z. Zhang, Classification of quiver Hopf algebras and pointed Hopf algebras of type one, Bull. Aust. Math. Soc. 87 (2013), 216-237. Also in arXiv:0802.3488.
* [ZZ12] Shouchuan Zhang, Yao-Zhong Zhang, Pointed Hopf Algebras with classical Weyl Groups, International Journal of Mathematics, 23 (2012) 1250066.
|
arxiv-papers
| 2013-07-31T05:45:07 |
2024-09-04T02:49:48.816740
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Shouchuan Zhang, Weicai Wu, Zhengtang Tan, Yao-Zhong Zhang",
"submitter": "Shouchuan Zhang",
"url": "https://arxiv.org/abs/1307.8227"
}
|
1307.8269
|
Serge Abiteboul and Émilien Antoine INRIA Saclay & ENS Cachan
[email protected]
Gerome Miklau INRIA Saclay & UMass Amherst [email protected]
Julia Stoyanovich and Vera Zaychik Moffitt Drexel University
[email protected] and [email protected]
# Introducing Access Control in Webdamlog††thanks: This work has been
partially funded by the European Research Council under the European
Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grantWebdam,
agreement 226513. http://webdam.inria.fr/
###### Abstract
We survey recent work on the specification of an access control mechanism in a
collaborative environment. The work is presented in the context of the
WebdamLog language, an extension of datalog to a distributed context. We
discuss a fine-grained access control mechanism for intentional data based on
provenance as well as a control mechanism for delegation, i.e., for deploying
rules at remote peers.
## 1 Introduction
The personal _data_ and favorite _applications_ of a Web user are typically
distributed across many heterogeneous devices and systems, e.g., residing on a
smartphone, laptop, tablet, TV box, or managed by Facebook, Google, etc.
Additional data and computational resources are also available to the user
from relatives, friends, colleagues, possibly via social network systems. Web
users are thus increasingly at risk of having private data leak and in general
of losing control over their own information. In this paper, we consider a
novel _collaborative access control mechanism_ that provides users with the
means to control access to their data by others and the functioning of
applications they run.
Our focus is information management in environments where both data and
programs are distributed. In such settings, there are four essential
requirements for access control:
Data access
Users would like to control who can read and modify their information.
Application control
Users would like to control which applications can run on their behalf, and
what information these applications can access.
Data dissemination
Users would like to control how pieces of information are transferred from one
participant to another, and how they are combined, with the owner of each
piece keeping some control over it.
Declarativeness
The specification of the exchange of data, applications, and of access control
policies should be declarative. The goal is to enable anyone to specify access
control.
To illustrate each of these requirements, let us consider the functionalities
of a social network such as Facebook, in which users interact by exchanging
data and applications. First, a user who wants to control who sees her
information, can use a classic access control mechanism, such as the one
currently employed by Facebook, based on groups of friends. Next, let us
consider a user who installs an application. This typically involves opening
much of her data to a server that is possibly managed by an unknown third
party. Many Facebook users see this as unreasonable, and would like to control
what the application can do on their behalf, and what information the
application can access. Third, with respect to data dissemination, users would
like to specify what other users can do with their data, e.g., whether their
friends are allowed to show their pictures to their respective friends.
Finally, the users want to specify access control on their data without having
to write programs. Thus, this simple example already demonstrates the need for
each one of the four above requirements.
From a formal point of view, we define an access control mechanism for
WebdamLog, a declarative _datalog_ -style language that emphasizes cooperation
between autonomous peers webdamlog . We obtain a language that allows for
declaratively specifying both data exchange and access control policies
governing this exchange. There are different aspects to our access control:
* •
For extensional data, the mechanism is standard, based on access control lists
at each peer, specifying who owns and who can read/write data in each relation
of that peer.
* •
For intentional data, the mechanism is more sophisticated and fine-grained. It
is based on _provenance_. In brief, only users with read access to all the
tuples that participated in the derivation of a fact can read this fact.
* •
The previous two mechanisms are used by default, and we also support the means
of overriding them.
* •
Finally, we introduce a mechanism for controlling the use of _delegation_ in
WebdamLog, which allows peers to delegate work to remote peers by installing
rules, and is one of the main originalities of WebdamLog.
Note that access control is implemented natively as part of the WebdamLog
framework. The main idea is to use extensional relations to specify access to
extensional data. Thus accessibility of extensional facts is itself recorded
as extensional facts, which can then be used by a WebdamLog program to _derive
access_ to intentional data.
### Organization
This short paper is organized as follows. The Webdamlog language is presented
in Section 2. In Section 3, we present some aspects of access control. We
conclude in Section 4.
## 2 Webdamlog
In webdamlog , we introduced Webdamlog, a novel Datalog-style rule-based
language. In Webdamlog, each piece of information belongs to a principal. We
distinguish between two kinds of principals: peer and virtual principal. A
peer, e.g., $\mathsf{AlicePhone}$ or $\mathsf{Picasa}$, has storage and
processing capabilities, and can receive and handle queries and update
requests. A virtual principal, e.g., $\mathsf{Alice}$ or
$\mathsf{RockClimbingClub}$, represents a user or a group of users, and relies
on peers for storage and processing. We further distinguish between facts,
representing local tuples and messages between peers, and rules, which may be
evaluated locally or delegated to other peers.
Webdamlog is primarily meant to be used in a distributed setting. Perhaps the
main novelty of the language is the notion of delegation, which amounts to a
peer installing a rule on another peer. In its simplest form, delegation is a
remote materialized view. In its general form, it allows peers to exchange
knowledge beyond simple facts, providing the means for a peer to delegate work
to other peers. We will not describe Webdamlog in detail here, but will
illustrate it with examples, referring the interested reader to webdamlog .
The following are examples of Webdamlog facts:
$\mathsf{agenda}$@$\mathsf{AlicePhone}$($\mathsf{12/12/2012,10:00,John,Orsay}$)
---
$\mathsf{photos}$@$\mathsf{Picasa}$$\mathsf{(fileName:picture34.jpg,}$
$\mathsf{date:09/12/2012,byteStream:010001)}$
$\mathsf{writeSecret}$@$\mathsf{Picasa}$($\mathsf{login:Alice,password:HG-
FT23}$)
The first fact represents a tuple in relation $\mathsf{agenda}$ on peer
$\mathsf{AlicePhone}$ with information about an upcoming meeting, and the
second, a photo in Alice’s Picasa account (a tuple in relation
$\mathsf{photos}$ on peer $\mathsf{Picasa}$). The third fact represents
Alice’s login credentials for her Picasa account (in relation
$\mathsf{writeSecret}$ on peer $\mathsf{Picasa}$). Suppose that Alice wishes
to retrieve, and store on her laptop, photos from Fontainbleau outings that
were taken by other members of her rock climbing group. To this effect, Alice
issues the following rule:
$\mathsf{outingPhotos}$@$\mathsf{AliceLaptop}$($\mathsf{\$pic}$) :-
---
$\mathsf{rockClimbingGroup}$@$\mathsf{Facebook}$($\mathsf{\$member}$),
$\mathsf{findPhoto}$@$\mathsf{AliceLaptop}$($\mathsf{\$member,\$photos,\$peer}$),
$\mathsf{\$photos}$@$\mathsf{\$peer}$($\mathsf{\$pic,\$meta}$),
$\mathsf{contains}$@$\mathsf{\$peer}$($\mathsf{\$meta,Fontainbleau}$)
This rule is a standard Webdamlog rule that illustrates various salient
features of the language. First, the rule is declarative. Second, the
assignment of values to peer names (e.g., $\mathsf{\$peer}$) and relation
names (e.g., $\mathsf{\$photos}$) is determined during rule evaluation. Third,
for $\mathsf{\$peer}$ assigned to a system other than $\mathsf{AliceLaptop}$
(e.g., $\mathsf{Picasa}$ or $\mathsf{Flickr}$), the activation of this rule
will result in activating rules (by delegation), or in some processing
simulating them in other systems. The evaluation of rules such as this one is
performed by the Webdamlog system, which is responsible for handing
communication and security protocols, and also includes a datalog evaluation
engine, namely the Bud system Hellerstein10 .
The semantics of a Webdamlog rule depends on the location of the relations
occurring in this rule. Let $\mathsf{p}$ be a particular peer. We say that a
rule is _local_ to $\mathsf{p}$ if the relations occurring in the body are all
in $\mathsf{p}$; intuitively, $\mathsf{p}$ can run such a rule. The effect of
a rule will also depend on whether the relation in the head of the rule is
local (to $\mathsf{p}$) or not and whether it is extensional or intentional.
Generally speaking, Webdamlog supports the following kinds of rules.
* •
A. Local rule with local intentional head. These rules, like classical datalog
rules, define local intentional relations, i.e., logical views.
* •
B. Local rule with local extensional head. These rules derive new facts that
are inserted into the local database. Note that, by default, as in Dedalus
dedalus , facts are not persistent. To have them persist, we use rules of the
form $\mathsf{m}$@$\mathsf{p}$($\mathsf{U}$) :-
$\mathsf{m}$@$\mathsf{p}$($\mathsf{U}$). Deletion can be captured by
controlling the persistence of facts.
* •
C. Local rule with non-local extensional head. Facts derived by such rules are
sent to other peers and stored in an extensional relation at that peer,
implementing a form of messaging.
* •
D. Local rule with non-local intentional head. Such a rule defines a new
intentional relation at a remote peer based on local relations of the defining
peer.
* •
E. Non-local. Rules of this kind allow a peer to install a rule at a remote
peer, which is itself defined in terms of relations of other remote peers.
This is the _delegation_ mechanism that enables the sharing of application
logic by peers, for instance, obtaining logic (rules) from other sites, and
deploying logic (rules) to other sites.
## 3 Access control in Webdamlog
We present three simple examples that highlight particular aspects of access
control in WebdamlLog.
### Fine-grained access control on intentional data
Suppose that an intentional relation
$\mathsf{\mathsf{allPhotos}}$@$\mathsf{Alice}$ has been specified by Alice.
Suppose that Alice gives the right to friends, say Bob and Sue, to insert
pictures into this relation. Alice’s friends can do this by defining the
rules:
[at Bob] | $\mathsf{\mathsf{allPhotos}}$@$\mathsf{Alice}$($\mathsf{\$f}$) :- $\mathsf{{\underline{\mathsf{bobPhotos}}}}$@$\mathsf{Bob}$($\mathsf{\$f}$)
---|---
[at Sue] | $\mathsf{\mathsf{allPhotos}}$@$\mathsf{Alice}$($\mathsf{\$f}$) :- $\mathsf{{\underline{\mathsf{suePhotos}}}}$@$\mathsf{Sue}$($\mathsf{\$f}$)
(Relation names $\underline{\mathsf{bobPhotos}}$ and
$\underline{\mathsf{suePhotos}}$ are underlined to indicate that they are
extensional.) $\mathsf{\mathsf{allPhotos}}$@$\mathsf{Alice}$ is intentional
and is now defined as the union of
$\mathsf{{\underline{\mathsf{bobPhotos}}}}$@$\mathsf{Bob}$ and
$\mathsf{{\underline{\mathsf{suePhotos}}}}$@$\mathsf{Sue}$.
The read privilege on $\mathsf{\mathsf{allPhotos}}$@$\mathsf{Alice}$ is a
prerequisite to having access to the contents of this relation, but access is
also controlled by the provenance of each fact, making read access fine-
grained. One can think of each intentional fact as carrying its provenance,
i.e., how it has been derived. In our simple example of Alice’s album, the
provenance of a photo coming from Bob will simply be the provenance token
associated with the corresponding fact at Bob. Then, to be able to read a fact
in $\mathsf{\mathsf{allPhotos}}$@$\mathsf{Alice}$ that is coming from
$\mathsf{{\underline{\mathsf{bobPhotos}}}}$@$\mathsf{Bob}$, Charlie will need
read access on $\mathsf{{\underline{\mathsf{bobPhotos}}}}$@$\mathsf{Bob}$. To
see a slightly more complicated example, suppose that a fact $F$ may be
obtained by taking the join of two base facts $F_{1},F_{2}$; and that the same
fact may be obtained alternatively by projection of a fact $F_{3}$. To access
$F$ a peer would need to have read access to its container (the relation that
contains it) as well as to facts that suffice to derive it, here $F_{3}$ or
the pair $(F_{1},F_{2})$. In other words, a peer that has read access to an
intentional fact must have sufficient rights to derive that fact.
### Overriding the default semantics
For intentional data, we use by default an access control based on the full
provenance of each fact. (If a fact is derived in several ways, each
derivation specifies a sufficient access right.) Access control based on full
provenance may be more restrictive than is needed in some applications, and we
provide the means to override it. Consider the following rule that Alice uses
to publish her own photos to her friends:
[at Alice] | $\mathsf{allPhotos}$@$\mathsf{\$x}$($\mathsf{\$f}$) :-
---|---
| $\underline{\mathsf{alicePhotos}}$@$\mathsf{Alice}$($f), [hide
$\mathsf{friends}$@$\mathsf{Alice}$($\mathsf{\$x}$)]
Ignore the hide annotation first. This rule is copying the photos of Alice’s
friends into their respective $\mathsf{allPhotos}$ relations. A friend, say
Pete, will be allowed to see one of Alice’s photos only if he is entitled to
read the relation $\mathsf{friends}$@$\mathsf{Alice}$. Now, it may be the case
that Alice does not want to share this relation with Pete, and so Pete will
not see her photos. The effect of the hide annotation is that the provenance
of facts coming from $\mathsf{friends}$@$\mathsf{Alice}$ is hidden. With this
annotation, Pete will be able to see the photos. This feature is indispensable
in preventing access control from becoming too restrictive.
### Controlling delegation
Recall that general delegation allows rules with non-local relations in the
body. This leads to significant flexibility for application development and is
the main distinguishing feature of the Webdamlog framework. It also creates
challenges for access control.
The following example illustrates the danger of a simplistic semantics for
non-local rules. Consider the two rules:
[at Bob] | $\mathsf{{\underline{\mathsf{message}}}}$@$\mathsf{Sue}$(“I hate you”) :- $\mathsf{date}$@$\mathsf{Alice}$($\mathsf{\$d}$)
---|---
|
$\mathsf{{\underline{\mathsf{aliceSecret}}}}$@$\mathsf{Bob}$($\mathsf{\$x}$)
:-
| $\mathsf{date}$@$\mathsf{Alice}$($\mathsf{\$d}$),
$\mathsf{{\underline{\mathsf{secret}}}}$@$\mathsf{Alice}$($\mathsf{\$x}$)
If we ignore access rights, by delegation, this results in running the
following two rules at Alice’s peer:
[at Alice] | $\mathsf{{\underline{\mathsf{message}}}}$@$\mathsf{Sue}$(“I hate you”) :- $\mathsf{date}$@$\mathsf{Alice}$($\mathsf{\$d}$)
---|---
|
$\mathsf{{\underline{\mathsf{aliceSecret}}}}$@$\mathsf{Bob}$($\mathsf{\$x}$)
:-
| $\mathsf{date}$@$\mathsf{Alice}$($\mathsf{\$d}$),
$\mathsf{{\underline{\mathsf{secret}}}}$@$\mathsf{Alice}$($\mathsf{\$x}$)
Assuming $\mathsf{date}$@$\mathsf{Alice}$($\mathsf{\$d}$) succeeds, then by
the first rule $\mathsf{Alice}$ sends some hate mail to $\mathsf{Sue}$, and by
the second it sends the contents of the relation
$\mathsf{{\underline{\mathsf{secret}}}}$@$\mathsf{Alice}$ to $\mathsf{Bob}$,
even if $\mathsf{Alice}$ did not give read access on this relation to
$\mathsf{Bob}$.
The main reason for this problem is that (by the standard semantics of
Webdamlog) we are running the delegation rules as if they were run by
$\mathsf{Alice}$. Under access control, we are going to run them in a
_sandbox_ with $\mathsf{Bob}$’s privileges. So with the first rule, the hate
message will be sent but marked as coming from $\mathsf{Bob}$. And with the
second, the data will be sent only if $\mathsf{Bob}$ has read access to
$\mathsf{{\underline{\mathsf{secret}}}}$@$\mathsf{Alice}$. So, for a client
$c$ delegating a rule to a server, the semantics of delegation under access
control policies guarantees that:
* •
If the rule has side effects (e.g., it results in the insertion of tuples in
the relation of another peer), the author of the update is $c$.
* •
The access privileges with which the rule executes are those of $c$.
Note that, in practice, $\mathsf{Alice}$ sends $\mathsf{Sue}$ a message saying
that the author of the message is $\mathsf{Bob}$. So, $\mathsf{Sue}$ may
question this fact and asks $\mathsf{Alice}$ to prove that this is indeed the
case. But if this is indeed the case, $\mathsf{Alice}$ has the delegation from
$\mathsf{Bob}$ to prove her good faith.
Delegation is at the heart of distributed processing. With delegation, a peer
$\mathsf{p}$ can ask another peer $\mathsf{q}$ to do some processing on its
behalf. A natural question is whether this will yield exactly the same
semantics (with possibly very different performance) as if $\mathsf{p}$ were
getting the data locally and running a local computation. It turns out that
the semantics is different. This is because $\mathsf{q}$ will use data that
(i) $\mathsf{q}$ has access to; and (ii) $\mathsf{p}$ has access to (because
of the sandboxing). On the other hand, a local computation at $\mathsf{p}$ is
limited by (ii) but not by (i).
## 4 Conclusion
The WebdamLog language has been introduced in webdamlog . The system has been
implemented and different aspects have been demonstrated in conferences
webdamexchange:demo ; AbiteboulAMST13 . The access control mechanism is
currently being implemented. The fine-grained mechanism for intentional data
raises various issues. In particular, the materialization of intentional
relations may generate lots of data if performed naively. This is the topic of
on-going research.
## References
* [1] S. Abiteboul, E. Antoine, G. Miklau, J. Stoyanovich, and J. Testard. [Demo] rule-based application development using WebdamLog. In SIGMOD, 2013.
* [2] S. Abiteboul, M. Bienvenu, A. Galland, and E. Antoine. A rule-based language for Web data management. In PODS, 2011.
* [3] P. Alvaro, W. R. Marczak, N. Conway, J. M. Hellerstein, D. Maier, and R. C. Sears. Dedalus: Datalog in Time and Space. Technical Report UCB/EECS-2009-173, EECS Department, University of California, Berkeley, December 2009.
* [4] E. Antoine, A. Galland, K. Lyngbaek, A. Marian, and N. Polyzotis. [Demo] Social Networking on top of the WebdamExchange System. In ICDE, 2011.
* [5] J. M. Hellerstein. The declarative imperative: experiences and conjectures in distributed logic. SIGMOD Record, 39(1):5–19, 2010.
|
arxiv-papers
| 2013-07-31T10:21:33 |
2024-09-04T02:49:48.826936
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serge Abiteboul, \\'Emilien Antoine, Gerome Miklau, Julia Stoyanovich,\n Vera Zaychik Moffitt",
"submitter": "\\'Emilien Antoine",
"url": "https://arxiv.org/abs/1307.8269"
}
|
1307.8292
|
# Three generated, squarefree, monomial ideals
Dorin Popescu and Andrei Zarojanu Dorin Popescu,
Simion Stoilow Institute of Mathematics of Romanian Academy, Research unit 5,
P.O.Box 1-764, Bucharest 014700, Romania
E-mail: dorin.popescu @ imar.ro Andrei Zarojanu,
Faculty of Mathematics and Computer Sciences, University of Bucharest,
Str. Academiei 14, Bucharest, Romania, and Simion Stoilow Institute of
Mathematics of Romanian Academy, Research group of the project ID-
PCE-2011-1023, P.O.Box 1-764, Bucharest 014700, Romania
E-mail: andrei_zarojanu @ yahoo.com
###### Abstract.
Let $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra
over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$
is generated by three monomials of degrees $d$. If the Stanley depth of $I/J$
is $\leq d+1$ then the usual depth of $I/J$ is $\leq d+1$ too.
> Key Words: Monomial Ideals, Depth, Stanley depth
>
> 2010 Mathematics Subject Classification: Primary 13C15,
> Secondary Secondary 13F20, 13F55, 13P10.
## 1\. Introduction
Let $S=K[x_{1},\ldots,x_{n}]$, $n\in{\bf N}$, be a polynomial ring over a
field $K$. Let $I\supsetneq J$ be two squarefree monomial ideals of $S$ and
$u\in I\setminus J$ a monomial in $I/J$. For
$Z\subset\\{x_{1},\ldots,x_{n}\\}$ with $(J:u)\cap K[Z]=0$, let $uK[Z]$ be the
linear $K$-subspace of $I/J$ generated by the elements $uf$, $f\in K[Z]$. A
presentation of $I/J$ as a finite direct sum of such spaces ${\mathcal{D}}:\
I/J=\bigoplus_{i=1}^{r}u_{i}K[Z_{i}]$ is called a Stanley decomposition of
$I/J$. Set
$\operatorname{sdepth}(\mathcal{D}):=\operatorname{min}\\{|Z_{i}|:i=1,\ldots,r\\}$
and
$\operatorname{sdepth}\ I/J:=\operatorname{max}\\{\operatorname{sdepth}\
({\mathcal{D}}):\;{\mathcal{D}}\;\text{is a Stanley decomposition
of}\;I/J\\}.$
Stanley’s Conjecture says that the Stanley depth
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$. The Stanley
depth of $I/J$ is a combinatorial invariant and does not depend on the
characteristic of the field $K$. If $J=0$ then this conjecture holds for
$n\leq 5$ by [12], or when $I$ is an intersection of four monomial prime
ideals by [11], [13], or an intersection of three monomial primary ideals by
[23], or a monomial almost complete intersection by [3]. The Stanley depth and
the Stanley’s Conjecture are similarly given when $I,J$ are not squarefree. In
the non squarefree monomial ideals a useful inequality is
$\operatorname{sdepth}I\leq\operatorname{sdepth}\sqrt{I}$ (see [8, Theorem
2.1]).
Suppose that $I$ is generated by squarefree monomials of degrees $\geq d$ for
some positive integer $d$. We may assume either that $J=0$, or $J$ is
generated in degrees $\geq d+1$ after a multigraded isomorphism. We have
$\operatorname{depth}_{S}I\geq d$ by [5, Proposition 3.1] and it follows
$\operatorname{depth}_{S}I/J\geq d$ (see [15, Lemma 1.1]). Depth of $I/J$ is a
homological invariant and depends on the characteristic of the field $K$. The
Stanley decompositions of $S/J$ corresponds bijectively to partitions into
intervals of the simplicial complex whose Stanley-Reisner ring is $S/J$. If
Stanley’s Conjecture holds then the simplicial complexes are partitionable
(see [4]). Using this idea an equivalent definition of Stanley’s depth of
$I/J$ was given in [5].
Let $P_{I\setminus J}$ be the poset of all squarefree monomials of $I\setminus
J$ with the order given by the divisibility. Let ${\mathcal{P}}$ be a
partition of $P_{I\setminus J}$ in intervals $[u,v]=\\{w\in P_{I\setminus
J}:u|w,w|v\\}$, let us say $P_{I\setminus J}=\cup_{i}[u_{i},v_{i}]$, the union
being disjoint. Define
$\operatorname{sdepth}{\mathcal{P}}=\operatorname{min}_{i}\operatorname{deg}v_{i}$.
Then
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{\mathcal{P}}\operatorname{sdepth}{\mathcal{P}}$,
where ${\mathcal{P}}$ runs in the set of all partitions of $P_{I\setminus J}$
(see [5], [21]).
For more than thirty years the Stanley Conjecture was a dream for many people
working in combinatorics and commutative algebra. Many people believe that
this conjecture holds and tried to prove directly some of its consequences.
For example in this way a lower bound of depth given by Lyubeznik [10] was
extended by Herzog at al. [6] for sdepth.
Some numerical upper bounds of sdepth give also upper bounds of depth, which
are independent of char $K$. More precisely, write $\rho_{j}(I\setminus J)$
for the number of all squarefree monomials of degrees $j$ in $I\setminus J$.
###### Theorem 1.1.
(Popescu [16, Theorem 1.3]) Assume that $\operatorname{depth}_{S}(I/J)\geq t$,
where $t$ is an integer such that $d\leq t<n$. If $\rho_{t+1}(I\setminus
J)<\alpha_{t}:=\sum_{i=0}^{t-d}(-1)^{t-d+i}\rho_{d+i}(I\setminus J)$, then
$\operatorname{depth}_{S}(I/J)=t$ independently of the characteristic of $K$.
The proof uses Koszul homology and is not very short. An extension is given
below.
###### Theorem 1.2.
( Shen [20, Theorem 2.4]) Assume that $\operatorname{depth}_{S}(I/J)\geq t$,
where $t$ is an integer such that $d\leq t<n$. If for some $k$ with $d+1\leq
k\leq t+1$ it holds $\rho_{k}(I\setminus
J)<\sum_{j=d}^{k-1}(-1)^{k-j+1}{t+1-j\choose k-j}\rho_{j}(I\setminus J)$, then
$\operatorname{depth}_{S}(I/J)=t$ independently of the characteristic of $K$.
Shen’s proof is very short, based on a strong tool, namely the Hilbert depth
considered by Bruns-Krattenhaler-Uliczka [2] (see also [22], [7]). Thus it is
important to have the right tool.
Let $r$ be the number of the squarefree monomials of degrees $d$ of $I$ and
$B$ (resp. $C$) be the set of the squarefree monomials of degrees $d+1$ (resp.
$d+2$) of $I\setminus J$. Set $s=|B|$, $q=|C|$. If $r>s$ then Theorem 1.1 says
that $\operatorname{depth}_{S}I/J=d$, namely the minimum possible. This was
done previously in [15] (the idea started in [14]). Moreover, Theorem 1.1
together with Hall’s marriage theorem for bipartite graphs gives the
following:
###### Theorem 1.3.
(Popescu [15, Theorem 4.3]) If $\operatorname{sdepth}_{S}I/J=d$ then
$\operatorname{depth}_{S}I/J=d$, that is Stanley’s Conjecture holds in this
case.
The purpose of our paper is to study the next step in proving Stanley’s
Conjecture namely the following weaker conjecture.
###### Conjecture 1.4.
Suppose that $I\subset S$ is minimally generated by some squarefree monomials
$f_{1},\ldots,f_{k}$ of degrees $d$, and a set $H$ of squarefree monomials of
degrees $\geq d+1$. Assume that $\operatorname{sdepth}_{S}I/J=d+1$. Then
$\operatorname{depth}_{S}I/J\leq d+1$
The following theorem is a partial answer.
###### Theorem 1.5.
The above conjecture holds in each of the following two cases:
1. (1)
$k=1$,
2. (2)
$1<k\leq 3$, $H=\emptyset$.
When $k=1$ and $s\not=q+1$ the result was stated in [17] and [18]. The theorem
follows from Proposition 3.1 and Theorems 2.3, 3.4.
We owe thanks to the Referee, who noticed some mistakes in a previous version
of this paper, especially in the proof of Lemma 3.3.
## 2\. Cases $r=1$ and $d=1$
Let $I\supsetneq J$ be two squarefree monomial ideals of $S$. We assume that
$I$ is generated by squarefree monomials of degrees $\geq d$ for some
$d\in{\bf N}$. We may suppose that either $J=0$, or is generated by some
squarefree monomials of degrees $\geq d+1$. As above $B$ (resp. $C$) denotes
the set of the squarefree monomials of degrees $d+1$ (resp. $d+2$) of
$I\setminus J$.
###### Lemma 2.1.
Suppose that $I\subset S$ is minimally generated by some square free monomials
$\\{f_{1},\ldots,f_{r}\\}$ of degrees $d$, and a set $E$ of square free
monomials of degrees $\geq d+1$. Assume that $\operatorname{sdepth}_{S}I/J\leq
d+1$ and the above Conjecture 1.4 holds for $k<r$ and for $k=r$, $|H|<|E|$ if
$E\not=\emptyset$. If either $C\not\subset(f_{2},\ldots,f_{r},E)$, or
$C\not\subset(f_{1},\ldots,f_{r},E\setminus\\{a\\})$ for some $a\in E$ then
$\operatorname{depth}_{S}I/J\leq d+1$.
###### Proof.
Let $c\in(C\setminus(f_{2},\ldots,f_{r},E))$. Then $c\in(f_{1})$, let us say
$c=f_{1}x_{t}x_{p}$. Set
$I^{\prime}=(f_{2},\ldots,f_{r},E,B\setminus\\{f_{1}x_{t},f_{1}x_{p}\\})$,
$J^{\prime}=I^{\prime}\cap J$. In the following exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/(J+I^{\prime})\rightarrow 0$
the last term is isomorphic with $(f_{1})/(J+I^{\prime})\cap(f_{1})$ and has
depth and sdepth $\geq d+2$ because $c\not\in(J+I^{\prime})$ (here it is
enough that depth $\geq d+1$, which is easier to see). By [19, Lemma 2.2] we
get $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$. It follows that
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$ by hypothesis and so
the Depth Lemma gives $\operatorname{depth}_{S}I/J\leq d+1$.
Now, let $I^{\prime\prime}=(f_{1},\ldots,f_{r},E\setminus\\{a\\})$ for some
$a\in E$ and $c\in C\setminus I^{\prime\prime}$. In the following exact
sequence
$0\rightarrow I^{\prime\prime}/I^{\prime\prime}\cap J\rightarrow
I/J\rightarrow I/(J+I^{\prime\prime})\rightarrow 0$
the last term is isomorphic with $(a)/(a)\cap(J+I^{\prime\prime})$ and has
depth and sdepth $\geq d+2$ because $c\not\in J+I^{\prime\prime}$ and as above
we get $\operatorname{depth}_{S}I/J\leq d+1$.
The following lemma could be seen somehow as a consequence of [17, Theorem
1.10], but we give here an easy direct proof.
###### Lemma 2.2.
Suppose that $r=1$, let us say $I=(f)$ and $E=\emptyset$. If
$\operatorname{sdepth}_{S}I/J=d+1$, $d=\operatorname{deg}f$ then
$\operatorname{depth}_{S}I/J\leq d+1$.
###### Proof.
First assume that $d>0$. Note that $I/J\cong S/(J:f)$. We have
$\operatorname{sdepth}_{S}I/J=\operatorname{sdepth}_{S}S/(J:f)$ and
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}S/(J:f)$. It is enough to
treat the case $d=1$. We may assume that $x_{1}|f$ and using [5, Lemma 3.6]
after skipping the variables of $f/x_{1}$ we may reduce our problem to the
case $d=1$.
Therefore we may assume that $d=1$. If $C=\emptyset$ then $x_{1}x_{t}x_{k}\in
J$ for all $1<t<k\leq n$ and so $(J:x_{1})$ contains all squarefree monomials
of degree two in $x_{t}$, $t>1$, that is the annihilator of the element
induced by $x_{1}$ in $I/J$ has dimension $\leq 2$. It follows that
$\operatorname{depth}_{S}I/J\leq 2$.
If let us say $c=x_{1}x_{2}x_{3}\in C$ then in the exact sequence
$0\rightarrow(B\setminus\\{x_{1}x_{2},x_{1}x_{3}\\})/J\cap(B\setminus\\{x_{1}x_{2},x_{1}x_{3}\\})\rightarrow
I/J\rightarrow I/J+(B\setminus\\{x_{1}x_{2},x_{1}x_{3}\\})\rightarrow 0$
the last term is isomorphic with
$(x_{1})/(J,(B\setminus\\{x_{1}x_{2},x_{1}x_{3}\\})$ and it has depth $\geq 2$
and sdepth $3$ because it has just the interval $[x_{1},c]$. The first term is
not zero since otherwise $\operatorname{sdepth}_{S}I/J=3$, which is false.
Then the first term has sdepth $\leq 2$ by [19, Lemma 2.2] and so it has depth
$\leq 2$ by[15, Theorem 4.3]. Now it is enough to apply the Depth Lemma.
Now assume that $d=0$, that is $I=S$. Set $S^{\prime}=S[x_{n+1}]$,
$I^{\prime}=(x_{n+1})$, $J^{\prime}=x_{n+1}J$. We have
$\operatorname{sdepth}_{S^{\prime}}I^{\prime}/J^{\prime}=\operatorname{sdepth}_{S^{\prime}}S^{\prime}/JS^{\prime}=1+\operatorname{sdepth}_{S}S/J=2$
using [5, Proposition 3.6]. From above we get
$\operatorname{depth}_{S^{\prime}}I^{\prime}/J^{\prime}\leq 2$ and it follows
$\operatorname{depth}_{S}I/J\leq 1$.
The following theorem extends the above lemma and [18], its proof is given in
the last section.
###### Theorem 2.3.
Suppose that $I\subset S$ is minimally generated by a squarefree monomial
$\\{f\\}$, of degree $d$ and a set $E\not=\emptyset$ of monomials of degrees
$d+1$. Assume that $\operatorname{sdepth}_{S}I/J\leq d+1$. Then
$\operatorname{depth}_{S}I/J\leq d+1$.
###### Lemma 2.4.
Suppose that $I=(x_{1},x_{2})$, $E=\emptyset$. If
$\operatorname{sdepth}_{S}I/J=2$ then
$\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
By [17, Proposition 1.3] we may suppose that $C\not\subset(x_{2})$. Then apply
Lemma 2.1, its hypothesis is given by Theorem 2.3.
We need the following lemma, its proof is given in the next section.
###### Lemma 2.5.
Suppose that $I\subset S$ is minimally generated by some squarefree monomials
$\\{f_{1},f_{2},f_{3}\\}$ of degree $d$ and that
$\operatorname{sdepth}I/J=d+1$. If there exists $c\in
C\cap((f_{3})\setminus(f_{1},f_{2}))$ then $\operatorname{depth}_{S}I/J\leq
d+1$.
###### Proposition 2.6.
Suppose that $I=(x_{1},x_{2},x_{3})$, $E=\emptyset$. If
$\operatorname{sdepth}_{S}I/J=2$ then $\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
By [17, Proposition 1.3] we may suppose that $C\not\subset(x_{1},x_{2})$. Then
we may apply Lemma 2.5.
###### Remark 2.7.
When $J=0$ the above proposition follows quickly from [1] (see also [5]).
## 3\. Case $r,d>1$
###### Proposition 3.1.
Suppose that $I\subset S$ is generated by two squarefree monomials
$\\{f_{1},f_{2}\\}$ of degrees $d$. Assume that
$\operatorname{sdepth}_{S}I/J\leq d+1$. Then $\operatorname{depth}_{S}I/J\leq
d+1$.
###### Proof.
We may suppose that $I$ is minimally generated by $f_{1},f_{2}$ because
otherwise apply the Theorem 2.3. Let $w$ be the least common multiple of
$f_{1},f_{2}$. First suppose that $C\not\subset(w)$. This is the case when
$w\in J$, or $\operatorname{deg}w>d+2$, or $w\in C$ and $q>1$. Then it is
enough to apply Lemma 2.1, the case $r=1$ being done in the Theorem 2.3. If
$q=1$ then $r>q$ and by [20, Corollary 2.6] (see also Theorem 1.2) we get
$\operatorname{depth}_{S}I/J\leq d+1$. Assume that $w\in B$. After renumbering
the variables $x_{i}$ we may suppose that $C=\\{wx_{i}:1\leq i\leq q\\}$ and
so in $B$ we have at least the elements of the form $w,f_{1}x_{i},f_{2}x_{i}$,
$1\leq i\leq q$ . Thus $s\geq 2q+1>q+2$ when $q>1$ and by [16, Theorem 1.3]
(see Theorem 1.1) we are done.
###### Lemma 3.2.
Suppose that $I\subset S$ is generated by three squarefree monomials
$\\{f_{1},f_{2},f_{3}\\}$ of degrees $d$, $\operatorname{sdepth}_{S}I/J=d+1$
and let $w_{ij}$ be the least common multiple of $f_{i},f_{j}$, $1\leq i<j\leq
3$. If $w_{12},w_{13},w_{23}\in B$ and are different then
$\operatorname{depth}_{S}I/J\leq d+1$.
###### Proof.
After renumbering the variables $x_{i}$ we may assume that $f_{1}=x_{1}\cdots
x_{d}$ and $f_{2}=x_{1}\cdots x_{d-1}x_{d+1}$. We see that $f_{3}$ must have
$d-1$ variables in common with $f_{1}$ and also with $f_{2}$. If
$f_{3}\notin(x_{1}...x_{d-1})$ then we may suppose that
$f_{3}=x_{2}...x_{d}x_{d+1}$ and $w_{12}=w_{13}$, which is false. It remains
that $f_{3}\in(x_{1}\cdots x_{d-1})$ so $f_{3}=x_{1}\cdots x_{d-1}x_{d+2}$.
But this case may be reduced to $d=1$ which is done in Proposition 2.6.
###### Lemma 3.3.
If $C\subset(w_{12},w_{13},w_{23})$ and $\operatorname{sdepth}_{S}I/J\leq d+1$
then $\operatorname{depth}_{S}I/J\leq d+1$.
###### Proof.
Note that if $q<r=3$ then $\operatorname{depth}_{S}I/J\leq d+1$ by [20,
Corollary 2.6] (see here Theorem 1.2). Suppose that $q>2$.
Now assume that all $w_{ij}\in B$. Set $C_{ij}=C\cap(w_{ij})$,
$q_{ij}=|C_{ij}|$ and $B_{ij}$ the set of all $b\in B$ which divide some $c\in
C_{ij}$. If all $w_{ij}$ are equal, let us say $w_{ij}=w$, then after
renumbering the variables $x_{i}$ the monomials of $C$ have the form $wx_{t}$,
$1\leq t\leq q$. Thus $B$ contains $w$ and $f_{j}x_{t}$ for $j\in[3]$ and
$t\in[q]$. It follows that $s\geq 3q+1>q+3$ for $q>1$ and so
$\operatorname{depth}_{S}I/J\leq d+1$ by [16, Theorem 1.3]. Then we may
suppose that all $w_{ij}$ are different and we may apply Lemma 3.2.
Next assume that $w_{12},w_{13}\in B$ and $w_{23}\in C$. As above we can
assume that $f_{2}=x_{1}\cdots x_{d}$, $f_{3}=x_{3}\cdots x_{d+2}$ and
$f_{1}=x_{2}\cdots x_{d+1}$. We have $C\subset C_{12}\cap C_{13}$, and
$q=q_{12}+q_{13}-1$ because $w_{23}\in C_{12}\cap C_{13}$. As in the case of
$r=2$ we have $|B_{12}|=2q_{12}+1$ and $|B_{13}\setminus B_{12}|\geq
2q_{13}-\operatorname{min}\\{q_{12},q_{13}\\}$. It follows that $s\geq
2q+4-\operatorname{min}\\{q_{12},q_{13}\\}>q+3$, which implies
$\operatorname{depth}_{S}I/J\leq d+1$ by [16, Theorem 1.3]. Note that if
$w_{23}\in J$, or $\operatorname{deg}w_{23}>d+2$ then $q=q_{12}+q_{13}$ and we
get in the same way that $s\geq 2q+2-\operatorname{min}\\{q_{12},q_{13}\\}\geq
q+3$. Thus $\operatorname{depth}_{S}I/J\leq d+1$ unless $q_{12}=q_{13}=1$. The
last case is false because $q>2$.
Suppose that all $w_{ij}$ are different, $w_{12}\in B$ and $w_{23},w_{13}\in
C$. We may assume that $f_{2}=x_{1}\cdots x_{d}$, $f_{3}=x_{3}\cdots x_{d+2}$
and $f_{1}=x_{2}\cdots x_{d}\cdot x_{d+3}$. We have $q=q_{12}+2$, $B_{12}\cap
B_{13}\subset\\{x_{d+1}f_{1},x_{d+2}f_{1}\\}$ and so $|B_{13}\setminus
B_{12}|\geq 2$. Also note that $B_{23}\cap(B_{12}\cup
B_{13})\subset\\{x_{d+1}f_{2},x_{d+2}f_{2},x_{2}f_{3}\\}$ and so
$|B_{23}\setminus(B_{12}\cup B_{13})|\geq 1$. It follows that $s\geq
2q_{12}+1+2+1=2q$. If $q>3$ we get $s>q+3$ and so
$\operatorname{depth}_{S}I/J\leq d+1$ by [16]. If $q=3$ then $q_{12}=1$ and so
$B_{12}=\\{w_{12},x_{t}f_{1},x_{t}f_{2}\\}$ for some $x_{t}\not|f_{1}$,
$x_{t}\not|f_{2}$. If $t=d+1$ or $t=d+2$ then we see that $|B_{13}\setminus
B_{12}|\geq 3$ and so $s>6=r+q$, which is enough. If $t>d+3$ then $s$ is even
bigger than $7$. If let us say $w_{23}\in J$, or
$\operatorname{deg}w_{23}>d+2$ then $q=q_{12}+1$ and as above $s\geq
2q_{12}+1+2=2q+1>q+3$ because $q\geq 3$, which is again enough. If also
$w_{13}\in J$, or $\operatorname{deg}w_{13}>d+2$ then $q=q_{12}$ and as above
$s\geq 2q_{12}+1=2q+1>q+3$ because $q\geq 3$.
Suppose that $w_{12}\in B$ and $w_{23}=w_{13}\in C$. We may assume that
$f_{2}=x_{1}\cdots x_{d}$, $f_{3}=x_{3}\cdots x_{d+2}$ and $f_{1}=x_{1}\cdots
x_{d-1}\cdot x_{d+2}$. We have $q=q_{12}$ and $B_{12}\supset B_{13}$. Thus
$s\geq 2q_{12}+1=2q+1>q+3$ and so again $\operatorname{depth}_{S}I/J\leq d+1$.
Finally if all $w_{ij}$ are in $C$ (they must be different, otherwise $q\leq
2$ which is false) then $q=3$ , $q_{ij}=1$ and we get $s\geq 12>q+3$ which is
again enough.
###### Theorem 3.4.
Suppose that $I\subset S$ is generated by three squarefree monomials
$\\{f_{1},f_{2},f_{3}\\}$ of degrees $d$, and
$\operatorname{sdepth}_{S}I/J=d+1$. Then $\operatorname{depth}_{S}I/J\leq
d+1$.
###### Proof.
We may suppose that $I$ is minimally generated by $f_{1},f_{2},f_{3}$ because
otherwise apply Proposition 3.1. If $C\not\subset(w_{12},w_{13},w_{23})$ then
apply Lemma 2.5. Thus we may suppose that $C\subset(w_{12},w_{13},w_{23})$ and
we may apply Lemma 3.3.
## 4\. Proof of Lemma 2.5
Let $c=f_{3}x_{i_{3}}x_{j_{3}}$ and set
$I^{\prime}=(f_{1},f_{2},B\setminus\\{f_{3}x_{i_{3}},f_{3}x_{j_{3}}\\}),J^{\prime}=I^{\prime}\cap
J$. Consider the following exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/(I^{\prime}+J)\rightarrow 0.$
The last term has $\operatorname{sdepth}=d+2$ so by [19, Lemma 2.2] we get
that the first term has $\operatorname{sdepth}\leq d+1$. If
$\operatorname{depth}I^{\prime}/J^{\prime}\leq d+1$ then by Depth Lemma we are
done. It is enough to show that
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}=d+1$ implies
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$, or directly
$\operatorname{depth}_{S}I/J\leq d+1$. Note that if
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}=d$ then
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=d$ by Theorem 1.3. Let
$B^{\prime}$, $C^{\prime}$, $E^{\prime}$ be similar to $B$, $C$, $E$ in the
case of $I^{\prime}/J^{\prime}$.
We see that $E^{\prime}\subset(f_{3})$. We may suppose that
$C^{\prime}\subset((f_{1})\cap(f_{2}))\cup(E^{\prime})$ and
$E^{\prime}\neq\emptyset$, otherwise apply Lemma 2.1 with the help of Theorem
2.3.
Set $I^{\prime}_{E}=(f_{1},f_{2}),J^{\prime}_{E}=I^{\prime}_{E}\cap
J^{\prime}$ and for all $i\in[n]\setminus\operatorname{supp}f_{1}$ such that
$f_{1}x_{i}\in B^{\prime}\setminus(f_{2})$ set
$I^{\prime}_{i}=(f_{2},B\setminus\\{f_{1}x_{i}\\})$,
$J^{\prime}_{i}=I^{\prime}_{i}\cap J^{\prime}$. We may suppose that
$\operatorname{sdepth}_{S}I^{\prime}_{E}/J^{\prime}_{E}\geq d+2$ and
$\operatorname{sdepth}_{S}I^{\prime}_{i}/J^{\prime}_{i}\geq d+2$. Indeed,
otherwise one of the left terms from the following exact sequences
$0\rightarrow I^{\prime}_{E}/J^{\prime}_{E}\rightarrow
I^{\prime}/J^{\prime}\rightarrow
I^{\prime}/I^{\prime}_{E}+J^{\prime}\rightarrow 0,$ $0\rightarrow
I^{\prime}_{i}/J^{\prime}_{i}\rightarrow I^{\prime}/J^{\prime}\rightarrow
I^{\prime}/I^{\prime}_{i}+J^{\prime}\rightarrow 0,$
have depth $\leq d+1$ by Proposition 3.1 and Theorem 2.3. With the Depth Lemma
we get $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$ since the right
terms above have depth $\geq d+1$. Let ${\mathcal{P}}_{E}$,
${\mathcal{P}}_{i}$ be partitions of $I^{\prime}_{E}/J^{\prime}_{E}$,
$I^{\prime}_{i}/J^{\prime}_{i}$ with sdepth $d+2$. We may choose
${\mathcal{P}}_{E}$ and ${\mathcal{P}}_{i}$ such that each interval starting
with a squarefree monomial of degree $\leq d+1$ ends with a monomial from
$C^{\prime}$.
Our goal is mainly to reduce our problem to the case when $w_{13},w_{12}\in
B^{\prime}\cup C^{\prime}$. Case 1
$C^{\prime}\not\subset(f_{1},f_{3})\cap(f_{2},f_{3})$
Let for example $c=f_{1}x_{u}x_{v}\in C^{\prime}\setminus(f_{2},f_{3})$, set
$I^{\prime\prime}=(f_{2},B^{\prime}\setminus\\{f_{1}x_{u},f_{1}x_{v}\\}),J^{\prime\prime}=I^{\prime\prime}\cap
J^{\prime}$ and consider the exact sequence:
$0\rightarrow I^{\prime\prime}/J^{\prime\prime}\rightarrow
I^{\prime}/J^{\prime}\rightarrow
I^{\prime}/(I^{\prime\prime}+J^{\prime})\rightarrow 0.$
The last term has sdepth $d+2$ so by [19, Lemma 2.2] we see that the first
term has $\operatorname{sdepth}\leq d+1$. Using Theorem 2.3 we have
$\operatorname{depth}_{S}I^{\prime\prime}/J^{\prime\prime}\leq d+1$ and then
by the Depth lemma we get $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq
d+1$ ending Case 1.
Let $f_{1}=x_{1}...x_{d}$ , in ${\mathcal{P}}_{E}$ we have the intervals
$[f_{1},c_{1}],[f_{2},c_{2}]$ and so at least one of $c_{1},c_{2}$, let us say
$c_{1}=f_{1}x_{i}x_{j}$, is not a multiple of $w_{12}$. In ${\mathcal{P}}_{i}$
we have the interval $[b,c_{1}]$ for some $b\in E^{\prime}$, otherwise
replacing the interval $[f_{1}x_{j},c_{1}]$ or the interval $[c_{1},c_{1}]$
with the interval $[f_{1},c_{1}]$ we get a partition ${\mathcal{P}}$ for
$I^{\prime}/J^{\prime}$ with $\operatorname{sdepth}=d+2$.
Case 2 There exists $t\in[n]$, $t\not\in\operatorname{supp}f_{1}\cup\\{i\\}$
such that ${\mathcal{P}}_{i}$ contains the interval
$[f_{1}x_{t},f_{1}x_{t}x_{i}]$, or $[f_{1}x_{t}x_{i},f_{1}x_{t}x_{i}]$.
In this case changing in ${\mathcal{P}}_{i}$ the hinted interval with
$[f_{1},f_{1}x_{t}x_{i}]$ we get a partition of $I^{\prime}/J^{\prime}$ with
sdepth $\geq d+2$ which is false.
As we have seen above we may suppose that in ${\mathcal{P}}_{i}$ there exists
an interval $[b,c_{1}]$ with $c_{1}\in(f_{1})\cap(E^{\prime})\subset(w_{13})$.
It follows that $w_{13}\in B^{\prime}\cup C^{\prime}$. We may assume that if
$w_{13}\in B^{\prime}$ then $x_{i}\not|w_{13}$, otherwise change $i$ by $j$.
Thus $c_{1}=x_{i}w_{13}$ or $c_{1}=w_{13}$. If
$C^{\prime}\cap(f_{1}x_{i})=\\{c_{1}\\}$ then in ${\mathcal{P}}_{j}$ we have
the interval $[f_{1}x_{i},c_{1}]$, that is we are in Case 2. Then there exists
another monomial $c^{\prime}\in C^{\prime}\cap(f_{1}x_{i})$. We may suppose
that $[c^{\prime},c^{\prime}]$ is not in ${\mathcal{P}}_{i}$, because
otherwise we are in Case 2. If we have $[u,c^{\prime}]$ in ${\mathcal{P}}_{i}$
for some $u\in E^{\prime}$ then $c^{\prime}\in(w_{13})$ and so
$c^{\prime}=c_{1}$ if $w_{13}\in C^{\prime}$, otherwise
$c^{\prime}=x_{i}w_{13}=c_{1}$ because $x_{i}\not|w_{13}$. Contradiction! Then
in ${\mathcal{P}}_{i}$ we have the interval $[f_{2},c^{\prime}]$ or the
interval $[f_{2}x_{k},c^{\prime}]$ for some $k$. Thus $c^{\prime}\in(w_{12})$
and so $w_{12}\in B^{\prime}\cup C^{\prime}$. Note that $w_{12}\not=w_{13}$
because $c_{1}\in(w_{13})\setminus(f_{2})$.
Case 3 $w_{12},w_{13}\in C^{\prime}$.
In this case $c_{1}=w_{13}$, $c^{\prime}=w_{12}$ and so
$f_{2},f_{3}\in(x_{i})$. Then in ${\mathcal{P}}_{i}$ we have the interval
$[f_{1}x_{j},f_{1}x_{j}x_{u}],u\neq i$ and
$f_{1}x_{j}x_{u}\notin(f_{2},f_{3})$ because $f_{1}x_{j}x_{u}\notin(x_{i})$,
that is we are in Case 1.
Case 4 $w_{12}\in B^{\prime},w_{13}\in C^{\prime}$.
Thus $c_{1}=w_{13}$. We may assume that
$w_{12}=x_{1}...x_{d+1},f_{2}=x_{2}...x_{d+1},i\neq d+1\neq j$ and
$c^{\prime}=x_{1}...x_{d+1}x_{i}$. We also see that $f_{3}\in(x_{i}x_{j})$
because $c_{1}=w_{13}$. In ${\mathcal{P}}_{i}$ we have the interval
$[f_{1}x_{j},f_{1}x_{j}x_{u}],u\neq i$. If $u\neq d+1$ then
$f_{1}x_{j}x_{u}\notin(f_{2},f_{3})$, that is we are in Case 1. Otherwise
$u=d+1$, and so $x_{j}w_{12}\in C^{\prime}$, in particular $f_{2}x_{j}\in
B^{\prime}$. We see that in ${\mathcal{P}}_{i}$ we can have either
$w_{12}\in[f_{2},c^{\prime}]$ or there exists an interval
$[w_{12},w_{12}x_{k}]$. If $k=j$ then $w_{12}x_{k}$ is the end of the interval
starting with $f_{1}x_{j}$, which is false. If $k=i$ then we are in Case 2.
Thus $i\neq k\neq j$.
When in ${\mathcal{P}}_{i}$ there exists the interval $[w_{12},w_{12}x_{k}]$
then there exists also the interval $[f_{1}x_{k},f_{1}x_{k}x_{t}]$. If
$f_{1}x_{k}x_{t}\in(f_{2})$ then $t=d+1$ and so $f_{1}x_{k}x_{t}=x_{k}w_{12}$
which is not possible because $x_{k}w_{12}$ is in $[w_{12},x_{k}w_{12}]$. If
$f_{1}x_{k}x_{t}\in(f_{3})$ then $\\{k,t\\}=\\{i,j\\}$ which is not possible
since $k\not\in\\{i,j\\}$. Then $f_{1}x_{k}x_{t}\notin(f_{2},f_{3})$, that is
we are in Case 1. It remains the case when $w_{12}$ is in the interval
$[f_{2},c^{\prime}]$. In ${\mathcal{P}}_{i}$ we have an interval
$[f_{2}x_{j},f_{2}x_{l}x_{j}]$ for some $l$. If $f_{2}x_{j}x_{l}\in(f_{1})$
then $l=1$ and so $f_{2}x_{j}x_{l}=x_{j}w_{12}$ which is already the end of
the interval starting with $f_{1}x_{j}$. Contradiction ! Thus
$f_{2}x_{l}x_{j}\in(f_{3})$, otherwise we are in Case 1. We get $l=i$ and
changing $[f_{2},c^{\prime}],[f_{2}x_{j},f_{2}x_{i}x_{j}]$ with
$[f_{2},f_{2}x_{i}x_{j}],[w_{12},c^{\prime}]$ we arrive in Case 2.
Case 5 $w_{12}\in C^{\prime},w_{13}\in B^{\prime}$.
Thus we may assume that
$w_{12}=c^{\prime}=x_{1}...x_{d+1}x_{i},f_{2}=x_{3}...x_{d+1}x_{i}$. As
$c_{1}\in(w_{13})$ we have $w_{13}\in\\{f_{1}x_{i},f_{1}x_{j}\\}$. If
$w_{13}=f_{1}x_{i}$ then in ${\mathcal{P}}_{i}$ we have an interval
$[f_{1}x_{j},f_{1}x_{j}x_{u}]$. If $f_{1}x_{j}x_{u}\in(f_{2})$ then $u=i$.
Also if $f_{1}x_{j}x_{u}\in(f_{3})$ we get $f_{1}x_{j}x_{u}\in(w_{13})$ and we
get again $u=i$, that is we are in Case 2. Thus
$f_{1}x_{j}x_{u}\notin(f_{2},f_{3})$ and we arrive in Case 1.
Then, we may suppose that $w_{13}=f_{1}x_{j}$. Since $f_{1}x_{d+1}|c^{\prime}$
we see that $f_{1}x_{d+1}\in B^{\prime}$. In ${\mathcal{P}}_{i}$ we can have
the interval $[f_{1}x_{d+1},f_{1}x_{d+1}x_{m}]$. If
$f_{2}x_{d+1}x_{m}\in(f_{2})$ then $m=i$, that is we are in Case 2. Then
$f_{2}x_{d+1}x_{m}\in(f_{3})$ because otherwise we are in Case 1. It follows
that $m=j$ and we have $[f_{1}x_{d+1},f_{1}x_{d+1}x_{j}]$ in
${\mathcal{P}}_{i}$. Then the interval $[f_{1}x_{j},f_{1}x_{j}x_{p}]$ existing
in ${\mathcal{P}}_{i}$ has $p\not=j$ and also $p\not=i$ because otherwise we
are in Case 2. Thus we must also have an interval
$[f_{1}x_{p},f_{1}x_{p}x_{k}]$ with $k\not=j$ and also $k\not=i$, otherwise we
are in Case 2. Then $f_{1}x_{p}x_{k}\notin(f_{2},f_{3})$, that is we are in
Case 1.
Case 6 $w_{12},w_{13}\in B^{\prime}$.
We may assume that $w_{12}=x_{1}...x_{d+1},f_{2}=x_{2}...x_{d+1}$ and
$c^{\prime}=x_{1}...x_{d+1}x_{i}$. If $w_{23}\in B^{\prime}$ then all $w_{ij}$
are different and by Lemma 3.2 we get $\operatorname{depth}_{S}I/J\leq d+1$.
Thus we may suppose that $w_{23}\in C^{\prime}$. We may choose
$f_{3}=x_{1}x_{3}...x_{d}x_{i}$ or $f_{3}=x_{1}x_{3}...x_{d}x_{j}$. If
$f_{3}=x_{1}x_{3}...x_{d}x_{i}$ then in ${\mathcal{P}}_{i}$ we have as above
the interval $[f_{1}x_{j},f_{1}x_{d+1}x_{j}]$. Indeed, if we have
$[f_{1}x_{j},f_{1}x_{m}x_{j}]$ then $f_{1}x_{m}x_{j}\not\in(f_{3})$ and so
$f_{1}x_{m}x_{j}\in(f_{2})$, otherwise we are in Case 1. It follows $m=d+1$.
As $f_{2}x_{j}|x_{j}w_{12}=f_{1}x_{d+1}x_{j}$ we get $f_{2}x_{j}\in
B^{\prime}$. Let $[f_{2},f_{2}x_{j}x_{k}]$ or $[f_{2}x_{j},f_{2}x_{j}x_{k}]$
be the existing interval of ${\mathcal{P}}_{i}$ containing $f_{2}x_{j}$. Note
that $f_{2}x_{j}x_{k}\not\in(f_{3})$ and if $f_{2}x_{j}x_{k}\in(f_{1})$ then
$f_{2}x_{j}x_{k}=x_{j}w_{12}$ which appeared already in the previous interval.
Thus $f_{2}x_{j}x_{k}\notin(f_{1},f_{3})$, that is we are in Case 1.
It remains that $f_{3}=x_{1}x_{3}...x_{d}x_{j}$ and, as before, we have in
${\mathcal{P}}_{i}$ the interval $[f_{1}x_{j},f_{1}x_{d+1}x_{j}]$. We see then
$f_{2}x_{j}\in B^{\prime}$ and we must have also an interval
$[f_{2},f_{2}x_{j}x_{k}]$ or $[f_{2}x_{j},f_{2}x_{j}x_{k}]$. If
$f_{2}x_{j}x_{k}\in(f_{1})\cup(f_{3})$ then we get $k=1$ and so
$f_{2}x_{j}x_{k}=x_{j}w_{12}$ which appeared in the previous interval. It
follows that $f_{2}x_{j}x_{k}\notin(f_{1},f_{3})$, that is we are in Case 1.
## 5\. Proof of Theorem 2.3
Suppose that $E\not=\emptyset$ and $s\leq q+1$. We may assume that
$|B\setminus E|\geq 2$ because otherwise $\operatorname{depth}_{S}I/J\leq d+1$
since the element induced by $f$ in $I/J$ is annihilated by all variables but
one and those from $\operatorname{supp}f$. For $b=fx_{i}\in B$ set
$I_{b}=(B\setminus\\{b\\})$, $J_{b}=J\cap I_{b}$. If
$\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$ then let ${\mathcal{P}}_{b}$ be
a partition on $I_{b}/J_{b}$ with sdepth $d+2$. We may choose
${\mathcal{P}}_{b}$ such that each interval starting with a squarefree
monomial of degree $d$, $d+1$ ends with a monomial of $C$. In
${\mathcal{P}}_{b}$ we have some intervals for all $b^{\prime}\in
B\setminus\\{b\\}]$ an interval $[b^{\prime},c_{b^{\prime}}]$. We define
$h:B\setminus\\{b\\}\rightarrow C$ by $b^{\prime}\rightarrow c_{b^{\prime}}$.
Then $h$ is an injection and $|\operatorname{Im}h|=s-1\leq q$ (if $s=1+q$ then
$h$ is a bijection). We may suppose that all intervals of ${\mathcal{P}}_{b}$
starting with a monomial $v$ of degree $\geq d+2$ have the form $[v,v]$.
###### Lemma 5.1.
Suppose that the following conditions hold:
1. (1)
$s\leq q+1$,
2. (2)
$\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$, for a $b\in B\cap(f)$,
3. (3)
$C\subset((f)\cap(E))\cup(\cup_{a,a^{\prime}\in
E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$.
Then either $\operatorname{sdepth}_{S}I/J\geq d+2$, or there exists a nonzero
ideal $I^{\prime}\subsetneq I$ generated by a subset of $\\{f\\}\cup B$ such
that $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for
$J^{\prime}=J\cap I^{\prime}$ and
$\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$.
###### Proof.
Consider $h$ as above for a partition ${\mathcal{P}}_{b}$ with sdepth $d+2$ of
$I_{b}/J_{b}$ which exists by (2). A sequence $a_{1},\ldots,a_{k}$ is called a
path from $a_{1}$ to $a_{k}$ if $a_{i}\in B\setminus\\{b\\}$, $i\in[k]$,
$a_{i}\not=a_{j}$ for $1\leq i<j\leq k$, $a_{i+1}|h(a_{i})$ for $1\leq i<k$,
and $h(a_{i})\not\in(b)$ for $1\leq i<k$. This path is bad if $h(a_{k})\in(b)$
and it is maximal if all divisors from B of $h(a_{k})$ are in
$\\{b,a_{1},\ldots,a_{k}\\}$. If $a=a_{1}$ we say that the above path starts
with $a$. Since $|B\setminus E|\geq 2$ there exists $a_{1}\in
B\setminus\\{b\\}$. Set $c_{1}=h(a_{1})$. If $c_{1}\in(b)$ then the path
$\\{a_{1}\\}$ is maximal and bad. By recurrence choose if possible $a_{p+1}$
to be a divisor from $B$ of $c_{p}$ which is not in
$\\{b,a_{1},\ldots,a_{p}\\}$ and set $c_{p}=h(a_{p})$, $p\geq 1$. This
construction ends at step $p=e$ if all divisors from $B$ of $c_{e-1}$ are in
$\\{b,a_{1},\ldots,a_{e-1}\\}$. If $c_{i}\not\in(b)$ for $1\leq i<e-1$ then
$\\{a_{1},\ldots,a_{e-1}\\}$ is a maximal path. If $c_{e-1}\in(b)$ then this
path is also bad. We have two cases:
1) there exist no maximal bad path starting with $a_{1}$,
2) there exists a maximal bad path starting with $a_{1}$.
In the first case, set $T_{1}=\\{b^{\prime}\in B:\mbox{there\ exists\ a \
path}\ a_{1},\ldots,a_{k}\ \mbox{with}\ a_{k}=b^{\prime}\\}$,
$G_{1}=B\setminus T_{1}$ and $I^{\prime}_{1}=(f,G_{1})$,
$I^{\prime\prime}_{1}=(G_{1})$, $J^{\prime}_{1}=I^{\prime}_{1}\cap J$,
$J^{\prime\prime}_{1}=I^{\prime\prime}_{1}\cap J$. Note that
$I^{\prime\prime}_{1}\not=0$ because $b\in I^{\prime\prime}_{1}$. Consider the
following exact sequence
$0\rightarrow I^{\prime}_{1}/J^{\prime}_{1}\rightarrow I/J\rightarrow
I/(J,I^{\prime}_{1})\rightarrow 0.$
If $T_{1}\cap(f)=\emptyset$ then the last term has depth $\geq d+1$ and sdepth
$\geq d+2$ using the restriction of ${\mathcal{P}}_{b}$ since
$h(b^{\prime})\not\in I^{\prime}_{1}$, for all $b^{\prime}\in T_{1}$. If the
first term has sdepth $\geq d+2$ then by [19, Lemma 2.2] the middle term has
sdepth $\geq d+2$. Otherwise, the first term has sdepth $\leq d+1$ and we may
take $I^{\prime}=I^{\prime}_{1}$.
If let us say $a\in(f)$ for some $a\in T_{1}$ then in the following exact
sequence
$0\rightarrow I^{\prime\prime}_{1}/J^{\prime\prime}_{1}\rightarrow
I/J\rightarrow I/(J,I^{\prime\prime}_{1})\rightarrow 0$
the last term has sdepth $\geq d+2$ and depth $\geq d+1$ since $h(a)\not\in
I^{\prime\prime}_{1}$ and we may substitute the interval $[a,h(a)]$ from the
restriction of ${\mathcal{P}}_{b}$ to $(T_{1})$ by $[f,h(a)]$, the second
monomial from $[f,h(a)]\cap B$ being also in $T_{1}$. As above we get either
$\operatorname{sdepth}_{S}I/J=d+2$, or
$\operatorname{sdepth}_{S}I^{\prime\prime}_{1}/J^{\prime\prime}_{1}\leq d+1$,
$\operatorname{depth}_{S}I/(J,I^{\prime\prime}_{1})\geq d+1$.
In the second case, let $a_{1},\ldots,a_{t_{1}}$ be a maximal bad path
starting with $a_{1}$. Set $c_{j}=h(a_{j})$, $j\in[t_{1}]$. Then
$c_{t_{1}}=bx_{u_{1}}$ for some $u_{1}$ and let us say $b=fx_{i}$. If
$a_{t_{1}}\in(f)$ then changing in ${\mathcal{P}}_{b}$ the interval
$[a_{t_{1}},c_{t_{1}}]$ by $[f,c_{t_{1}}]$ we get a partition on $I/J$ with
sdepth $d+2$. Thus we may assume that $a_{t_{1}}\in E$. If
$fx_{u_{1}}\in\\{a_{1},\ldots,a_{t_{1}-1}\\}$, let us say $fx_{u_{1}}=a_{v}$,
$1\leq v<t_{1}$ then we may replace in ${\mathcal{P}}_{b}$ the intervals
$[a_{p},c_{p}]$, $v\leq p\leq t_{1}$ with the intervals $[a_{v},c_{t_{1}}]$,
$[a_{p+1},c_{p}]$, $v\leq p<t_{1}$. Now we see that we have in
${\mathcal{P}}_{b}$ the interval $[fx_{u_{1}},fx_{i}x_{u_{1}}]$ and switching
it with the interval $[f,fx_{i}x_{u_{1}}]$ we get a partition with sdepth
$\geq d+2$ for $I/J$. Thus we may assume that
$fx_{u_{1}}\not\in\\{a_{1},\ldots,a_{t_{1}}\\}$. Now set
$a_{t_{1}+1}=fx_{u_{1}}$. Let $a_{t_{1}+1},\ldots,a_{k}$ be a path starting
with $a_{t_{1}+1}$ and set $c_{j}=h(a_{j})$, $t_{1}<j\leq k$. If $a_{p}=a_{v}$
for $v<t_{1}$, $p>t_{1}$ then change in ${\mathcal{P}}_{b}$ the intervals
$[a_{j},c_{j}]$, $v\leq j\leq p$ with the intervals $[a_{v},c_{p}]$,
$[a_{j+1},c_{j}]$, $v\leq j<p$. We have in ${\mathcal{P}}_{b}$ an interval
$[fx_{u_{1}},fx_{i}x_{u_{1}}]$ and switching it to $[f,fx_{i}x_{u_{1}}]$ we
get a partition with sdepth $\geq d+2$ for $I/J$. Thus we may suppose that in
fact $a_{p}\not\in\\{b,a_{1},\ldots,a_{p-1}\\}$ for any $p>t_{1}$ (with
respect to any path starting with $a_{t_{1}+1}$). We have again two subcases:
$1^{\prime})$ there exist no maximal bad path starting with $a_{t_{1}+1}$,
$2^{\prime})$ there exists a maximal bad path starting with $a_{t_{1}+1}$.
In $1^{\prime})$ set $T_{2}=\\{b^{\prime}\in B:\mbox{there\ exists\ a \ path}\
a_{t_{1}+1},\ldots,a_{k}\ \mbox{with}\ a_{k}=b^{\prime}\\}$, $G_{2}=B\setminus
T_{2}$ and $I^{\prime}_{2}=(f,G_{2})$, $I^{\prime\prime}_{2}=(G_{2})$,
$J^{\prime}_{2}=I^{\prime}_{2}\cap J$,
$J^{\prime\prime}_{2}=I^{\prime\prime}_{2}\cap J$. As above, we see that if
$T_{2}\cap(f)=\emptyset$ then we may take $I^{\prime}=I^{\prime}_{2}$ and if
$T_{2}\cap(f)\not=\emptyset$ then $I^{\prime}=I^{\prime\prime}_{2}$ works.
In the second case, let $a_{t_{1}+1},\ldots,a_{t_{2}}$ be a maximal bad path
starting with $a_{t_{1}+1}$ and set $c_{j}=h(a_{j})$ for $j>t_{1}$. As we saw
the whole set $\\{a_{1},\ldots,a_{t_{2}}\\}$ has different monomials. As above
$c_{t_{2}}=bx_{u_{2}}$ and we may reduce to the case when
$fx_{u_{2}}\not\in\\{a_{1},\ldots,a_{t_{1}}\\}$. Set $a_{t_{2}+1}=fx_{u_{2}}$
and again we consider two subcases, which we treat as above. Anyway after
several such steps we must arrive in the case $p=t_{m}$ when $b|c_{t_{m}}$ and
again a certain $fx_{u_{m}}$ is not among $\\{a_{1},\ldots,a_{t_{m}}\\}$ and
taking $a_{t_{m}+1}=fx_{u_{m}}$ there exist no maximal bad path starting with
$a_{t_{m}+1}$. This follows since we may reduce to the case when the set
$\\{a_{1},\ldots,a_{t_{m}}\\}$ has different monomials and so the procedures
should stop for some m. Finally, using $T_{m}=\\{b^{\prime}\in B:\mbox{there\
exists\ a \ path}\ a_{t_{m}+1},\ldots,a_{k}\ \mbox{with}\ a_{k}=b^{\prime}\\}$
as $T_{1}$ above we are done.
Now Theorem 2.3 follows from the next proposition, the case $s>q+1$ being a
consequence of [16] (see here Theorem 1.1).
###### Proposition 5.2.
Suppose that $I\subset S$ is minimally generated by a squarefree monomial $f$
of degree $d$, and a set $E$ of squarefree monomials of degrees $\geq d+1$.
Assume that $\operatorname{sdepth}_{S}I/J=d+1$ and $s\leq q+1$. Then
$\operatorname{depth}_{S}I/J\leq d+1$.
###### Proof.
Apply induction on $|E|$, the case $E=\emptyset$ follows from Lemma 2.2.
Suppose that $|E|>0$. We may assume that $E$ contains just monomials of
degrees $d+1$ by [17, Lemma 1.6]. Using Theorem 1.3 and induction on $|E|$
apply Lemma 2.1. Thus we may suppose that
$C\subset((f)\cap(E))\cup(\cup_{a,a^{\prime}\in
E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$.
Let $b\in(B\cap(f))$ and $I^{\prime}_{b}=(B\setminus\\{b\\})$. Set
$J^{\prime}_{b}=I^{\prime}_{b}\cap J$. Clearly $b\not\in I^{\prime}_{b}$. As
in Case 1 from the previous section we see that if
$\operatorname{sdepth}_{S}I^{\prime}_{b}/J^{\prime}_{b}\leq d+1$ then
$\operatorname{depth}_{S}I^{\prime}_{b}/J^{\prime}_{b}\leq d+1$ by Theorem 1.3
and so $\operatorname{depth}_{S}I/J\leq d+1$ by the Depth Lemma. Thus we may
suppose that $\operatorname{sdepth}_{S}I^{\prime}_{b}/J^{\prime}_{b}\geq d+2$.
Applying Lemma 5.1 we get either $\operatorname{sdepth}_{S}I/J\geq d+2$
contradicting our assumption, or there exists a nonzero ideal
$I^{\prime}\subsetneq I$ generated by a subset of $\\{f\\}\cup B$ such that
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for $J^{\prime}=J\cap
I^{\prime}$ and $\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$. In the
last case we see that $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$
by induction hypothesis on $|E|$ and so $\operatorname{depth}_{S}I/J\leq d+1$
by the Depth Lemma applied to the following exact sequence
$0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow
I/(J,I^{\prime})\rightarrow 0.$
Acknowledgement. Research partially supported by grant ID-PCE-2011-1023 of
Romanian Ministry of Education, Research and Innovation.
Andrei Zarojanu was supported by the strategic grant POSDRU/159/1.5/S/137750,
Project Doctoral and Postdoctoral programs support for increased
competitiveness in Exact Sciences research cofinanced by the European Social
Found within the Sectorial Operational Program Human Resources Development
$2007-2013$.
## References
* [1] C. Biro, D.M. Howard, M.T. Keller, W.T. Trotter, S.J. Young, Interval partitions and Stanley depth, J. Combin. Theory Ser. A 117 (2010), 475-482.
* [2] W. Bruns, C. Krattenthaler, J. Uliczka, Stanley decompositions and Hilbert depth in the Koszul complex, J. Commutative Alg., 2 (2010), 327-357.
* [3] M. Cimpoeas, The Stanley conjecture on monomial almost complete intersection ideals, Bull. Math. Soc. Sci. Math. Roumanie, 55(103), (2012), 35-39.
* [4] J. Herzog, A. Soleyman Jahann, S. Yassemi, Stanley decompositions and partitionable simplicial complexes, J. Algebr. Comb., 27, (2008), 113-125.
* [5] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [6] J. Herzog, D. Popescu, M. Vladoiu, Stanley depth and size of a monomial ideal, Proc. Amer. Math. Soc., 140 (2012), 493-504.
* [7] B. Ichim, J. J. Moyano-Fernandez, How to compute the multigraded Hilbert depth of a module, to appear in Math. Nachr., arXiv:AC/1209.0084.
* [8] M. Ishaq, Upper bounds for the Stanley depth, Comm. Algebra, 40(2012), 87-97.
* [9] M. Ishaq, Values and bounds of the Stanley depth, Carpathian J. Math. 27, (2011), 217-224, arXiv:AC/1010.4692.
* [10] G. Lyubeznik, On the Arithmetical Rank of Monomial ideals, J. Algebra 112, 86-89 (1988).
* [11] A. Popescu, Special Stanley Decompositions, Bull. Math. Soc. Sc. Math. Roumanie, 53(101), no 4 (2010), arXiv:AC/1008.3680.
* [12] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), (2009), 377-382,
* [13] D. Popescu, Stanley conjecture on intersections of four monomial prime ideals, Communications in Alg., 41 (2013), 1-12, arXiv:AC/1009.5646.
* [14] D. Popescu, Depth and minimal number of generators of square free monomial ideals, An. St. Univ. Ovidius, Constanta, 19 (2), (2011), 187-194.
* [15] D. Popescu, Depth of factors of square free monomial ideals, Proceedings of AMS 142 (2014), 1965-1972, arXiv:AC/1110.1963.
* [16] D. Popescu, Upper bounds of depth of monomial ideals, J. Commutative Algebra, 5, 2013, 323-327, arXiv:AC/1206.3977.
* [17] D. Popescu, A. Zarojanu, Depth of some square free monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, 56(104), 2013,117-124.
* [18] D. Popescu, A. Zarojanu, Depth of some special monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, 56(104), 2013, 365-368, arXiv:AC/1301.5171v1.
* [19] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010),773-784.
* [20] Y.H. Shen, Lexsegment ideals of Hilbert depth 1, (2012), arXiv:AC/1208.1822v1.
* [21] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
* [22] J. Uliczka, Remarks on Hilbert series of graded modules over polynomial rings, Manuscripta Math., 132 (2010), 159-168.
* [23] A. Zarojanu, Stanley Conjecture on three monomial primary ideals, Bull. Math. Soc. Sc. Math. Roumanie, 55(103),(2012), 335-338.
|
arxiv-papers
| 2013-07-31T11:46:39 |
2024-09-04T02:49:48.837305
|
{
"license": "Public Domain",
"authors": "Dorin Popescu and Andrei Zarojanu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1307.8292"
}
|
1307.8349
|
# Synthetic gauge fields in synthetic dimensions
A. Celi ICFO – Institut de Ciències Fotòniques, Mediterranean Technology
Park, E-08860 Castelldefels (Barcelona), Spain P. Massignan ICFO – Institut
de Ciències Fotòniques, Mediterranean Technology Park, E-08860 Castelldefels
(Barcelona), Spain J. Ruseckas Institute of Theoretical Physics and
Astronomy, Vilnius University, A. Goštauto 12, Vilnius 01108, Lithuania N.
Goldman Center for Nonlinear Phenomena and Complex Systems - Université Libre
de Bruxelles, 231, Campus Plaine, B-1050 Brussels, Belgium I. B. Spielman
Joint Quantum Institute, University of Maryland, College Park, Maryland
20742-4111, USA National Institute of Standards and Technology, Gaithersburg,
Maryland 20899, USA G. Juzeliūnas Institute of Theoretical Physics and
Astronomy, Vilnius University, A. Goštauto 12, Vilnius 01108, Lithuania M.
Lewenstein ICFO – Institut de Ciències Fotòniques, Mediterranean Technology
Park, E-08860 Castelldefels (Barcelona), Spain ICREA – Institució Catalana de
Recerca i Estudis Avançats, E-08010 Barcelona, Spain
###### Abstract
We describe a simple technique for generating a cold-atom lattice pierced by a
uniform magnetic field. Our method is to extend a one-dimensional optical
lattice into the “dimension” provided by the internal atomic degrees of
freedom, yielding a synthetic 2D lattice. Suitable laser-coupling between
these internal states leads to a uniform magnetic flux within the 2D lattice.
We show that this setup reproduces the main features of magnetic lattice
systems, such as the fractal Hofstadter butterfly spectrum and the chiral edge
states of the associated Chern insulating phases.
###### pacs:
37.10.Jk, 03.75.Hh, 05.30.Fk
Intense effort is currently devoted to the creation of gauge fields for
electrically neutral atoms Lewenstein2007 ; Bloch2008a ; Dalibard2011 ;
Goldman2013RPP . Following a number of theoretical proposals in presence
Ruostekoski:2002 ; Jaksch2003 ; Mueller2004 ; Sorosen2005 ; Eckart2005 ;
Osterloh2005 ; Gerbier2010 ; Kitagawa2010 ; Kolovsky2011 or in absence of
optical lattices Dum1996 ; Visser1998 ; Juzeliunas2004 ; Ruseckas2005 ;
Juzeliunas2006 ; Spielman2009 ; Gunter2009 , synthetic magnetic fields have
been engineered both in vacuum Lin2009b ; Lin2009a ; Spielman-Hall-effect ;
Wang2012 ; Cheuk2012 and in periodic lattices Bloch2011 ; Struck2013 ;
Bloch2013 ; Ketterle2013 . The addition of a lattice offers the advantage to
engineer extraordinarily large magnetic fluxes, typically of the order of one
magnetic flux quantum per plaquette Ruostekoski:2002 ; Jaksch2003 ;
Mueller2004 ; Osterloh2005 ; Gerbier2010 , which are out of reach using real
magnetic fields in solid-state systems (e.g. artificial magnetic fields
recently reported in graphene Columbia2013 ; Manchester2013 ; MIT2013 ). Such
cold-atom lattice configurations will enable one to access striking
properties, such as Hofstadter-like fractal spectra Hofstadter1976 and Chern
insulating phases, in a controllable manner. Existing schemes for creating
uniform magnetic fluxes require several laser fields and/or additional
ingredients, such as tilted potentials Jaksch2003 ; Osterloh2005 ,
superlattices Gerbier2010 , or lattice-shaking methods Eckart2005 ;
Zenesini2009 ; Eckart2010 ; Kolovsky2011 ; Struck2011 ; Hauke2012 .
Experimentally, strong staggered magnetic flux configurations have been
reported Bloch2011 ; Struck2013 , and very recently also uniform ones
Bloch2013 ; Ketterle2013 . Besides, an alternative route is offered by optical
flux lattices Dudarev2004 ; Cooper2011a ; Cooper2011 ; Juz-Spielm2012 .
Figure 1: (a) Proposed experimental layout with ${}^{87}\rm{Rb}$. A pair of
counter-propagating $\lambda=1064{\ {\rm nm}}$ lasers provide a $5{{E_{L}}}$
deep optical lattice lattice with period $a=\lambda/2$. A pair of “Raman”
laser beams with wavelength $\lambda_{R}=790{\ {\rm nm}}$, at angles
$\pm\theta$ from ${\mathbf{e}}_{x}$, couple the internal atomic states with
recoil wavevector ${{k_{R}}}=2\pi\cos(\theta)/\lambda_{R}$. The laser beams’
polarizations – all linear – are marked by symbols at their ends. (b) Raman
couplings in the $F=1$ manifold. The transitions are induced by the beams
depicted in (a). (c) Synthetic 2D lattice with magnetic flux
$\Phi=\gamma/2\pi$ per plaquette ($\gamma=2{{k_{R}}}a$). Here $n=x/a$ ($m$)
labels the sites along ${\mathbf{e}}_{x}$ (Zeeman sublevels).
In all of these lattice schemes, the sites are identified by their location in
space. This need not be the case: the available spatial degrees of freedom can
be augmented by employing the internal atomic “spin” degrees of freedom as an
extra, or synthetic, lattice-dimension Boada:2012 . Here we demonstrate that
this extra dimension can support a uniform magnetic flux, and we propose a
specific scheme using a 1D optical lattice along with Raman transitions within
the atomic ground state manifold (Fig. 1). The flux is produced by a
combination of ordinary tunneling in real space and laser-assisted tunneling
in the extra dimension creating the necessary Peierls phases. Our proposal
therefore extends the toolbox of existing techniques to create gauge
potentials for cold atoms.
The proposed scheme distinguished by the naturally sharp boundaries in the
extra dimension, a feature which greatly simplifies the detection of chiral
edge states resulting from the synthetic magnetic flux Goldman2010a ;
Stanescu:2010 ; Goldman:2012prl ; Buchhold:2012 ; Goldman:2013PNAS . We
demonstrate that the chiral motion of these topological edge states can be
directly visualized using in situ images of the cloud, and we explicitly show
their robustness against impurity scattering. We also show that by using
additional Raman and radio frequency transitions one can connect the edges in
the extra dimension, providing a remarkably simple way to realize the fractal
Hofstadter butterfly spectrum Hofstadter1976 .
Model. For specificity, consider ${}^{87}{\rm Rb}$’s $F=1$ ground state
hyperfine manifold Note1 , composed of three magnetic sublevels $m_{F}=0,\pm
1$, illuminated by the combination of optical lattice and Raman laser beams
depicted in Fig. 1(a) (additional lattice potentials along ${\mathbf{e}}_{y}$
and ${\mathbf{e}}_{z}$, confining motion to ${\mathbf{e}}_{x}$ are not shown;
${\bf e}_{xyz}$ are the three Cartesian unit vectors). In the schematic, the
counter-propagating $\lambda=1064{\ {\rm nm}}$ lasers beams define the lattice
with period $a=\lambda/2$, recoil momentum ${{k_{L}}}=2\pi/\lambda$, and
energy ${{E_{L}}}=\hbar^{2}{{k_{L}}}^{2}/2m$ (where $m$ is the atomic mass).
We consider a sufficiently deep lattice $V_{\rm lat}=5{{E_{L}}}$ for the tight
binding approximation to be valid, but shallow enough to avoid Mott-insulator
physics. For these parameters, the tunneling amplitude is
$t=0.065{{E_{L}}}=h\times 133{\ {\rm Hz}}$. The Raman lasers at wavelength
$\lambda_{R}\approx 790{\ {\rm nm}}$ intersect with opening angle $\theta$,
giving an associated Raman recoil momentum
${{k_{R}}}=2\pi\cos(\theta)/\lambda_{R}$. The Raman couplings recently
exploited in experiment Lin2009a ; Lin2009b , between the three magnetic
sublevels $m_{F}=0,\pm 1$ of the $F=1$ ground-state manifold of 87Rb are shown
in Fig. 1(b). The Raman transitions provide the hopping in the synthetic
dimension which require a minimum amount of laser light (less than 1% required
for existing schemes Spielman2009 ), minimizing spontaneous emission. In
addition, periodic boundary conditions in the synthetic direction can be
created by coupling $m_{F}=+1$ to $m_{F}=-1$ using an off-resonant Raman
transition from $|{F=1,m_{F}=+1}\rangle$ to an ancillary state, e.g.,
$|{F=2,m_{F}=0}\rangle$ (detuned by $\delta_{\rm pbc}$ and coupled with
strength $\Omega_{R,{\rm pbc}}$), completed by a radio-frequency transition to
$|{F=1,m_{F}=1}\rangle$ with strength $\Omega_{RF}$, giving a $\Lambda$-like
scheme with strength $\Omega_{\rm pbc}=-\Omega_{R,{\rm
pbc}}\Omega_{RF}/2\delta_{\rm pbc}$.
A constant magnetic field $B_{0}{\mathbf{e}}_{z}$ Zeeman splits the magnetic
sublevels $|{m_{F}=\pm 1}\rangle$ by $\mp\hbar\omega_{0}=g_{F}\mu_{\rm
B}B_{0}$, where $g_{F}$ is the Landé $g$-factor and $\mu_{\rm B}$ is the Bohr
magneton, see Fig. 1(ab). The Raman spin-flip transitions, detuned by $\delta$
from two-photon resonance, impart a $2{{k_{R}}}$ recoil momentum along
${\mathbf{e}}_{x}$. Taking $\hbar=1$, the laser fields can be described via a
spatially periodic effective magnetic field
${\boldsymbol{\Omega}}_{T}=\delta\mathbf{e}_{z}+\Omega_{R}\left[\cos\left(2{{k_{R}}}x\right)\mathbf{e}_{x}-\sin\left(2{{k_{R}}}x\right)\mathbf{e}_{y}\right]\,,$
(1)
which couples the hyperfine ground-states giving the effective atom-light
Hamiltonian Goldman2013RPP ; Dudarev2004 ; Deutsch1998 ; Juz-Spielm2012
$H_{\rm al}={\boldsymbol{\Omega}}_{T}\cdot\mathbf{F}=\delta
F_{z}+(F_{+}e^{i{{k_{R}}}x}+F_{-}e^{-i{{k_{R}}}x})\Omega_{R}/2\,,$ (2)
where the operators $F_{\pm}=F_{x}\pm iF_{y}$ act as
$F_{+}\left|m\right\rangle=g_{F,m}\left|m+1\right\rangle$ with
$g_{F,m}=\sqrt{F\left(F+1\right)-m\left(m+1\right)}$. Thus the Raman beams
sequentially couple states $m=-F,\,\ldots\,,F$, with each transition
accompanied by an $x$-dependent phase. This naturally generates Peierls phases
for “motion” along the $m$ (spin) direction, denoted as ${\bf e}_{m}$.
The combination of the optical lattice along ${\mathbf{e}}_{x}$ and the Raman-
induced hopping along ${\bf e}_{m}$ yield an effective 2D lattice with one
physical and one synthetic dimension, as depicted in Fig. 1(c) for $F=1$. For
a system of length $L_{x}$ along ${\mathbf{e}}_{x}$, the lattice has
$N=L_{x}/a$ sites along ${\mathbf{e}}_{x}$, and a width of $W=2F+1$ sites
along ${\bf e}_{m}$. For $\delta=0$ the system is described by the Hamiltonian
$H=\sum_{n,m}\left(-ta_{n+1,m}^{{\dagger}}+\Omega_{m-1}e^{-i\gamma
n}a_{n,m-1}^{{\dagger}}\right)a_{n,m}+{\rm H.c.}\,,$ (3)
where $n$ labels the spatial index and $m$ labels the spin index;
$\gamma=2{{k_{R}}}a$ sets the magnetic flux; $\Omega_{m}=\Omega_{R}g_{F,m}/2$
is the synthetic tunneling strength; and $a_{n,\,m}^{{\dagger}}$ is the atomic
creation operator in the dimensionally extended lattice. This two-dimensional
lattice is pierced by a uniform synthetic magnetic flux
$\Phi=\gamma/2\pi={{k_{R}}}a/\pi$ per plaquette (in units of the Dirac flux
quantum). The quantity $g_{F,m}$ is independent of $m$ for $F=1/2$ and $F=1$,
but for larger $F$ hopping along ${\bf e}_{m}$ is generally non-uniform.
Open boundaries. Since $\Omega_{m}\neq 0$ only when $m\in\\{-F,\ldots,F-1\\}$,
Eq. (3) has open boundary conditions along ${\bf e}_{m}$, with sharp edges at
$m=\pm F$. By gauge-transforming $a_{n,m}$ and $a^{\dagger}_{n,m}$, the
hopping phase $\exp(i2{{k_{R}}}x)$ can be transferred to the hopping along
${\mathbf{e}}_{x}$. Combining this with a Fourier transformation along
${\mathbf{e}}_{x}$,
$b_{q,\,m}^{{\dagger}}=N^{-1/2}\sum_{n=1}^{N}a_{n,\,m}^{{\dagger}}e^{i\left(q+\gamma
m\right)n}$, splits the Hamiltonian $H=\sum_{q}H_{q}$ into momentum components
$\displaystyle H_{q}$ $\displaystyle=\sum_{m=-F}^{F}\varepsilon_{q+\gamma
m}b_{q,\,m}^{{\dagger}}b_{q,\,m}+\left(\Omega_{m}b_{q,\,m+1}^{{\dagger}}b_{q,\,m}+{\rm
H.c.}\right),$
where $\varepsilon_{k}=-2t\cos(k)$, $q\equiv 2\pi l/N$, and
$l\in\\{1,\dots,N\\}$. Figure 2 shows the resulting band structure for $F=1$.
Away from the avoided crossings, the lowest band describes the propagation of
“edge states” localized in spin space at $m=\pm F$ (blue and red arrows):
these states propagate along ${\mathbf{e}}_{x}$ in opposite directions. In the
physical system, these give rise to a spin current
$j_{s}(x)=j_{\uparrow}-j_{\downarrow}$. When $W=2F+1\gg 1$, these edge states
become analogous to those in quantum Hall systems SuppMat ; Hugel:2013 . The
$F=9/2$ manifold of ${}^{40}{\rm K}$ allows experimental access to this
large-$W$ limit Hatsugai:1993 , since its 10 internal states reproduce the
Hofstadter-butterfly topological band structure.
Figure 2: Spectrum for open boundary conditions: $F=1$, and
$\Phi=\gamma/2\pi=1/2\pi$ flux per plaquette. Colors specify the spin state
$m$, as indicated. The ground state branch displays “edges” corresponding to
$m=\pm 1$.
The edge-state propagation can be directly visualized by confining a multi-
component Fermi gas to a region $x\in[-L_{x}/2,L_{x}/2]$ and by setting the
Fermi energy $E_{\text{F}}$ within the Raman-induced gap (dashed line in Fig.
2) Note2 . In this configuration, different types of states are initially
populated: (a) edge states localized at $m=\pm F$ with opposite group
velocities, and (b) bulk states delocalized in spin space with small group
velocities (the central or bulk region of the lowest band is almost
dispersionless for small flux $\Phi\ll 1$). When the confining potential along
${\mathbf{e}}_{x}$ is suddenly released, the edge states at $m=\pm F$
propagate along $\pm{\mathbf{e}}_{x}$. Figure 3 depicts such dynamics, where
we allowed tightly confined atoms (as above) to expand into a harmonic
potential $V_{\text{harm}}(x)$. This potential limits the propagation of the
edge states along ${\mathbf{e}}_{x}$ and leads to chiral dynamics around the
synthetic 2D lattice: when an edge state localized at $m=+F$ reaches the Fermi
radius $x=R_{\text{F}}$, it cannot backscatter because of its chiral nature,
and thus, it is obliged to jump on the other edge located at $m=-F$ and
counter-propagate. The edge-state dynamics of the $F=9/2$ lattice is presented
in SuppMat .
Figure 3: (a) Initial condition: a Fermi gas is trapped in the central region
$x\in[-13a,13a]$ and the Fermi energy is set to populate only the lowest
energy band. The occupied edge states localized at $m=\pm F$ have opposite
group velocities (for simplicity we sketch the “F=1” case). An additional
harmonic potential limits the edge-states propagation, leading to chiral
dynamics around the synthetic 2D lattice. (b) Dynamics after releasing the
cloud into the harmonic potential, for $\Omega_{0}=0.5t$, $\Phi=1/2\pi$,
$V_{\text{harm}}(x)=t(x/50a)^{2}$ and $E_{\text{F}}\\!=\\!-1.4t$. Dashed lines
represent the Fermi radius $R_{\text{F}}$ at which the edge states localized
at $m\\!=\\!\pm F$ jump to the opposite edge $m\\!=\\!\mp F$.
An interesting feature of edge states is their robustness against local
perturbations. To check this in the context of our proposal, we consider the
effects of a spatially localized impurity on the transmission probability. The
Hamiltonian with an impurity localized at $n=0$ is
$H_{\mathrm{imp}}=H+V\,,\quad V=\sum_{m}V_{m}a^{{\dagger}}_{0,m}a_{0,m}\,,$
(4)
where the zero-th order Hamiltonian $H$ is given by Eq. (3), and $V_{m}$ is
the interaction potential between the impurity and atoms in state $m$. The
perturbation may be generated, e.g., by a tightly focused laser, or by a
distinguishable atom, deeply trapped by a species selective optical lattice
Massignan2006 ; Nishida2008 ; Lamporesi2010 If the impurity scatters equally
strongly with all spin components, it corresponds to an extended obstacle
along ${\bf e}_{m}$: a “roadblock” in the synthetic 2D lattice. On the other
hand, if the impurity interacts significantly only with a given spin
component, it yields a localized perturbation in the synthetic 2D lattice. In
particular, edge perturbations can be engineered by choosing an impurity that
only scatters strongly the $m=F$ or $m=-F$ states.
For $F=1$ there are 3 dispersion branches, as shown in Fig. 2, so there are 9
possible scattering channels. However, here we focus to the energy range lying
inside the bulk-gap (around the dashed lines in Fig. 2), where there is only
one available scattering channel, i.e., scattering to the opposite edge state.
The transmission probability as a function of the energy of the incident atom
is calculated in SuppMat , and shown in Fig. 4. For spin-independent
collisions with the impurity ($V_{m}=U$), the transmission probability goes to
zero at two values of the energy within the gap. In analogy with Fano
resonances Fano1961 ; Satanin2005 , these zeros are associated with two quasi-
bound states localized around the impurity potential due to two local
parabolic minima (for $F=1$) in the upper dispersion branches. Outside of the
resonant regions the transmission probability is close to 1. On the other
hand, an impurity which scatters only the $m=0$ component
($V_{m}=U\delta_{m,0}$) is effectively localized in the central chain of the
synthetic 2D lattice. As such, it can couple resonantly two oppositely
propagating edge states, leading to a single sharp minimum in the transmission
probability. Instead, an impurity which is localized at the edge of the
synthetic dimension (e.g., $V_{m}=U\delta_{m,1}$) does not lead to a resonant
behavior of the transmission probability. For such spin-dependent impurity the
transmission probability is always close to 1, since the edge state can go
around the impurity in the synthetic dimension.
Figure 4: Edge-state transmission probability. Black: a spin-independent
impurity. Blue: only $m=0$ scatters. Red: only $m=1$ scatters. Parameters are
the same as in Fig. 2(a) and the scattering strength is $U=-t$.
Cyclic couplings. In our $F=1$ example, periodic boundary conditions along
${\bf e}_{m}$ can be induced with an extra coupling (with a Rabi frequency
$\Omega_{1}=\Omega_{\rm pbc}=\Omega_{0}$) from $|{m=1}\rangle$ to
$|{m=-1}\rangle$ accompanied by the momentum recoil $k$ along
${\mathbf{e}}_{x}$. The system becomes periodic only provided the flux
$\gamma$ per plaquette is rational, i.e., $\gamma=2\pi P/Q$ with $P,Q$ co-
prime integers. Note that the number of loops in the synthetic dimension
required to have an integral number of flux quanta, i.e. periodicity, is $l/M$
where $l=LCM(M,Q)$, thus, for M=3, $Q$ or $Q/3$ loops.
In this cyclic scheme, the system reproduces the Hofstadter problem defined in
the infinite plane: its spectrum $E=E(p)$ is obtained by solving the Harper
equation along ${\mathbf{e}}_{x}$ Hatsugai:1993 , where $p$ is the quasi-
momentum associated with the closed synthetic dimension ${\mathbf{e}}_{y}$.
The conserved momentum along ${\mathbf{e}}_{y}$ can only take three values:
$p_{j}=2\pi j/3$ with $j\in\\{-1,0,1\\}$. Exploiting the fact that the
Hamiltonian (3) with closed b.c. is translationally invariant in the spin
dimension, we perform the Fourier transform
$a_{n,m}^{\dagger}=3^{-1/2}\sum_{j=-1}^{1}e^{i2\pi mj/3}c_{n,j}^{\dagger}$,
giving
$H=\sum_{j,n}\epsilon(2\pi
j/3+n\gamma)c^{{\dagger}}_{n,j}c_{n,j}-(tc^{{\dagger}}_{n+1,j}c_{n,j}+{\rm
H.c.}),$ (5)
and $\epsilon(k)=-2\Omega_{0}\cos(k)$. Its spectrum is plotted in Fig. 5.
There are $l$ points in each band associated with the rational flux $\gamma$:
enough to be visible. For our finite chain of length $N$, the infinite-chain
result will be accurate only for $Q\ll N$, while for $Q$ approaching $N$ the
system is far from periodic in $Q$ and the butterfly gets blurred.
Figure 5: The spectrum of Eq. (5) on an infinite 1D chain, for a three-level
system with closed b.c. has the typical Hofstadter butterfly characteristics.
Interactions. We wish to consider the effects of repulsive interactions. We
focus here on the case where the interactions are SU(W)-invariant (this
amounts to negleglecting the spin-dependent contribution to the interaction; a
very good approximation for $F=1$ ${}^{87}\rm{Rb}$). In our lattice, the
resulting interaction Hamiltonian
$\displaystyle H_{\rm int}$
$\displaystyle=\frac{\mathcal{U}}{2}\sum_{n}\mathcal{N}_{n}(\mathcal{N}_{n}-1)\,,\qquad\mathcal{N}_{n}\equiv\sum_{m}a_{n,\,m}^{\dagger}a_{n,\,m},$
is local along ${\mathbf{e}}_{x}$, but infinite in range along ${\bf e}_{m}$.
We exploit the SU(W)-invariance of $H_{\rm int}$ by adopting the Fock basis
$c_{n,j}$ in which the hopping along ${\bf e}_{m}$ is diagonal, as in Eq. (5)
(a similar basis exists for open boundary conditions in the synthetic
dimension). Let us denote its eigenvalues by $\epsilon_{n,j}$. It follows that
we can minimize the energy for fixed $\langle H_{\rm int}\rangle$ by
populating only the states associated to $c_{n,j_{n}}$ with lowest
$\epsilon_{n,j_{n}}$, as this minimizes the kinetic term $\langle H\rangle$.
Two cases are possible: i) $j_{n}$ is unique, i.e. the local ground state is
not degenerate; ii) $\epsilon_{j,n}$ is minimal for two of the three possible
values of $j$. The latter case can occur only for closed b.c. in the synthetic
dimension and for rational values of the flux $\gamma/(2\pi)=P/Q$. In presence
of open b.c., it is indeed easy to show that the eigenvalues are always
independent of $\gamma$ (and as such as $n$), and never degenerate. In case
i), the ground state can be mapped to the one of a 1D uniform Bose-Hubbard
chain. In case ii) instead, the 1D Hubbard chain will possess a primitive cell
containg $Q$ consecutive lattice points, as well known from the non-
interacting Hofstadter problem. Interactions which are non-SU(N)-invariant
lead to considerably more complicated situations, with the ground state
possessing a complex, fully 2D character.
Conclusions. Our proposal for creating strong synthetic gauge fields using a
synthetic 2D lattice is well suited to directly observe chiral edge-states
dynamics, by using spin-sensitive detection of the different edge modes. This
platform also allows to test the edge states’ robustness against impurities.
To detect the full spectrum, interaction effects must be minimized, for
example using a fermionic band insulator or a dilute thermal Bose gas. The
spectrum may also be probed by transport measurements: wavepackets of atoms
with narrow energy dispersion can be prepared and brought into the lattice
using a waveguide, and their transmission through the region of effective
magnetic field observed Lauber2011 ; Cheiney2013 .
###### Acknowledgements.
We acknowledge enlightening discussions with E. Anisimovas, F. Chevy, J.
Dalibard, L. Fallani, F. Gerbier and C. Salomon and support from FRS-FNRS
(Belgium), ERC AdG QUAGATUA, EU IP SIQS, Spanish MINCIN (FIS2008-00784
TOQATA), ESF POLATOM network, the European Social Fund under the Global Grant
measure. IBS acknowledges the support of the ARO with funding from DARPA’s OLE
program and the Atomtronics-MURI; and the NSF through the PFC at JQI.
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## I Supplemental material
### I.1 Edge states in thin stripes: Hofstadter square lattice vs Hofstadter
ladder
In the main text, we have concentrated on the spectra and edge-state dynamics
for spin 1 atoms ($F=1$). In that case a synthetic 2D lattice is constituted
of $N\times 3$ lattice sites, where $3=W=2F+1$ is the number of sites along
the synthetic (spin) direction and $N$ is the number of sites along the
spatial direction $x$ (see Figs. 1 – 3 in the main text). Such a lattice has
natural open boundaries along the spin direction at $y=\pm Fa$ (where $a$ is
the lattice spacing), while $N$ can be arbitrarily large. In this Appendix, we
illustrate how the edge-state properties discussed in the main text can be
related to the topological band structure and chiral edge states of the
standard Hofstadter square lattice Hatsugai:1993 , namely, a square lattice of
$N\times W$ sites, with $N,W\gg 1$, subjected to a uniform magnetic flux
$\Phi$ per plaquette. The number of lattice sites along the $y$ direction is
denoted $W$, so as to refer to the width of the stripe.
To do so, we consider an extrapolation between the Hofstadter lattice (size
$N\times W$) and the thin stripe considered in the main text (size $N\times
3$), by progressively reducing the number of lattice sites along the $y$
direction $W$, while applying periodic boundary conditions along the $x$
direction, see Fig. 6 (a). The first spectrum shown in Fig. 6 (b), obtained
for $W=50$, shows the usual band structure of the Hofstadter model, where a
clear distinction between the bulk bands and the edge states dispersions is
observed. To highlight this edge/bulk picture, we simultaneously represent the
energies $E=E(q)$ together with the mean position $\langle y\rangle$ of the
eigenstates along the spin direction, see the color code in Fig. 6 (a). The
many bulk states progressively disappear, as the number of inequivalent
lattice sites is reduced to $W=5$, while the dispersion branches of the edge
states are only slightly modified. In fact, for $\Phi=p/q\in\mathbb{Q}$, the
edge-state branches remain remarkably robust for $W\rightarrow q$. When $W$ is
further reduced such that $W<q$, the edge-state branches are altered, but they
retain their general characteristics: in the thin stripe (“double-ladder”)
limit $W=3$ considered in the main text, the lowest energy band describes
edges states localized on opposite edges (at $y=\pm a$) of the double-ladder,
propagating in opposite directions. Therefore, we can conclude that the edge-
state structure present in the double-ladder lattice ($W=3$) is reminiscent of
the chiral (topological) edge states present in the standard Hofstadter square
lattice (see also Ref. Hugel:2013, for a detailed study of the Hofstadter
ladder with $W=2$ corresponding to $F=1/2$).
Figure 6: (a) Hofstadter model on a stripe of width $W$, and definition of the
color code: dark blue (resp. red) dots correspond to states localized at the
bottom (resp. top) edge of the system, whereas green-yellow dots correspond to
bulk states. (b) Energy spectrum $E=E(q)$ of the Hofstadter model with the
flux $\Phi=1/5$, for different stripe widths $W$. Here, the modulus of the
hopping amplitude is taken equal to $t$ along both directions, and $q$ denotes
the quasi-momentum. The double-ladder configuration used in the main text
corresponds to $W=3$ (i.e., $F=1$ and $\Omega_{0}=t$).
### I.2 The $F=9/2$ case
In the main text, we focused on the study of the $F=1$ case, which is widely
investigated in current cold-atom experiments Lin2009a ; Lin2009b . This leads
to the double-ladder lattice, whose connection with the standard Hofstadter
model has been described in the previous Section of this Supplementary
material. However, it would be desirable to engineer a synthetic 2D lattice
with more internal states to make this connection even more visible. For
example, considering the ground-state manifold of 40K, where $F=9/2$, would
allow to engineer a lattice of size $N\times 10$, which according to Fig. 6
(b) would clearly display the topological band structure of the Hofstadter
model. We note that using other atomic species (such as 173Yb) could also lead
to similar configurations with $W>5$, both for bosonic and fermionic systems.
One important aspect of the present proposal is the fact that for $F>1$ the
magnitude of hopping along the $y$ (spin) direction is not constant. Indeed,
the hopping from a lattice site $m$ to a lattice site $m+1$ is given by the
frequency
$t_{m\rightarrow m+1}=\Omega g_{F,m}=\Omega\sqrt{F(F+1)-m(m+1)},$ (6)
where we remind that $m=m$ refers to the internal states of the atom and $F$
is the total angular momentum. This inhomogenous hopping, shown in Fig. 7 (a)
for $F=9/2$, is not present in the standard Hofstadter model, where the tight-
binding hopping amplitude $t$ is constant. To illustrate this effect, we show
the band structure of a synthetic lattice engineered with $F=9/2$ atoms (Fig.
7 (b)), and we compare it with the band structure of the homogenous Hofstadter
model with $W=10$ (Fig. 7 (c)). We observe that the bulk/edge band structure
is well conserved, when choosing $\Omega=t/\langle g_{F,m}\rangle$, where
$\langle g_{F,m}\rangle=\sum_{m}g_{F,m}/2F$. However, we note that the states
corresponding to the edge-state dispersions are no longer perfectly localized
at the edges: close to the lowest bulk band, there are dispersive states with
$|\langle m\rangle|<9/2$. We also note that the states with the highest
velocity $v\\!\sim\\!\partial_{q}E$ are those that are the most localized at
the edges.
Figure 7: (a) Synthetic lattice for $F=9/2$ atoms. The hopping amplitude $t$
along the $x$ (spatial) direction is constant, while the hopping amplitude
along the $y$ (spin) direction, $\Omega g_{F,m}$, is given by Eq. (6). (b) The
energy spectrum for the $F=9/2$ synthetic lattice, setting $\Phi=1/5$ and
$\Omega=t/\langle g_{F,m}\rangle=0.24t$. (c) The energy spectrum for the
homogenous Hofstadter lattice with $W=10$ lattice sites along the $y$
direction and $\Phi=1/5$, see also Fig. 6b. Note that the edge states are more
spatially localized in the homogeneous case [(c)] than in the inhomogeneous
synthetic lattice [(b)].
In Fig. 8, we show the edge-state dynamics for a fermionic system with $F=9/2$
atoms (e.g. ${}^{40}K$), confined by a harmonic potential
$V_{\text{harm}}(x)=t(x/50a)^{2}$. We clearly observe a chiral motion in the
2D synthetic lattice, which is due to the populated edge states lying within
the lowest bulk gap (Fig. 7 (b)). As already described above, these edge
states are not perfectly localized at $m=\pm 9/2$, due to the inhomogeneity of
the hopping along the spin direction. As a result, the dynamics show the
rotation of the cloud in the 2D lattice, instead of a clear edge-state motion.
Figure 8: Edge-states dynamics for a fermionic system with $F=9/2$ atoms (e.g.
${}^{40}K$): the Fermi gas is trapped in the central region $x\in[-13a,13a]$
and the Fermi energy is set such as to populate only the lowest energy band.
The populated “edge” states localized at $m=\pm F$ have opposite group
velocities. An additional harmonic potential limits the edge-states
propagation, leading to chiral dynamics around the synthetic 2D lattice. The
parameters are $\Omega=t/\langle g_{F,m}\rangle=0.24t$, $\Phi=1/5$,
$V_{\text{harm}}(x)=t(x/50a)^{2}$ and $E_{\text{F}}\\!=\\!-2t$. Dashed lines
represent the Fermi radius $R_{\text{F}}$ at which the edge states localized
at $m\\!=\\!\pm F$ jump unto the opposite edge $m\\!=\\!\mp F$. The time steps
are $\Delta_{t}=37.5\hbar/J$.
### I.3 Scattering on a localized impurity
#### I.3.1 Formulation
Our aim here is to calculate the transmission probability for an atom in the
1D physical lattice affected by an impurity localized at $n=0$ and thus
described by the Hamiltonian
$H_{\mathrm{imp}}=H+V\,,\quad
V=\sum_{m,m^{\prime}}V_{m,m^{\prime}}a^{{\dagger}}_{0,m}a_{0,m^{\prime}}\,,$
(7)
where $H$ is an unperturbed Hamiltonian for the 1D array of atoms is given by
Eq.(3) of the main text, and $m$ refers to the spin levels representing a
synthetic degree of freedom.
We shall make use of the Green’s operator $G=[E-H_{\mathrm{imp}}+i0^{+}]^{-1}$
of the full Hamiltonian $H_{\mathrm{imp}}$. The Green’s operator of the
complete system will be expressed in terms of the Green’s operator
$G_{0}=[E-H+i0^{+}]^{-1}$ of the unperturbed system using the Dyson equation
Economou2006 $G=G_{0}+G_{0}VG$. On the other hand, the zero-order Green’s
operator $G_{0}$ will be presented via the eigenfunctions and eigen-energies
of the unperturbed Hamiltonian $H$. Having the complete Green’s operator $G$
we will determine the scattering T-matrix $T=V+VGV$ from which the
transmission probabilities will be calculated.
#### I.3.2 Spectrum of the Hamiltonian without impurity
Applying a gauge transformation $\tilde{a}_{n,m}=a_{n,m}e^{-i\gamma nm}$ we
transfer the phases featured in the hopping elements to the hopping in the
physical direction in the Hamiltonian $H$ defined by Eq. (3) in the main text,
giving:
$H=\sum_{n,m}\left(-te^{-i\gamma
m}\tilde{a}_{n+1,m}^{{\dagger}}+\Omega_{m-1}\tilde{a}_{n,m-1}^{{\dagger}}\right)\tilde{a}_{n,m}+\mathrm{h.c.}\,.$
(8)
From now on we will express all energies in the units of the hopping integral
$t$; therefore, we will set $t=1$. The atomic center-of mass wave function
satisfies the Schrödinger equation
$H\Psi=E\Psi\,.$ (9)
We search for the eigenvectors of the Hamiltonian (8) in the form of plane
waves (Bloch states) by taking the probability amplitudes to find an atom in
the site $n,m$ as
$\Psi_{m}(n)=\chi_{q,m}e^{iqn}\,.$ (10)
We will interpret the index $m$ as a row number and consider $\Psi$ and
$\chi_{q}$ as columns. Equation (9) yields the following eigenvalue equations
$H_{q}\chi_{q}=E_{q}\chi_{q}\,.$
Here $H_{q}$ is $(2F+1)\times(2F+1)$ matrix with the diagonal matrix elements
$(H_{q})_{m,m}=-2\cos(q+\gamma m)$ and nonzero non-diagonal elements
$(H_{q})_{m,m^{\prime}}=\Omega_{m}\delta_{m^{\prime},m+1}$ and
$(H_{q})_{m,m^{\prime}}=\Omega_{m-1}\delta_{m^{\prime},m-1}$. In particular,
when $F=1$ the matrix $H_{q}$ reduces to
$H_{q}=\left(\begin{array}[]{ccc}-2\cos(q-\gamma)&\Omega&0\\\
\Omega&-2\cos(q)&\Omega\\\ 0&\Omega&-2\cos(q+\gamma)\end{array}\right)\,.$
(11)
By solving an eigenvalue problem we get a set of $2F+1$ algebraic equations.
It has has $2F+1$ solutions to be labelled with an index $\nu$.
#### I.3.3 Green’s function of the system without impurity
Given the eigenfunctions $\Psi_{q,s}(n)$, the general expression for the
retarded zero-order Green’s function is
$G_{0}(n,n^{\prime};E)=\sum_{\nu=1}^{2F+1}\int_{-\pi}^{\pi}\frac{\Psi_{q,\nu}(n)\Psi_{q,\nu}^{*}(n^{\prime})}{E-E_{q,\nu}+i\eta}dq\,,$
(12)
where $\eta\rightarrow+0$. Zeros in the denominator can be obtained from the
eigen-energy equation
$\det[E-H_{q}]=0\,,$ (13)
which generally has $2F+1$ solutions. For each eigen-energy $E$ and wave
vector $q_{\nu}$, the analytical expressions for the eigenvectors
$\chi_{q_{\nu},\nu}$ can be obtained from the equation
$[H_{q}-E]\chi_{q_{\nu},\nu}=0$ by setting the first element of
$\chi_{q_{\nu},\nu}$ to unity and dropping one of the resulting equations.
Using Eq. (12) and performing the integration we obtain the retarded zero-
order Green’s function
$G_{0}(n,n^{\prime};E)=-i\sum_{\nu}\frac{1}{v_{\nu}}\begin{cases}\chi_{q_{\nu},\nu}\chi_{q_{\nu},\nu}^{T}e^{iq_{\nu}(n-n^{\prime})}\,,&n>n^{\prime},\\\
\chi_{-q_{\nu},\nu}\chi_{-q_{\nu},\nu}^{T}e^{-iq_{\nu}(n-n^{\prime})}\,,&n<n^{\prime},\end{cases}$
(14)
Here
$v_{\nu}\equiv\left.\frac{\partial}{\partial q}E_{q,\nu}\right|_{q=q_{\nu}}$
(15)
is the group velocity. It can be calculated from the equation
$v_{\nu}=-\left.\frac{\frac{\partial}{\partial
q}\det[E-H_{q}]}{\frac{\partial}{\partial
E}\det[E-H_{q}]}\right|_{q=q_{\nu}}\,.$ (16)
Note that we do not have complex conjugation in Eq. (14) since for real wave
vectors $q_{\nu}$ the colums $\chi_{q_{\nu},\nu}$ are real. This is because
the Hamiltonian $H_{q}$ has real matrix elements.
#### I.3.4 Green’s function for the system with localized impurity
Combining the Dyson equation $G=G_{0}+G_{0}VG$ with Eq. (4) for $V$, one has
$\displaystyle G(n,n^{\prime})$ $\displaystyle=$ $\displaystyle
G_{0}(n,n^{\prime})+\sum_{n^{\prime\prime}}G_{0}(n,n^{\prime\prime})V\delta_{n^{\prime\prime},0}G(n^{\prime\prime},n^{\prime})$
(17) $\displaystyle=$ $\displaystyle
G_{0}(n,n^{\prime})+G_{0}(n,0)VG(0,n^{\prime})\,.$
Taking $n=0$ in Eq. (17) we get
$G(0,n^{\prime})=G_{0}(0,n^{\prime})+G_{0}(0,0)VG(0,n^{\prime})\,.$ (18)
From here we obtain
$G(0,n^{\prime})=[1-G_{0}(0,0)V]^{-1}G_{0}(0,n^{\prime})\,.$ (19)
Substituting Eq. (19) back into Eq. (17) we get the required expression for
the Green’s function
$G(n,n^{\prime})=G_{0}(n,n^{\prime})+G_{0}(n,0)V[1-G_{0}(0,0)V]^{-1}G_{0}(0,n^{\prime})\,.$
(20)
#### I.3.5 Transmission probabilities
The scattering is described by $T$ matrix
$T=V+VGV\,.$ (21)
Using Eq. (20), $T$ matrix reads
$T(n,n^{\prime})=V[1-G^{(0)}(0,0)V]^{-1}\delta_{n,0}\delta_{n^{\prime},0}\,.$
(22)
For transmitted waves the matrix element of the scattering matrix is
$S_{\nu,\nu^{\prime}}^{t}=\delta_{\nu,\nu^{\prime}}-i\frac{1}{\sqrt{v_{\nu}v_{\nu^{\prime}}}}\sum_{n,n^{\prime}}\chi_{q_{\nu},\nu}^{{\dagger}}e^{-iq_{\nu}n}T(n,n^{\prime})\chi_{q_{\nu^{\prime}},\nu^{\prime}}e^{iq_{\nu^{\prime}}n^{\prime}}\,.$
(23)
Using Eq. (22) we obtain
$S_{\nu,\nu^{\prime}}^{t}=\delta_{\nu,\nu^{\prime}}-\sqrt{\frac{v_{\nu}}{v_{\nu^{\prime}}}}i\frac{1}{v_{\nu}}\chi_{q_{\nu},\nu}^{{\dagger}}V\left[1+i\sum_{\nu^{\prime\prime}}\frac{1}{v_{\nu^{\prime\prime}}}\chi_{q_{\nu^{\prime\prime}},\nu^{\prime\prime}}\chi_{q_{\nu^{\prime\prime}},\nu^{\prime\prime}}^{{\dagger}}V\right]^{-1}\chi_{q_{\nu^{\prime}},\nu^{\prime}}\,.$
(24)
Transmission probability from the propagating mode $\nu^{\prime}$ to the mode
$\nu$ is
$T_{\nu,\nu^{\prime}}=|S_{\nu,\nu^{\prime}}^{t}|^{2}\,.$ (25)
These equations are used in calculating the transmission probabilities in the
main text.
|
arxiv-papers
| 2013-07-31T15:12:15 |
2024-09-04T02:49:48.865165
|
{
"license": "Public Domain",
"authors": "A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I.B. Spielman, G.\n Juzeliunas, and M. Lewenstein",
"submitter": "Alessio Celi",
"url": "https://arxiv.org/abs/1307.8349"
}
|
1307.8362
|
# Beam heat load in superconducting wigglers
S. Casalbuoni [email protected]
ANKA Karlsruhe Institute of Technology Karlsruhe Germany
###### Abstract
The beam heat load is a fundamental input parameter for the design of
superconducting wigglers since it is needed to specify the cooling power. In
this presentation I will review the possible beam heat load sources and the
measurements of beam heat load performed and planned onto the cold vacuum
chambers installed at different synchrotron light sources.
## 1 INTRODUCTION
Superconducting (SC) wigglers are used worldwide in low and middle energy (1-3
GeV) storage rings to increase the flux in the harder part of the X-ray
spectrum from 20 to 100 keV used for material science, biology, medical
diagnostics and therapy [1]. In order to satisfy similar demands and further
increase the brilliance SC undulators are under development for middle and
high energy storage rings [2, 3]. Free electron lasers would also benefit of
superconducting undulators (elliptically polarised) [4]. SC technology has
been proposed also to be applied in undulators and wigglers for high energy
physics projects as for the positron source of the International Linear
Collider [5] and the damping wigglers for the Compact Linear Collider [6].
All these devices consist of a cryostat with SC NbTi coils kept at about 4 K
and of a beam vacuum chamber to let the beam through the coils. The beam
vacuum chamber also referred to as liner is kept at about 10-20 K. In order to
maximize the peak magnetic field, the space between the liner and the coils
should be minimized. Because of the necessity to impregnate the SC coils, they
must be located out of the ultrahigh vacuum (UHV) where the beam is confined.
The liner intercepts the beam heat load and it is ideally thermally
disconnected from the coils to avoid a degradation of their performance. In
reality the liner and the coils have always some thermal connection. A proper
cryogenic design of all the devices described above requires the knowledge of
the beam heat load to the beam vacuum chamber. In the following section I
describe some of the possible beam heat load sources. I then report on the
measurements of the beam heat load to cold vacuum chambers performed at
different synchrotron light sources with the installed SC undulators and
wigglers. Afterwards I present dedicated experiments to measure the beam heat
load to a cold vacuum chamber which will hopefully be useful also to
understand the underlying mechanism. The last section contains conclusions and
outlook.
## 2 POSSIBLE BEAM HEAT LOAD SOURCES
Possible beam heat load sources are: synchrotron radiation, RF effects due to
geometrical and resistive wall impedance, and electron and/or ion bombardment.
### 2.1 Synchrotron Radiation Heating
The power of the synchrotron radiation emitted from the upstream bending
magnet hitting the upper and lower surfaces of the vertical gap of the SC
undulator or wiggler is [7, 8]:
$P_{\rm
syn}=2P_{0}\frac{21}{32}\int_{\psi_{0}}^{\psi_{1}}\frac{\gamma}{(1+\gamma^{2}\psi^{2})^{5/2}}\Biggl{[}1+\frac{5}{7}\frac{\gamma^{2}\psi^{2}}{(1+\gamma^{2}\psi^{2})}\Biggr{]}d\psi$
(1)
where $\psi_{0}$ and $\psi_{1}$ are the lower and upper values of $\psi$
indicated in Fig. 1,
$\gamma=E/m_{e}c^{2}~{},$ $P_{0}=\frac{eI\gamma^{4}}{6\pi\epsilon_{0}\rho},$
$e$ is the electron charge, $I$ is the average beam current, $\epsilon_{0}$ is
the vacuum permittivity, $E$ is the beam energy, $\rho$ is the radius of
curvature of the electron trajectory in the bending magnet, $m_{e}$ is the
electron mass and $c$ is the speed of light. The factor two in front of the
integral takes into account of the upper and lower surfaces of the vacuum
chamber of the SC undulator or wiggler (see Fig. 1).
Figure 1: Scheme of the synchrotron radiation from the upstream bending magnet
hitting the upper and lower surfaces of the liner of a SC undulator or
wiggler.
The beam heat load contribution from synchrotron radiation depends linearly on
the stored average beam current, it depends on the electron beam energy and on
the geometry, that is on the relative position of the bending magnet, of the
collimator and the liner. It is however independent on the filling pattern and
on the bunch length.
### 2.2 RF Heating
The total power $P_{RF}$ lost by the beam due to the wake fields of $N_{b}$
equally spaced bunches can be obtained by using the relation [9]:
$P_{RF}=I^{2}\sum\limits_{n=-\infty}^{\infty}ReZ_{||}(nN_{b}\omega_{0})\left|S(nN_{b}\omega_{0})\right|^{2}$
(2)
$S$ being the single bunch spectrum, $ReZ_{||}$ the real part of the
longitudinal component of the coupling impedance. Assuming bunches with
Gaussian shape and length $\sigma_{z}=3$ mm a schematic representation of the
multibunch spectrum not in scale is shown by the blue vertical lines in Fig.
2. If we would plot the lines spaced in scale they would be indistinguishable
from the single bunch spectrum. As obtained from Eq. (2), when $N_{b}$ times
the revolution frequency $f_{0}$ (for a 300 m circumference storage ring
$f_{0}=\omega_{0}/(2\pi)\sim 1$ MHz) is much smaller than the inverse of the
bunch duration $c/(\sqrt{2}\pi\sigma_{z})\sim$ few 10 GHz ($c=$ speed of
light), the multibunch spectrum is well approximated by the single bunch
spectrum
$S(\omega)=\exp{-\frac{\sigma^{2}_{z}\omega^{2}}{2c^{2}}}.$
In this case and in absence of resonant modes $N_{b}\omega_{0}\rightarrow
d\omega$ and $nN_{b}\omega_{0}\rightarrow\omega$, so Eq. (2) becomes:
$P_{RF}=\frac{I^{2}T_{0}}{N_{b}}k_{l}$ (3)
where $k_{l}$ is the loss factor, $T_{0}=2\pi/\omega_{0}$ the revolution
period and $I=N_{b}Q/T_{0}$ the average beam current with $Q$ the average
bunch charge. The loss factor is given by:
$k_{l}=\frac{1}{\pi}\int\limits_{0}^{\infty}\left|S(\omega)\right|^{2}ReZ_{||}(\omega)d\omega~{}.$
(4)
Figure 2: Single bunch spectrum with two different bunch lengths (solid red
and blue lines) and sketch of multibunch spectrum (solid blue vertical lines
modulated by the single bunch spectrum) [10].
The total power $P_{RF}$ lost by the beam is an upper limit of the power
dissipated in the structure since, excluding the case of resistive wall, it is
unknown where this power is deposited. In case of resistive wall the power is
deposited in the first few $\mu$m of the vacuum chamber. For wakes induced by
geometrical changes of the cross section of the vacuum chamber the power could
be deposited in the chamber itself, or could be exchanged in the interaction
with other bunches and be deposited somewhere else in the accelerator [10].
Geometrical changes in the cross section are almost unavoidable at the
transitions, where tapers and RF fingers and bellows are employed. Even
flanges connecting parts with the same aperture contribute to cross section
changes due to the finite mechanical accuracy of the manufactured components.
Since however these parts are far away from the coils, towards the entrance
and exit of the cryostat and thermally anchored to a radiation shield at $\sim
50-80$ K, their contribution to heat the central liner close to the coils is
expected to be negligible. In case of high ordered modes excited by the beam
and trapped in the liner it should be possible to considerably remove the
losses by changing the inverse of the bunch spacing by the bandwidth of the
resonance [11].
The contribution of the surface roughness to the impedance is relevant for
bunches with a length of the order of magnitude of the surface corrugations.
The different theoretical models developed to calculate the coupling impedance
of a beam pipe with a rough surface are reviewed in Ref. [12, 13]. An
important parameter to determine the impedance is the so called aspect ratio,
that is the ratio between the average peak heights and the average distance
between the peaks.
In order to reduce the losses due to resistive wall heating the material
chosen for the surface ( of few $\mu$m) of the liner exposed to the beam is a
high conductivity material as copper. Aluminum is also used, for example in
the SC undulator under development at the Argonne Photon Source (APS) [2]. The
real part of the longitudinal impedance due to resistive wall is given by:
$ReZ_{||}(\omega)=\frac{L}{\pi 2b}R_{surf}$ (5)
where $L$ is the length of the considered portion of vacuum chamber,
$R_{surf}(\omega)$ is the surface resistance, and in case of a circular beam
pipe $2b$ is the diameter [14] while in case of a rectangular beam pipe is the
gap [7]. For copper at low temperatures and $RRR>7$ the anomalous skin effect
[7, 8, 14, 15] has to be considered:
$R_{surf}(\omega)=R_{\infty}(\omega)(1+1.157\alpha^{-0.276}),~{}~{}~{}\mbox{for}~{}\alpha\geq
3$ (6)
with
$\alpha=\frac{3}{2}\left[\frac{\ell}{\delta(\omega)}\right]^{2}=\frac{3}{4}\mu_{r}\mu_{0}\sigma\omega\ell^{2}$
where $\ell$ is the mean free path,
$\delta(\omega)=\sqrt{\frac{2}{\mu_{r}\mu_{0}\sigma\omega}}$
the skin depth, $\mu_{r}$ the relative permeability, $\mu_{0}$ the vacuum
permeability and $\sigma$ the electrical conductivity at room temperature (for
copper $\sigma=6.45\times 10^{7}$ S/m), and with
$R_{\infty}(\omega)=\left(\frac{\sqrt{3}}{16\pi}\frac{\ell}{\sigma}(\mu_{r}\mu_{0}\omega)^{2}\right)^{1/3}~{}.$
The beam heat load due to RF effects depends quadratically on the stored
average beam current. It depends on the bunch length and on the filling
pattern, in particular on the number of bunches and on the bunch spacing, and
on the position of the bunch in the vacuum chamber. It does not depend on the
beam energy.
### 2.3 Electrons and/or Ions Bombardment Heating
Ions and electrons created by ionization and photodesorption, and accelerated
against the wall by the passing beam will also contribute to the beam heat
load. The beam dynamics involved is unknown and might be quite complicated. It
is however likely that it is dominated by the beam properties and by the
chamber surface characteristics, as secondary emission yield, photoemission
yield, photoemission induced electron energy distribution, etc…, which are
only partially measured for a cryosorbed gas layer.
Since the beam dynamics is unknown we do not know for this source of beam heat
load its dependence on the different beam parameters as filling pattern, beam
energy, average stored beam current, bunch length, bunch spacing and number of
bunches. Taking this into account we cannot then state that an observed linear
or quadratic dependence of the beam heat load on the average beam current is
sufficient to prove that the main contribution to the beam heat load comes
from synchrotron radiation or RF effects, respectively.
## 3 OBSERVATIONS WITH SC WIGGLERS AND UNDULATORS
Cold bore SC wigglers and an undulator installed in different storage rings
have been used also to measure the beam heat load. The interpretation of the
measurements is not straightforward since these devices have not been designed
to perform beam heat load diagnostics. In all cases the beam heat load
measured is higher than the one expected from the synchrotron radiation of the
upper bending magnet and from resistive wall heating. In the following I
summarize the results from the measurements performed with SC wigglers at MAX
II, at the Diamond Light Source (DLS), and with a SC undulator at ANKA
(Ångstrom source Karlsruhe).
### 3.1 Experience at MAX II: SC Wiggler
The two SC wigglers designed and manufactured at MAX-lab successfully
operating for almost a decade showed both a higher helium consumption than
predicted [1]. For one of the wigglers the beam heat load has been measured to
be 0.86 W instead of the predicted 0.17 W. The beam heat load measured as a
function of the stored average beam current, can be fitted by the sum of a
linear and of a quadratic component, respectively made responsible of
synchrotron radiation and resistive wall losses. The contribution from the
synchrotron radiation is double than the one predicted. This discrepancy has
been attributed to a misalignment of the bending magnet-collimator-liner
system. The contribution to the beam induced heating from the image currents
is 0.59 W, about 10 times larger than expected from the calculations [7], is
not understood.
### 3.2 Experience at DLS: SC Wigglers
Two SC wigglers from the Budker Institute for Nuclear Physics are installed at
the DLS. The beam heat load is extrapolated by using the temperature rise in
the liner and the heat shields to deduce the extra cooling power of the
cryocoolers plus the additional liquid helium boil off [16]. The uncertainty
in the measurements is up to $30\%$. A quadratic dependence on the bunch
charge and on the stored average beam current is observed, and also in this
case the predicted values are smaller than the measured ones.
### 3.3 Experience at ANKA: SC Undulator
A cold bore superconducting undulator built by ACCEL Instr. GmbH, Bergisch
Gladbach, Germany [17], was installed in one of the four straight sections of
the ANKA storage ring in March 2005 and removed in July 2012. The performance
of this device was limited by the too high beam heat load. Namely, the
superconducting coils performance was reduced during users operation from 750
A to 300 A meaning a reduction in the peak magnetic field on axis from 0.42 T
to 0.26 T. The observed beam heat load up to 2.5 W [18] at a gap of 8 mm and
at 100 mA stored average beam current is much higher than the predicted values
of 63 mW from the synchrotron radiation of the upstream bending magnet and of
22 mW from the image currents [8].
A simple model of electron bombardment appears to be consistent with the large
variation of beam heat load and of pressure rise values as a function of the
average beam current for different gaps [8, 18] observed in the cold bore of
the SC undulator. Still to be understood is the mechanism responsible for the
electron multipacting and the role played by the cryosorbed gas layer. A
common cause of electron bombardment is the buildup of an electron cloud,
which strongly depends on the chamber surface properties. The surface
properties as secondary electron yield, photoemission yield, photoemission
induced electron energy distribution, needed in the simulation codes to
determine the possible occurrence and size of an electron cloud buildup, have
only partly been measured for a cryosorbed gas layer. Even using uncommonly
large values for these parameters, the heat load inferred from the ECLOUD
simulations [19] is about one order of magnitude lower than the measurements
[20]. While electron cloud buildup models have been well benchmarked in
machines with positively charged beams, in electron machines they do not
reproduce the observations satisfactory. This has been shown at the ECLOUD10
workshop also by K. Harkay [21] and by J. Calvey [22] comparing the RFA data
taken with electron beams in the APS and in CesrTA, respectively, with the
simulations performed using the electron cloud buildup codes POSINST [23] and
ECLOUD [19]. From these comparisons it seems that the electron cloud buildup
codes do not contain all the physics going on for electron beams. In order to
fit the data with the simulations, the approach at APS and CesrTA is to change
the photoelectron model. At ANKA we tried to study if the presence of a smooth
ion background (i.e. a partially neutralized electron beam) can change the
photoelectron dynamics so that the photo-electrons can receive a significant
amount of kinetic energy from the ion cloud plus electron beam system.
Following preliminary analytical results by P. F. Tavares (MAX-lab), S. Gerstl
(ANKA) has included an ion cloud potential in the ECLOUD code: preliminary
simulations are encouraging.
## 4 Dedicated experiments
### 4.1 LBNL-SINAP Calorimeter
A calorimeter to measure the beam heat load in a storage ring via temperature
gradients has been proposed by the Lawrence Berkeley National Laboratory
(LBNL) [24]. Two proposals with different cooling concepts have been made: one
using a He boiler and the other conduction cooling. This last concept will be
realized in collaboration with the Shanghai Institute of Applied Physics
(SINAP) and the device is planned to be installed in the Shanghai Light Source
[4]. The LBNL-SINAP calorimeter, shown in Fig. 3, will allow to measure the
beam heat load at different gaps. It will be provided with heaters to permit
constant temperature operation and in situ calibration checks. Measurements
for different materials will be possible by changing the substrate of the
liner which faces the beam.
Figure 3: Sketch of the LBNL-SINAP calorimeter [4].
### 4.2 COLDDIAG
With the aim of measuring the beam heat load on a cold bore and in order to
gain a deeper understanding in the beam heat load mechanisms, a cold vacuum
chamber for diagnostics (COLDDIAG) has been proposed [25] and built [26]. This
project led by ANKA is in collaboration with CERN, DLS, Frascati National
Laboratory, Rome University “La Sapienza”, STFC Daresbury Laboratory, STFC
Rutherford Appleton Laboratory, University of Manchester, Cockcroft Institute
of Science and Technology and Lund University MAX-lab. The vacuum chamber is
being designed and fabricated in collaboration with Babcock Noell GmbH.
COLDDIAG consists of a cold vacuum chamber (see cryostat in Fig. 4) located
between two warm sections. This will allow to observe the influence of
synchrotron radiation on the beam heat load and a direct comparison between
the cryogenic and room temperature regions, with and without a cryosorbed gas
layer, respectively. The same suite of diagnostics is used in both the cold
and warm regions. The diagnostics being implemented are: i) retarding field
analyzers to measure the electron flux, ii) temperature sensors to measure the
total heat load, iii) pressure gauges, iv) and mass spectrometers to measure
the gas content. In addition, to suppress charged particles from hitting the
chamber wall a solenoid is installed on the downstream half of the cold liner
section. The magnet reaches on axis a magnetic field of around 10 mT with a
current of 1 A. The inner vacuum chamber will be removable in order to test
different geometries and materials. COLDDIAG is built to fit in a short
straight section at ANKA, but ANKA is proposing its installation in different
synchrotron light sources with different energies and beam characteristics.
Figure 4: Overview of the cryostat and the diagnostics installed in COLDDIAG.
[28].
A successful final acceptance test has been performed with the liner reaching
a temperature of 4 K and the beam vacuum a pressure of 10-9 mbar [27].
COLDDIAG was installed in the storage ring at the DLS in November 2011. Due to
a mechanical failure of the thermal transition of the cold beam tube, the
cryostat had to be removed after one week of operation. Preliminary results
show a quadratic behaviour of the beam heat load as a function of average beam
current. The measured beam heat load of $\sim$8 W at 250 mA is almost two
orders of magnitude larger than the predicted value from resistive wall
heating $\sim 0.1-0.2$ W. Even if more statistics is needed, the almost random
temperature distribution on the liner and the small but visible effect of the
solenoid on the temperature distribution point out to electron bombardment as
at least one component of the beam heat load observed [28]. Currently the
design of the liner thermal transition is changed and a second installation at
the Diamond Light Source is under discussion.
During a longer installation in the DLS it is planned to monitor the
temperature, the electron flux, the pressure and the gas composition changing
[26]:
* •
the average beam current to compare the beam heat load data with synchrotron
radiation and resistive wall heating predictions,
* •
the bunch length to compare with resistive wall heating predictions,
* •
the filling pattern in particular the bunch spacing to test the relevance of
the electron cloud as heating mechanisms,
* •
beam position to test the relevance of synchrotron radiation and the gap
dependence of the beam heat load,
* •
inject different gases naturally present in the beam vacuum (H2, CO, CO2, CH4)
to understand the influence of the cryosorbed gas layer on the beam heat load,
and eventually identify the gases to be reduced in the beam vacuum.
## 5 CONCLUSIONS AND OUTLOOK
The beam heat load measurements performed with cold bore SC wigglers and an
undulator installed in different storage rings are not yet understood.
Two upcoming dedicated experimental setups, the LBNL-SINAP calorimeter and the
COLDDIAG, will be able to measure the beam heat load with high accuracies
$<0.05$ W and hopefully help to understand the beam heating mechanism. Even if
both setups are designed to measure the beam heat load, they are nicely
complementary. While the LBNL-SINAP calorimeter will allow beam heat load
measurements at different gaps, the COLDDIAG has one cold and two warm
sections, and it is equipped with additional diagnostics as retarding field
analyzers, pressure gauges and mass spectrometers to shed light on the role
played by the cryogenic layer in the beam heating mechanism. Preliminary
measurements performed with the COLDDIAG installed at the DLS indicate a value
of the beam heat load of $\sim 8$ W at 250 mA, which is almost two orders of
magnitude larger than the predicted value from resistive wall heating $\sim$
0.1 - 0.2 W.
Additional studies on the beam heat load can come from the SC wigglers
installed in many different storage rings and from the new SC undulators to be
installed at ANKA and at the APS.
## 6 ACKNOWLEDGMENT
I would like to thank M. Migliorati, A. Mostacci (University of Rome “La
Sapienza” and LNF, Frascati, Italy) and B. Spataro (LNF, Frascati, Italy) for
useful discussions on RF heating.
## References
* [1] N. Mezentsev and E. Wallén, “Superconducting Wigglers,” Synch. Rad. News 24 No.3 (2011) 3.
* [2] Y. Ivanyushenkov, M. Abliz, K. Boerste, T. Buffington, C. Doose, J. Fuerst, Q. Hasse, M. Kasa, S.H. Kim, R.L. Kustom, E.R. Moog, D. Skiadopoulos, E.M. Trakhtenberg, I.B. Vasserman, “STATUS OF THE FIRST PLANAR SUPERCONDUCTING UNDULATOR FOR THE ADVANCED PHOTON SOURCE,” IPAC’12, New Orleans, USA (2012), MOPPP078, http://www.JACoW.org
* [3] S. Casalbuoni, T. Baumbach, S. Gerstl, A. Grau, M. Hagelstein, C. Heske, T. Holubek, D. Saez de Jauregui, C. Boffo and W. Walter, “RECENT PROGRESS WITH SUPERCONDUCTING UNDULATORS AT ANKA,” Proc. ICFA Workshop on Future Light Sources, Newport News, Virginia, USA (2012).
* [4] S. Prestemon, http://accelconf.web.cern.ch
/AccelConf/FEL2011/talks/thoai2$\\_$talk.pdf
* [5] D. J. Scott, J. A. Clarke, D. E. Baynham, V. Bayliss, T. Bradshaw, G. Burton, A. Brummitt, S. Carr, A. Lintern, J. Rochford, O. Taylor, Y. Ivanyushenkov, “Demonstration of a High-Field Short-Period Superconducting Helical Undulator Suitable for Future TeV-Scale Linear Collider Positron Sources,” Phys. Rev. Lett. 107 (2011) 174803.
* [6] D. Schoerling, F. Antoniou, A. Bernhard, A. Bragin, M. Karppinen, R. Maccaferri, N. Mezentsev, Y. Papaphilippou, P. Peiffer, R. Rossmanith, G. Rumolo, S. Russenschuck, P. Vobly, and K. Zolotarev, “Design and system integration of the superconducting wiggler magnets for the Compact Linear Collider damping rings,” Phys. Rev. ST Accel. Beams 15 (2012) 042401.
* [7] E. Wallén, G. LeBlanc, “Cryogenic system of the MAX-Wiggler,” Cryogenics 44 (2004) 879.
* [8] S. Casalbuoni, A. Grau, M. Hagelstein, R. Rossmanith, F. Zimmermann, B. Kostka, E. Mashkina, E. Steffens, A. Bernhard, D. Wollmann, and T. Baumbach, “Beam heat load and pressure rise in a cold vacuum chamber,” Phys. Rev. ST Accel. Beams 10 (2007) 093202.
* [9] S. Heifets, K. Ko, C. Ng, X. Lin, A. Chao, G. Stupakov, M. Zolotorev, J. Seeman, U. Wienands, C. Perkins, M. Nordby, E. Daly, /N. Kurita, D. Wright, E. Henestroza, G. Lambertson, J. Corlett, J. Byrd, M. Zisman, T. Weiland, W. Stoeffl, and C. Belser, “Impedance Study for the PEP-II B-factory,” SLAC/AP-99 (1995).
* [10] S. Casalbuoni, M. Migliorati, A. Mostacci, L. Palumbo, B. Spataro, “Computation of the beam heatload contribution to RF effects to COLDDAIG,” submitted for publication.
* [11] B. Spataro, private communication.
* [12] A. Mostacci, L. Palumbo, D. Alesini, “Review of surface roughness effect on beam quality,” in The Physics and Applications of High Brightness Electron Beams, Ed. J. Rosenzweig, L. Serafini and G. Travish (World Scientific, 2003).
* [13] K. L. F. Bane, “Wakefields of Sub-Picosecond Electron Bunches,” Int. J. Mod. Phys. A22 (2007) 3736.
* [14] W. Chou and F. Ruggiero, “Anomalous skin effect and resistive wall heating,” LHC Project Note 2 (SL/AP) (1995).
* [15] H. London, “ The high frequency resistance of superconducting tin,” Proc. R. Soc. A 176 (1940) 522; A.B. Pippard, “ The anomalous skin effect in normal metals,” Proc. R. Soc. A 191 (1947) 385; G.E.H. Reuter and E.H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A 195 (1948) 336; R.G. Chambers, “ The anomalous skin effect,” Proc. R. Soc. A 215 (1952) 481.
* [16] J.C. Schouten and E.C.M. Rial, “ELECTRON BEAM HEATING AND OPERATION OF THE CRYOGENIC UNDULATOR AND SUPERCONDUCTING WIGGLERS AT DIAMOND,” IPAC’11, San Sebasti n, Spain (2011), THPC179, http://www.JACoW.org
* [17] S. Casalbuoni, M. Hagelstein, B. Kostka, R. Rossmanith, M. Weisser, E. Steffens, A. Bernhard, D. Wollmann, and T. Baumbach, “Generation of x-ray radiation in a storage ring by a superconductive cold-bore in-vacuum undulator,”, Phys. Rev. ST Accel. Beams 9 (2006) 010702.
* [18] S. Casalbuoni, S. Gerstl, A. Grau, T. Holubek, D. Saez de Jauregui, “BEAM HEAT LOAD AND PRESSURE IN THE SUPERCONDUCTING UNDULATOR INSTALLED AT ANKA,” IPAC’12, New Orleans, Louisiana, USA (2012), MOPPP068, http://www.JACoW.org
* [19] G. Rumolo and F. Zimmermann, CERN SL-Note-2002-016.
* [20] U. Iriso, S. Casalbuoni, G. Rumolo, F. Zimmermann, “ELECTRON CLOUD SIMULATIONS FOR ANKA,” PAC’09, Vancouver, Canada (2009), TH5PFP052, http://www.JACoW.org
* [21] K. Harkay, ECLOUD10, Ithaca, New York USA, 2010.
* [22] J. Calvey, ECLOUD10, Ithaca, New York USA, 2010.
* [23] M.A. Furman and M.T. Pivi, “Probabilistic model for the simulation of secondary electron emission,” Phys. Rev. ST Accel. Beams 5 (2002) 124404.
* [24] F. Trillaud, S. Prestemon, R. D. Schlueter, and S. Marks, “ Design of a Cryogenic Calorimeter for Synchrotron Light Source Beam-Based Heating,” IEEE Trans. on Appl. Supercond. Vol. 21-3 (2011) 1756.
* [25] S. Casalbuoni, T. Baumbach, A. Grau, M. Hagelstein, R. Rossmanith, V. Baglin, B. Jenninger, R. Cimino, M. Cox, E. Mashkina, and E. Wallén, “DESIGN OF A COLD VACUUM CHAMBER FOR DIAGNOSTICS,” EPAC’08, Genoa, Italy (2008), WEPC103, http://www.JACoW.org
* [26] S. Casalbuoni, T. Baumbach, S. Gerstl , A. Grau, M. Hagelstein, D. Saez de Jauregui, C. Boffo, G. Sikler, V. Baglin, R. Cimino, M. Commisso, B. Spataro, A. Mostacci, M. Cox, J. Schouten, E. Wallén, R. Weigel, J. Clarke, D. Scott, T. Bradshaw, I. Shinton, R. Jones, “COLDDIAG: A Cold Vacuum Chamber for Diagnostics,” IEEE Trans. on Appl. Supercond. Vol. 21-3 (2011) 2300.
* [27] S. Gerstl, T. Baumbach, S. Casalbuoni, A. Grau, M. Hagelstein, T. Holubek, D. Saez de Jauregui, V. Baglin, C. Boffo, G. Sikler, T. Bradshaw, R. Cimino, M. Commisso, A. Mostacci, B. Spataro, J. Clarke, R. Jones, D. Scott, M. Cox, J. Schouten, I. Shinton, E. Wallén, R. Weigel, “FACTORY ACCEPTANCE TEST OF COLDDIAG: A COLD VACUUM CHAMBER FOR DIAGNOSTICS,” IPAC’11, San Sebastian, Spain (2011), THPC159, http://www.JACoW.org
* [28] S. Gerstl, T. Baumbach, S. Casalbuoni, A. W. Grau, M. Hagelstein, D. Saez de Jauregui, T. Holubek, R. Bartolini, M. P. Cox, J. C. Schouten, R. Walker, M. Migliorati, B. Spataro, I. R. R. Shinton, “FIRST MEASUREMENTS OF COLDDIAG: A COLD VACUUM CHAMBER FOR DIAGNOSTICS,” IPAC’12, New Orleans, Louisiana, USA (2012), MOPPP069, http://www.JACoW.org
|
arxiv-papers
| 2013-07-31T15:50:08 |
2024-09-04T02:49:48.873980
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Casalbuoni (ANKA, KIT, Karlsruhe)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1307.8362"
}
|
1308.0090
|
# Resistive threshold logic
A. P. James, L.V.J. Francis and D. Kumar
###### Abstract
We report a resistance based threshold logic family useful for mimicking brain
like large variable logic functions in VLSI. A universal Boolean logic cell
based on an analog resistive divider and threshold logic circuit is presented.
The resistive divider is implemented using memristors and provides output
voltage as a summation of weighted product of input voltages. The output of
resistive divider is converted into a binary value by a threshold operation
implemented by CMOS inverter and/or Opamp. An universal cell structure is
presented to decrease the overall implementation complexity and number of
components. When the number of input variables become very high, the proposed
cell offers advantages of smaller area and design simplicity in comparison
with CMOS based logic circuits.
## 1 Introduction
Logic gates implement boolean algebraic expressions obtained from truth
tables. Increase in functional requirements of digital IC’s such as in
microprocessors and ASIC’s results in complex logic state implementations. A
complex set of logic states when represented as a truth table would have large
number of input and output variables. As the number of input variables
increases, it is often not possible to manually reduce the boolean logic
expressions to reduce the number of components required for its
implementation. The most common approach to reduce the number of components
required with a large number of variables is by using logic minimisation based
on prime implicant logics. Technique such as Karnaugh map[1],
QuineMcCluskey[2], Petrick’s method, Buchberger’s algorithm [3] and Espresso
minimization algorithm [4], are the widely used approaches. However, when the
number of inputs increases significantly, logic minimisation methods become
inefficient. In addition, implementations using existing logic families become
challenging as they are often restricted by the gate delays, the number of
inputs and the number of components.
The common approach employed to implement boolean algebra with a large number
(>10) of variables, is to apply the minimization techniques for standard gates
with a limited number of inputs (<10). This always results in more number of
circuit components than that was possible with gates that could support as
many number of inputs as the number of variables. In addition to this issue,
the number of components required to implement a gate vary from one boolean
logic to another, which results in increased structural complexity and results
in increased investment in production scale verification and testing cycles.
Generic digital circuits such as a single 2n to 1 multiplexer can be used to
implement $n$-input boolean logic function in canonical sum-of-products form.
As the number of inputs to the multiplexer increases, a typical AND-OR logic
would have large number of inputs per gate for its implementation. In order to
implement large variable boolean logic functions such as using multiplexers,
we introduce the concept of resistance threshold logic that minimises number
of components and design complexity. The proposed resistive threshold logic is
made up of a resistive divider and a threshold logic circuit. The idea of such
an analog-binary cell is inspired from the implementation challenges of the
long established theory and practices of neuron cell modelling and logic
circuits [5]. Conventional neuron inspired logic gate implementations[6] are
complex due to the requirements of multi-valued weights and neuron like
threshold functions. In addition, they fail to meet the original aim of having
large input logic gates useful for mimicking brain like logic functions. In
contrast, the resistive threshold logic is aimed to be simple in structure
having the ability to realise large variable logic functions, and is intended
to be used as a new standard cell universal logic family with a possible
ability to mimic brain logic.
## 2 Proposed Cell
Figure 1: The circuit diagram of the proposed resistive divider boolean logic
cell that consists of a two input resistive divider and a variable threshold
CMOS inverter is presented.
The proposed logic cell shown in Fig. 1 consists of a resistive divider and a
variable threshold inverter. In contrast to the earlier reported work on
cognitive memory network [7], in this work, we propose a significantly
different configuration, implementation and application of the structurally
similar and conceptually different cell. The input to the resistive divider
are the digital values that can be equated to the logic inputs of a digital
logic gate. Based on the output of the resistive divider and a predefined
inverter threshold, we propose to implement the basic boolean logic functions.
The selection of the threshold and the use of resistive logic in designing a
generalized logic cell is the primary contribution of this research.
An $N$-input resistance divider circuit consists of $N$ input resistors
$R_{i}$ and one reference resistor $R_{0}$. The output voltage $V_{0}$ for
$N$-input voltages $V_{i}$ can be represnted as
$V_{0}={\sum_{i=1}^{N}\frac{V_{i}}{R_{i}}}/{(\frac{1}{R_{0}}+\sum_{i=1}^{N}\frac{1}{R_{i}})}$.
The inputs $V_{i}$ have either of the two logical levels $V_{H}$ or $V_{L}$,
representing a binary logic [1,0]. We keep equal values to $R_{i}^{\prime}s$
and $R_{0}=mR_{i}$, which results in:
$V_{0}=\frac{\sum_{i=1}^{N}{V_{i}}}{\frac{1}{m}+{N}}$.
A straight forward approach to implement resistors is by using semiconductor
resistors. Semiconductor resistors consist of a resistive body that is
surrounded by an insulator often developed over a substrate, and two terminal
contacts implemented using conductive metallic strips. The value of
semiconductor resistance can be obtained from the expression, $\frac{\rho
L}{x_{j}W}$, where $\rho$ is the resistivity, $L$ is the length, $x_{j}$ is
the layer thickness and $W$ is the width of the resistive body.
Figure 2: The impact of change in input resistance on the output voltage
$V_{0}$ of the resistive divider is graphically illustrated. The results are
demonstrated for 100 input resistive divider, with each line showing the
relative change in $V_{0}$ for the corresponding number of resistors are
uniformly perturbated within a $\pm 10\%$ tolerance level of resistor values.
Note: here we keep $V_{i}=1$.
A concern while using resistance devices (such as semiconductor resistors) is
the impact of change in resistance value due to second order implementation
effects, such as improper junctions and defects. Figure 2 shows a simulated
study of the impact of change in resistance values on the output voltage of a
resistive divider circuit. It is assumed here that the changes in the resistor
values are limited within a tolerance level of $\pm 10\%$ of the actual
resistive values. It can be seen that a maximum of $\pm 10\%$ resistive values
introduces only about $.0894\%$ change in output voltage, which makes the
practical implementation of the resistive divider feasible even under
realistic conditions. While using semiconductor resistors, when the number of
inputs increase, the leakage current through the semiconductor resistance
becomes prohibitively high. This drawback is overcome by replacing
semiconductor resistors with memristors [8], which has negligible amount of
leakage current.
The proposed resistive divider circuit uses the memristor modeled by HP [8].
The device has a thin film of titanium dioxide (TiO2) sandwiched between two
platinum terminals. The titanium dioxide layer is doped on one side with
oxygen vacancies, TiO2-x. The doped region has lower resistance than that of
the insulated undoped region. The boundary between doped and undoped region
determines the effective resistance of the device. Let $D$ be the total width
of the TiO2 layer and $W$ be the width of the doped TiO2 layer. When a
positive voltage is applied at the doped side, the oxygen vacancies moves
towards the undoped region, increasing the width of the doped region, $W$ and
hence the effective resistance of the memristor decreases. The effective
resistance $M_{eff}$ of the memristor is
$M_{eff}=\frac{W}{D}R_{ON}+(1-\frac{W}{D})R_{OFF}$, where, $R_{ON}$ (=1
k$\Omega$) is the resistance of the memristor if it is completely doped and
$R_{OFF}$ (=100 k$\Omega$) is the resistance of the memristor if it is
undoped. When input voltage is withdrawn or when there is no potential
difference between the terminals, the memristor maintains the boundary between
the doped and undoped region, since the oxygen ions remain immobile after
removal of the input voltage. Thus the resistance will be maintained at the
same value before withdrawing the input voltage. From the equation,
$i=\frac{v}{M(q)}$ [9], where $v$ and $i$ are the voltage and current across
the memristor, and $M(q)$ is charge dependent resistance of the memristor, we
can see that when the voltage difference across the memristor is $0$, the
current through the memristor is $0$. If there is a reverse potential across
the memristor, the width of the undoped region increases, resulting in an
increase in the effective resistance of the memristor. This high resistance
will block the reverse leakage current through the memristor. When the number
of inputs increases, the collective forward current through the circuit does
not increase significantly, since the effective resistance in the memristor is
constant. Table 1 shows the effect of increase in number of inputs on the
collective current flowing through the circuit.
Table 1: Effect of increase in number of inputs on the forward current flowing through the memristor in the circuit. Number of inputs | Current through a single memristor | Current through the potential divider circuits
---|---|---
2 | 3.33$\mu$A | 6.66$\mu$A
10 | 0.909$\mu$A | 9.09$\mu$A
100 | 0.99099nA | 9.90099$\mu$A
Table 2: Truth Table of Two Input Resistive Divider Logic Cell When Used as NAND and NOR Gates Input Voltage ($V_{i}$) | Output Voltage | NANDa | NORb
---|---|---|---
$V_{1}$ | $V_{2}$ | $V_{0}$ | |
$V_{L}$ | $V_{L}$ | $\frac{2V_{L}}{3}$ | $V_{H}$ | $V_{H}$
$V_{L}$ | $V_{H}$ | $\frac{V_{L}+V_{H}}{3}$ | $V_{H}$ | $V_{L}$
$V_{H}$ | $V_{L}$ | $\frac{V_{L}+V_{H}}{3}$ | $V_{H}$ | $V_{L}$
$V_{H}$ | $V_{H}$ | $\frac{2V_{H}}{3}$ | $V_{L}$ | $V_{L}$
a NAND threshold range $\frac{V_{L}+V_{H}}{3}<V_{th}<\frac{2V_{H}}{3}$
b NOR threshold range $\frac{2V_{L}}{3}<V_{th}<\frac{V_{L}+V_{H}}{3}$
Table 2 shows the truth table of the two input resistive divider logic cell,
that implements the NAND and NOR gates using a predefined inverter threshold
$V_{th}$. Assuming that $V_{dd}=1V,V_{H}=1V,V_{L}=0V$ it is clear from Table 2
that if the threshold voltage of the inverter is set between $0V$ and $1/3V$,
the cell will work as NOR logic and if it is between $2/3V$ and $1/3V$ the
cell will work as NAND logic. That means by varying the threshold voltage of
the inverter, NAND and NOR logic can be implemented using a single cell. In
general, the range of threshold voltage, $V_{th}$ of NOR gate is
$\frac{NmV_{L}}{1+Nm}\leq V_{th}\leq\frac{(V_{H}+(N-1)V_{L})m}{Nm+1}$ , and
NAND gate is, $\frac{m(V_{L}+(N-1)V_{H})}{(Nm+1)}\leq
V_{th}\leq\frac{mNV_{H}}{Nm+1}$. To find the $m$ value, the lower limit of
NAND gate threshold range $(\frac{m(V_{L}+(N-1)V_{H})}{(Nm+1)})$ is equated to
$\frac{V_{H}+V_{L}}{2}$. Now if we assume $V_{L}$ as $0V$ then we get the $m$
value as$\frac{1}{N-2}$ and we can say that the threshold voltage of NAND gate
must be between $\frac{V_{H}+V_{L}}{2}$ and $\frac{mNV_{H}}{Nm+1}$.
The threshold voltage of the MOSFET is dependent on several parameters such as
substrate bias voltage $V_{bs}$, the surface potential $\phi_{s}$, and
substrate doping concentration [10]. The threshold voltage $V_{tn}$ of the
MOSFET can be varied by changing its substrate bias, $V_{bs}$. The dependence
of substrate bias and the threshold voltage is expressed as,
$V_{tn}=V_{tn0}+K_{1}(\sqrt{\phi_{s}-V_{bs}}-\sqrt{\phi_{s}})+C$ , where,
$V_{tn0}$ is the zero bias threshold voltage, the surface potential
$\phi_{s}=2\frac{k_{B}T}{q}\ln(\frac{N_{a}}{n_{i}})$, $K_{1}$ is a parameter
derived by considering non-uniform doping and short channel effects
$K_{1}=\gamma_{2}-2K_{2}\sqrt{\phi_{s}-V_{bm}}$ where
$K_{2}=\frac{(\gamma_{1}-\gamma_{2})(\sqrt{\phi_{s}-V_{b}x}-\sqrt{\phi_{s}})}{2\sqrt{\phi_{s}}(\sqrt{\phi_{s}-V_{bm}}-\sqrt{\phi_{s}})+V_{bm}}$
$\gamma_{1}$ and $\gamma_{2}$ are body bias coefficient when substrate doping
concentration are equal to $N_{ch}$ and $N_{sub}$ respectively.
$\gamma_{1}=\frac{\sqrt{2q\epsilon_{Si}N_{ch}}}{C_{ox}},\gamma_{2}=\frac{\sqrt{2q\epsilon_{Si}N_{sub}}}{C_{ox}}$
and $V_{bm}$ is the maximum substrate bias voltage. And $C$ shows the effect
of narrow channel on threshold voltage. The threshold voltage of the inverter
can be represented as,
$V_{th}=\left({(V_{tn}+(V_{DD}-|V_{tp}|))\sqrt{\frac{\mu_{p}W_{p}}{\mu_{n}W_{n}}}}\right)/\left({1+\sqrt{\frac{\mu_{p}W_{p}}{\mu_{n}W_{n}}}}\right)$,
which shows the role of the threshold voltages of the MOSFETs in determining
the threshold of the inverter.
Figure 3: The relation between Output voltage of the inverter and Output
voltage of the resistive divider, for 10 input and 20 input boolean logic,
when it is working as a NOR gate is shown
Fig. 3 shows the relationship between the output voltage of the resistive
divider cell (input to the inverter) and the output voltage of an inverter,
for 10 input and 20 input situations, when the cell is working in NOR logic.
$V_{0}$ value when the inputs are $V_{1}=1$ and $V_{2}=V_{3}=..V_{10}=0$ is
$0.0556V$, and when $V_{1}=V_{2}=..V_{10}=0$ is $0$, so the threshold voltage
of the inverter must be between $0$ and $0.0556$, to work as a NOR logic.
Similarly for 20 input boolean logic, the threshold voltage of the inverter
must be between $0$ and $0.026$. This shows that if the threshold voltage of
the inverter can be lowered to a very small value we can implement resistive
threshold logic with large number of inputs.
In order to reduce the threshold voltage, here we introduced three inverters
with three different $V_{DD}$’s. Fig. 4 shows a universal gate structure which
can be used to implement AND, NAND, OR, NOR and NOT logic. For the cell to
work as a NAND logic, the switches $S_{1}$ and $S_{4}$ are closed, and the
output is taken from $V_{out}$. So in this case, three inverters will be
enabled. To implement AND logic, the switches $S_{1}$ and $S_{3}$ are closed,
and the output is taken from $\overline{V_{out}}$. For the AND logic, two
inverters need to be enabled. If the switches $S_{2}$ and $S_{4}$ are closed,
we get a NOR logic from $V_{out}$, here only one inverter has to be enabled.
If both $S_{2}$ and $S_{3}$ are closed, OR logic can be implemented, here two
inverters are used. The approach shown in Fig. 4, demonstrates the concept of
generalization of resistive threshold logic cell to implement the most basic
boolean logic functions. To maintain practical relevance of the approach all
the results reported are based on device parameters from 0.25$\mu m$ TSMC
process. Note that as $V_{DD}$ decreases $V_{th}$ also decreases. When
$V_{DD}$ changes the $V_{GS}$ of PMOS in the CMOS inverter will also change.
As a result, in the case of the proposed cell with 10 inputs, the PMOS will be
in cut off state when the input condition is $V_{1}=1$ and
$V_{2}=V_{3}=..V_{10}=0$ and we get a low level output from the 1st inverter.
Since the 1st inverter can only provide a high value of $0.25V$, we use other
two inverters in order to get a high value of $1V$. The working of the
proposed cell in Fig. 4 as a NAND or NOR gate purely rests on the values of
$V_{tn}$ and $V_{th}$ of the inverter, for a given number of inputs.
Figure 4: The circuit diagram to implement NAND, NOR, AND, OR and NOT logic
functions consisting of memristive resistance divider and CMOS inverters with
three different power supply values.
If $V_{H}$ is set as $1V$ and $V_{L}$ as $0$, then the threshold voltage
$V_{th}$ range for NAND gate must be between $0.5V$ and the $V_{0}$ value
obtained when all inputs are $V_{H}$. Figure 5 shows the relationship that
exists between $V_{tn}$ and $V_{th}$ to implement the proposed cell as NAND
gate, as the number of inputs changes from 3 to 100. For each number of inputs
the $V_{th}$ is calculated for a particular $V_{tn}$ and with a fixed
$V_{tp}$, $W_{p}$, $\mu_{p}$, $W_{n}$, $\mu_{n}$ and $V_{DD}$ values. For a
given number of inputs the threshold voltage is above $0.5V$, so by using a
single inverter with $V_{DD}$ as $1V$, NAND logic can be implemented. That
means NAND logic can be implemented using the proposed cell with one inverter
such as in Fig. 1. Using three inverters with different $V_{DD}$, a 100 input
NOR logic can be realised. For implementing NOR logic, for larger number of
inputs, the threshold voltage of the inverter circuit has to be reduced to a
very low value. This problem can be overcome by boosting the signal, using an
Opamp amplifier, before applying to the inverter. Table 3 shows the leakage
power and the spectral noise due to Johnson, shot and flicker noise in
multi-$V_{DD}$ logic proposed in Fig 4. The the maximum noise levels are very
low (ie in nV) relative to signal reference of 1V range.
Table 3: Leakage power and noise spectral density for 100 input gate proposed multi-$V_{DD}$ gate configuration in Fig 4 Performance measure | NAND | AND | NOR | OR
---|---|---|---|---
Noise spectral density per unit square root bandwidth ($nV/Hz^{1/2}$) | 7.94 | 9.75 | 75.71 | 10.15
Leakage power ($nW$) | 0.014 | 0.017 | 0.967 | 0.971
Figure 5: A graph indicating the dependence of threshold voltage of the CMOS
inverter and threshold voltage of the NMOS. The threshold values shown in the
graph is a result of changing the number of inputs from 3 to 100 and
calculating the minimum inverter threshold voltages required to implement the
circuit as a NAND gate Figure 6: The universal gate structure that implements
NAND, NOR, AND, OR and NOT logic functions using memristive resistance divider
and Opamp threshold circuit.
The universal circuit in Fig 4 is modified to incorporate Opamp threshold
logic as shown in Fig. 6. The threshold logic when implemented using Opamp
[11], offers the advantage of scalability over increase in number of inputs.
The Opamp is designed using 8 MOSFETs and in the same technology as that of
the CMOS NOT gate. The Opamp reference voltage for NOR logic, $V_{REF}$ is
fixed as $V_{L}+\delta$ and for NAND logic, $V_{REF}$ is fixed as
$V_{H}-\delta$, where $\delta$ is small voltage defined to ensure the bounds
of $V_{th}$. The Opamp shifts the voltage to a high value or low value
depending on the input voltage, $V_{0}$. It also acts as a buffer helping to
isolate the inputs from the output enabling realistic implementations of very
large of inputs per gate.
### 2.1 Comparisons
Fig. 7 indicates the area required to implement NOR and NAND universal logic
gates for 2, 10, and 1000 input logic gates implemented using CMOS logic, and
that using the resistive threshold logic. In implementing CMOS logic the
maximum number of inputs per gate is taken as 5. The Fan in of the proposed
cell using Opamp is very high ($=14.498\times 10^{6}$), indicating that we can
implement a large variable boolean logic using a single resistive divider
cell. For increasing number of inputs, the proposed cells contain lesser
number of components and area, when compared to the CMOS logic. Since CMOS
based logic gates are practically limited to small number of inputs, we have
used a layered combination of 5 input gates to implement gates with 10 or more
inputs. Table 4 compares the power dissipation of the proposed logic with that
of CMOS logic for NAND and NOR gates. CMOS gates dissipates lesser power as
against its memristive counterparts. The use of low power memristive
devices[12] would be required to reduce the power dissipation. Table 5 shows
the comparison of the noise margin of the logic families for single input NAND
and NOR logic, indicating that the proposed logic has comparable noise
tolerance levels to that with the existing techniques. In addition, the
averaging nature of the potential divider can further help to increase the
noise tolerance levels than specified through noise margins. Table 6 shows a
comparison of propagation delay when a square pulse with 40$\mu$s time period
and 50% duty cycle is applied. The resistive threshold logic shows better
response when the number of inputs become very high, and when with lower
number of inputs show comparable delays.
Figure 7: The bar graph shows the area comparison of CMOS with that of Resistive Threshold Logic (with Opamp threshold circuit, Fig. 6), using NAND and NOR gate implementations. Table 4: Comparison of the Resistive Logic with CMOS Logic Logic familya | Logic function | Power Dissipation
---|---|---
| | 10 i/p | 100i/p
CMOS logic | | 0.009nW | 0.036nW
Resistive logic (Opamp threshold) | NOR | 10.6$\mu$W | 11.49$\mu$W
CMOS logic | | 0.062nW | 0.753nW
Resistive logic (Opamp threshold) | NAND | 9.2$\mu$W | 10.09$\mu$W
aThe technology size of all the components in the circuit is kept same for all
the gates for fairness in comparison.
As the resistance elements does not significantly introduce the delay with
increase in number of inputs, a large number of inputs (>100) is practically
possible for the proposed cell. In contrast with the existing technologies
that are practically limited to about 5-10 inputs per gate, the ability of the
proposed resistive threshold logic to handle large number of inputs reduces
the complexity of the design and layout of the large variable digital
circuits.
Table 5: Noise margin of different logic families Logic families | NAND | NOR
---|---|---
| NML | NMH | NML | NMH
CMOS | 0.363V | 0.587V | 0.233V | 0.616V
Pseudo NMOS | 0.429V | 0.413V | 0.276V | 0.461V
Domino CMOS | 0.407V | 0.376V | 0.104V | 0.43V
Resistive logic | 0.369V | 0.558V | 0.132V | 0.777V
Table 6: Propagation delay of different logic families for different number of inputs Logic families | NAND delay | NOR delay
---|---|---
| 3i/p | 10i/p | 1000i/p | 3i/p | 10i/p | 1000i/p
CMOS | 0.47$\mu$s | 0.54$\mu$s | 0.65$\mu$s | 0.50$\mu$s | 0.52$\mu$s | 0.66$\mu$s
Pseudo NMOS | 0.48$\mu$s | 0.60$\mu$s | 0.85$\mu$s | 0.51$\mu$s | 0.58$\mu$s | 0.72$\mu$s
Domino CMOS | 0.48$\mu$s | 0.51$\mu$s | 0.75$\mu$s | 0.51$\mu$s | 0.58$\mu$s | 0.75$\mu$s
Resistive logic (Opamp threshold) | 0.45$\mu$s | 0.45$\mu$s | 0.45$\mu$s | 0.60$\mu$s | 0.60$\mu$s | 0.60$\mu$s
### 2.2 Example Circuits
The proposed logic is compared with the CMOS implementation using a 16 bit
adder and a 16x1 MUX. The simulation were performed in spice using feature
size of 0.25$\mu$m TSMC process BSIM models and HP memristor model. A ripple
carry adder without applying reduction technique is implemented using 16
single bit adders. The single bit adder require 3 NOT, 3 two input AND, 1
three input OR, 4 three input AND and 1 four input OR gates. Hence, a total of
48 NOT, 24 AND, 16 OR, 64 AND and 16 OR gates are required for the 16 bit
adder. Figure 8 shows an example of 16th output bit of the adder simulated
using input pulses with initial start delay of 10$\mu$s, rise and fall time of
5$n$s, and ON period of either 20$\mu$s or 10$\mu$s with 50% duty cycle.
Figure 8: The signal output of the 16th bit of the designed ripple adder using
the proposed resistive threshold logic. $V_{in}$’s is the inputs,$C_{in}$ and
$C_{out}$ is the input and output carry, and $V_{out}$ the output sum bit.
The 16 bit MUX when using the proposed logic required 16 input OR gate and 5
input AND gates, while CMOS logic required 2, 4 and 5 input AND/OR gates. In
the case of adder, CMOS logic has lesser area in comparison to the resistive
threshold logic, while in 16x1 MUX implementation proposed logic result in
lesser area when compared to CMOS logic. Table 7 demonstrates that when the
number inputs for the AND and OR gates are increased, the proposed logic
require lesser area than its CMOS counterpart. Power dissipation on the other
hand is higher for the proposed logic due to higher forward currents in
memristor as compared with CMOS. This issue can be addressed by using low
power memristors [12] and low power Opamps.
Table 7: Comparison of Circuit Implemented using Resistive Threshold Logic with that of CMOS logic Logic families | 16 bit full adder | 16x1 MUX
---|---|---
| Power | Area | Power | Area
CMOS logic | 2.5nW | 4.557$\mu$m2 | 0.189nW | 1.070$\mu$m2
Resistive logic (Opamp threshold) | 3.277mW | 8.081$\mu$m2 | 0.447mW | 0.825$\mu$m2
Note:The power dissipation for Opamps in the 16 bit full adder is 2.47mW.
## 3 Conclusion
The concept of resistive threshold logic was presented in an application to
implement conventional digital logic gates. The presented resistive threshold
logic family due to its ability to support large number of inputs can
significantly help reduce the design complexity. Although, the presented
resistive threshold outperforms the conventional CMOS logic implementations in
large input gates in terms of performance parameters such as area, delay and
power, for small input gates further developments on low power and high speed
Opamp designs are required. The CMOS - Resistance Threshold Logic co-design
can optimise the circuit design of conventional CMOS based large variable
boolean logic problems. A disadvantage of the proposed threshold logic using
the memristor technology in [8] as compared with CMOS logic is the higher
power dissipation. However, with the advancements of newer low power
memresitive devices such as [12], the problem of lowering power dissipation to
the levels of CMOS, can be a realistic task. The proposed logic can be
extended to technologies such as carbon nanotubes and organic circuits. In
addition, the ability of the proposed logic to develop large number of input
gates can be seen as an early step in achieving the goal of mimicking brain
like large variable boolean logic applications in VLSI.
## Acknowledgment
The authors would like to thank the anonymous reviewers for their time and
thoughtful review comments, which has resulted in the improvement of overall
quality of the brief.
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|
arxiv-papers
| 2013-08-01T04:17:30 |
2024-09-04T02:49:48.888250
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. P. James and L.R.V.J. Francis and D. Kumar",
"submitter": "Alex James Dr",
"url": "https://arxiv.org/abs/1308.0090"
}
|
1308.0116
|
¡html¿¡head¿ ¡meta http-equiv=”content-type” content=”text/html;
charset=ISO-8859-1”¿
¡title¿CERN-2013-001¡/title¿
¡/head¿
¡body¿
¡h1¿¡a href=”http://cas.web.cern.ch/cas/Bilbao-2011/Bilbao-advert.html”¿CAS -
CERN Accelerator School: Course on High Power Hadron Machines¡/a¿¡/h1¿
¡h2¿Bilbao, Spain, 24 May - 2 Jun 2011¡/h2¿
¡h2¿Proceedings - CERN Yellow Report
¡a href=”https://cds.cern.ch/record/1312630”¿CERN-2013-001¡/a¿¡/h2¿
¡h3¿editors: R. Bailey¡/h3¿
These proceedings collate lectures given at the twenty-fifth specialized
course organised by the CERN Accelerator School (CAS). The course was held in
Bilbao, Spain from 24 May to 2 June 2011, in collaboration with ESS Bilbao.
The course covered the background accelerator physics, different types of
particle accelerators and the underlying accelerator systems and technologies,
all from the perspective of high beam power. The participants pursued one of
six case studies in order to get ”hands-on” experience of the issues connected
with high power machines.
¡h2¿Lectures¡/h2¿
¡p¿
Title: ¡a href=”http://cdsweb.cern.ch/record/1543883”¿Beam dynamics in
linacs¡/a¿ ¡br¿ Author: Letchford, Alan¡br¿ Journal-ref: CERN Yellow Report
CERN-2013-001, pp. 1-16¡br¿ ¡br¿
LIST:arXiv:1302.1001¡br¿
LIST:arXiv:1302.2026¡br¿
LIST:arXiv:1303.1355¡br¿
LIST:arXiv:1302.5264¡br¿
LIST:arXiv:1303.1358¡br¿
LIST:arXiv:1303.1360¡br¿
LIST:arXiv:1303.6552¡br¿
LIST:arXiv:1303.6514¡br¿
LIST:arXiv:1303.6762¡br¿
LIST:arXiv:1303.6766¡br¿
LIST:arXiv:1303.6767¡br¿ Title: ¡a
href=”http://cdsweb.cern.ch/record/1542476”¿Vacuum I¡/a¿ ¡br¿ Author:
Franchetti, G¡br¿ Journal-ref: CERN Yellow Report CERN-2013-001, pp.
309-326¡br¿ ¡br¿ Title: ¡a href=”http://cdsweb.cern.ch/record/1542478”¿Vacuum
II¡/a¿ ¡br¿ Author: Franchetti, G¡br¿ Journal-ref: CERN Yellow Report
CERN-2013-001, pp. 327-347¡br¿
LIST:arXiv:1307.8286¡br¿
LIST:arXiv:1302.3745¡br¿
LIST:arXiv:1307.8301¡br¿
LIST:arXiv:1303.6519¡br¿
LIST:arXiv:1303.6520¡br¿
LIST:arXiv:1303.1365¡br¿
LIST:arXiv:1307.8304¡br¿
¡/p¿ ¡/body¿¡/html¿
|
arxiv-papers
| 2013-08-01T08:04:27 |
2024-09-04T02:49:48.895156
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Bailey (ed.) (CERN)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1308.0116"
}
|
1308.0249
|
# The Dark Mass Problem
Solved ?
Ll. Bel e-mail: [email protected]
###### Abstract
I discuss some of the basic properties of a potential theory derived from a
modified Newton’s law of action at a distance that includes a $1/r$ attractive
force.
## 1 Point particles
Let us start assuming that two point particles with masses $m_{1}$ and $m_{2}$
located at positions $x^{i}$ and $y^{i}$ attract each other with a force
$F^{i}$:
$F_{y|}^{i}(x)=-\frac{Gm_{1}m_{2}}{r^{3}}(x^{i}-y^{i})-\frac{G^{\prime}m_{1}m_{2}}{r^{2}}(x^{i}-y^{i}),\quad
i,j,\cdots=1,2,3$ (1)
where $r^{2}=|\overrightarrow{x}-\overrightarrow{y}|^{2}$, $G$ is Newton’s
constant and $G^{\prime}$ is a free positive physical constant with dimensions
M-1L2T-2.
This law of attraction has been considered, more or less directly, as a
candidate to solve what is known today as the Dark mass problem (more on that
below).
In considering this new law of force it is important to keep in mind that
space has three dimensions and not two, and as a consequence of this only the
Newtonian component of (1) is solenoidal.
Since:
$\frac{\partial F_{y|i}(x)}{\partial x^{j}}-\frac{\partial
F_{y|j}(x)}{\partial x^{i}}=0,$ (2)
introducing the potential function $V$ defined by:
$F_{y|i}(x)=-m_{1}\frac{\partial V_{y|}(x)}{\partial x^{i}}$ (3)
from (1) there follows that:
$\Delta V_{y|}(x)=4\pi G\rho_{y|}(x),\quad\rho_{y|}(x)\equiv
m_{2}\delta(r)+\frac{\alpha
m_{2}}{r^{2}},\quad\alpha=\frac{1}{4\pi}\frac{G^{\prime}}{G}$ (4)
Let us assume that the point particle with mass $m_{2}$ is kept fixed at the
location $y^{i}$ and that the particle $m_{1}$ is free to move under its
attraction. This simple formula above tells us that this particle will move
without friction as it would do across a cloud of dark matter with an always
positive effective density $\rho$, while obeying a potential theory formally
identical to Newton’s one.
## 2 Extended sources
If instead of a point particle with mass $m_{2}$ we consider a continuous
distribution of mass with density $\mu(y)$ then we shall have:
$F^{i}(x)=-Gm_{1}\int_{D}\mu(y)\frac{(x^{i}-y^{i})}{r^{3}}\,d^{3}y-G^{\prime}m_{1}\int_{D}\mu(y)\frac{(x^{i}-y^{i})}{r^{2}}\,d^{3}y,$
(5)
where $D$ is the domain where $\mu\neq 0$, and:
$V(x)=-G\int_{D}\mu(y)\frac{1}{r}\,d^{3}y+G^{\prime}\int_{D}\mu(y)\ln(r)\,d^{3}y,$
(6)
from where we get:
$\Delta V(x)=4\pi
G\rho(x),\quad\rho(x)\equiv\mu(x)+\alpha\int_{D}\frac{\mu(y)}{r^{2}})\,d^{3}y,$
(7)
Let us consider in particular the case where the density $\mu$ is constant
inside a sphere of radius $a$ and zero otherwise (Figure 1). In this case, $r$
being now $|\overrightarrow{x}|$, the effective density $\rho$ will be:
$\rho(r)=\mu
H(a-r)+\alpha\int_{0}^{2\pi}d\phi\int_{0}^{a}du\int_{0}^{\pi}d\theta\frac{\mu
u^{2}\sin\theta}{r^{2}+u^{2}-2ru\cos\theta}$ (8)
$H$ being the Heaviside function; or:
$\rho(r)=\mu H(a-r)+2\pi\alpha\mu\sigma(r,a)$ (9)
with:
$\sigma(r,w)=w+\frac{1}{2r}(r^{2}-w^{2})\ln\left(\frac{|r-w|}{r+w}\right)$
(10)
from where, to calculate the central potential or the force, we could proceed
as usual integrating the Poisson’s equation (7), or use their corresponding
integral definitions.
If instead we have an spherical sector with inner radius $b$ (Figure 2) then
the effective density $\rho(r)$ is:
$\rho(r)=\mu H(a-r)H(r-b)+2\pi\alpha\mu(\sigma(r,a)-\sigma(r,b))$ (11)
Notice that in the cavity defined by $r<b$, where the raw matter density is
zero, the effective density remains positive and therefore the force remains
attractive towards the center. This means that it is a legitimate speculation
to point out that if gravity includes a contribution of the type that I have
considered here, then we could sometimes be fooled to believe that a massive
object gravitates around a source that actually does not exists at all.
## 3 Comments
Notable consequences of the proposed new law are:
i) If $G^{\prime}\neq 0$ and there is ordinary matter somewhere then there is
Dark matter everywhere. Therefore this new law has a lot to say about the
rotation curves of spiral galaxies as well as micro and macro lensing. The
price that we pay for it is the non locality of the theory since the effective
density is not a point function.
ii) It predicts new surprising effects like massive bodies orbiting central
un-existing matter sources in empty cavities (Empty cavity effect), and beyond
that it promises an interesting development of the physics of voids in
cosmology..
iii) Last but not least, the transport of this proposal to General relativity
is quite natural. It suffices to use the effective density $\rho(x)$ instead
of the pure matter one $\mu$ in the source energy-momentum tensor of
Einstein’s equations. The first conceptually important implications are: 1)
that the gravitational field of a point particle becomes more singular than
what it is in Einstein’s theory and 2) that, so to speak, all exact vacuum
solutions become approximate approximate solutions of the modified theory. On
the other hand cosmology only needs to sort out what part of the total density
is matter density and what part it is dark matter.
Reference [1] considers a theory based from the beginning on the potential:
$V(r)=-\frac{Gm_{1}m_{2}}{r}+G^{\prime}\ln r$ (12)
Some aspects of this point of point view bear a similitude with mine but it is
by no means equivalent to it. It does not predict any Empty cavity effect.
References [2] and [3] are based on a Modified quantum field theory that leads
them to describe the gravity of point particles by a potential:
$V(r)=\frac{1}{2\pi^{2}}\int_{0}^{\infty}(V(\omega)\omega^{3})\frac{\sin(\omega
r)}{\omega r}\frac{d\omega}{\omega}.$ (13)
They claim that this potential can be approximated by a $1/r^{2}$ or a $1/r$
one depending on the scale of distances. They also note ”that from a dynamical
point of view the modification of the Newton’s law of gravity can be
interpreted as if point sources lose their point-like character and acquire an
additional distribution in space”. This is also the main point in my very much
simpler proposition.
Reference [4] starts with the introduction of the potential function (6) but
the only source model that is considered is that of a thin disk of matter that
can be dealt with in the framework of potential theory in dimension 2.
## References
* [1] W. H. Kinney and M. Brisudova, arXiv:astro-ph/0006453v1
* [2] A. A. Kirillov, D. Turaev, Physics Letters B 532, 185-192 (2002)
* [3] A. A. Kirillov, arXiv:astro-ph/0405623v1
* [4] J. C. Fabris and J. Pereira Campos, arXiv:0710.3683v1 [astro-ph]
|
arxiv-papers
| 2013-03-21T07:59:29 |
2024-09-04T02:49:48.907766
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ll. Bel",
"submitter": "Llu\\'is Bel",
"url": "https://arxiv.org/abs/1308.0249"
}
|
1308.0259
|
# Reservoir engineering of a mechanical resonator: generating a macroscopic
superposition state and monitoring its decoherence
Muhammad Asjad School of Science and Technology, Physics Division, University
of Camerino, Camerino (MC), Italy David Vitali School of Science and
Technology, Physics Division, University of Camerino, Camerino (MC), and INFN,
Sezione di Perugia, Italy
###### Abstract
A deterministic scheme for generating a macroscopic superposition state of a
nanomechanical resonator is proposed. The nonclassical state is generated
through a suitably engineered dissipative dynamics exploiting the
optomechanical quadratic interaction with a bichromatically driven optical
cavity mode. The resulting driven dissipative dynamics can be employed for
monitoring and testing the decoherence processes affecting the nanomechanical
resonator under controlled conditions.
## I Introduction
Quantum reservoir engineering generally labels a strategy which exploits the
non-unitary evolution of a system in order to generate robust quantum coherent
states and dynamics Diehl _et al._ (2008). The idea is in some respect
challenging the intuitive expectation that in order to obtain quantum coherent
dynamics one should guarantee that the evolution is unitary at all stages. Due
to the noisy and irreversible nature of the processes which generate the
target dynamics, strategies based on quantum reservoir engineering are in
general more robust against variations of the parameters than protocols solely
based on unitary evolution Diehl _et al._ (2008); Verstraete _et al._
(2009). A prominent example of quantum reservoir engineering is laser cooling,
achieving preparation of atoms and molecules at ultralow temperatures by means
of an optical excitation followed by radiative decay Wineland _et al._
(1978). The idea of quantum reservoir engineering has been formulated in Ref.
Poyatos _et al._ (1996), and further pursued in Ref. Carvalho _et al._
(2001). Proposals for quantum reservoir engineering of many-body systems have
been then discussed in the literature Diehl _et al._ (2008); Verstraete _et
al._ (2009) and first experimental realizations have been reported Syassen
_et al._ (2008); Krauter _et al._ (2011).
In particular reservoir engineering has been proposed and already used Krauter
_et al._ (2011) for the generation of steady state nonclassical states, such
as linear superposition (Schrödinger cat) states Poyatos _et al._ (1996);
Carvalho _et al._ (2001) or entangled states in microwave cavities Pielawa
_et al._ (2007, 2010). In this case one has the advantage that the desired
target state is largely independent of the specific initial states, and at the
same time is robust with respect to a large class of decoherence processes.
These ideas have been recently extended also to the field of cavity
optomechanics for the generation of entangled states of two cavity modes Wang
and Clerk (2013), and of two mechanical modes Tan _et al._ (2013).
Here we apply reservoir engineering for the deterministic generation of robust
macroscopic superpositions of coherent states of a mechanical resonator (MR).
First proposals for the generation of superposition states exploited the
intrinsic nonlinearity of radiation pressure interaction Bose _et al._
(1999); Marshall _et al._ (2003) but are hard to realize due to the extremely
weak nonlinear coupling. More recent proposals focused on the conditional
generation of those linear superposition states Paternostro (2011); Vanner
_et al._ (2013), exploiting for example the effective measurement of the MR
position _squared_ in order to generate a superposition of two spatially
separated states Romero-Isart _et al._ (2011, 2011); Jacobs _et al._ (2009,
2011). These latter schemes do not suffer from weak radiation pressure
nonlinearities, but are probabilistic and strongly dependent upon the
efficiency of the conditional measurement on the optical mode. As underlined
above, the generation of a linear superposition state through reservoir
engineering is instead deterministic and extremely robust, because the state
is reached asymptotically as a result of a dissipative irreversible evolution,
and is less sensitive to the details of the preparation process.
Here we propose to generate a superposition of coherent states of a MR by
exploiting the nonlinearity associated with the _quadratic_ interaction of the
MR with an optical cavity mode, appropriately driven by a bichromatic field
(see also Ref. Tan _et al._ (2013)). We study the resulting dynamics,
determined by the joint action of the engineered reservoir realized by the
driven cavity mode and of the standard (and unavoidable) thermal reservoir of
the MR. We show that a high-quality superposition state can be generated in a
transient time interval, which then decoheres at longer times due to the
action of thermal reservoir. The present scheme is particularly useful for
monitoring decoherence processes affecting nanomechanical resonators,
similarly to what has been done for cavities Deléglise _et al._ (2008) or
trapped ions Myatt _et al._ (2000), and could also be useful for testing
alternative decoherence models (see Ref. Romero-Isart _et al._ (2011) and
references therein).
In Sec. II we describe the properties of the required engineered reservoir. In
Sec. III we show how to engineer such a reservoir by tailoring the
optomechanical interaction with a bichromatically driven cavity mode. Sec. IV
describes the resulting dynamics under realistic scenarios, and we verify that
a superposition state can be efficiently generated in a transient time
interval and that its decoherence can be monitored. Sec. V is for concluding
remarks.
## II The desired dissipative evolution
Let $\rho$ be the reduced density matrix of the MR, and
$\rho_{\infty}=|\psi_{\infty}\rangle\langle\psi_{\infty}|$ the target linear
superposition state we want to generate in the steady state of the MR, the so-
called even Schrödinger cat state
$|\psi_{\infty}\rangle=(|\beta\rangle+|-\beta\rangle)/\mathcal{N},$ (1)
where $|\beta\rangle$ denotes a coherent state of the MR with complex
amplitude $\beta$ and $\mathcal{N}=\sqrt{2[1+\exp(-2|\beta|^{2})]}$ is the
normalization constant. Reservoir engineering means in the present case
tailoring the interaction with the optical cavity mode in order to have an
effective reduced dynamics of the MR described by the master equation
$\frac{\partial}{\partial t}\rho=\mathcal{L}\rho\,,$ (2)
for which $\rho_{\infty}$ is a fixed point, namely,
$\mathcal{L}\rho_{\infty}=0\,.$ (3)
A simple solution is to take the Lindbladian ${\mathcal{L}}$
$\displaystyle\mathcal{L}\rho$ $\displaystyle=$
$\displaystyle\Gamma{\mathcal{D}}(C)\rho$ (4)
$\displaystyle{\mathcal{D}}(C)\rho$ $\displaystyle=$
$\displaystyle\left(2C\rho C^{\dagger}-C^{\dagger}C\rho-\rho
C^{\dagger}C\right),$ (5)
with $\Gamma$ a model-dependent rate, and with the operator $C$ such that
$|\psi_{\infty}\rangle$ is eigenstate of $C$ with zero eigenvalue. Such a
condition is satisfied by
$C=(b^{2}-\beta^{2}),$ (6)
where $b$ is the annihilation operator of the MR, and $\beta$ is just the
complex amplitude of the target linear superposition state.
Notice that state $\rho_{\infty}$ is not the unique solution of Eq. (3)
because any coherent or incoherent superposition of $|\beta\rangle$ and
$|-\beta\rangle$ solves Eq. (3). However for our purposes it is sufficient
that, at least for a physically realizable class of initial states of the MR,
the dissipative evolution asymptotically drives it only to $\rho_{\infty}$ and
not to other states of the convex set of states ${\mathcal{C}_{L}}$ defined by
Eq. (3). In this respect one can profit from an additional symmetry of the
Lindbladian of Eq. (4), i.e., the fact that it commutes with the parity
operator ${\mathcal{P}}=(-1)^{b^{\dagger}b}$, and therefore ${\mathcal{P}}$ is
conserved as long as the dynamics is driven by $\mathcal{L}$ only or at least
parity non-conserving terms are negligible in the time evolution generator. In
such a case, since $|\psi_{\infty}\rangle$ is the unique pure state of
${\mathcal{C}_{L}}$ which is even, that is, eigenstate of ${\mathcal{P}}$ with
eigenvalue $+1$, the asymptotic steady state of the MR will also have parity
$+1$. In particular it is possible to see that if the initial state is pure
and even, the asymptotic state will be $|\psi_{\infty}\rangle$. A natural case
of this kind is provided by a vacuum initial state $|0\rangle\langle 0|$,
which is obtained if the MR is initially cooled to its ground state.
Therefore our goal is to generate the effective reduced dynamics of the MR
driven by the above Lindbladian of Eqs. (4)-(6) when the cavity mode is
adiabatically eliminated. In practice however, the MR dynamics will be
affected not only by the cavity mode “engineered reservoir” but also by the
standard thermal reservoir. Therefore we have to establish the effect of these
undesired latter terms, and to determine if and when they are negligible.
## III Engineering the dissipative process
Our starting point is the Hamiltonian of an optomechanical system formed by a
driven cavity mode interacting _quadratically_ with a MR. Such a quadratic
interaction is achieved for example in a membrane-in-the-middle (MIM) setup,
when the membrane is placed at a node, or exactly at an avoided crossing point
within the cavity Thompson _et al._ (2008); Sankey _et al._ (2010); Karuza
_et al._ (2013). Alternatively, such a quadratic coupling can be obtained by
trapping levitating nanoparticles around an intensity maximum of a cavity mode
Barker (2010); Li _et al._ (2011); Gieseler _et al._ (2012); Kiesel _et
al._ (2013). We assume that the cavity is bichromatically driven, that is
$\displaystyle H=\hbar\omega_{m}b^{\dagger}b+\hbar\omega_{c}a^{\dagger}a+\hbar
g_{2}a^{\dagger}a(b+b^{\dagger})^{2}$ (7)
$\displaystyle+\mathrm{i}\hbar\left[(E_{0}e^{-\mathrm{i}\omega_{\mathrm{L}}t}+E_{1}e^{-\mathrm{i}(\omega_{\mathrm{L}}+\Omega)t})a^{\dagger}\right.$
$\displaystyle\left.-(E_{0}e^{\mathrm{i}\omega_{\mathrm{L}}t}+E_{1}e^{\mathrm{i}(\omega_{\mathrm{L}}+\Omega)t})a\right],$
where $\omega_{m}$ is the resonance frequency of the MR, $\omega_{c}$ the
cavity mode frequency,
$E_{0}=\sqrt{2P_{0}\kappa_{0}/\hbar\,\omega_{\mathrm{L}}}$,
$E_{1}=\sqrt{2P_{1}\kappa_{0}/\hbar\,(\omega_{\mathrm{L}}+\Omega)}$, with
$\kappa_{0}$ the cavity decay rate through the input mirror, and $P_{0}$ and
$P_{1}$ (with $P_{0}\gg P_{1}$) the respective input power at the two driving
frequencies. $g_{2}$ is the quadratic optomechanical coupling rate, which is
equal to $g_{2}=\Theta(\partial^{2}\omega_{c}/\partial
z_{0}^{2})(\hbar/2m\omega_{m})$ in the MIM case, with $\Theta$ the transverse
overlap between the mechanical and optical mode at the membrane, and $m$ the
membrane mode effective mass Karuza _et al._ (2013).
We then move to the frame rotating at the main laser frequency
$\omega_{\mathrm{L}}$, where the system Hamiltonian becomes
$\displaystyle H=\hbar\omega_{m}b^{\dagger}b+\hbar\Delta_{0}a^{\dagger}a+\hbar
g_{2}a^{\dagger}a(b+b^{\dagger})^{2}$ (8)
$\displaystyle+\mathrm{i}\hbar\left[(E_{0}+E_{1}e^{-\mathrm{i}\Omega
t})a^{\dagger}-(E_{0}+E_{1}e^{\mathrm{i}\Omega t})a\right],$
where $\Delta_{0}=\omega_{c}-\omega_{\mathrm{L}}$ is the cavity mode detuning.
The dynamics is however also driven by fluctuation-dissipation processes
associated with the coupling of the cavity mode with the optical vacuum field
outside the cavity, and of the MR with its thermal reservoir characterized by
a temperature $T$ and a mean thermal phonon number
$\bar{n}=\left[\exp(\hbar\omega_{m}/k_{B}T)-1\right]^{-1}$. In the usual
Markovian approximation Gardiner and Zoller (2000), we have that optical
dissipation is described by
$\kappa_{\rm T}{\mathcal{D}}(a)\rho_{om},$ (9)
where $\rho_{om}$ is the density matrix of the total optomechanical system,
and $\kappa_{\rm T}$ is the total cavity decay rate, while mechanical
fluctuation-dissipation effects are described by the following terms in the
master equation Gardiner and Zoller (2000)
$\frac{\gamma_{\rm
m}}{2}(\bar{n}+1){\mathcal{D}}(b)\rho_{om}+\frac{\gamma_{\rm
m}}{2}\bar{n}{\mathcal{D}}(b^{\dagger})\rho_{om},$ (10)
where $\gamma_{m}=\omega_{m}/Q_{m}$ is the mechanical damping, and $Q_{m}$ is
the mechanical quality factor. Therefore the time evolution of the system is
described by the following general master equation
$\displaystyle\frac{\partial}{\partial
t}\rho_{om}=-\frac{\mathrm{i}}{\hbar}\left[H,\rho_{om}\right]+\kappa_{\rm
T}{\mathcal{D}}(a)\rho_{om}$ (11) $\displaystyle+\frac{\gamma_{\rm
m}}{2}(\bar{n}+1){\mathcal{D}}(b)\rho_{om}+\frac{\gamma_{\rm
m}}{2}\bar{n}{\mathcal{D}}(b^{\dagger})\rho_{om},$
with $H$ given by Eq. (8).
The intense driving associated with the laser field at the carrier frequency
$\omega_{\mathrm{L}}$ generates a stationary intracavity state of the cavity
mode with large coherent amplitude
$\alpha_{s}=\frac{E_{0}}{\kappa_{\rm T}+\mathrm{i}\Delta_{0}},$ (12)
and it is convenient to look at the dynamics of the quantum fluctuations of
the cavity mode, performing the displacement $a=\alpha_{s}+\delta a$. After
some algebra and using Eq. (12), the master equation of Eq. (11) becomes
$\displaystyle\frac{\partial}{\partial
t}\rho_{om}=-\frac{\mathrm{i}}{\hbar}\left[H_{\delta},\rho_{om}\right]+\kappa_{\rm
T}{\mathcal{D}}(\delta a)\rho_{om}$ (13) $\displaystyle+\frac{\gamma_{\rm
m}}{2}(\bar{n}+1){\mathcal{D}}(b)\rho_{om}+\frac{\gamma_{\rm
m}}{2}\bar{n}{\mathcal{D}}(b^{\dagger})\rho_{om},$
with the modified Hamiltonian
$\displaystyle H_{\delta}=\hbar\tilde{\omega}_{m}b^{\dagger}b+\hbar
g_{2}|\alpha_{s}|^{2}\left(b^{2}+b^{\dagger\,2}\right)+\hbar\Delta_{0}\delta
a^{\dagger}\delta a$
$\displaystyle+\mathrm{i}\hbar\left[E_{1}e^{-\mathrm{i}\Omega t}\delta
a^{\dagger}-E_{1}e^{\mathrm{i}\Omega t}\delta a\right]$ (14)
$\displaystyle+\hbar g_{2}\left(\alpha_{s}^{*}\delta a+\alpha_{s}\delta
a^{\dagger}\right)(b+b^{\dagger})^{2}+\hbar g_{2}\delta a^{\dagger}\delta
a(b+b^{\dagger})^{2},$
where $\tilde{\omega}_{m}=\omega_{m}+2g_{2}|\alpha_{s}|^{2}$ is the
renormalized mechanical frequency.
We now take $\Omega=\Delta_{0}$, i.e., we assume that the second, less intense
beam is exactly resonant with the cavity mode, and move to the interaction
picture with respect to
$H_{0}=\hbar\tilde{\omega}_{m}b^{\dagger}b+\hbar\Delta_{0}\delta
a^{\dagger}\delta a.$ (15)
Within such a picture, the dissipative terms in the master equation of Eq.
(13) remain unchanged, while the Hamiltonian becomes
$\displaystyle H_{\delta}^{int}=\hbar
g_{2}|\alpha_{s}|^{2}\left(b^{2}e^{-2\mathrm{i}\tilde{\omega}_{m}t}+b^{\dagger\,2}e^{2\mathrm{i}\tilde{\omega}_{m}t}\right)+\mathrm{i}\hbar\left[E_{1}\delta
a^{\dagger}-E_{1}\delta a\right]$ $\displaystyle+\hbar
g_{2}\left(\alpha_{s}^{*}\delta ae^{-\mathrm{i}\Delta_{0}t}+\alpha_{s}\delta
a^{\dagger}e^{\mathrm{i}\Delta_{0}t}\right)(be^{-\mathrm{i}\tilde{\omega}_{m}t}+b^{\dagger}e^{\mathrm{i}\tilde{\omega}_{m}t})^{2}$
$\displaystyle+\hbar g_{2}\delta a^{\dagger}\delta
a(be^{-\mathrm{i}\tilde{\omega}_{m}t}+b^{\dagger}e^{\mathrm{i}\tilde{\omega}_{m}t})^{2}.$
(16)
We have made no approximation up to now. We now take the following _resonance
condition_ , $\Delta_{0}=\Omega=2\tilde{\omega}_{m}$, which means that the
second driving beam is resonant not only with the cavity, but also with the
second order sideband of the carrier beam at $\omega_{\mathrm{L}}$, and make
the two following approximations: i) we neglect the last, higher order
interaction term $\hbar g_{2}\delta a^{\dagger}\delta
a(be^{-\mathrm{i}\tilde{\omega}_{m}t}+b^{\dagger}e^{\mathrm{i}\tilde{\omega}_{m}t})^{2}$,
which is justified whenever $|\delta a|\ll|\alpha_{s}|$; ii) we make the
rotating wave approximation (RWA) and neglect all the fast-oscillating terms
at $\tilde{\omega}_{m}$ and $2\tilde{\omega}_{m}$, which is justified in the
weak coupling limit $g_{2}|\alpha_{s}|\ll\tilde{\omega}_{m}$. The effective
interaction picture Hamiltonian of Eq. (16) reduces to
$H_{\rm eff}=\hbar g_{2}\alpha_{s}^{*}\delta
a\left(b^{\dagger\,2}-\mathrm{i}E_{1}/g_{2}\alpha_{s}^{*}\right)+{\rm H.C.},$
(17)
and therefore under the conditions specified above, the dynamics of the
optomechanical system is described by Eq. (13) with $H_{\delta}$ replaced by
$H_{\rm eff}$. An analogous effective optomechanical dynamics has been
considered in Ref. Tan _et al._ (2013), where however the renormalization of
the mechanical frequency $\omega_{m}\to\tilde{\omega}_{m}$ has been neglected.
### III.1 Reduced dynamics of the mechanical resonator
In the bad cavity limit, i.e., when $\kappa_{\rm T}\gg
g_{2}|\alpha_{s}|,\gamma_{m}\bar{n}$, the optical mode fluctuations $\delta a$
can be adiabatically eliminated because they quickly decay and their state
always remains close to the vacuum state (see for example Ref. Gardiner and
Zoller (2000), pag. 147 and Ref. Wiseman and Milburn (1993)). One gets the
following final effective master equation for the reduced density matrix of
the MR, $\rho$,
$\frac{\partial}{\partial t}\rho=\Gamma{\mathcal{D}}(C)\rho+\frac{\gamma_{\rm
m}}{2}(\bar{n}+1){\mathcal{D}}(b)\rho+\frac{\gamma_{\rm
m}}{2}\bar{n}{\mathcal{D}}(b^{\dagger})\rho,$ (18)
where $\Gamma=g_{2}^{2}|\alpha_{s}|^{2}/\kappa_{\rm T}$ and $C$ is just given
by Eq. (6), with the superposition state amplitude
$\beta^{2}=E_{1}/{\mathrm{i}}g_{2}\alpha_{s}$. The first term is just the
desired term, i.e., the engineered dissipative evolution able to drive the MR
asymptotically to the target superposition state $|\psi_{\infty}\rangle$.
However, the time evolution of the MR state is also driven by the second and
third terms which are due to the coupling with the thermal reservoir at
temperature $T$. The latter are “undesired” terms, because they drive the MR
to a thermal state rather than the desired even Schrödinger cat state, and
also because they do not conserve the parity. Due to the joint action of these
two dissipative evolutions, the asymptotic state achieved by the MR at long
times will be different from the desired even superposition state
$|\psi_{\infty}\rangle$. However, if $\Gamma\gg\gamma_{m}\bar{n}$ so that the
effect of the thermal reservoir is negligible, we expect that the target state
can be generated at least for a reasonable transient time interval around
$\bar{t}\sim 1/\Gamma$. This condition, together with the conditions
$|\alpha_{s}|\gg 1$, and $\kappa_{\rm T},\tilde{\omega}_{m}\gg
g_{2}|\alpha_{s}|$ which are needed for deriving Eq. (18), represent the
parameter conditions for realizing the robust generation of a superposition
state of the MR.
## IV Results
Let us now we verify if and when the proposal is implementable in a state-of-
the-art optomechanical setup and a nanomechanical resonator can be prepared
with high fidelity, at least for a long-lived transient, in the macroscopic
superposition state $|\psi_{\infty}\rangle$. We consider parameter values
achievable in state-of-the-art MIM setups Thompson _et al._ (2008); Sankey
_et al._ (2010); Karuza _et al._ (2013); Wilson _et al._ (2009); Purdy _et
al._ (2012); Flowers-Jacobs _et al._ (2012); Karuza _et al._ (2012); Purdy
_et al._ (2013). For the mechanical resonator we take $\omega_{m}=10$ MHz,
$\gamma_{m}=0.1$ Hz (implying $Q_{m}=10^{8}$), $m=1$ ng and we can take
$\partial^{2}\omega_{c}/\partial z_{0}^{2}=2\pi\times 20$ GHz/nm2 Flowers-
Jacobs _et al._ (2012), yielding $g_{2}\simeq 5$ Hz. We then take a laser
with frequency $\omega_{\rm L}=1.77\times 10^{15}$ Hz (corresponding to a
wavelength $\lambda=1064$ nm) and input power $P_{0}=40$ mW. We also choose a
cavity with total decay rate $\kappa_{\rm T}=10^{5}$ Hz and with decay rate
through the input mirror $\kappa_{0}\sim\kappa_{\rm T}/2$, yielding $E_{0}\sim
1.5\times 10^{11}$ Hz. The corresponding value of the intracavity amplitude
from Eq. (12) is $|\alpha_{s}|\sim 3.45\times 10^{3}$. As a consequence
$g_{2}|\alpha_{s}|\sim 1.7\times 10^{4}$ Hz, which therefore agrees with the
assumptions made. Moreover we have an effective decay rate
$\Gamma=g_{2}^{2}|\alpha_{s}|^{2}/\kappa_{\rm T}\sim 2.98$ kHz which is
reasonably larger than the thermal decay rate $\gamma_{m}\bar{n}$ as long as
$\bar{n}\lesssim 100$. Even though nontrivial, this latter condition is
achievable in current optomechanical experiments because cryogenic
environments at temperatures $T\simeq 10$ mK are feasible and, with the chosen
value $\omega_{m}=10$ MHz, this corresponds just to $\bar{n}\simeq 100$.
The amplitude $\beta$ of the target state is determined by $E_{1}$ and
therefore by the input power $P_{1}$. Assuming $P_{1}\sim 1$ pW, one gets
$E_{1}\sim 10^{6}$ Hz and therefore $|\beta|\sim 23.6$, which corresponds to a
quite macroscopic superposition state; here however, in order to verify
numerically the proposal in a not too large operational Hilbert space, we have
taken $P_{1}\sim 0.01$ pW, yielding $E_{1}\sim 10^{5}$ Hz and therefore
$|\beta|\sim 2.36$.
### IV.1 Cat state generation starting from the mechanical ground state
As discussed above, we expect to generate a long-lived transient even
Schrödinger cat state of the MR when $\Gamma\gg\gamma_{m}\bar{n}$ and if we
start from the mechanical ground state, which is pure and even. Since in the
considered scenario it is very hard to go below $\bar{n}\sim 100$ with
cryogenic techniques only, this initial state could be achieved, at least in
principle, by first laser cooling the MR to its ground state, i.e., by first
considering a _linear_ optomechanical interaction with a cavity mode and
driving it on its first red sideband Marquardt _et al._ (2007); Wilson-Rae
_et al._ (2007); Genes _et al._ (2008); Chan _et al._ (2011); Verhagen _et
al._ (2012). Then, one should switch to the quadratic optomechanical
interaction (either by displacing the membrane or by driving a different
appropriate cavity mode) soon after ground state cooling is attained.
We have numerically solved the time evolution of the optomechanical system
density matrix $\rho_{om}$ as described by Eq. (13) with the Hamiltonian of
Eq. (17), starting from the mechanical ground state and the vacuum state for
the cavity mode fluctuations. Plots of the Wigner representation of the
reduced state $\rho$ of the MR at different times are shown in Fig. 1, which
refers to the set of parameters described above, and $\bar{n}=100$. These
plots confirm our expectations and that the state $|\psi_{\infty}\rangle$ with
$|\beta|\sim 2.36$ is generated in the transient regime $t\sim 1/\Gamma$ due
to the appropriate bichromatic driving and the quadratic optomechanical
interaction. The superposition state then decoheres on a time scale governed
by $\gamma_{m}\left(2\bar{n}+1\right)$. These results are consistent with
those of Ref. Tan _et al._ (2013) which also studies the generation of a cat
state of a MR starting from the ground state in a bichromatically driven
quadratic optomechanical system by means of the Wigner function of the reduced
MR state.
Figure 1: Time evolution of the Wigner function of the reduced state of the MR
starting from an initial factorized state in which both the optical cavity
fluctuations and the mechanical mode are in their ground state. The set of
parameters is given in the text, and $\bar{n}=100$. (a) Wigner function of the
initial state of the MR; (b) Wigner function of the MR state at at time
$t=0.71/\Gamma=2.39\times 10^{-4}$ s; (c) Wigner function of the MR state at
time $t=100/\Gamma=0.03355$ s. The even superposition state is successfully
generated in a short time of the order of $1/\Gamma$, and it slowly loses its
nonclassical interference fringes at a longer timescale, of the order of
$\left[\gamma_{m}\left(2\bar{n}+1\right)\right]^{-1}$.
This qualitative analysis based on the Wigner function is confirmed by a
quantitative analysis based on the time evolution of the fidelity of the state
with respect to the target state $|\psi_{\infty}\rangle$. Rather than the more
common Uhlmann fidelity Uhlmann (1976); Jozsa (1994), in order to simplify the
numerical calculation, here we use the Hilbert-Schmidt fidelity introduced in
Ref. Wang _et al._ (2008)
${\mathcal{F}}(\rho_{0},\rho_{1})=\frac{\left|{\rm
Tr}\left\\{\rho_{0}\rho_{1}\right\\}\right|}{\sqrt{{\rm
Tr}\left\\{\rho_{0}^{2}\right\\}{\rm Tr}\left\\{\rho_{1}^{2}\right\\}}}.$ (19)
When $\rho_{0}$ is pure, this fidelity coincides with the probability of
finding the state $\rho_{0}$ being in $\rho_{1}$, divided by the square root
of the purity $\sqrt{{\rm Tr}\left\\{\rho_{1}^{2}\right\\}}$. In Fig. 2 we
plot the time evolution of
${\mathcal{F}}(t)={\mathcal{F}}(\rho_{\infty},\rho(t))$ corresponding to the
same parameter condition of Fig. 1. The fidelity reaches a maximum
${\mathcal{F}}\simeq 0.9992$ at $t\simeq 1/\Gamma$ when an almost perfect cat
state is generated, which then decays so that ${\mathcal{F}}\simeq
1/\sqrt{2}\simeq 0.7$.
Figure 2: Plot of fidelity
${\mathcal{F}}(t)={\mathcal{F}}(\rho_{\infty},\rho(t))$ as a function of time.
The inset shows the behavior at short times. Parameters are those given in the
text and coinciding with those of Fig. 1.
### IV.2 Cat state generation after two-phonon cooling
Fast switching from the linear optomechanical interaction needed for cooling
to the mechanical ground state to the quadratic optomechanical interaction
necessary for generating the even cat state is quite challenging in practical
experimental situations. However, one could exploit the quadratic interaction
also for pre-cooling the MR and avoid using a different cavity mode and
different driving field. To be more specific one could use the same
interaction Hamiltonian and parameter conditions described in the previous
section and consider the special case $E_{1}=\beta=0$, i.e., with the weak
resonant field turned off. In this case, the engineered interaction with the
cavity mode induces a two-phonon cooling process driving the MR to its ground
state. The joint dynamics in the presence of nonlinear two-phonon damping and
standard decay to the thermal equilibrium with $\bar{n}$ thermal phonons has
been already studied in Ref. Nunnenkamp _et al._ (2010), where it is shown
that in the limit $\Gamma\gg\gamma_{m}\bar{n}$ we are considering, cooling is
good even though not perfect, being the MR steady state a mixture of the zero
and one phonon state, with probabilities $\rho_{11}(\infty)=n_{\rm
eff}=(4+1/\bar{n})^{-1}$ and $\rho_{00}(\infty)=1-\rho_{11}(\infty)$.
Therefore a feasible cat state generation protocol is to first cool the MR
with the two-phonon cooling process with $E_{1}=0$, and then switch on the
weak resonant field with $E_{1}\neq 0$ for generating the even cat state as
discussed above. We now see that despite the initial approximate $25\%$
probability of being in the odd one phonon state, the cat state generation
process is still quite efficient, showing that such a robust macroscopic
superposition can be generated in achievable quadratic optomechanical setups.
We have in fact numerically solved the master equation for the optomechanical
system density matrix $\rho_{om}$ of Eq. (13) with the Hamiltonian of Eq.
(17), now taking as initial state the vacuum state for the cavity mode
fluctuations and the above mixture of the zero and one phonon state for the
MR, using the same set of parameters of the previous subsection (we have
verified that with this set of parameters one actually cools the MR to this
mixture of states). Plots of the Wigner representation of the reduced state
$\rho$ of the MR at different times are shown in Fig. 3, which refer to
$\bar{n}=10$ and in Fig. 4, which refers to $\bar{n}=100$. In both cases the
target cat state is generated with high fidelity at $t\sim 1/\Gamma$, despite
the residual excitation in the one-phonon state. This is confirmed by time
evolution of the fidelity of the state with respect to the target state
$|\psi_{\infty}\rangle$, which is shown in Fig. 5 for $\bar{n}=10$ (a) and
$\bar{n}=100$ (b). The fidelity reaches a maximum ${\mathcal{F}}\simeq 0.94$
at $t\simeq 1/\Gamma$ which does not depend upon $\bar{n}$ and then decays to
${\mathcal{F}}\simeq 1/\sqrt{2}\simeq 0.7$. The superposition state decoheres
to a mixture of two Gaussian states on a time scale governed by the thermal
decoherence rate given by $\gamma_{\rm
dec}=2\gamma_{m}|\beta|^{2}\left(2\bar{n}+1\right)$ Kennedy and Walls (1988);
Kim and Buzěk (1992); Brune _et al._ (1996); Deléglise _et al._ (2008).
Figure 3: Time evolution of the Wigner function of the reduced state of the MR
with mean phonon number $\bar{n}(0)=10$. (a) Wigner function of the initial
state of the MR at time t=0; (b) Wigner function of the MR state at time
$t=1/\Gamma=3.3547\times 10^{-4}$ s; (c) Wigner function of the MR state at
time $t=1000/\Gamma=0.3355$ s. The other parameters are given in the text and
coincide with those of Fig. 2.
Figure 4: Time evolution of the Wigner function of the reduced state of the MR
with mean phonon number $\bar{n}(0)=100$. (a) Wigner function of the initial
state of the MR at time t=0; (b) Wigner function of the MR state at time
$t=1/\Gamma=3.3547\times 10^{-4}$ s; (c) Wigner function of the MR state at
time $t=100/\Gamma=0.03355$ s. The other parameters are given in the text and
coincide with those of Fig. 2.
Figure 5: Plot of fidelity
${\mathcal{F}}(t)={\mathcal{F}}(\rho_{\infty},\rho(t))$ as a function of time
for (a) $\bar{n}=10$ and (b) $\bar{n}=100$. The other parameters are given in
the text and coincide with those of Fig. 2. The insets show the behavior at
short times.
### IV.3 Approximate description of the progressive decoherence of the
generated superposition state
The above analysis shows that the combined action of the engineered reservoir
term with rate $\Gamma$ and the thermal reservoir terms with rate
$\gamma_{m}\bar{n}$, when $\Gamma\gg\gamma_{m}\bar{n}$, generates a
superposition state at time $t\simeq 1/\Gamma$ which then decoheres with
decoherence rate $2\gamma_{m}|\beta|^{2}\left(2\bar{n}+1\right)$. In
particular, Figs. 1-4 suggest that the MR decoheres to an asymptotic state
given by the mixture of the two coherent states
$|\pm\beta\rangle\langle\pm\beta|$, with $\beta=\sqrt{E_{1}/ig_{2}\alpha_{s}}$
just the amplitude of the target superposition state. To state it in other
words, the combined action of the engineered and “natural” reservoirs tends to
stabilize such a mixture of coherent states emerging after the decoherence
process. Taking into account the well-established theory of decoherence of
superposition of two coherent state in the presence of a thermal reservoir
Kennedy and Walls (1988); Kim and Buzěk (1992); Brune _et al._ (1996);
Deléglise _et al._ (2008), one is led to approximate the time evolution of
the reduced MR state after a transient time $t\geq t_{0}\simeq 1/\Gamma$ with
the following expression
$\displaystyle\rho_{\rm app}(t>t_{0})={\cal
N}(t-t_{0})^{-1}\left\\{|\beta\rangle\langle\beta|+|-\beta\rangle\langle-\beta|\right.$
(20)
$\displaystyle\left.+e^{-\left(1+2\bar{n}\right)\gamma_{m}\left(t-t_{0}\right)}\left[|\beta\rangle\langle-\beta|+|-\beta\rangle\langle\beta|\right]\right\\},$
with ${\cal
N}(t)=2\left[1+e^{-2|\beta|^{2}}e^{-\left(1+2\bar{n}\right)\gamma_{m}t}\right]$,
describing a decohering cat state, which decoheres to its corresponding
mixture just at the rate $2\gamma_{m}|\beta|^{2}\left(2\bar{n}+1\right)$.
We can check the validity of this approximate description at $t>t_{0}$ by
using again the Hilbert-Schmidt fidelity of Eq. (19) for measuring the overlap
between the actual reduced MR state $\rho(t)$ given by the solution of the
master equation of Eq. (13) and the approximate solution $\rho_{\rm app}$ of
Eq. (20). In Fig. 6 we plot the “distance” between the two states,
$D(t)=1-{\mathcal{F}}\left(\rho(t),\rho_{\rm app}(t)\right)$ for the same set
of parameters of Fig. 2, and we find a very good agreement for the proposed
solution. Therefore the generated Schrödinger cat state can be used for
verifying experimentally the decoherence processes affecting the
nanomechanical resonator and eventually testing alternative decoherence
models, as suggested in Ref. Romero-Isart _et al._ (2011).
Figure 6: Plot of the distance between the actual solution of the master
equation $\rho(t)$ and the approximate MR state of Eq. (20) as a function of
time for (a)$\bar{n}=100$ and initial ground state for the MR; (b)
$\bar{n}=10$ and the mixture of zero and one phonon state as initial state of
the MR; (c) $\bar{n}=100$ and the mixture of zero and one phonon state as
initial state of the MR. The other parameters are given in the text and
coincide with those of Fig. 2.
### IV.4 Cat state decoherence as decay of non-Gaussianity
The decoherence process affecting the MR state can also be described as a
dynamical “Gaussification” process in which the non-Gaussian even cat state
generated at short times by the engineered two-phonon reservoir becomes at
long times a convex mixture of Gaussian state, i.e., the equal-weight
incoherent superposition of the two coherent states $|\pm\beta\rangle$. This
suggests an alternative quantitative description of the above loss of quantum
coherence caused by the interplay between the engineered and natural reservoir
in terms of a measure of _quantum non-Gaussianity_ recently proposed in Refs.
Genoni _et al._ (2013); Palma _et al._ (2013).
A state is quantum non-Gaussian if it cannot be written as a convex sum of
Gaussian states, and a simple sufficient condition for non-Gaussianity can be
given in terms of the value of the Wigner function of the state at the phase
space origin $W[\rho](0)$ Genoni _et al._ (2013): $\rho$ is quantum non-
Gaussian if $W[\rho](0)<(2/\pi)\exp[-2\langle n\rangle(\langle n\rangle+1)]$,
where $\langle n\rangle={\rm Tr}\left\\{\rho b^{\dagger}b\right\\}$ is the
mean number of excitations. However this condition does not detect many
quantum non-Gaussian states (for example even cat states) and a more efficient
condition for detecting quantum non-Gaussian states has been derived in Ref.
Palma _et al._ (2013): $\rho$ is quantum non-Gaussian if there is a Gaussian
map ${\mathcal{E}}$ such that
$NG=W[{\mathcal{E}}(\rho)](0)-\frac{2}{\pi}\exp[-2\langle
n_{\mathcal{E}}\rangle(\langle n_{\mathcal{E}}\rangle+1)<0,$ (21)
where ${\mathcal{E}}(\rho)$ is the state transformed by the Gaussian map and
$n_{\mathcal{E}}$ is the mean excitation number of the transformed state.
Figure 7: Plot of the non-Gaussianity $NG$ of Eq. 21 versus time (soon after
the cat state generation) for (a) $\bar{n}=100$ starting from the mechanical
ground state, (b) $\bar{n}=10$ starting from the two-phonon cooling initial
state; (c) $\bar{n}=100$ starting from the two-phonon cooling initial state.
The other parameters are given in the text and coincide with those of Fig. 2.
We have calculated the quantity $NG$ quantifying non-Gaussianity by
restricting to Gaussian unitary maps formed by a composition of the phase
space displacement operator $D(\alpha)=\exp\left[\alpha
b^{\dagger}-\alpha^{*}b\right]$ and of the squeezing operator
$S(s)=\exp\left[(s/2)(b^{\dagger})^{2}-(s^{*}/2)b^{2}\right]$, and minimizing
$NG$ over $\alpha$ and $s$. The values $\alpha=0.35i$ and $s=0.01$ work very
well at all time instants after the cat state generation, either when starting
from the mechanical ground state and when starting from the mixture of the
vacuum and one phonon state obtained with two-phonon cooling. Plot of the time
evolution of $NG$ soon after the cat state generation, in the three cases
studied above, i.e., starting from the ground state and $\bar{n}=100$ (a),
starting from two-phonon cooling and $\bar{n}=10$ (b), and $\bar{n}=100$ (c),
are shown in Fig. 7. In all cases we see an exponential-like “decay” of non-
Gaussianity to the Gaussian limit $NG=0$, as expected, which is faster in the
cases when $\bar{n}=100$; the non-Gaussianity decay rate is in good agreement
with the usual decoherence rate
$2\gamma_{m}|\beta|^{2}\left(2\bar{n}+1\right)$. Therefore the measure of non-
Gaussianity of Eq. (21) proposed in Ref. Palma _et al._ (2013) detects very
well the non-Gaussian property, and for the present even cat state the
dynamics of non-Gaussianity provides a satisfactory description of the
decoherence process.
## V Conclusions
We have proposed a scheme for the deterministic generation of a linear
superposition of two coherent states of a MR based on the implementation of an
engineered reservoir realized by a bad cavity mode, bichromatically driven and
coupled _quadratically_ with the MR. The proposal extends in various aspects
the proposal of Ref. Tan _et al._ (2013) and is feasible adopting either MIM
optomechanical setups or levitated nanospheres trapped around an intensity
maximum of the optical cavity mode. The interplay between the engineered
reservoir and the natural thermal reservoir of the MR allows the efficient
generation of the linear superposition state in a transient regime if the rate
of the engineered reservoir $\Gamma$ is larger than $\gamma_{m}\bar{n}$, which
is experimentally achievable in cryogenic environments at about $T\sim 10$ mK.
The generation of an even superposition of two coherent states of opposite
phases is almost ideal when starting from the MR ground state. This initial
condition could be obtained by laser pre-cooling the MR through a linear
optomechanical interaction, which however must be then suddenly switched to a
quadratic interaction, by shifting for example the membrane to a node of the
cavity mode. However the cat state generation is very efficient also when
precooling is realized by exploiting only the two-phonon relaxation processes
associated with the quadratic interaction Nunnenkamp _et al._ (2010), which
is much easier to implement since it is based on the same configuration
allowing the cat state generation.
At longer times, the thermal reservoir is responsible for the progressive
decoherence of the generated superposition state, which asymptotically tends
to a steady state given by the incoherent mixture of the two coherent states
of the superposition, and which can be satisfactorily approximated by a simple
analytical expression. For this reason, the present protocol is ideal for
testing decoherence models acting on nanomechanical resonators.
An important issue is also the development of an efficient detection of the
generated MR state. A satisfactory detection could be obtained by realizing a
homodyne tomography D Ariano _et al._ (1994) of the Wigner function of the
generated state. Homodyne tomography of the MR state could be obtained by
first transferring such state to an auxiliary cavity mode, weakly linearly
coupled to the MR, as suggested in Ref. Vitali _et al._ (2007) or adopting
the pulsed homodyne measurement scheme of Ref. Vanner _et al._ (2011). When
the auxiliary cavity mode is driven on its first red sideband and can be
adiabatically eliminated, its output field $a_{2}^{\rm out}$ is proportional
to the MR annihilation operator $b$ plus additional noise Vitali _et al._
(2007), and therefore a calibrated homodyne detection of this output field at
various phases could be exploited for a tomographic reconstruction of the MR
Wigner function. The presence of the driven, weakly linearly coupled detection
cavity mode affects the two-phonon processes creating the engineered
reservoir, and therefore the detection process should be turned on only after
the cat state generation has been completed.
## VI Acknowledgments
This work has been supported by the European Commission (ITN-Marie Curie
project cQOM), and by MIUR (PRIN 2011).
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|
arxiv-papers
| 2013-08-01T16:21:40 |
2024-09-04T02:49:48.913355
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Muhammad Asjad and David Vitali",
"submitter": "Muhammad Asjad Mr.",
"url": "https://arxiv.org/abs/1308.0259"
}
|
1308.0291
|
# Schrödinger Equation on Fractals Curves Imbedding in $R^{3}$
Alireza Khalili Golmankhaneh a†
Dumitru Baleanu b,c,d 111Tel:+903122844500, Fax:+903122868962
E-mail addresses: [email protected]
aDepartment of Physics, Islamic Azad University, Urmia Branch,
PO Box 969, Urmia, Iran
†E-mail:[email protected]
bDepartment of Mathematics and Computer Science
$\c{C}ankaya$ University, 06530 Ankara, Turkey
cInstitute of Space Sciences,
P.O.BOX, MG-23, R76900, Magurele-Bucharest, Romania
dDepartment of Chemical and Materials Engineering, Faculty of Engineering,
King Abdulaziz University, P.O. Box: 80204, Jeddah, 21589, Saudi Arabia
###### Abstract
In this paper we have generalized the quantum mechanics on fractal time-space.
The time is changing on Cantor-set like but space is considered as fractal
curve like Von-Koch curve. The Feynman path method in quantum mechanics has
been suggested on fractal curve. Using $F^{\alpha}$-calculus and Feynman path
method we found the Schrëdinger on fractal time-space. The Hamiltonian
operator and momentum operator has been derived. More, the continuity equation
and the probability density is given in generalized formulation.
Keywords:Feynman path method, Schrëdinger on fractal time-space,continuity
equation
## 1 Introduction
Fractal is objects that are very fragmented and irregular at all scales. Their
important properties are non-differentiability and having non-integer
dimension. Fractal has topological dimension less than Hausdorff-Besicovitch,
box-counting, and similarity dimensions. In general, dimension of fractal can
be integer or not well-defined dimension[1, 2, 3, 4, 5, 6, 7]. Fractional
local calculus and nonlocal has applied to model the process with memory and
fractal structure[8, 9, 10, 11, 12, 13, 14]. The electric and magnetic fields
are derived using fractional integrals as a approximation method on fractals
[15]. The quantum space-time on the basis of relativity principle and
geometrical concept of fractals is introduced [16].The probability density of
quantum wave function with by Dirichlet boundary conditions in a D-dimensional
spaces has been studied [17]. The fractal concept to quantum physics and the
relationships between fractional integral and Feynman path integral method is
developed [18, 19]. The generalized wave functions is introduced to fractal
dimension, a wide class of quantum problems, including the infinite potential
well, harmonic oscillator, linear potential, and free particle [20].Fratal
path in quantum mechanics and their contributing in Feynman path integral is
investigated [21]. The classical mechanics is derived without the need of the
least-action principle using path-integral approach [22]. The calculus on the
fractals has been studied in different methods like probabilistic approach
method, sequence of discrete Laplacians, measure-theoretical method, time
scale calculus [23].Riemann integration like method has been studied since
that is useful and algorithmic [24, 25, 26, 27, 28, 29].Using the calculus on
fractals the Newtonian mechanics, Lagrange and Hamilton mechanics, and Maxwell
equations has been generalized [30, 31, 32]. As a pursue theses research we
generalized the quantum mechanics on fractals.
The plan of this paper is as following:
Section 2 we review the fractal calculus. In section 3 we defined the
gradient, divergent and Laplacian on fractal space. Section 4 is explained the
quantum mechanics on fractals curves. In section 5 we suggested the
probability density and continuity equation on the generalized quantum
formalism. Finally, section 6 is devoted to conclusion.
## 2 A Summery of the calculus on fractal curves
We review the $F^{\alpha}$-calculus on fractal curves [24, 25, 26, 27, 28, 29,
30, 31, 32]. Suppose fractal curve $F\subset R^{3}$ which is continuously
parameterizable i.e there exists a function
$\textbf{w}:[a_{0},b_{0}]\rightarrow F\subset R^{3}$ which is continuous. We
also assume w to be invertible. A subdivision $P_{[a,b]}$ of interval
$[a,b],a<b,$ is a finite set of points $\\{a=v_{0}<v_{1},...<v_{n}=b\\}$. For
$a_{0}\leq a<b<b_{0}$ and appropriate $\alpha$ to be chosen, therefore let
$\gamma^{\alpha}(F,a,b)=\lim_{\delta\rightarrow
0}\inf_{\\{P_{[a,b]}:|P|\leq\delta\\}}\sum_{i=0}^{n-1}\frac{|\textbf{w}(v_{i+1})-\textbf{w}(v_{i})|^{\alpha}}{\Gamma(\alpha+1)},$
(1)
where $|.|$ denotes the Euclidean norm on $R^{3}$ and
$|P|=\max\\{v_{i+1}-v_{i};i=0,...,n-1\\}$. A $\gamma$-dimension of $F$, which
is defined as
$\textmd{dim}_{\gamma}(F)=\inf\\{\alpha:\gamma^{\alpha}(F,a,b)=0\\}=\sup\\{\alpha:~{}\gamma^{\alpha}(F,a,b)=\infty\\}.$
(2)
After this defintion $\alpha$ is equal to $dim_{\gamma}(F).$ The staircase
function $S_{F}^{\alpha}:[a_{0},b_{0}]\rightarrow R$ of order $\alpha$ for a
set $F$, is defined as
$S_{F}^{\alpha}(v)=\begin{cases}\gamma^{\alpha}(F,p_{0},v)~{}~{}~{}v\geq
p_{0}\\\ -\gamma^{\alpha}(F,v,p_{0})~{}~{}~{}v<p_{0},\end{cases}$ (3)
where $a_{0}\leq p_{0}\leq b_{0}$ is arbitrary but fixed, and
$v\in[a_{0},b_{0}].$ It is monotonic function. The $\theta=\textbf{w}(v),$
denote a point on fractal curve $F$
$J(\theta)=S_{F}^{\alpha}(\textbf{w}^{-1}(\theta)),~{}~{}~{}\theta\in F.$ (4)
We suppose that fractal curves whose $S_{F}^{\alpha}$ is finite and invertible
on $[a,b]$. The $F^{\alpha}$-derivative of the bounded function
$f:F\rightarrow R$ $(f\in B(F))$ at $\theta\in F$ is defined.
Then the directional $F^{\alpha}$-derivative of function $f$ at $\theta\in F$
is defined as
$^{w_{j}}\mathfrak{D}_{F}^{\alpha}f(\textbf{w}(v))=F-\lim_{t^{\prime}\rightarrow
t}\frac{f(w_{1}(v),w_{2}(v),...w_{j}(v^{\prime}),...w_{i}(v))-f(\textbf{w}(v))}{S_{F}^{\alpha}(v^{\prime})-S_{F}^{\alpha}(v)},$
(5)
where $w_{j}$ is shows direction of $F^{\alpha}$-derivative, if the limit
exists [27].
Let $f\in B(F)$ is an $F$-continuous function on $C(a,b)$ which is the segment
$\\{\textbf{w}(v):v\in[a,b]\\}$ of $F$. Now let $g:f\rightarrow R$ be define
as
$g(w(v))=\int_{C(a,v)}f(\theta)d_{F}^{\alpha}\theta,$ (6)
for all $v\in[a,b]$. So that
$\mathfrak{D}_{F}^{\alpha}g(\theta)=f(\theta)$ (7)
Note: Let $\gamma^{\alpha}(F,a,b)$ be finite and $f(\theta)=1$, $\theta\in F$
denote the constant function. Then
$\int_{C(a,b)}f(\theta)d_{F}^{\alpha}\theta=\int_{C(a,b)}1d_{F}^{\alpha}\theta=S_{F}^{\alpha}(b)-S_{F}^{\alpha}(a)=J((w(b))-J((w(a)).$
(8)
Remark: $F^{\alpha}$-derivative and $F^{\alpha}$-integral is a linear
operation.
1) Let $f:F\rightarrow R$, $f(\theta)=k\in R$ then
$\mathfrak{D}_{F}^{\alpha}f=0$.
2) IF $f:F\rightarrow R$ be a $F$-continuous function such that
$\mathfrak{D}_{F}^{\alpha}f=0$. Then $f=k$ where $k$ on $C(a,b).$
Suppose $f:F\rightarrow R$ be $F^{\alpha}$-differentiable function and
$h:F\rightarrow R$ be $F$-continuous such that
$h(\theta)=\mathfrak{D}_{F}^{\alpha}f(\theta)$, so
$\int_{C(a,b)}h(\theta)d_{F}^{\alpha}\theta=f(w(b))-f(w(a)).$ (9)
Analogue Taylor series is defined for $h(\theta)\in B(F)$ as
$f(\textbf{w}(v))=\sum_{n=0}^{\infty}\frac{(S_{F}^{\alpha}(v)-S_{F}^{\alpha}(v^{\prime}))^{n}}{n!}(\mathfrak{D}_{F}^{\alpha})^{n}f(\textbf{w}(v^{\prime})),$
(10)
where $h(\theta)$ is $F^{\alpha}$-differentiable any number of times on
$C(a,b)$. That is $(\mathfrak{D}_{F}^{\alpha})^{n}h\in B(F)$, $\forall n>0$.
## 3 Gradient, Divergent, Curl and Laplacian on Fractal Curves
In this section we generalized the $F^{\alpha}$-calculus by defining the
gradient, divergent, curl and Laplacian on fractal curves imbedding in
$R^{3}$.
### 3.1 Gradient on fractal curves
Let us consider the $f\in B(F)$ as an $F$-continuous function on
$C(a,b)\subset F$ and $\textbf{w}(v,w_{i}(v)):R\rightarrow R^{3},i=1,2,3$, so
the gradient of the $f(\textbf{w}):F\rightarrow R$ is
$\mathfrak{\nabla}_{F}^{\alpha}f(\textbf{w})=~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha}f(\textbf{w})\hat{e}^{i}~{}~{}~{}i=1,2,3,$
(11)
where the $\hat{e}^{i}$ is the basis of $R^{n}$.
### 3.2 Divergent on fractal curves
Let the
$\textbf{f}(\textbf{w}(v))=f_{i}(\textbf{w}(v))~{}\hat{e}^{i}~{}~{}i=1,2,3$,
be a vector field on fractal curve. Then we define the divergent of the
$\textbf{f}:F\rightarrow R^{n}$ as follows:
$\mathfrak{\nabla}_{F}^{\alpha}.\textbf{f}(\textbf{w}(v))=~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha}f_{i}(\textbf{w}(v)),$
(12)
where $f_{i}(\textbf{w}(v))$ are components of vector field.
### 3.3 Laplacian on fractal curves
Consider the $\textbf{w}(v,w_{i}(v)):R\rightarrow R^{3}$ on the fractal curve,
therefore the Laplacian is defined as
$\triangle_{F}^{\alpha}f=(\mathfrak{\nabla}_{F}^{\alpha})^{2}f=(^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}f(\textbf{w}(v))$
(13)
where the $\triangle_{F}^{\alpha}$ is called Laplacian on fractal curve.
## 4 Quantum mechanics on fractal curve
The classical mechanics is reformulated in terms of a minimum principle. The
Euler-Lagrange equations of motion is derived from the least action. The
Feynman paths for a particle in quantum mechanics are fractals with dimension
2 [33]. In this section, we obtain the Schrödinger equation on fractal curves.
### 4.1 Generalized Feynman path integral method
Feynman method for studying quantum mechanics using classical Lagrangian and
action is presented in Ref [34, 35]. Now we want to generalized Feynman method
using Lagrangian on fractals curves. Consider generalized action as
$\mathfrak{A}^{\alpha}_{F}=\int_{t_{1}}^{t_{2}}L_{F}^{\alpha}(t,\textbf{w}(v),~{}^{t}D_{F}^{\alpha}\textbf{w}(v))d_{F}^{\alpha}v~{}d_{F}^{\alpha}t~{}~{}~{}~{}L_{F}^{\alpha}:F\times
F\times F\rightarrow R.$ (14)
In view of Feynman method, if wave function on fractal in
$t_{1},\textbf{w}_{a}(v_{1})$ is
$\psi_{F}^{\alpha}(t_{1},\textbf{w}_{a}(v_{1}))$. So it gives the total
probability amplitude at $t_{2},\textbf{ w}_{b}(v_{2})$ as
$\psi_{F}^{\alpha}(t_{2},\textbf{w}_{b}(v_{2}))=\int_{-\infty}^{\infty}K_{F}^{\alpha}(t_{2},\textbf{w}_{b}(v_{2}),t_{1},\textbf{w}_{a}(v_{1}))(\psi_{F}^{\alpha}(t_{1},\textbf{w}_{a}(v_{1}))d_{F}^{\alpha}\textbf{w}(v),$
(15)
where $K_{F}^{\alpha}$ is the propagator which is defined as follows:
$K_{F}^{\alpha}(t_{2},\textbf{w}_{b}(v_{2}),t_{1},\textbf{w}_{a}(v_{1}))=\int_{w_{a}}^{w{b}}\exp[\frac{i}{\hbar}\mathfrak{A}^{\alpha}_{F}]\mathcal{D}_{F}^{\alpha}\textbf{w}(v).$
(16)
Where symbol $\mathcal{D}_{F}^{\alpha}$ indicates the integration over all
fractal paths from $\textbf{w}_{a}(v_{1})$ to $\textbf{w}_{b}(v_{2})$.
Now we derive the Schrödinger equation for a free particle on fractal curve,
which is describes the evolution of the wave function from
$\textbf{w}_{a}(v_{1})$ to $\textbf{w}_{b}(v_{2})$ , when $t_{2}$ differs from
$t_{1}$ an infinitesimal amount $\epsilon$. Supposing
$S_{F}^{\alpha}(v_{2})=S_{F}^{\alpha}(v_{1})+\epsilon$, leads to Lagrangian
for free particle as
$L_{F}^{\alpha}(t,\textbf{w}(v),~{}^{t}D_{F}^{\alpha}\textbf{w}(v))\simeq\frac{m(\textbf{w}(v)-\textbf{w}(v_{0}))^{2}}{2(S_{F}^{\alpha}(v_{2})-S_{F}^{\alpha}(v_{1}))}.$
(17)
The generalized action on fractal $\mathfrak{A}^{\alpha}_{F}$ is approximately
$\mathfrak{A}^{\alpha}_{F}\sim\epsilon
L_{F}^{\alpha}=\frac{m(\textbf{w}(v)-\textbf{w}(v_{0}))^{2}}{2\epsilon}.$ (18)
As a consequence, we obtain
$\psi_{F}^{\alpha}(t+\epsilon,\textbf{w}(v))=\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m(\textbf{w}(v)-\textbf{w}_{0}(v_{0}))^{2}}{2\epsilon}]\psi_{F}^{\alpha}(t,\textbf{w}_{0}(v_{0}))\mathcal{D}_{F}^{\alpha}\textbf{w}_{0}(v_{0}).$
(19)
Here, because of properties of exponential function only those fractal paths
give significant contributions which are very close to $\textbf{w}(v)$.
Changing the variable in the integral
$\delta=\textbf{w}(v)-\textbf{w}_{0}(v_{0})$ we have
$\psi_{F}^{\alpha}(t,\textbf{w}_{0}(v_{0}))=\psi_{F}^{\alpha}(t,\textbf{w}(v)+\delta)$.
Since both $\epsilon$ and $\delta$ are small quantities, so that
$\psi_{F}^{\alpha}(t,\textbf{w}(v)+\delta)$ and
$\psi_{F}^{\alpha}(t+\epsilon,\textbf{w}(v))$ can be expanded using Eq. (10).
We only keep up to terms of second order of the $\epsilon$ and $\delta$. As a
result we get
$\displaystyle\psi_{F}^{\alpha}(t,\textbf{w}(v))+\epsilon(~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))\simeq\chi_{F}(t)\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}](\psi_{F}^{\alpha}(t,\textbf{w}(v))+\delta(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v)))$
$\displaystyle+\frac{\delta^{2}}{2}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))d_{F}^{\alpha}\delta,$
(20)
where $\chi_{F}(t)$ is the characteristic function for Cantor like sets. The
second term in the right hand side vanishes on integration. It follows by
equating the leading terms on both sides we obtain
$\psi_{F}^{\alpha}(t,\textbf{w}(v))=\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}]\psi_{F}^{\alpha}(t,\textbf{w}(v))d_{F}^{\alpha}\delta.$
(21)
Also, we arrive at
$A=\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}]d_{F}^{\alpha}\delta=\sqrt{\frac{2i\pi\hbar\epsilon}{m}},$
(22)
and
$\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}](\frac{\delta^{2}}{2}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))d_{F}^{\alpha}\delta=\epsilon\frac{i\hbar}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v))).$
(23)
Finally, equating the remaining terms, we get Schrödinger equation on fractal
curves for free particle as
$(i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))=~{}\chi_{F}(t)\frac{-\hbar^{2}}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v))).$
(24)
The Eq. (24) leads to the definition of the Hamiltonian and momentum operator
on fractal curves as
$\hat{H}_{F}^{\alpha}=i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha}~{}~{}~{}\hat{P}_{F}^{\alpha}=-i\hbar\mathfrak{\nabla}_{F}^{\alpha}$
(25)
The solution of Eq. (24) can be find using conjugate equation as
$i\hbar~{}\frac{\partial\theta(t,\xi)}{\partial
t}=\frac{-\hbar^{2}}{2m}\frac{\partial^{2}}{\partial\xi^{2}}\theta(t,\xi)~{}~{}~{}\theta(\xi,t)=\phi[\psi_{F}^{\alpha}(t,\textbf{w}(v)))].$
(26)
Since the solution Eq. (26) is
$\theta(t,\xi)=(Ae^{ik\xi}+Be^{-ik\xi})e^{-i\beta t},$ (27)
where $k=\frac{\sqrt{2mE}}{\hbar}$ and $\omega=\frac{E}{\hbar}$ are constants.
Now by applying $\phi^{-1}$ we have the solutions as
$\psi_{F}^{\alpha}(t,\textbf{w}(v)))=(Ae^{ikS_{F}^{\alpha}(v)}+Be^{-ikS_{F}^{\alpha}(v)})e^{-i\beta
S_{F}^{\alpha}(t)}.$ (28)
It is straight forward to extended to the case of a free particle to the
motion involving the potential. In this case the Lagrangian will be
$L_{F}^{\alpha}=T_{F}^{\alpha}-V_{F}^{\alpha}(t,\textbf{w}(v))$. By
substituting the Lagrangian in the Eq. (4.1) one can derive the Schrödinger
equation as
$\displaystyle\psi_{F}^{\alpha}(t,\textbf{w}(v))+\epsilon(~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))\simeq\chi_{F}(t)\int_{-\infty}^{+\infty}\frac{1}{A}\exp[\frac{i}{\hbar}\frac{m\delta^{2}}{2\epsilon}][1-\frac{i\epsilon}{\hbar}V_{F}^{\alpha}(t,\textbf{w}(v))](\psi_{F}^{\alpha}(t,\textbf{w}(v))$
$\displaystyle+\delta(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v)))+\frac{\delta^{2}}{2}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))d_{F}^{\alpha}\delta.$
(29)
The same manner we worked out above the Eq. (4.1) becomes
$(i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha})\psi_{F}^{\alpha}(t,\textbf{w}(v))=~{}\chi_{F}(t)\frac{-\hbar^{2}}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}\psi_{F}^{\alpha}(t,\textbf{w}(v)))+\chi_{F}(t)V_{F}^{\alpha}(t,\textbf{w}(v))\psi_{F}^{\alpha}(t,\textbf{w}(v))$
(30)
## 5 Continuity equation and probability on fractal
It is well known that the continuity equation is a important concept in
quantum mechanics. Therefor, the probability density on the fractal for a
particle is defined as
$\rho_{F}^{\alpha}(t,\textbf{w}(v))=(~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v)))~{}\psi_{F}^{\alpha}(t,\textbf{w}(v)).$
(31)
The complex conjugate wave function of Eq. (30) is
$(-i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha})~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))=~{}\chi_{F}(t)\frac{-\hbar^{2}}{2m}(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v)))+\chi_{F}(t)V(t,\textbf{w}(v))~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))$
(32)
where
$V_{F}^{\alpha}(t,\textbf{w}(v))=~{}^{*}V_{F}^{\alpha}(t,\textbf{w}(v))$.
Applying this identity is given below
$^{t}\mathfrak{D}_{F}^{\alpha}(\psi_{F}^{\alpha}(t,\textbf{w}(v))~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v)))=~{}^{t}\mathfrak{D}_{F}^{\alpha}(\psi_{F}^{\alpha}(t,\textbf{w}(v)))^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))+\psi_{F}^{\alpha}(t,\textbf{w}(v))~{}^{t}\mathfrak{D}_{F}^{\alpha}(~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))),$
(33)
and substituting Eq. (30) and Eq. (32), into Eq.(33) yield us
$\displaystyle
i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha}\rho_{F}^{\alpha}(t,\textbf{w}(v))=\chi_{F}(t)\frac{\hbar^{2}}{2m}[\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))$
$\displaystyle-~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}\psi_{F}^{\alpha}(t,\textbf{w}(v))].$
(34)
As a consequence the definition of a probability current density on fractal
curve is
$\displaystyle
J_{F}^{\alpha}(t,\textbf{w}(v))=\chi_{F}(t)\frac{\hbar}{2mi}[\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))$
$\displaystyle-~{}^{*}\psi_{F}^{\alpha}(t,\textbf{w}(v))(~{}^{w_{i}}\mathfrak{D}_{F}^{\alpha})^{2}~{}\psi_{F}^{\alpha}(t,\textbf{w}(v))]$
(35)
In the following table correspondence between standard quantum mechanics and
generalized quantum framework is presented.
Comparison between Standard Quantum and Quantum on Fractals |
---|---
Postulates | Standard Quantum | Quantum on Fractals
State | $\psi(t,x)$ | $\psi_{F}^{\alpha}(t,\textbf{w}(v))$
Hamiltonian | $i\hbar\frac{\partial}{\partial x}$ | $i\hbar~{}^{t}\mathfrak{D}_{F}^{\alpha}$
Momentum | $-i\hbar\nabla$ | $-i\hbar\mathfrak{\nabla}_{F}^{\alpha}$
## 6 Conclusion
The calculus on sets, vector space and manifold is used in the classical,
quantum mechanics and general relativity respectively. The geometry has
important role in this generalization and modeling the physical phenomena.
Recently, fractal geometry has been suggested by Mandelbrot. So the calculus
on them has been suggest by many researcher but it is still an open problem.
In this work we have studied the calculus on fractal curves. Since the path
integral in Feynman formulation is fractal so that is motivated us to suggest
this generalization. This framework can suggest correct way for obtaining
Schrödinger equation from Fyenman path quantum mechanics.
## Acknowledgements
One of the authors (AKG) would like to thank Professor A. D. Gangal for useful
discussion on this topic during the period of time he was in Pune University.
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arxiv-papers
| 2013-08-01T18:28:29 |
2024-09-04T02:49:48.921725
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alireza Khalili Golmankhaneh, Dumitru Baleanu",
"submitter": "Alireza Khalili Golmankhaneh",
"url": "https://arxiv.org/abs/1308.0291"
}
|
1308.0455
|
# New RIC Bounds via $\ell_{q}$-minimization with $0<q\leq 1$ in Compressed
Sensing
Shenglong Zhou†, Lingchen Kong†, Ziyan Luo‡, Naihua Xiu† July 28, 2013.
${\dagger}$ Department of Applied Mathematics, Beijing Jiaotong University,
Beijing 100044, P. R. China; ${\ddagger}$ State Key Laboratory of Rail Traffic
Control and Safety, Beijing Jiaotong University, Beijing 100044, P. R. China
(e-mail: [email protected], [email protected], [email protected],
[email protected]).
###### Abstract
The restricted isometry constants (RICs) play an important role in exact
recovery theory of sparse signals via $\ell_{q}(0<q\leq 1)$ relaxations in
compressed sensing. Recently, Cai and Zhang [6] have achieved a sharp bound
$\delta_{tk}<\sqrt{1-1/t}$ for $t\geq\frac{4}{3}$ to guarantee the exact
recovery of $k$ sparse signals through the $\ell_{1}$ minimization. This paper
aims to establish new RICs bounds via $\ell_{q}(0<q\leq 1)$ relaxation. Based
on a key inequality on $\ell_{q}$ norm, we show that (i) the exact recovery
can be succeeded via $\ell_{1/2}$ and $\ell_{1}$ minimizations if
$\delta_{tk}<\sqrt{1-1/t}$ for any $t>1$, (ii)several sufficient conditions
can be derived, such as for any $q\in(0,\frac{1}{2})$, $\delta_{2k}<0.5547$
when $k\geq 2$, for any $q\in(\frac{1}{2},1)$, $\delta_{2k}<0.6782$ when
$k\geq 1$, (iii) the bound on $\delta_{k}$ is given as well for any $0<q\leq
1$, especially for $q=\frac{1}{2},1$, we obtain $\delta_{k}<\frac{1}{3}$ when
$k(\geq 2)$ is even or $\delta_{k}<0.3203$ when $k(\geq 3)$ is odd.
###### Index Terms:
compressed sensing, restricted isometry constant, bound, $\ell_{q}$
minimization, exact recovery
## I Introduction
The concept of compressed sensing (CS) was initiated by Donoho [13],
Cand$\grave{\textmd{e}}$s, Romberg and Tao [7] and Cand$\grave{\textmd{e}}$s
and Tao [8] with the involved essential idea–recovering some original
$n$-dimensional but sparse signal$\setminus$image from linear measurement with
dimension far fewer than $n$. Large numbers of researchers, including applied
mathematicians, computer scientists and engineers, have paid their attention
to this area owing to its wide applications in signal processing,
communications, astronomy, biology, medicine, seismology and so on, see, e.g.,
survey papers [1, 19] and a monograph [14].
To recover a sparse solution $x\in\mathbb{R}^{n}$ of the underdetermined
system of the form $\Phi x=y$, where $y\in\mathbb{R}^{m}$ is the available
measurement and $\Phi\in\mathbb{R}^{m\times n}$ is a known measurement matrix
(with $m\ll n$ ), the underlying model is the following $\ell_{0}$
_minimization_ :
$\displaystyle\textup{min}~{}\|x\|_{0},~{}~{}\textup{s.t.}~{}\Phi x=y,$ (1)
where $\|x\|_{0}$ is $\ell_{0}$-norm of the vector $x\in\mathbb{R}^{n}$, i.e.,
the number of nonzero entries in $x$ (this is not a true norm, as
$\|\cdot\|_{0}$ is not positive homogeneous). However (1) is combinatorial and
computationally intractable.
One natural approach is to solve (1) via convex _$\ell_{1}$ minimization_:
$\displaystyle\textup{min}~{}\|x\|_{1},~{}~{}\textup{s.t.}~{}\Phi x=y.$ (2)
The other way is to relax (1) through the nonconvex _$\ell_{q}(0 <q<1)$
minimization_:
$\displaystyle\textup{min}~{}\|x\|_{q}^{q},~{}~{}\textup{s.t.}~{}\Phi x=y,$
(3)
where $\|x\|_{q}^{q}=\sum_{j}|x_{j}|^{q}$. Motivated by the fact
$\lim\limits_{q\rightarrow 0^{+}}\|x\|_{q}^{q}=\|x\|_{0}$, it is shown that
there are several advantages of using this approach to recover the sparse
signal [18]. This model for recovering the sparse solution is widely
considered, see [9, 10, 11, 12, 15, 16, 17, 18, 20].
One of the most popular conditions for exact sparse recovery via $\ell_{1}$ or
$\ell_{q}$ minimization is related to the _Restricted Isometry Property_ (RIP)
introduced by Cand$\grave{\textmd{e}}$s and Tao [8], which was recalled as
follows.
###### Definition I.1.
For $k\in\\{1,2,\cdots,n\\}$, the restricted isometry constant is the smallest
positive number $\delta_{k}$ such that
$\displaystyle(1-\delta_{k})\|x\|_{2}^{2}\leq\|\Phi
x\|_{2}^{2}\leq(1+\delta_{k})\|x\|_{2}^{2}$ (4)
holds for all $k$-sparse vector $x\in\mathbb{R}^{n}$, i.e., $\|x\|_{0}\leq k$.
It is known that $\delta_{k}$ has the monotone property for $k$ (see, e.g.,
[2, 3]), i.e.,
$\displaystyle\delta_{k_{1}}\leq\delta_{k_{2}},~{}~{}\textrm{if}~{}~{}k_{1}\leq
k_{2}\leq n.$ (5)
Current upper bounds on the restricted isometry constants (RICs) via
$\ell_{q}(0<q<1)$ minimization for exact signal recovery were emerged in many
studies [9, 12, 15, 16, 17, 18, 20], such as $\delta_{2k}<0.4531$ for any
$q\in(0,1]$ in [16], $\delta_{2k}<0.4531$ for any $q\in(0,q_{0}]$ with some
$q_{0}\in(0,1]$ in [18] and $\delta_{2k}<0.5$ for any $q\in(0,0.9181]$ in
[20]. Comparing with those RIC bounds, Cai and Zhang [6] recently have given a
sharp bound $\delta_{2k}<\frac{\sqrt{2}}{2}$ via $\ell_{1}$ minimization.
Motivated by results above, we make our concentrations on improving RIC bounds
via $\ell_{q}$ relaxation with $0<q\leq 1$. The main contributions of this
paper are the following three aspects:
(i) If the restricted isometry constant of $\Phi$ satisfies
$\delta_{tk}<\sqrt{(t-1)/t}$ for $t>1$, which implies
$\delta_{2k}<\frac{\sqrt{2}}{2}$, then exact recovery can be succeeded via
$\ell_{\frac{1}{2}}$ and $\ell_{1}$ minimizations.
(ii) For any $k\geq 1$, the bound for $\delta_{2k}$ is an nondecreasing
function on $q\in(0,\frac{1}{2})$ and $q\in(\frac{1}{2},1)$. Moreover, several
sufficient conditions are derived, such as for any $q\in(0,\frac{1}{2})$,
$\delta_{2k}<0.5547$ when $k\geq 2$, for any $q\in(\frac{1}{2},1)$,
$\delta_{2k}<0.6782$ when $k\geq 1$. The detailed can be seen in Tab. 2 of the
Section III, which are all better bounds than current ones in terms of
$\ell_{q}(0<q<1)$ minimization.
(iii) The bound on $\delta_{k}$ is given as well for any $0<q\leq 1$.
Especially for $q=\frac{1}{2},1$, we obtain $\delta_{k}<\frac{1}{3}$ when $k$
is even or $\delta_{k}<0.3203$ when $k(\geq 3)$ is odd.
The organization of this paper is as follows. In the next section, we
establish several key lemmas. Our main results on $\delta_{tk}$ with $t>1$ and
$\delta_{k}$ will be presented in Sections III and IV respectively. We make
some concluding remarks in Section V and give the proofs of all lemmas and
theorems in the last section.
## II Key Lemmas
This section will propose several technical lemmas, which play an important
role in the sequel analysis. We begin with recalling the lemma of the sparse
representation of a polytope stated by Cai and Zhang [6]. Here, we define
$\|x\|_{\infty}:=\textmd{max}_{i}~{}\\{|x_{i}|\\}$ and
$\|x\|_{-\infty}:=\textmd{min}_{i}~{}\\{|x_{i}|\\}$ (In fact,
$\textit{l}_{-\infty}$ is not a norm since the triangle inequality fails).
###### Lemma II.1.
For a positive number $\alpha$ and a positive integer $s$, define the polytope
$T(\alpha,s)\subset\mathbb{R}^{n}$ by
$T(\alpha,s)=\left\\{v\in\mathbb{R}^{n}\large~{}|~{}\|v\|_{\infty}\leq\alpha,\|v\|_{1}\leq
s\alpha\right\\}.$
For any $v\in\mathbb{R}^{n}$, define the set
$U(\alpha,s,v)\subset\mathbb{R}^{n}$ of sparse vectors by
$\displaystyle
U(\alpha,s,v)=\\{u\in\mathbb{R}^{n}\large~{}|~{}supp(u)\subseteq
supp(v),\|u\|_{0}\leq s,$
$\displaystyle\|u\|_{1}=\|v\|_{1},\|u\|_{\infty}\leq\alpha\\}.$
Then $v\in T(\alpha,s)$ if and only if $v$ is in the convex hull of
$U(\alpha,s,v)$. In particular, any $v\in T(\alpha,s)$ can be expressed as
$v=\sum_{i=1}^{N}\lambda_{i}u_{i},$ where
$0\leq\lambda_{i}\leq 1,\sum_{i=1}^{N}\lambda_{i}=1,u_{i}\in
U(\alpha,s,v),i=1,2,\cdots,N.$
Next we establish an interesting and important inequality in the following
lemma, which gives a sharpened estimation of $\ell_{1}$ with
$\ell_{0},\ell_{q},\ell_{\infty}$ and $\ell_{-\infty}$.
###### Lemma II.2.
For $q\in(0,1]$ and $x\in\mathbb{R}^{n}$, we have
$\displaystyle\|x\|_{1}\leq\frac{\|x\|_{q}}{n^{1/q-1}}+p_{q}n(\|x\|_{\infty}-\|x\|_{-\infty}),$
(6)
where
$\displaystyle p_{q}:=q^{\frac{q}{1-q}}-q^{\frac{1}{1-q}}.$ (7)
Moreover, $p_{q}$ is a nonincreasing and convex function of $q\in[0,1]$ with
$p_{0}:=\lim_{q\rightarrow 0^{+}}p_{q}=1~{}and~{}p_{1}:=\lim_{q\rightarrow
1^{-}}p_{q}=0.$
Figure 1: Plot of $p_{q}\in[0,1]$ as a function of $q\in[0,1]$, and
$p_{\frac{1}{2}}=\frac{1}{4}$.
###### Remark II.3.
Actually, we can substitute $n$ with $\|x\|_{0}$ in inequality (6), which
leads to
$\displaystyle\|x\|_{1}\leq\frac{\|x\|_{q}}{\|x\|_{0}^{1/q-1}}+p_{q}\|x\|_{0}(\|x\|_{\infty}-\|x\|_{-\infty}).$
(8)
Moreover, combining with the H$\ddot{o}$lder Inequality and
$\left(\ref{l1q0}\right)$, we have
###### Proposition II.4.
For $q\in(0,1]$ and $x\in\mathbb{R}^{n}$, we have
$\displaystyle\|x\|_{0}^{1-\frac{1}{q}}\|x\|_{q}\leq\|x\|_{1}\leq\left(\|x\|_{0}^{1-\frac{1}{q}}+p_{q}\|x\|_{0}\right)\|x\|_{q}.$
(9)
Here, $\left(\ref{1q}\right)$ is an interesting inequality. Although
$\left(\ref{1q}\right)$ will not be applied in our proof, it manifests the
relationship between $\ell_{1}$ and $\ell_{q}$ norm.
In order to analyze a sequent useful function more clearly, we first observe
the function $q^{\frac{q}{q-1}}$ of $q\in(0,1)$, whose figure is plotted
below.
Figure 2: Plot of $q^{\frac{q}{q-1}}$ as a function of $q\in[0,1]$.
It is easy to check that
$\displaystyle\lim_{q\rightarrow
0^{+}}q^{\frac{q}{q-1}}=1,~{}~{}\lim_{q\rightarrow 1^{-}}q^{\frac{q}{q-1}}=e.$
(10)
So $q^{\frac{q}{q-1}}$ can be defined as a function of $q$ on $[0,1]$, and it
is a nondecreasing function.
In addition, for any given integer $k\geq 1$, it is trivial that if
$q^{\frac{q}{q-1}}$ is an integer, then $q^{\frac{q}{q-1}}k$ apparently is an
integer as well for instance $q=1/2$. However, the integrity of
$q^{\frac{q}{q-1}}$ is not necessary to ensure the integrity of
$q^{\frac{q}{q-1}}k$, such as $q=2/3$ and $k=4$.
Based on analysis above, we now define a real valued function
$g(q,k):(0,1)\times\\{1,2,3,\cdots\\}\rightarrow\mathbb{R}$ by
$\displaystyle g(q,k):=\lceil
q^{\frac{q}{q-1}}k\rceil^{1-1/q}k^{1/q}+p_{q}\lceil q^{\frac{q}{q-1}}k\rceil,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q\in(0,1),~{}k\in\\{1,2,3,\cdots\\},$
(11)
where $p_{q}$ is defined as in (7) and $\lceil a\rceil$ denotes the smallest
integer that is no less than $a$.
###### Lemma II.5.
Let $g(q,k)$ be defined as in $\left(\ref{g}\right)$. Then $g(q,k)=k$ when
$q^{\frac{q}{q-1}}k$ is an integer and otherwise $g(q,k)\leq k+p_{q}$.
Moreover,
$\displaystyle g(0,k):=\lim\limits_{q\rightarrow 0^{+}}g(q,k)=k+1,$
$\displaystyle g(1,k):=\lim\limits_{q\rightarrow 1^{-}}g(q,k)=k.$
Therefore, $g(q,k)$ can be regarded as a function of $q$ on $[0,1]$, and the
image of $g(q,k)$ with the special case $k=1$, where
$g(0,1)=2,g(\frac{1}{2},1)=1,g(1,1)=1$, is plotted in Fig.II.
Figure 3: Plot of $g(q,1)$ as a function of $q\in[0,1]$.
Another two useful functions are introduced and analyzed in the following
lemma, which will ease sequent analysis of our main results.
###### Lemma II.6.
For $t>1$ and $\theta\geq 0,\rho\geq 0$, we define
$\displaystyle\mu(t,\theta)$ $\displaystyle:=$
$\displaystyle\frac{\sqrt{(t+\theta-1)(t-1)}+1-t}{\theta},$ (12)
$\displaystyle\gamma(\rho,\theta)$ $\displaystyle:=$
$\displaystyle\frac{\rho-\rho^{2}}{\frac{1}{2}-\rho+\rho^{2}(1+\frac{\theta}{2(t-1)})}.$
(13)
Then $\gamma\left(\mu\left(t,\theta\right),\theta\right)$ is a nonincreasing
function on $\theta$ when $t$ is fixed while a nondecreasing function on $t$
when $\theta$ is fixed.
## III Main Results on $\delta_{tk}$ with $t>1$
Now we give our main results on $\delta_{tk}$ with $t>1$:
###### Theorem III.1.
For any $q\in(0,1]$, if
$\displaystyle\delta_{g(q,k)(t-1)+k}<\gamma\left(\mu\left(t,\frac{g(q,k)}{k}\right),\frac{g(q,k)}{k}\right)$
(14)
holds for some $t>1$, then each $k$-sparse minimizer of the $\ell_{q}$
minimization $(\ref{lq})$ is the sparse solution of $(\ref{l0})$. Furthermore,
setting $t=1+\frac{(\tau-1)k}{g(q,k)}$ with $\tau>1$, then the sufficient
condition $\left(\ref{dertagk}\right)$ of exact signal recovery can be
reformulated as
$\displaystyle\delta_{\tau
k}<\gamma\left(\mu\left(1+\frac{(\tau-1)k}{g(q,k)},\frac{g(q,k)}{k}\right),\frac{g(q,k)}{k}\right).$
(15)
From Lemma II.5, when $q=1$ or $q^{\frac{q}{q-1}}k$ is an integer (such as
$q=\frac{1}{2}$), it follows that $g(q,k)=k$. Associating with (14) in Theorem
III.1, we have
$\delta_{tk}=\delta_{g(q,k)(t-1)+k}<\gamma\left(\mu\left(t,1\right),1\right)=\sqrt{\frac{t-1}{t}}$.
Therefore, a corollary can be elicited as below.
###### Corollary III.2.
For $q=1$ or $q\in(0,1)$ such that $q^{\frac{q}{q-1}}k$ is an integer, if
$\delta_{tk}<\sqrt{\frac{t-1}{t}}$ holds with some $t>1$ and $k\geq 1$, then
each $k$-sparse minimizer of the $\ell_{q}$ minimization
$\left(\ref{lq}\right)$ is the sparse solution of $\left(\ref{l0}\right)$.
In particular, taking $t=2,3,4$, we obtain
$\delta_{2k}<\frac{\sqrt{2}}{2}\approx 0.7071$,
$\delta_{3k}<0.8164,~{}\delta_{4k}<0.8660$ respectively. It is worth
mentioning that $\delta_{tk}<\sqrt{\frac{t-1}{t}}$ is the sharp bound for
$\ell_{1}$ minimization which has been proved by Cai and Zhang [6]. Because
exact recovery can fail for any $q\in(0,1]$ if the bound of $\delta_{2k}$ is
no less than $\frac{\sqrt{2}}{2}$ (see [12]), $\delta_{2k}<\frac{\sqrt{2}}{2}$
is also the sharp bound for $\ell_{\frac{1}{2}}$ minimization.
Actually, besides $q=\frac{1}{2}$, $k\geq 1$, there are several other $(q,k)$s
satisfying that $q^{\frac{q}{q-1}}k$ are integers, for instance $(0.2025,2)$,
$(\frac{2}{3},4)$. Thus $\delta_{tk}<\sqrt{\frac{t-1}{t}}$ is also a sharp RIC
bound for such $(q,k)$s.
###### Remark III.3.
(i) For any $k\geq 1$, we can check
$g(q,1)\geq\frac{g(q,k)}{k}.$
Then from Lemma II.6 and $\left(\ref{taok0}\right)$ in Theorem III.1, for
$k\geq 1$ and any $q\in(0,1]$, it yields that
$\displaystyle\delta_{\tau
k}<\gamma\left(\mu\left(1+\frac{\tau-1}{g(q,1)},g(q,1)\right),g(q,1)\right),$
(16)
whose figure (with $\tau=2$) is plotted as follows.
Figure 4: Plot of bounds on $\delta_{2k}$ as a function of $q\in(0,1]$ when
$k\geq 1$.
(ii) Moreover, under some assumptions $k\geq k_{0}(k_{0}=1,2,3,\cdots)$, since
for $q\in(\frac{1}{2},1]$
$\lim_{q\rightarrow\frac{1}{2}^{+}}\frac{g(q,k_{0})}{k_{0}}\geq\max\\{\lim_{q\rightarrow\frac{1}{2}^{+}}\frac{g(q,k)}{k},~{}\frac{g(q,k_{0})}{k_{0}}\\}$
and for $q\in(0,\frac{1}{2}]$
$\lim_{q\rightarrow
0^{+}}\frac{g(q,k_{0})}{k_{0}}\geq\max\\{\lim_{q\rightarrow
0^{+}}\frac{g(q,k)}{k},~{}\frac{g(q,k_{0})}{k_{0}}\\}.$
Then from Lemma II.6, we have Tab. 2 by calculating limits for cases
$q\rightarrow 0^{+}$ and $q\rightarrow\frac{1}{2}^{+}$ of the right-hand side
of $(\ref{taok0})$ with $k=k_{0}$.
_Tab. 2: Bounds on $\delta_{2k},\delta_{3k},\delta_{4k}$ for any
$q\in(0,\frac{1}{2})$ and $q\in(\frac{1}{2},1)$._
## IV Main Results on $\delta_{k}$
In this section, we state the bound on $\delta_{k}$ for any $q\in(0,1]$ in the
following results:
###### Theorem IV.1.
For any $q\in(0,1]$, if
$\displaystyle\delta_{k}<$ $\displaystyle~{}~{}~{}~{}\frac{1}{1+2\lceil
g(q,k)\rceil/k},~{}~{}~{}~{}~{}~{}\text{for even number}~{}k\geq 2,$
$\displaystyle\delta_{k}<$ $\displaystyle\frac{1}{1+2\lceil
g(q,k)\rceil/\sqrt{k^{2}-1}},~{}~{}\text{for odd number}~{}k\geq 3,$
holds, then each $k$-sparse minimizer of the $\ell_{q}$ minimization
$(\ref{lq})$ is the sparse solution of $(\ref{l0})$.
Particularly, for the case $q=1$ or $q^{\frac{q}{q-1}}k$ to be an integer
(such as $q=\frac{1}{2}$ ), we have the corollary below by applying Lemma
II.5.
###### Corollary IV.2.
For $q=1$ or $q\in(0,1)$ such that $q^{\frac{q}{q-1}}k$ is an integer, if
$\displaystyle\delta_{k}<$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}1/3,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for
even number}~{}k\geq 2,$ $\displaystyle\delta_{k}<$
$\displaystyle\frac{1}{1+2k/\sqrt{k^{2}-1}},~{}~{}\text{for odd
number}~{}k\geq 3,$
hold, then each $k$-sparse minimizer of the $\ell_{q}$ minimization
$(\ref{lq})$ is the sparse solution of $(\ref{l0})$.
Taking $q=\frac{1}{2},1$, then $g(q,k)=k$ from Lemma II.5, which produces the
bound $\delta_{k}<\frac{1}{3}$ if $k\geq 2$ is even. Meanwhile
$\delta_{k}<\frac{1}{3}$ for $k\geq 2$ is the sharp bound for $\ell_{1}$
minimization that has been gotten by Cai and Zhang [4]. From Theorem IV.1 and
Corollary IV.2, we list the following table.
Tab. 3: Upper bounds on $\delta_{k}$ for different $q$.
## V Concluding Remarks
In this paper, we have generalized the upper bounds for RICs from $\ell_{1}$
minimization to $\ell_{q}(0<q\leq 1)$ minimization, and established new RIC
bounds through $\ell_{q}$ minimization with $q\in(0,1]$ for exact sparse
recovery. An interesting issue which deserves future research would be: how to
improve these new bounds for some $q\in(0,1]$ when $q^{\frac{q}{q-1}}k$ is not
an integer.
## VI Proofs
Proof of Lemma II.2
Stimulated by the approach in [20], without loss of generality, we only need
to prove the case $x\in\Omega:=\\{(x_{1},x_{2},\cdots,x_{n})\neq
0~{}|~{}x_{1}\geq x_{2}\geq\cdots\geq x_{n}\geq 0\\}$ due to the symmetry of
components $|x_{1}|,|x_{2}|,\cdots,|x_{n}|$. Clearly, $x_{1}\neq 0$. Notice
that if the inequality (6) holds for any $(1,x_{2},\cdots,x_{n})\in\Omega$,
then we can immediately generalize the conclusion to all $x\in\Omega$ through
substituting $x/x_{1},x\in\Omega$ into (6) and eliminating the common factor
$1/x_{1}$. Henceforth, it remains to show
$\displaystyle\|x\|_{1}\leq\frac{\|x\|_{q}}{n^{1/q-1}}+p_{q}n(1-x_{n}),$ (17)
with $x\in\\{(1,x_{2},\cdots,x_{n})~{}|~{}1\geq x_{2}\geq\cdots\geq x_{n}\geq
0\\}$, where $p_{q}$ is a function of $q$ specified in (7).
First, for any given $q\in(0,1]$ define that
$f(x):=\|x\|_{1}-n^{1-1/q}\|x\|_{q}.$
It is easy to verify that $f(x)$ is a convex function on $\mathbb{R}^{n}_{+}$.
Since the maximum of a convex function always arrives on the boundary, we have
$\displaystyle h(x_{n}):$ $\displaystyle=$ $\displaystyle\max_{1\geq x_{2}\geq
x_{3}\geq\cdots\geq x_{n}}~{}f(1,x_{2},x_{3},\cdots,x_{n})$ $\displaystyle=$
$\displaystyle f(1,\cdots,1,x_{n},\cdots,x_{n}),~{}~{}x_{n}\in[0,1]$
Letting the distribution of $1$ appear for $r$ times ($1\leq r\leq n$) in the
maximum solution of $f$, we have
$h(x_{n})=r(1-x_{n})+nx_{n}-\frac{\left(r(1-x_{n}^{q})+nx_{n}^{q}\right)^{1/q}}{n^{1/q-1}}.$
By the convexity of $h$ and $h(1)=0$, it follows that
$h(x_{n})\leq(1-x_{n})h(0)+x_{n}h(1)=(1-x_{n})h(0).$
Then it holds that
$\displaystyle f(x)$ $\displaystyle\leq$ $\displaystyle
h(x_{n})\leq(1-x_{n})h(0)$ $\displaystyle=$
$\displaystyle(1-x_{n})(r-n^{1-1/q}r^{1/q})$ $\displaystyle\leq$
$\displaystyle(1-x_{n})\max_{r\in\\{1,2,\cdots,n\\}}\\{r-n^{1-1/q}r^{1/q}\\}$
$\displaystyle\leq$ $\displaystyle(1-x_{n})\max_{0<r_{1}\leq
n}\\{r_{1}-n^{1-1/q}r_{1}^{1/q}\\}$ $\displaystyle=$
$\displaystyle(1-x_{n})p_{q}n,$
where $p_{q}$ is defined as (7) and the last equality holds when
$r_{1}=q^{\frac{q}{1-q}}n\in(0,n]$ for any $q\in(0,1]$.
By computing the first and second order partial derivatives of $p_{q}$ on $q$,
it is easy to verify that $p_{q}$ is a nonincreasing convex function of
$q\in(0,1]$ and
$\lim_{q\rightarrow 0^{+}}p_{q}=1~{}\textmd{and}~{}\lim_{q\rightarrow
1^{-}}p_{q}=0.$
Thus the proof is completed. ∎
Proof of Lemma II.5
If $q^{\frac{q}{q-1}}k$ is an integer, then
$\displaystyle g(q,k)$ $\displaystyle=$
$\displaystyle(q^{\frac{q}{q-1}}k)^{1-1/q}k^{1/q}+p_{q}(q^{\frac{q}{q-1}}k)$
$\displaystyle=$ $\displaystyle
qk^{1-1/q}k^{1/q}+(q^{\frac{q}{1-q}}-q^{\frac{1}{1-q}})(q^{\frac{q}{q-1}}k)$
$\displaystyle=$ $\displaystyle qk+(1-q)k=k.$
If $q^{\frac{q}{q-1}}k$ is not an integer, then
$\displaystyle g(q,k)$ $\displaystyle\leq$
$\displaystyle(q^{\frac{q}{q-1}}k)^{1-1/q}k^{1/q}+p_{q}(q^{\frac{q}{q-1}}k+1)$
$\displaystyle=$ $\displaystyle
qk^{1-1/q}k^{1/q}+(q^{\frac{q}{1-q}}-q^{\frac{1}{1-q}})(q^{\frac{q}{q-1}}k+1)$
$\displaystyle=$ $\displaystyle qk+(1-q)k+p_{q}=k+p_{q}.$
Due to $\lim_{q\rightarrow 1^{-}}q^{\frac{q}{q-1}}=e$ and $\lim_{q\rightarrow
1^{-}}p_{q}=0$ , we have
$\displaystyle g(1,k):$ $\displaystyle=$ $\displaystyle\lim_{q\rightarrow
1^{-}}g(q,k)$ $\displaystyle=$ $\displaystyle\lim_{q\rightarrow
1^{-}}\left\\{\lceil q^{\frac{q}{q-1}}k\rceil^{1-1/q}k^{1/q}+p_{q}\lceil
q^{\frac{q}{q-1}}k\rceil\right\\}$ $\displaystyle=$ $\displaystyle k+0=k.$
Now we prove the remaining part $\lim_{q\rightarrow 0^{+}}g(q,k)=k+1$. Since
$\lim_{q\rightarrow 0^{+}}q^{\frac{q}{q-1}}=1$ and $q^{\frac{q}{q-1}}\in(1,e]$
is a nondecreasing function on $q\in(0,1]$, for any fixed $k$, we can set
$q^{\frac{q}{q-1}}=1+\varepsilon(q)$ with sufficient small
$0<\varepsilon(q)<\frac{1}{k}$. Thus
$\lceil
q^{\frac{q}{q-1}}k\rceil=\lceil(1+\varepsilon(q))k\rceil=k+1,~{}~{}\text{as}~{}q(\neq
0)\rightarrow 0^{+},$
It follows readily that
$\displaystyle g(0,k):$ $\displaystyle=$ $\displaystyle\lim_{q\rightarrow
0^{+}}g(q,k)$ $\displaystyle=$ $\displaystyle\lim_{q\rightarrow
0^{+}}\left\\{\lceil q^{\frac{q}{q-1}}k\rceil^{1-1/q}k^{1/q}+p_{q}\lceil
q^{\frac{q}{q-1}}k\rceil\right\\}$ $\displaystyle=$
$\displaystyle\lim_{q\rightarrow
0^{+}}\left\\{(k+1)^{1-1/q}k^{1/q}+p_{q}(k+1)\right\\}$ $\displaystyle=$
$\displaystyle\lim_{q\rightarrow
0^{+}}\left\\{(k+1)\left(\frac{k}{k+1}\right)^{1/q}+p_{q}(k+1)\right\\}$
$\displaystyle=$ $\displaystyle 0+k+1=k+1.$
The whole proof is finished.∎
Proof of Lemma II.6
We verify $\gamma\left(\mu\left(t,\theta\right),\theta\right)$ is a
nonincreasing function on $\theta\geq 0$ and a nondecreasing function on
$t>1$. By directly computing the first order partial derivative of
$\gamma\left(\mu\left(t,\theta\right),\theta\right)$ on $\theta\geq 0$, it
yields
$\frac{\partial}{\partial\theta}\gamma\left(\mu\left(t,\theta\right),\theta\right)=\frac{-\sqrt{(t+\theta-1)(t-1)}}{2(t+\theta-1)^{2}}\leq
0.$
Likewise, by computing the first order partial derivative of
$\gamma\left(\mu\left(t,\theta\right),\theta\right)$ on $t>1$, we have
$\frac{\partial}{\partial
t}\gamma\left(\mu\left(t,\theta\right),\theta\right)=\frac{\theta}{2\sqrt{(t-1)(t+\theta-1)^{3}}}\geq
0.$
Then the desired conclusions hold immediately.∎
Before proving Theorem III.1, we introduce hereafter several notations. For
$h\in\mathbb{R}^{n}$, we denote hereafter $h_{T}$ the vector equal to $h$ on
an index set $T$ and zero elsewhere. Especially, we denote $h_{max(k)}$ as $h$
with all but the largest $k$ entries in absolute value set to zero, and
$h_{-max(k)}:=h-h_{max(k)}$.
Proof of Theorem III.1
The approach of this proof is similar as [6]. First we consider the case that
$g(k,q)(t-1)$ is an integer. By the Null Space Property [18] in $\ell_{q}$
minimization case, we only need to check for all
$h\in\mathcal{N}(\Phi)\setminus\\{0\\}$,
$\|h_{max(k)}\|_{q}^{q}<\|h_{-max(k)}\|_{q}^{q}.$
Suppose on the contrary that there exists
$h\in\mathcal{N}(\Phi)\setminus\\{0\\}$, such that
$\|h_{max(k)}\|_{q}^{q}\geq\|h_{-max(k)}\|_{q}^{q}$. Set
$\alpha=k^{-1/q}\|h_{max(k)}\|_{q}$ and decompose $h_{-max(k)}$ into a sum of
vectors $h_{T_{1}},h_{T_{2}},\ldots$, where $T_{1}$ corresponds to the
locations of the $\lceil q^{\frac{q}{q-1}}k\rceil$ largest coefficients of
$h_{-max(k)}$ ; $T_{2}$ to the locations of the $\lceil
q^{\frac{q}{q-1}}k\rceil$ largest coefficients of $h_{-max(k)T_{1}^{C}}$, and
so on. That is
$\displaystyle h_{-max(k)}=h_{T_{1}}+h_{T_{2}}+h_{T_{3}}+\cdots.$
Here, the sparsity of $h_{T_{j}}(j\geq 1)$ is at most $\lceil
q^{\frac{q}{q-1}}k\rceil$.
Clearly,
$k\|h_{-max(k)}\|_{\infty}^{q}\leq\|h_{max(k)}\|_{q}^{q}=k\alpha^{q}$, which
generates $\|h_{-max(k)}\|_{\infty}\leq\alpha$. From Lemma II.2, for $j\geq
1$,
$\displaystyle\|h_{T_{j}}\|_{1}$ $\displaystyle\leq$ $\displaystyle\lceil
q^{q/(q-1)}k\rceil^{1-1/q}\|h_{T_{j}}\|_{q}$ (18) $\displaystyle+p_{q}\lceil
q^{\frac{q}{q-1}}k\rceil(\|h_{T_{j}}\|_{\infty}-\|h_{T_{j}}\|_{-\infty}).~{}~{}~{}~{}~{}~{}$
Then we sum $\|h_{T_{j}}\|_{1}$ for $j\geq 1$ to obtain that
$\displaystyle\|h_{-max(k)}\|_{1}=\sum_{j\geq 1}\|h_{T_{j}}\|_{1}$ (19)
$\displaystyle\leq$ $\displaystyle\lceil
q^{\frac{q}{q-1}}k\rceil^{1-1/q}\sum_{j\geq 1}\|h_{T_{j}}\|_{q}$
$\displaystyle+p_{q}\lceil q^{\frac{q}{q-1}}k\rceil\sum_{j\geq
1}\left(\|h_{T_{j}}\|_{\infty}-\|h_{T_{j}}\|_{-\infty}\right)$
$\displaystyle\leq$ $\displaystyle\lceil
q^{\frac{q}{q-1}}k\rceil^{\frac{q-1}{q}}(\sum_{j\geq
1}\|h_{T_{j}}\|_{q}^{q})^{1/q}+p_{q}\lceil
q^{\frac{q}{q-1}}k\rceil\|h_{T_{1}}\|_{\infty}.$ $\displaystyle\leq$
$\displaystyle\lceil
q^{\frac{q}{q-1}}k\rceil^{\frac{q-1}{q}}k^{1/q}\alpha+p_{q}\lceil
q^{\frac{q}{q-1}}k\rceil\alpha=g(q,k)\alpha.$
We again divide $h_{-max(k)}$ into two parts, $h_{-max(k)}=h^{(1)}+h^{(2)}$,
where
$h^{(1)}:=h\cdot\textbf{1}_{\\{i:|h_{-max(k)}(i)|>\frac{\alpha}{t-1}\\}},$
$h^{(2)}:=h\cdot\textbf{1}_{\\{i:|h_{-max(k)}(i)|\leq\frac{\alpha}{t-1}\\}}.$
Therefore $h^{(1)}$ is $g(q,k)(t-1)$-sparse as a result of facts that
$\|h^{(1)}\|_{1}\leq\|h_{-max(k)}\|_{1}\leq g(q,k)\alpha$ and all non-zero
entries of $h^{(1)}$ has magnitude larger than $\frac{\alpha}{t-1}$. Let
$\|h^{(1)}\|_{0}=m$, then
$\displaystyle\|h^{(2)}\|_{1}$ $\displaystyle=$
$\displaystyle\|h_{max(k)}\|_{1}-\|h^{(1)}\|_{1}$ (20) $\displaystyle\leq$
$\displaystyle\left[g(q,k)(t-1)-m\right]\frac{\alpha}{t-1},$
$\displaystyle\|h^{(2)}\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\frac{\alpha}{t-1}.$ (21)
Applying Lemma II.1 with $s=g(q,k)(t-1)-m$, it makes $h^{(2)}$ be expressed as
a convex combination of sparse vectors:
$h^{(2)}=\sum_{i=1}^{N}\lambda_{i}u_{i}$, where $u_{i}$ is $s$-sparse,
$\|u_{i}\|_{1}=\|h^{(2)}\|_{1},\|u_{i}\|_{\infty}\leq\frac{\alpha}{t-1}$.
Henceforth,
$\|u_{i}\|_{2}\leq\sqrt{g(q,k)(t-1)-m}\|u_{i}\|_{\infty}\leq\sqrt{\frac{g(q,k)}{t-1}}\alpha.$
For any $\mu\geq 0$, denoting $\eta_{i}=h_{max(k)}+h^{(1)}+\mu u_{i}$, we
obtain
$\displaystyle\sum_{j=1}^{N}\lambda_{j}\eta_{j}-\frac{1}{2}\eta_{i}=h_{max(k)}+h^{(1)}+\mu
h^{(2)}-\frac{1}{2}\eta_{i}~{}~{}~{}~{}$ (22) $\displaystyle=$
$\displaystyle(\frac{1}{2}-\mu)\left(h_{max(k)}+h^{(1)}\right)-\frac{1}{2}\mu
u_{i}+\mu h,$
where $\eta_{i},\sum_{i=1}^{N}\lambda_{i}\eta_{i}-\frac{1}{2}\eta_{i}-\mu h$
are all $\left(g(q,k)(t-1)+k\right)$-sparse vectors from the sparsity of
$\|h_{max(k)}\|_{0}\leq k$, $\|h^{(1)}\|_{0}=m$ and $\|u_{i}\|_{0}\leq s$.
It is easy to check the following identity,
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\|\Phi(\sum_{j=1}^{N}\lambda_{j}\eta_{j}-\frac{1}{2}\eta_{i})\|_{2}^{2}=\frac{1}{4}\sum_{i=1}^{N}\lambda_{i}\|\Phi\eta_{i}\|_{2}^{2}.$
(23)
Since $\Phi h=0$, together with (22), we have
$\Phi(\sum_{j=1}^{N}\lambda_{j}\eta_{j}-\frac{1}{2}\eta_{i})=\Phi((\frac{1}{2}-\mu)(h_{max(k)}+h^{(1)})-\frac{1}{2}\mu
u_{i}).$
Setting $\mu=\mu\left(t,g(q,k)/k\right)>0$, if (14) holds, that is
$\displaystyle\delta:=\delta_{g(q,k)(t-1)+k}<\gamma\left(\mu\left(t,\frac{g(q,k)}{k}\right),\frac{g(q,k)}{k}\right),$
(24)
then combining (23) with (24), we get
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\|\Phi((\frac{1}{2}-\mu)(h_{max(k)}+h^{(1)})-\frac{1}{2}\mu
u_{i})\|_{2}^{2}$
$\displaystyle-\frac{1}{4}\sum_{i=1}^{N}\lambda_{i}\|\Phi\eta_{i}\|_{2}^{2}$
$\displaystyle\leq$
$\displaystyle(1+\delta)\sum_{i=1}^{N}\lambda_{i}[(\frac{1}{2}-\mu)^{2}\|h_{max(k)}+h^{(1)}\|_{2}^{2}+\frac{\mu^{2}}{4}\|u_{i}\|_{2}^{2}]$
$\displaystyle-\frac{1-\delta}{4}\sum_{i=1}^{N}\lambda_{i}(\|h_{max(k)}+h^{(1)}\|_{2}^{2}+\mu^{2}\|u_{i}\|_{2}^{2})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}[((1+\delta)(\frac{1}{2}-\mu)^{2}-\frac{1-\delta}{4})\cdot$
$\displaystyle\|h_{max(k)}+h^{(1)}\|_{2}^{2}+\frac{1}{2}\delta\mu^{2}\|u_{i}\|_{2}^{2}]$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\|h_{max(k)}+h^{(1)}\|_{2}^{2}\cdot$
$\displaystyle\left[\mu^{2}-\mu+\delta(\frac{1}{2}-\mu+(1+\frac{g(q,k)}{2k(t-1)})\mu^{2})\right]$
$\displaystyle=$ $\displaystyle\|h_{max(k)}+h^{(1)}\|_{2}^{2}\cdot$
$\displaystyle\left[\mu^{2}-\mu+\delta(\frac{1}{2}-\mu+(1+\frac{g(q,k)}{2k(t-1)})\mu^{2})\right]$
$\displaystyle=$
$\displaystyle\|h_{max(k)}+h^{(1)}\|_{2}^{2}(\frac{1}{2}-\mu+(1+\frac{g(q,k)}{2k(t-1)})\mu^{2})\cdot$
$\displaystyle\left[\delta-\gamma(\mu\left(t,\frac{g(q,k)}{k}),\frac{g(q,k)}{k}\right)\right]$
$\displaystyle<$ $\displaystyle 0,$
where the inequality (VI) is derived from the following facts:
$\displaystyle\|h_{max(k)}\|_{2}^{2}$ $\displaystyle\geq$ $\displaystyle
k^{1-2/q}\|h_{max(k)}\|_{q}^{2}$ (26) $\displaystyle=$ $\displaystyle
k^{1-2/q}(k\alpha^{q})^{2/q}=k\alpha^{2},$ $\displaystyle\|u_{i}\|_{2}$
$\displaystyle\leq$
$\displaystyle\sqrt{\frac{g(q,k)}{t-1}}\alpha\leq\sqrt{\frac{g(q,k)}{k}}\frac{\|h_{max(k)}\|_{2}}{\sqrt{t-1}}~{}~{}~{}~{}~{}~{}$
(27) $\displaystyle\leq$
$\displaystyle\sqrt{\frac{g(q,k)}{k}}\frac{\|h_{max(k)}+h^{(1)}\|_{2}}{\sqrt{t-1}}.$
Obviously, this is a contradiction.
When $g(k,q)(t-1)$ is not an integer, by setting
$t^{\prime}=\frac{\lceil g(k,q)(t-1)\rceil}{g(k,q)}+1,$
we have $t^{\prime}>t$ and $g(k,q)(t^{\prime}-1)$ is an integer. Utilizing the
nondecreasing monotonicity of
$\gamma\left(\mu\left(t,\theta\right),\theta\right)$ on $t\geq 0$ for fixed
$\theta$ presented in Lemma II.6, we can get
$\displaystyle\delta_{g(k,q)(t^{\prime}-1)+k}$ $\displaystyle=$
$\displaystyle\delta_{g(k,q)(t-1)+k}$ $\displaystyle<$
$\displaystyle\gamma\left(\mu\left(t,\frac{g(q,k)}{k}\right),\frac{g(q,k)}{k}\right)$
$\displaystyle<$
$\displaystyle\gamma\left(\mu\left(t^{\prime},\frac{g(q,k)}{k}\right),\frac{g(q,k)}{k}\right),$
which can be deduced to the former case. Hence we complete the proof. ∎
In order to prove the result Theorem IV.1, we need another important concept
in the RIP framework the restricted orthogonal constants (ROC) proposed in
[8].
###### Definition VI.1.
Suppose $\Phi\in\mathbb{R}^{m\times n}$, define the restricted orthogonal
constants (ROC) of order $k_{1},k_{2}$ as the smallest non-negative number
$\theta_{k_{1},k_{2}}$ such that
$\displaystyle\left|\left\langle\Phi h_{1},\Phi
h_{2}\right\rangle\right|\leq\theta_{k_{1},k_{2}}\|h_{1}\|_{2}\|h_{2}\|_{2}$
(28)
for all $k_{1}$-sparse vector $h_{1}\in\mathbb{R}^{n}$ and $k_{2}$-sparse
vector $h_{2}\in\mathbb{R}^{n}$ with disjoint supports.
Proof of Theorem IV.1
Similar to the proof of Theorem III.1, we only need to check for all
$h\in\mathcal{N}(\Phi)\setminus\\{0\\}$,
$\|h_{max(k)}\|_{q}^{q}<\|h_{-max(k)}\|_{q}^{q}.$
Suppose there exists $h\in\mathcal{N}(\Phi)\setminus\\{0\\}$, such that
$\|h_{max(k)}\|_{q}^{q}\geq\|h_{-max(k)}\|_{q}^{q}$. Set
$\alpha=k^{-1/q}\|h_{max(k)}\|_{q}$. From the proof of Theorem III.1, we have
$\|h_{-max(k)}\|_{1}\leq g(q,k)\alpha\leq\lceil g(q,k)\rceil\alpha$ and
$\|h_{-max(k)}\|_{\infty}\leq\alpha$. Then it follows from Lemma 5.1 in [5]
that
$\displaystyle\left|\left\langle\Phi h_{max(k)},\Phi
h_{-max(k)}\right\rangle\right|$ $\displaystyle\leq$
$\displaystyle\theta_{k,\lceil g(q,k)\rceil}\|h_{max(k)}\|_{2}\sqrt{\lceil
g(q,k)\rceil}\alpha$ $\displaystyle\leq$
$\displaystyle\theta_{k,k}\sqrt{\frac{\lceil
g(q,k)\rceil}{k}}\|h_{max(k)}\|_{2}\sqrt{\lceil g(q,k)\rceil}\alpha$
$\displaystyle\leq$ $\displaystyle\theta_{k,k}\frac{\lceil
g(q,k)\rceil}{k}\|h_{max(k)}\|_{2}^{2},$
where the first inequality holds by Lemma 5.4 in [5] and the second inequality
by (26). Thus from the condition
$\delta_{k}+\theta_{k,k}\frac{\lceil g(q,k)\rceil}{k}<1,$
it follows that
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\left|\left\langle\Phi
h_{max(k)},\Phi h\right\rangle\right|$ $\displaystyle\geq$
$\displaystyle\left|\left\langle\Phi h_{max(k)},\Phi
h_{max(k)}\right\rangle\right|-\left|\left\langle\Phi h_{max(k)},\Phi
h_{-max(k)}\right\rangle\right|$ $\displaystyle\geq$
$\displaystyle(1-\delta_{k})\|h_{max(k)}\|_{2}^{2}-\theta_{k,k}\frac{\lceil
g(q,k)\rceil}{k}\|h_{max(k)}\|_{2}^{2}$ $\displaystyle=$
$\displaystyle(1-\delta_{k}-\theta_{k,k}\frac{\lceil
g(q,k)\rceil}{k})\|h_{max(k)}\|_{2}^{2}$ $\displaystyle>$ $\displaystyle 0.$
Obviously, this is a contradiction. By Lemma 3.1 in [5],
$\displaystyle\theta_{k,k}<$
$\displaystyle~{}~{}~{}~{}2\delta_{k},~{}~{}~{}~{}~{}~{}\text{for any
even}~{}k\geq 2,$ $\displaystyle\theta_{k,k}<$
$\displaystyle\frac{2k}{\sqrt{k^{2}-1}}\delta_{k}~{}~{}\text{for any
odd}~{}k\geq 3.$
Hence, when $k\geq 2$ is even, it yields that
$\delta_{k}+\frac{g(q,k)}{k}\theta_{k,k}<\left(1+\frac{2\lceil
g(q,k)\rceil}{k}\right)\delta_{k},$
and when $k\geq 3$ is odd, it generates that
$\delta_{k}+\frac{g(q,k)}{k}\theta_{k,k}<\left(1+\frac{2\lceil
g(q,k)\rceil}{\sqrt{k^{2}-1}}\right)\delta_{k}.$
Therefore the theorem is proved. ∎
## Acknowledgement
The work was supported in part by the National Basic Research Program of China
(2010CB732501), and the National Natural Science Foundation of China
(11171018, 71271021).
## References
* [1] A.M. Bruckstein, D.L. Donoho, and A. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images. _SIAM Rev._ , vol. 51, pp. 34-81, 2009.
* [2] T. Cai, L. Wang and G. Xu, New bounds for restricted isometry constants, _IEEE Trans. Inform. Theory_ , vol. 56, pp. 4388-4394, 2010.
* [3] T. Cai, L. Wang and G. Xu, Shifting inequality and recovery of sparse signals, _IEEE Trans. Signal Process._ , vol. 58, pp. 1300-1308, 2010.
* [4] T. Cai and A. Zhang, Sharp RIP bound for sparse signal and low-rank matrix recovery, _Appl. and Comput. Harmon. Anal._ , vol. 35, pp. 74-93, 2013.
* [5] T. Cai and A. Zhang, Compressed sensing and affine rank minimization under restricted isometry, to appear in _IEEE Trans. Signal Process._ , 2013.
* [6] T. Cai, and A. Zhang, Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices, to appear, 2013.
* [7] E.J. Cand$\grave{e}$s, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, _IEEE Trans.Inf. Theory_ , vol. 52, pp. 489-509, 2006.
* [8] E.J. Cand$\grave{e}$s and T. Tao, Decoding by linear programming, _IEEE Trans. Inf. Theory_ , vol. 51, pp. 4203-4215, 2005.
* [9] R. Chartrand, Exact reconstruction of sparse signals via nonconvex minimization, _IEEE Signal Process. Lett._ , vol. 14, pp. 707-710, 2007.
* [10] X. Chen, D. Ge, Z. Wang and Y. Ye, Complexity of Unconstrained $\ell_{2}-\ell_{p}$ Minimization, to appear in Mathematical Programming.
* [11] X.Chen, F. Xu and Y. Ye, Lower bound theory of nonzero entries in solutions of $\ell_{2}-\ell_{p}$ minimization, _SIAM J. Scientific Computing_ , vol. 32, pp. 2832-2852, 2010.
* [12] M.E. Davies and R. Gribonval, Restricted isometry constants where $\ell_{p}$ sparse recovery can fail for $0<p\leq 1$, _IEEE Trans. Inf. Theory_ , vol. 55, pp. 2203-2214, 2010.
* [13] D.L. Donoho, Compressed sensing, _IEEE Trans. Inf. Theory_ , vol. 52, pp. 1289-1306, 2006.
* [14] Y.C. Eldar and G.Kutyniok, Compressed Sensing: Theory and Applications, Cambridge University Press, 2012.
* [15] S. Foucart, A note on guaranteed sparse recovery via $\ell_{q}$-minimization, _Appl. Comput. Harmon. Anal._ , vol. 29, pp. 97-103, 2010.
* [16] S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $\ell_{q}$-minimization for $0<q\leq 1$, _Appl. Comput. Harmon. Anal._ , vol. 26, pp. 395-407, 2009.
* [17] R. Gribonval and M. Nielsen, Sparse decompositions in unions of bases, _IEEE Trans.Inf. Theory_ , vol. 49, pp. 3320-3325 ,2003.
* [18] M.J. Lai and J. Wang, An unconstrained $\ell_{q}$ minimization for sparse solution of under determined linear systems, _SIAM J. Optimization_ , vol. 21, pp. 82-101, 2011.
* [19] H. Rauhut, Compressive sensing and structured random matrices, _Radon Series Comp. Appl. Math._ , vol. 9, pp. 1-92. 2010.
* [20] Y. Hsia and R.L. Sheu, On RIC bounds of Compressed Sensing Matrices for Approximating Sparse Solutions Using $\ell_{q}$ Quasi Norms, to appear, 2012.
Shenglong Zhou is a PhD student in Department of Applied Mathematics, Beijing
Jiaotong University. He received his BS degree from Beijing Jiaotong
University of information and computing science in 2011. His research field is
theory and methods for optimization. Lingchen Kong is an associate Professor
in Department of Applied Mathematics, Beijing Jiaotong University. He received
his PhD degree in Operations Research from Beijing Jiaotong University in
2007. From 2007 to 2009, he was a Post-Doctoral Fellow of Department of
Combinatorics and Optimization, Faculty of Mathematics, University of
Waterloo, Canada. His research interests are in sparse optimization,
mathematics of operations research. Ziyan Luo is a lecturer in the State Key
Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University.
She received her PhD degree in Operations Research from Beijing Jiaotong
University in 2010. From 2011 to 2012, she was a visiting scholar in
Management Science and Engineering, School of Engineering, Stanford
University, USA. Her research interests are in sparse optimization,
semidefinite programming and interior point methods. Naihua Xiu is a Professor
in Department of Applied Mathematics, Beijing Jiaotong University. He received
his PhD degree in Operations Research from Academy Mathematics and System
Science of the Chinese Academy of Science in 1997. He was a Research Fellow of
City University of Hong Kong from 2000 to 2002, and he was a Visiting Scholar
of University of Waterloo from 2006 to 2007. His research interest includes
variational analysis, mathematical optimization, mathematics of operations
research.
|
arxiv-papers
| 2013-08-02T10:20:18 |
2024-09-04T02:49:48.934938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shenglong Zhou, Lingchen Kong, Ziyan Luo, Naihua Xiu",
"submitter": "Shenglong Zhou",
"url": "https://arxiv.org/abs/1308.0455"
}
|
1308.0755
|
# Leading-order temporal asymptotics of the Fokas-Lenells Equation without
solitons
Jian Xu School of Mathematical Sciences
Fudan University
Shanghai 200433
People’s Republic of China [email protected] and Engui Fan School of
Mathematical Sciences, Institute of Mathematics and Key Laboratory of
Mathematics for Nonlinear Science
Fudan University
Shanghai 200433
People’s Republic of China correspondence author:[email protected]
###### Abstract.
We use the Deift-Zhou method to obtain, in the solitonless sector, the leading
order asymptotic of the solution to the Cauchy problem of the Fokas-Lenells
equation as $t\rightarrow+\infty$ on the full-line .
###### Key words and phrases:
Riemann-Hilbert problem, Fokas-Lenells equation, Initial value problem, Deift-
Zhou method
## 1\. Introduction
The Fokas-Lenells equation (FL equation shortly) is a completely integrable
nonlinear partial differential equation which has been derived as an
integrable generalization of the nonlinear Schrödinger equation (NLS equation)
using bi-Hamiltonian methods [1]. In the context of nonlinear optics, the FL
equation models the propagation of nonlinear light pulses in monomode optical
fibers when certain higher-order nonlinear effects are taken into account [2].
The FL equation is related to the NLS equation in the same way as the Camassa-
Holm equation associated with the KdV equation. The soliton solutions of the
FL equation have been constructed via the Riemann-Hilbert method in [4]. And
The initial-boundary value problem for the FL equation on the half-line was
studied in [5]. A simple N-bright-soliton solution was given by Lenells [3]
and the N-dark soliton solution was obtained by means of Bäcklund
transformation [6]. And Matsuno get the bright and dark soliton solutions for
the FL equation in [7] and [8] by a direct method.
In this paper, we use the Riemann-Hilbert problem showed in [4] to get the
long-time asymptotics behavior of the solution of the FL equation (2.9) by the
nonlinear steepest descent method or Deift-Zhou method. The nonlinear steepest
descent method is introduced by Deift and Zhou in [9] in 1993, the history of
the long-time asymptotics problem also canbe found in [9], and it is the first
time to obtain the long-time asymptotics behavior of the solution rigorously,
for the MKdV equation. Then it becomes a most power tool for the long-time
asymptotics of the nonlinear evolution equations in complete integrable
system, for example, the non-focusing NLS equation [10], the Sine-Gordon
equation [15], the KdV equation [19], the Cammasa-Holm equation [20], and so
on. Deift and his collaborators extend this method to analyse the small-
dispersion problem for the KdV equation and the semiclassical problem of the
focusing NLS equation. And this is also a very usefull tool in the asymptotics
problem in orthogonal polynomials and large $n$ limit problem in random matrix
theory.
For several soliton-bearing equations, for example, KdV, Landau-Lifshitz, and
NLS, and the reduced Maxwell-Bloch system, it is well known that the dominant
$O(1)$ asymptotic $t\rightarrow\infty$ effect of the continuous spectra on the
multisoliton solutions is a shift in phase and position of their constituent
solitons [12]. The purpose of our studies is to derive an explicit functional
form for the next-to-leading-order $O(t^{-\frac{1}{2}})$ term of the effect of
this interaction for the Fokas-Lenells equation. An asymptotic investigation
of the solution can be divided into two stages: (i) the investigation of the
continuum (solitonless) component of the solution [13]; and (ii) the inclusion
of the soliton component via the application of a dressing procedure [14] to
the continuum background. The purpose of this paper is to carry out,
systematically, stage (i) of the abovementioned asymptotic paradigm (since
this phase of the asymptotic procedure is rather technical and long in itself,
the completed results for stage (ii) are the subject of a forthcoming article
). The results obtained in this paper are formulated as theorems 3.18.
The outline of this paper is as follows: In Section 2 we recall some classic
definition of Riemann-Hilbert problem and then, we write down the Riemann-
Hilbert problem of the Fokas-Lenells equation. In Section 3 we analyse the
leading order asymptotics of the solution of the Fokas-Lenells equation as
$t\rightarrow+\infty$ via the Deift-Zhou method.
## 2\. The Riemann-Hilbert problem for the Fokas-Lenells equation
### 2.1. What a Riemann-Hilbert problem is
In this subsection, we first explain what a Riemann-Hilbert problem is
###### Definition 2.1.
Let the contour $\Gamma$ be the union of a finite number of smooth and
oriented curves (orientation means that each arc of $\Gamma$ has a positive
side and a negative side: the positive (respectively, negative) side lies to
the left (respectively, right) as one traverses the contour in the direction
of the arrow) on the Riemann sphere $\bar{\mathbb{C}}$ (i.e. the complex plane
with the point at infinity) such that $\bar{\mathbb{C}}\backslash\Gamma$ has
only a finite number of connected components. Let $V(k)$ be an $2\times 2$
matrix defined on the contour $\Gamma$. The Riemann-Hilbert problem
$(\Gamma,V)$ is the problem of finding an $2\times 2$ matrix-valued function
$M(k)$ that satisfies
1. (i)
$M(k)$ is analytic for $k\in\bar{\mathbb{C}}\backslash\Gamma$ and extends
continuously to the contour $\Gamma$.
2. (ii)
$M_{+}(k)=M_{-}(k)V(k),\quad k\in\Gamma$.
3. (iii)
$M(k)\rightarrow\mathbb{I},\quad as\quad k\rightarrow\infty.$
The Riemann-Hilbert problem can be solved as follows (see, [11]). Assume that
$V(k)$ admits some factorization
$V(k)=b_{-}^{-1}(k)b_{+}(k),$ (2.1)
where
$b_{+}(k)=\omega_{+}(k)-\mathbb{I},\quad b_{-}(k)=\mathbb{I}-\omega_{-}(k).$
(2.2)
And define
$\omega(k)=\omega_{+}(k)+\omega_{-}(k).$ (2.3)
Let
$(C_{\pm}f)(k)=\int_{\Gamma}\frac{f(\xi)}{\xi-k_{\pm}}\frac{d\xi}{2\pi
i},\quad k\in\Gamma,f\in L^{2}(\Gamma),$ (2.4)
denote the Cauchy operator on $\Gamma$. As is well known, the operator
$C_{\pm}$ are bounded from $L^{2}(\Gamma)$ to $L^{2}(\Gamma)$, and
$C_{+}-C_{-}=\@slowromancap i@$, here $\@slowromancap i@$ denote the identify
operator.
Define
$C_{\omega}f=C_{+}(f\omega_{-})+C_{-}(f\omega_{+})$ (2.5)
for $2\times 2$ matrix-valued functions $f$. Let $\mu$ be the solution of the
basic inverse equation
$\mu=\mathbb{I}+C_{\omega}\mu.$ (2.6)
Then
$M(k)=\mathbb{I}+\int_{\Gamma}\frac{\mu(\xi)\omega(\xi)}{\xi-k}\frac{d\xi}{2\pi
i},\quad k\in\bar{\mathbb{C}}\backslash\Gamma,$ (2.7)
is the solution of the Riemann-Hilbert problem. (See [9],P.322).
### 2.2. Riemann-Hilbert problem for FL equation
The Fokas-Lenells equation is
$iu_{t}-\nu u_{tx}+\gamma u_{xx}+\sigma|u|^{2}(u+i\nu u_{x})=0,\quad\sigma=\pm
1.$ (2.8)
where $\nu$ and $\gamma$ are constants.
If we replaced $u(x,t)$ by $u(-x,t)$, we can see the sign of $\nu$ is the same
as the $\gamma$’s. Hence, we can assume that $\alpha=\frac{\gamma}{\nu}>0$ and
$\beta=\frac{1}{\nu}$. Then we change the variable as follows:
$u\rightarrow\beta\sqrt{\alpha}e^{i\beta x}u,\qquad\sigma\rightarrow-\sigma$
the equation (2.8) can be changed into the desired form:
$u_{tx}+\alpha\beta^{2}u-2i\alpha\beta u_{x}-\alpha u_{xx}+\sigma
i\alpha\beta^{2}|u|^{2}u_{x}=0,\quad\sigma=\pm 1.$ (2.9)
This equation admits Lax pair
$\left\\{\begin{array}[]{l}\Phi_{x}+ik^{2}\sigma_{3}\Phi=kU_{x}\Phi\\\
\Phi_{t}+i\eta^{2}\sigma_{3}\Phi=[\alpha
kU_{x}+\frac{i\alpha\beta^{2}}{2}\sigma_{3}(\frac{1}{k}U-U^{2})]\Phi.\end{array}\right.$
(2.10)
where $U=\left(\begin{array}[]{cc}0&u\\\ v&0\end{array}\right)$,
$\eta=\sqrt{\alpha}(k-\frac{\beta}{2k})$ with $v=\sigma\bar{u}$. And in the
following of the paper we just consider $\sigma=1$.
According to the paper [4], we can get the Riemann-Hilbert problem of the
Fokas-Lenells equation (2.9) as follows:
$\left\\{\begin{array}[]{l}M_{+}(x,t,k)=M_{-}(x,t,k)J(x,t,k),\quad
k\in{\mathbb{R}}\cup i{\mathbb{R}},\\\ M(x,t,k)\rightarrow\mathbb{I},\qquad
k\rightarrow\infty.\end{array}\right.$ (2.11)
Figure 1. The jump contour in the complex $k-$plane.
where the function $M(x,t,k)$ is defined by (4.24) in [4] and the jump matrix
$J(x,t,k)$ is defined by
$J(x,t,k)=e^{(-ik^{2}x-i\eta^{2}t)\hat{\sigma}_{3}}\left(\begin{array}[]{cc}\frac{1}{a(k)\overline{a(\bar{k})}}&\frac{b(k)}{\overline{a(\bar{k})}}\\\
-\frac{\overline{b(\bar{k})}}{a(k)}&1\end{array}\right)$ (2.12)
with $\eta=\sqrt{\alpha}(k-\frac{\beta}{2k})$ and $a(k),b(k)$ are defined by
(4.26) in [4]. And $e^{\hat{\sigma}_{3}}A=e^{\sigma_{3}}Ae^{-\sigma_{3}}$,
here $A$ is a $2\times 2$ matrix. We can also know that
$a(k)\overline{a(\bar{k})}-b(k)\overline{b(\bar{k})}=1$ and
$a(k)=\overline{a(\bar{k})}$, $b(k)=-\overline{b(\bar{k})}$ from [4].
We introduce $r(k)=\frac{\overline{b(\bar{k})}}{a(k)}$, then the jump matrix
$J(x,t,k)$ can be transformed into the following form:
$J(x,t,k)=e^{(-ik^{2}x-i\eta^{2}t)\hat{\sigma}_{3}}\left(\begin{array}[]{cc}1-r(k)\overline{r(\bar{k})}&\overline{r(\bar{k})}\\\
-r(k)&1\end{array}\right)$ (2.13)
The solution of Fokas-Lenells equation (2.9) can be expressed by
$u_{x}(x,t)=2im(x,t)e^{4i\int_{-\infty}^{x}|m|^{2}(x^{\prime},t)dx^{\prime}}$
(2.14)
where
$m(x,t)=\lim_{k\rightarrow\infty}(kM(x,t,k))_{12}$ (2.15)
with $M(x,t,k)$ is the unique solution of the Riemann-Hilbert problem (2.11).
###### Remark 2.1.
In this paper, we consider the case when $a(k)$ has no zeros, that is without
solitons, the unique solvability of the Riemann-Hilbert problem in (2.11) is a
consequence of a vanishing lemma 4.2 in [4].
## 3\. The Long-time asymptotics for the Fokas-Lenells equation
In this section, we get the asymptotics behavior of the solution of the Fokas-
Lenells equation (2.9) as $t\rightarrow\infty$ by the Deift-Zhou method [9].
Let $F(x,t,k)=k^{2}x+\eta^{2}t$ and $\theta(k)=k^{2}\frac{x}{t}+\eta^{2}$,
then $F=t\theta$.
### 3.1. Case 1: $\frac{x}{t}+\alpha<0$
In this case, the real part of $i\theta(k)$ has the signature
$\mathrm{Re}{i\theta(k)}\left\\{\begin{array}[]{l}>0,\quad
if\quad\mathrm{Im}{k^{2}}>0,\\\ <0,\quad
if\quad\mathrm{Im}k^{2}<0.\end{array}\right.$ (3.1)
showed in Figure 2.
Figure 2. The signature table of $\mathrm{Re}(i\theta)$ in the case 1.
The jump matrix $J(x,t,k)$ ,i.e. (2.13), has an factorization
$J(x,t,k)=\left(\begin{array}[]{cc}1&0\\\
\frac{-r(k)}{1-r(k)\overline{r(\bar{k})}}e^{2it\theta(k)}&1\end{array}\right)\left(\begin{array}[]{cc}1-r(k)\overline{r(\bar{k})}&0\\\
0&\frac{1}{1-r(k)\overline{r(\bar{k})}}\end{array}\right)\left(\begin{array}[]{cc}1&\frac{\overline{r(\bar{k})}}{1-r(k)\overline{r(\bar{k})}}e^{-2it\theta(k)}\\\
0&1\end{array}\right),$ (3.2)
We find that the transformation
$\tilde{M}(x,t,k)=M(x,t,k)\tilde{\delta}^{-\sigma_{3}},$ (3.3)
leads to the Riemann-Hilbert problem
$\left\\{\begin{array}[]{l}\tilde{M}_{+}(x,t,k)=\tilde{M}_{-}(x,t,k)\tilde{J}(x,t,k),\qquad\mathrm{Im}k^{2}=0,\\\
\tilde{M}\rightarrow\mathbb{I},\qquad k\rightarrow\infty.\end{array}\right.$
(3.4)
with jump matrix $\tilde{J}(x,t,k)$ that admits the lower/upper factorization
$\tilde{J}(x,t,k)=\left(\begin{array}[]{cc}1&0\\\
\frac{-r(k)}{1-r(k)\overline{r(\bar{k})}}\frac{1}{\tilde{\delta}^{2}(k)_{-}}e^{2it\theta(k)}&1\end{array}\right)\left(\begin{array}[]{cc}1&\frac{\overline{r(\bar{k})}}{1-r(k)\overline{r(\bar{k})}}\tilde{\delta}^{2}(k)_{+}e^{-2it\theta(k)}\\\
0&1\end{array}\right)=\tilde{J}_{1}^{-1}\tilde{J}_{2}$ (3.5)
if the function $\tilde{\delta}(k)$ solves the scalar Riemann-Hilbert problem
$\left\\{\begin{array}[]{ll}\tilde{\delta}_{+}(k)=\tilde{\delta}_{-}(k)(1-r(k)\overline{r(\bar{k})}),&\mathrm{Im}k^{2}=0,\\\
\tilde{\delta}(k)\rightarrow 1,&k\rightarrow\infty.\end{array}\right.$ (3.6)
The solution for the Riemann-Hilbert problem for $\tilde{\delta}$ has the
explicit form
$\tilde{\delta}(k)=e^{\frac{1}{2\pi i}\int_{{\mathbb{R}}\cup
i{\mathbb{R}}}\frac{\log{(1-r(k^{\prime})\overline{r(\bar{k}^{\prime})})}}{k^{\prime}-k}dk^{\prime}}.$
(3.7)
Without loss of generality, we may assume that the left factor of (3.5)
extends analytically to the region $\mathrm{Im}k^{2}<0$ and continuous in the
closure of the region. Then the right factor extends the region
$\mathrm{Im}k^{2}>0$.
Our Riemann-Hilbert problem on ${\mathbb{R}}\cup i{\mathbb{R}}$ is equivalent
to a new Riemann-Hilbert problem on the contour
$\tilde{\Sigma}=e^{i\frac{\pi}{6}}{\mathbb{R}}\cup
e^{-i\frac{\pi}{6}}{\mathbb{R}}\cup e^{i\frac{\pi}{3}}{\mathbb{R}}\cup
e^{-i\frac{\pi}{3}}{\mathbb{R}},$ (3.8)
where the orientation of the contour $\tilde{\Sigma}$ and the new function
$\hat{M}(x,t,k)$ are given in the following
$\hat{M}(x,t,k)=\left\\{\begin{array}[]{ll}\tilde{M}(x,t,k),&k\in\hat{D}_{2}\cup\hat{D}_{5}\cup\hat{D}_{8}\cup\hat{D}_{11},\\\
\tilde{M}(x,t,k)\tilde{J}_{2}^{-1},&k\in\hat{D}_{1}\cup\hat{D}_{3}\cup\hat{D}_{7}\cup\hat{D}_{9},\\\
\tilde{M}(x,t,k)\tilde{J}_{1}^{-1},&k\in\hat{D}_{4}\cup\hat{D}_{6}\cup\hat{D}_{10}\cup\hat{D}_{12}.\end{array}\right.$
(3.9)
where the domains $\\{\hat{D}_{j}\\}_{1}^{12}$ are showed in Figure 3.
Figure 3. The contour $\tilde{\Sigma}$ and regions in case 1.
Then one can verify
$\left\\{\begin{array}[]{ll}\mbox{$\hat{M}$ is analytic off $\tilde{\Sigma}$
($\hat{M}$ is analytic across ${\mathbb{R}}\cup i{\mathbb{R}}$),}&\\\
\hat{M}_{+}(x,t,k)=\hat{M}_{-}(x,t,k)\hat{J}(x,t,k),&k\in\tilde{\Sigma},\\\
\hat{M}(x,t,k)\rightarrow\mathbb{I},&k\rightarrow\infty.\end{array}\right.$
(3.10)
where
$\hat{J}(x,t,k)=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}1&\frac{\overline{r(\bar{k})}}{1-r(k)\overline{r(\bar{k})}}\tilde{\delta}^{2}(k)_{+}e^{-2it\theta(k)}\\\
0&1\end{array}\right)^{-1},&k\in\tilde{\Sigma}\cap\\{\mathrm{Im}k^{2}>0\\},\\\
\left(\begin{array}[]{cc}1&0\\\
\frac{-r(k)}{1-r(k)\overline{r(\bar{k})}}\frac{1}{\tilde{\delta}^{2}(k)_{-}}e^{2it\theta(k)}&1\end{array}\right),&k\in\tilde{\Sigma}\cap\\{\mathrm{Im}k^{2}<0\\}.\end{array}\right.$
(3.11)
###### Theorem 3.1.
As $t\rightarrow\infty$,
$||\hat{M}_{\pm}(x,t,k)-\mathbb{I}||_{L^{2}(\tilde{\Sigma})}\rightarrow
0,\quad rapidly.$ (3.12)
$u_{x}(x,t)$ and therefore $u$ decay rapidly as $t\rightarrow\infty$.
###### Proof.
Since the Riemann-Hilbert problem for $M$ and the Riemann-Hilbert problem for
$\hat{M}$ are equivalent, the existence of the solution of $M$ implies the
existence of the solution of $\hat{M}$.
We make the trivial factorization
$\hat{J}(x,t,k)=b_{-}^{-1}b_{+},\quad b_{-}=\mathbb{I},b_{+}=\hat{J}.$
and define $\hat{\omega}$ as (2.3). Then as section 2 (also see, [11] or [9])
, we obtain the solution of the Riemann-Hilbert problem for $\hat{M}$,
$\hat{M}(x,t,k)=\mathbb{I}+\int_{\hat{\Sigma}}\frac{\hat{\mu}(x,t,\xi)\hat{\omega}(x,t,\xi)}{\xi-k}\frac{d\xi}{2\pi
i},\quad k\in{\mathbb{C}}\backslash\hat{\Sigma}.$ (3.13)
where $\hat{\mu}$ is the solution of the singular integral equation
$\hat{\mu}=\mathbb{I}+C_{\hat{\omega}}\hat{\mu}$, where $C_{\hat{\omega}}$
defined as (2.5) with $\omega$ replaced by $\hat{\omega}$. Since
$||\hat{J}(x,t,k)-\mathbb{I}||_{L^{2}(\hat{\Sigma})\cap
L^{\infty}(\hat{\Sigma})}\rightarrow 0,\quad exponentially,\quad as\quad
t\rightarrow\infty,$
by (3.13),
$||\hat{M}-\mathbb{I}||_{L^{2}(\hat{\Sigma})}\rightarrow 0,\quad rapidly,\quad
as\quad t\rightarrow\infty.$
Then, by (2.14) we get $u_{x}$ decays rapidly , and then $u$ decays rapidly,
as $t\rightarrow\infty$. ∎
### 3.2. Case 2: $\frac{x}{t}+\alpha>0$
In this case, the real part of $i\theta(k)$ has the signature as the Figure 3.
And we set
$k_{0}=(\frac{\alpha\beta^{2}}{4(\frac{x}{t}+\alpha)})^{\frac{1}{4}}$.
Figure 4. The signature table of $\mathrm{Re}(i\theta)$ in the case 2.
The jump matrix $J(x,t,k)$ has the following factorization
$J(x,t,k)=\left\\{\begin{array}[]{l}\left(\begin{array}[]{cc}1&\overline{r(\bar{k})}e^{-2it\theta(k)}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
-r(k)e^{2it\theta(k)}&1\end{array}\right),\\\ \left(\begin{array}[]{cc}1&0\\\
\frac{-r(k)}{1-r(k)\overline{r(\bar{k})}}e^{2it\theta(k)}&1\end{array}\right)\left(\begin{array}[]{cc}1-r(k)\overline{r(\bar{k})}&0\\\
0&\frac{1}{1-r(k)\overline{r(\bar{k})}}\end{array}\right)\left(\begin{array}[]{cc}1&\frac{\overline{r(\bar{k})}}{1-r(k)\overline{r(\bar{k})}}e^{-2it\theta(k)}\\\
0&1\end{array}\right).\end{array}\right.$ (3.14)
#### 3.2.1. The conjugate transform
Introducing a scalar function $\delta(k)$ which solves the Riemann-Hilbert
problem
$\left\\{\begin{array}[]{rll}\delta(k)_{+}&=\delta(k)_{-}(1-r(k)\overline{r(\bar{k})})&k\in\Sigma=(-k_{0},k_{0})\cup
i(-k_{0},k_{0}),\\\
&=\delta(k)_{-}=\delta(k)&k\in\\{\mathrm{Im}k^{2}=0\\}\backslash\Sigma.\\\
&\delta(k)\rightarrow 1&k\rightarrow\infty.\end{array}\right.$ (3.15)
The solution of this Riemann-Hilbert problem is given by
$\delta(k)=\left((\frac{k-k_{0}}{k})(\frac{k+k_{0}}{k})\right)^{i\vartheta}e^{\chi_{+}(k)}e^{\chi_{-}(k)}\left((\frac{k}{k-ik_{0}})(\frac{k}{k+ik_{0}})\right)^{i\tilde{\vartheta}}e^{\tilde{\chi}_{+}(k)}e^{\tilde{\chi}_{-}(k)},$
(3.16)
where
$\vartheta=-\frac{1}{2\pi}\ln{(1-|r(k_{0})|^{2})},$ (3.17a)
$\tilde{\vartheta}=-\frac{1}{2\pi}\ln{(1+|r(ik_{0})|^{2})},$ (3.17b)
$\chi_{\pm}(k)=\frac{1}{2\pi i}\int_{0}^{\pm
k_{0}}\ln{\left(\frac{1-|r(k^{\prime})|^{2}}{1-|r(k_{0})|^{2}}\right)}\frac{dk^{\prime}}{k^{\prime}-k},$
(3.17c) $\tilde{\chi}_{\pm}(k)=\frac{1}{2\pi i}\int_{\pm
ik_{0}}^{i0}\ln{\left(\frac{1-r(k^{\prime})\overline{r(\bar{k}^{\prime})}}{1+|r(ik_{0})|^{2}}\right)}\frac{dk^{\prime}}{k^{\prime}-k}.$
(3.17d)
Moreover, for all $k\in{\mathbb{C}}$, $|\delta|$ and $|\delta^{-1}|$ are
bounded.
The conjugate transform is that
$M^{(1)}(x,t,k)=M(x,t,k)\delta(k)^{-\sigma_{3}}.$ (3.18)
Figure 5. The jump contour $\Sigma^{(1)}$ for $M^{(1)}(x,t,k)$.
Then we can get the Riemann-Hilbert problem of $M^{(1)}(x,t,k)$
$\left\\{\begin{array}[]{ll}M^{(1)}(x,t,k)_{+}=M^{(1)}(x,t,k)_{-}J^{(1)}(x,t,k),&k\in{\mathbb{R}}\cup
i{\mathbb{R}}\\\
M^{(1)}(x,t,k)\rightarrow\mathbb{I},&k\rightarrow\infty.\end{array}\right.$
(3.19)
where
$J^{(1)}(x,t,k)=\left\\{\begin{array}[]{l}\left(\begin{array}[]{cc}1&0\\\
\frac{-r(k)}{1-r(k)\overline{r(\bar{k})}}\frac{1}{\delta^{2}(k)_{-}}e^{2it\theta(k)}&1\end{array}\right)\left(\begin{array}[]{cc}1&\frac{\overline{r(\bar{k})}}{1-r(k)\overline{r(\bar{k})}}\delta^{2}(k)_{+}e^{-2it\theta(k)}\\\
0&1\end{array}\right),k\in\Sigma^{(1)}=\Sigma,\\\
\left(\begin{array}[]{cc}1&\overline{r(\bar{k})}\delta^{2}(k)e^{-2it\theta(k)}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
-r(k)\frac{1}{\delta^{2}(k)}e^{2it\theta(k)}&1\end{array}\right),\quad
k\in\\{\mathrm{Im}{k^{2}=0}\\}\backslash\Sigma^{(1)}.\end{array}\right.$
(3.20)
Figure 6. The jump contour $\tilde{\Sigma}^{(1)}$ for
$\tilde{M}^{(1)}(x,t,k)$.
Then we reverse the direction of the part of
$\\{\mathrm{Im}{k^{2}=0}\\}\backslash\Sigma^{(1)}$, we have
$\left\\{\begin{array}[]{ll}M^{(1)}(x,t,k)_{+}=M^{(1)}(x,t,k)_{-}\tilde{J}^{(1)}(x,t,k),&k\in{\mathbb{R}}\cup
i{\mathbb{R}}\\\
M^{(1)}(x,t,k)\rightarrow\mathbb{I},&k\rightarrow\infty.\end{array}\right.$
(3.21)
$\tilde{J}^{(1)}(x,t,k)=\left\\{\begin{array}[]{l}\left(\begin{array}[]{cc}1&0\\\
\frac{-r(k)}{1-r(k)\overline{r(\bar{k})}}\frac{1}{\delta^{2}(k)_{-}}e^{2it\theta(k)}&1\end{array}\right)\left(\begin{array}[]{cc}1&\frac{\overline{r(\bar{k})}}{1-r(k)\overline{r(\bar{k})}}\delta^{2}(k)_{+}e^{-2it\theta(k)}\\\
0&1\end{array}\right),k\in\tilde{\Sigma}^{(1)}=\Sigma\\\
\left(\begin{array}[]{cc}1&0\\\
r(k)\frac{1}{\delta^{2}(k)}e^{2it\theta(k)}&1\end{array}\right)\left(\begin{array}[]{cc}1&-\overline{r(\bar{k})}\delta^{2}(k)e^{-2it\theta(k)}\\\
0&1\end{array}\right),\quad
k\in\\{\mathrm{Im}{k^{2}=0}\\}\backslash\tilde{\Sigma}^{(1)}.\end{array}\right.$
(3.22)
#### 3.2.2. The second transform
The main purpose of this section is to reformulate the original Riemann-
Hilbert problem (3.21) as an equivalent Riemann-Hilbert problem on the
augmented contour $\Sigma^{(2)}$ (see Figure 7),
$\Sigma^{(2)}=L\cup L_{0}\cup\bar{L}\cup\bar{L}_{0}\cup{\mathbb{R}}\cup
i{\mathbb{R}}.$ (3.23)
where $L=L_{1}\cup\tilde{L}_{1}\cup L_{2}\cup\tilde{L}_{2}$,
Denote the contour
$\begin{array}[]{ll}L_{1}=\\{k=k_{0}+uk_{0}e^{i\frac{3\pi}{4}},\quad
u\in(-\infty,\frac{1}{\sqrt{2}}]\\},&\tilde{L}_{1}=\\{k=ik_{0}+uk_{0}e^{-i\frac{\pi}{4}},\quad
u\in(-\infty,\frac{1}{\sqrt{2}}]\\}\\\
L_{2}=\\{k=-k_{0}+uk_{0}e^{-i\frac{\pi}{4}},\quad
u\in(-\infty,\frac{1}{\sqrt{2}}]\\},&\tilde{L}_{2}=\\{k=ik_{0}+uk_{0}e^{i\frac{\pi}{4}},\quad
u\in(-\infty,\frac{1}{\sqrt{2}}]\\}\end{array}$ (3.24)
Denote the contour
$L_{0}=\\{uk_{0}e^{i\frac{\pi}{4}},\quad
u\in[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}]\\}.$ (3.25)
Denote the contour
$\begin{array}[]{rrl}L_{\varepsilon}&=&L_{1\varepsilon}\cup\tilde{L}_{1\varepsilon}\cup
L_{2\varepsilon}\cup\tilde{L}_{2\varepsilon}\\\
&=&\\{k=k_{0}+uk_{0}e^{i\frac{3\pi}{4}},\quad\varepsilon<u\leq\frac{1}{\sqrt{2}}\\}\cup\\{k=ik_{0}+uk_{0}e^{-i\frac{\pi}{4}},\quad
u\in(\varepsilon,\frac{1}{\sqrt{2}}]\\}\\\
&&\cup\\{k=-k_{0}+uk_{0}e^{-i\frac{\pi}{4}},\quad
u\in(\varepsilon,\frac{1}{\sqrt{2}}]\\}\cup\\{k=ik_{0}+uk_{0}e^{i\frac{\pi}{4}},\quad
u\in(\varepsilon,\frac{1}{\sqrt{2}}]\\}\end{array}$ (3.26)
Following the method in [9], we can have
###### Proposition 3.2.
Let
$\rho(k)=\left\\{\begin{array}[]{ll}\rho_{1}(k)=\frac{\overline{r(\bar{k})}}{1-r(k)\overline{r(\bar{k})}},&k\in\Sigma\\\
\rho_{2}(k)=-\overline{r(\bar{k})},&k\in\\{\mathrm{Im}{k^{2}=0}\\}\backslash\Sigma.\end{array}\right.$
(3.27)
Then $\rho$ has a decomposition
$\rho(k)=h_{\@slowromancap i@}(k)+(h_{\@slowromancap ii@}(k)+R(k)),$ (3.28)
where $h_{\@slowromancap i@}(k)$ is small and $h_{\@slowromancap ii@}(k)$ has
an analytic continuation to $L$ and $L_{0}$. For example, if $\rho(k)=r(k)$ as
$k>k_{0}$, $h_{\@slowromancap ii@}(k)$ of this function $\rho(k)$ has an
analytic continuation to the first quadrant. And $R(k)$ is piecewise rational
($R(k)=0$, if $k\in L_{0}$) function.
And $R,h_{\@slowromancap i@},h_{\@slowromancap ii@}$ satisfy
$|e^{-2it\theta(k)}h_{\@slowromancap
i@}(k)|\leq\frac{c}{(1+|k|^{2})t^{l}},for\quad z\in{\mathbb{R}}\cup
i{\mathbb{R}},$ (3.29a)
$|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|\leq\frac{c}{(1+|k|^{2})t^{l}},\quad k\in L,\quad k_{0}<M.$ (3.29b)
$|e^{-2it\theta(k)}h_{\@slowromancap ii@}(k)|\leq
ce^{-t\frac{\alpha\beta^{2}}{4k_{0}^{2}}},\quad k\in L_{0},\quad k_{0}<M.$
(3.29c)
and
$|e^{-2it\theta(k)}R(k)|\leq
ce^{-\frac{\varepsilon^{2}\alpha\beta^{2}}{M^{2}}t},\quad k\in
L_{\varepsilon}.$ (3.29d)
for arbitrary natural number $l$, for sufficiently large constants $c$, for
some fixed positive constant $M$.
###### Proof.
See appendix. ∎
###### Remark 3.3.
Taking conjugate $\overline{\rho(k)}=\overline{h_{\@slowromancap
i@}(k)}+\overline{h_{\@slowromancap ii@}(\bar{k})}+\overline{R(\bar{k})}$
leads to the same estimates for $e^{2it\theta(k)}\overline{h_{\@slowromancap
i@}(k)},e^{2it\theta(k)}\overline{h_{\@slowromancap ii@}(\bar{k})}$ and
$e^{2it\theta(k)}\overline{R(\bar{k})}$ on ${\mathbb{R}}\cup
i{\mathbb{R}}\cup\bar{L}\cup\bar{L}_{0}$.
From the Riemann-Hilbert problem (3.21) and formula (3.22), the Riemann-
Hilbert problem across ${\mathbb{R}}\cup i{\mathbb{R}}$ oriented as Figure 6
is given by
$\left\\{\begin{array}[]{ll}M^{(1)}(x,t,k)_{+}=M^{(1)}(x,t,k)_{-}(b_{-})^{-1}b_{+},&k\in{\mathbb{R}}\cup
i{\mathbb{R}}\\\
M^{(1)}(x,t,k)\rightarrow\mathbb{I},&k\rightarrow\infty.\end{array}\right.$
(3.30)
where
$b_{+}=\mathbb{I}+\omega_{+}=\delta_{+}^{\hat{\sigma}_{3}}e^{-it\theta(k)\hat{\sigma}_{3}}\left(\begin{array}[]{cc}1&\rho(k)\\\
0&1\end{array}\right),$ (3.31)
$b_{-}=\mathbb{I}-\omega_{-}=\delta_{-}^{\hat{\sigma}_{3}}e^{-it\theta(k)\hat{\sigma}_{3}}\left(\begin{array}[]{cc}1&0\\\
\overline{\rho(\bar{k})}&1\end{array}\right),$ (3.32)
and $\rho$ is given by (3.27).
We write
$b_{+}=b^{o}_{+}b^{a}_{+}=(\mathbb{I}+\omega^{o}_{+})(\mathbb{I}+\omega^{a}_{+})=\left(\begin{array}[]{cc}1&h_{\@slowromancap
i@}(k)\delta_{+}^{2}e^{-2it\theta}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}1&(h_{\@slowromancap
ii@}(k)+R(k))\delta_{+}^{2}e^{-2it\theta}\\\ 0&1\end{array}\right),$ (3.33a)
$b_{-}=b^{o}_{-}b^{a}_{-}=(\mathbb{I}-\omega^{o}_{-})(\mathbb{I}-\omega^{a}_{-})=\left(\begin{array}[]{cc}1&0\\\
\overline{h_{\@slowromancap
i@}(\bar{k})}\frac{1}{\delta_{-}^{2}}e^{2it\theta}&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
(\overline{h_{\@slowromancap
ii@}(\bar{k})+R(\bar{k})})\frac{1}{\delta_{-}^{2}}e^{2it\theta}&1\end{array}\right),$
(3.33b)
Now we can use the signature table of $\mathrm{Re}{i\theta}$ showed in Figure
3 to open the jump contour for the Riemann-Hilbert problem of $M^{(1)}$ to the
contours in Figure 7.
Figure 7. The different regions $\\{D_{j}\\}_{1}^{20}$ of the complex
$k-$plane.
Introducing $M^{(2)}(x,t,k)=M^{(1)}(x,t,k)\phi$, where $\phi$ is defined as
follows:
$\phi=\left\\{\begin{array}[]{ll}\mathbb{I},&k\in D_{2},D_{5},D_{16},D_{19}\\\
(b^{a}_{-})^{-1},&k\in D_{1},D_{3},D_{15},D_{17},D_{9},D_{10},D_{11},D_{12}\\\
(b^{a}_{+})^{-1},&k\in
D_{4},D_{6},D_{18},D_{20},D_{7},D_{8},D_{13},D_{14}\end{array}\right.$ (3.34)
where the regions $\\{D_{j}\\}_{1}^{20}$ are showed in Figure 7.
Then the Riemann-Hilbert problem of $M^{(2)}(x,t,k)$ is defined
$M^{(2)}_{+}(x,t,k)=M_{-}^{(2)}(x,t,k)J^{(2)}(x,t,k)$ (3.35)
with
$J^{(2)}(x,t,k)=\left\\{\begin{array}[]{ll}(b^{o}_{-})^{-1}(b^{o}_{+}),&k\in
R\cup i{\mathbb{R}}\\\ \mathbb{I}^{-1}(b^{a}_{+}),&k\in L\cup{\bf L_{0}}\\\
(b^{a}_{-})^{-1}\mathbb{I},&k\in\bar{L}\cup{\bf\bar{L}_{0}}\end{array}\right.$
(3.36)
Using the symbol $J^{(2)}(x,t,k)=b^{-1}_{-}(x,t,k)b_{+}(x,t,k)$, and set
$\omega_{\pm}(x,t,k)=\pm(b_{\pm}(x,t,k)-\mathbb{I})$ ,
$\omega(x,t,k)=\omega_{+}(x,t,k)+\omega_{-}(x,t,k)$. From section 2, we have
$M^{(2)}(x,t,k)=\mathbb{I}+\int_{\Sigma^{(2)}}\frac{\mu(x,t,\xi)\omega(x,t,\xi)}{\xi-k}\frac{d\xi}{2\pi
i},\quad k\in{\mathbb{C}}\backslash\Sigma^{(2)}.$ (3.37)
And substituting (3.37) into (2.15), we learn that
$\begin{array}[]{rl}m(x,t)=&\frac{1}{2}\lim_{k\rightarrow\infty}(k[\sigma_{3},M^{(2)}(x,t,k)])_{12},\\\
=&-\frac{1}{2}([\sigma_{3},\int_{\Sigma^{(2)}}\mu(x,t,\xi)\omega(x,t,\xi)]\frac{d\xi}{2\pi
i})_{12},\\\
=&-\frac{1}{2}([\sigma_{3},\int_{\Sigma^{(2)}}((\mathbb{I}-C_{\omega})^{-1}\mathbb{I})(\xi)\omega(x,t,\xi)]\frac{d\xi}{2\pi
i})_{12}.\end{array}$ (3.38)
#### 3.2.3. Transform to the Riemann-Hilbert problem of $M^{(3)}(x,t,k)$
Follow the method of [9] P.323-329, we can reduce the Riemann-Hilbert problem
of $M^{(2)}(x,t,k)$ to the Riemann-Hilbert problem of $M^{(3)}(x,t,k)$.
Figure 8. The jump contour $\Sigma^{(3)}$ for $M^{(3)}(x,t,k)$.
Let $\omega^{e}$ be a sum of three terms
$\omega^{e}=\omega^{a}+\omega^{b}+\omega^{c}+\omega^{d}.$ (3.39)
We then have the following:
$\begin{array}[]{l}\omega^{a}=\omega\mbox{ is supported on the
${\mathbb{R}}\cup i{\mathbb{R}}$ and consists of terms of type
$h_{\@slowromancap i@}(k)$ and $\overline{h_{\@slowromancap i@}(k)}$}.\\\
\omega^{b}=\omega\mbox{ is supported on the $L\cup\bar{L}$ and consists of
terms of type $h_{\@slowromancap ii@}(k)$ and $\overline{h_{\@slowromancap
ii@}(\bar{k})}$}.\\\ \omega^{c}=\omega\mbox{ is supported on the
$L_{\varepsilon}\cup\bar{L}_{\varepsilon}$ and consists of terms of type
$R(k)$ and $\overline{R(\bar{k})}$}.\\\ \omega^{d}=\omega\mbox{ is supported
on the $L_{0}\cup\bar{L}_{0}$ }.\end{array}$ (3.40)
Set $\omega^{\prime}=\omega-\omega^{e}$. Then, $\omega^{\prime}=0$ on
$\Sigma^{(2)}\backslash\Sigma^{(3)}$. Thus, $\omega^{\prime}$ is supported on
$\Sigma^{(3)}$ with contribution to $\omega$ from rational terms $R$ and
$\bar{R}$.
###### Proposition 3.4.
For $0<k_{0}<M$, we have
$||\omega^{a}||_{L^{1}(R\cup i{\mathbb{R}})\cap L^{2}({\mathbb{R}}\cup
i{\mathbb{R}})\cap L^{\infty}({\mathbb{R}}\cup
i{\mathbb{R}})}\leq\frac{c}{t^{l}},$ (3.41a)
$||\omega^{b}||_{L^{1}(L\cup\bar{L})\cap L^{2}(L\cup\bar{L})\cap
L^{\infty}(L\cup\bar{L})}\leq\frac{c}{t^{l}},$ (3.41b)
$||\omega^{c}||_{L^{1}(L_{\varepsilon}\cup\bar{L}_{\varepsilon})\cap
L^{2}(L_{\varepsilon}\cup\bar{L}_{\varepsilon})\cap
L^{\infty}(L_{\varepsilon}\cup\bar{L}_{\varepsilon})}\leq
ce^{-\frac{\varepsilon^{2}\alpha\beta^{2}}{k_{0}^{2}}t},$ (3.41c)
$||\omega^{d}||_{L^{1}(L_{0}\cup\bar{L}_{0})\cap
L^{2}(L_{0}\cup\bar{L}_{0})\cap L^{\infty}(L_{0}\cup\bar{L}_{0})}\leq
ce^{-\frac{\alpha\beta^{2}}{4k_{0}^{2}}t},$ (3.41d)
Moreover,
$||\omega^{\prime}||_{L^{2}(\Sigma^{(3)})}\leq\frac{c}{t^{\frac{1}{4}}},\qquad||\omega^{\prime}||_{L^{1}(\Sigma^{(3)})}\leq\frac{c}{t^{\frac{1}{2}}}$
(3.42)
###### Proof.
Consequence of proposition 3.2, and analogous calculations as in lemma 2.13 of
[9]. Let us show equation (3.42).
From the appendix, we have
$|R(k)|\leq C(k_{0})(1+|k|^{5})^{-1}$ (3.43)
on the contour
$k=\\{k_{0}+uk_{0}e^{i\frac{3\pi}{4}},-\infty<u\leq\varepsilon\\}$,
$\varepsilon\leq\frac{1}{\sqrt{2}}$.
Since
$\begin{array}[]{rrl}\mathrm{Re}i\theta(k)&=&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{(u^{2}-\sqrt{2}u)^{2}(u^{2}-\sqrt{2}u+2)}{(u^{2}-\sqrt{2}u+1)^{2}}\\\
&\geq&\frac{\alpha\beta^{2}}{4k_{0}^{2}}Ku^{2}\end{array}$ (3.44)
on the contour
$k=\\{k_{0}+uk_{0}e^{i\frac{3\pi}{4}},-\varepsilon<u\leq\varepsilon\\}$, and
$\mathrm{Re}i\theta(k)\geq-\frac{\alpha\beta^{2}}{4k_{0}^{2}}K^{\prime}u$
(3.45)
on the contour
$k=\\{k_{0}+uk_{0}e^{i\frac{3\pi}{4}},-\infty<u\leq-\varepsilon\\}$, where $K$
and $K^{\prime}$ are positive constants.
We have the similar estimates on the other parts of the contour
$\Sigma^{(3)}$.
Moreover,
$\begin{array}[]{rrl}||\omega^{\prime}||^{2}_{L^{2}(\Sigma^{(2)})}&=&||\omega^{\prime}||^{2}_{L^{2}(\Sigma^{(3)})}\\\
&\leq&C_{1}(k_{0})\int_{\Sigma^{(3)}}\left(e^{-u^{2}K_{1}t\frac{\alpha\beta^{2}}{k_{0}^{2}}}+e^{-uK_{2}t\frac{\alpha\beta^{2}}{k_{0}^{2}}}(1+|k|^{5})^{-2}|dk|\right)\\\
&\leq&C_{2}(k_{0})\left(\int_{{\mathbb{R}}}e^{-u^{2}K_{1}t\frac{\alpha\beta^{2}}{k_{0}^{2}}}k_{0}du+\int_{{\mathbb{R}}}e^{-uK_{2}t\frac{\alpha\beta^{2}}{k_{0}^{2}}}k_{0}du\right)\\\
&\leq&C_{3}(k_{0})\left(\frac{k^{2}_{0}}{\alpha\beta^{2}t}\right)^{\frac{1}{2}},\end{array}$
(3.46)
where $K_{1},K_{2}$ are constants. ∎
###### Proposition 3.5.
As $t\rightarrow\infty$ and $0<k_{0}<M$,
$||(1-C_{\omega})^{-1}||_{L^{2}(\Sigma^{(2)})}\leq C$ is equalilent to
$||(1-C_{\omega^{\prime}})^{-1}||_{L^{2}(\Sigma^{(2)})}\leq C$.
###### Proof.
Consequence of the following inequality,
$||C_{\omega}-C_{\omega^{\prime}}||_{L^{2}(\Sigma^{(2)})}\leq
c||\omega^{e}||_{L^{2}(\Sigma^{(2)})}$, the fact that
$||\omega^{e}||_{L^{2}(\Sigma^{(2)})}\leq\frac{c}{t^{l}}$, and the second
resolvent identity. ∎
###### Proposition 3.6.
If $||(1-C_{\omega^{\prime}})^{-1}||_{L^{2}(\Sigma^{(2)})}\leq C$, then for
arbitrary positive integer $l$, as $t\rightarrow\infty$ such that $0<k_{0}<M$,
$m(x,t)=-\frac{1}{2}([\sigma_{3},\int_{\Sigma^{(2)}}((\mathbb{I}-C_{\omega^{{}^{\prime}}})^{-1}\mathbb{I})(\xi)\omega^{{}^{\prime}}(x,t,\xi)]\frac{d\xi}{2\pi
i})_{12}+O(\frac{c}{t^{l}}).$ (3.47)
###### Proof.
From the second resolvent identity, one can derive the following expression
(see equation (2.27) in [9]),
$\begin{array}[]{rrl}\int_{\Sigma^{(2)}}((1-C_{\omega})^{-1}\mathbb{I})\omega\frac{d\xi}{2\pi
i}&=&\int_{\Sigma^{(2)}}((1-C_{\omega^{\prime}})^{-1}\mathbb{I})\omega^{\prime}\frac{d\xi}{2\pi
i}+\int_{\Sigma^{(2)}}\omega^{e}\frac{d\xi}{2\pi i}\\\
&&+\int_{\Sigma^{(2)}}((1-C_{\omega^{\prime}})^{-1}(C_{\omega^{e}}\mathbb{I}))\omega\frac{d\xi}{2\pi
i}\\\
&&+\int_{\Sigma^{(2)}}((1-C_{\omega^{\prime}})^{-1}(C_{\omega^{\prime}}\mathbb{I}))\omega^{e}\frac{d\xi}{2\pi
i}\\\
&&+\int_{\Sigma^{(2)}}((1-C_{\omega^{\prime}})^{-1}C_{\omega^{e}}(1-C_{\omega})^{-1})(C_{\omega}\mathbb{I})\omega\frac{d\xi}{2\pi
i}\\\
&=&\int_{\Sigma^{(2)}}((1-C_{\omega^{\prime}})^{-1}\mathbb{I})\omega^{\prime}\frac{d\xi}{2\pi
i}+\@slowromancap i@+\@slowromancap ii@+\@slowromancap iii@+\@slowromancap
iv@.\end{array}$ (3.48)
For $0<k_{0}<M$, from Proposition (3.4) it follows that,
$\begin{array}[]{rrl}|\@slowromancap
i@|&\leq&||\omega^{a}||_{L^{1}({\mathbb{R}}\cup
i{\mathbb{R}})}+||\omega^{b}||_{L^{1}(L\cup\bar{L})}+||\omega^{c}||_{L^{1}(L_{\varepsilon}\cup\bar{L}_{\varepsilon})}+||\omega^{d}||_{L^{1}(L_{0}\cup\bar{L}_{0})}\\\
&\leq&ct^{-l},\end{array}$ (3.49) $\begin{array}[]{rrl}|\@slowromancap
ii@|&\leq&||(1-C_{\omega^{\prime}})^{-1}||_{L^{2}(\Sigma^{(2)})}||(C_{\omega^{e}}\mathbb{I})||_{L^{2}(\Sigma^{(2)})}||\omega||_{L^{2}(\Sigma^{(2)})}\\\
&\leq&c||\omega^{e}||_{L^{2}(\Sigma^{(2)})}(||\omega^{e}||_{L^{2}(\Sigma^{(2)})}+||\omega^{\prime}||_{L^{2}(\Sigma^{(2)})})\\\
&\leq&ct^{-l}(ct^{-l}+c)\leq ct^{-l},\end{array}$ (3.50)
$\begin{array}[]{rrl}|\@slowromancap
iii@|&\leq&||(1-C_{\omega^{\prime}})^{-1}||_{L^{2}(\Sigma^{(2)})}||(C_{\omega^{\prime}}\mathbb{I})||_{L^{2}(\Sigma^{(2)})}||\omega^{e}||_{L^{2}(\Sigma^{(2)})}\\\
&\leq&ct^{-l}\end{array}$ (3.51) $\begin{array}[]{lll}|\@slowromancap
iv@|&\leq&||(1-C_{\omega^{\prime}})^{-1}C_{\omega^{e}}(1-C_{\omega})^{-1})(C_{\omega}\mathbb{I})||_{L^{2}(\Sigma^{(2)})}||\omega||_{L^{2}(\Sigma^{(2)})}\\\
&\leq&||(1-C_{\omega^{\prime}})^{-1}||_{L^{2}(\Sigma^{(2)})}||C_{\omega^{e}}||_{L^{2}(\Sigma^{(2)})}||(1-C_{\omega})^{-1}||_{L^{2}(\Sigma^{(2)})}||(C_{\omega}\mathbb{I})||_{L^{2}(\Sigma^{(2)})}||\omega||_{L^{2}(\Sigma^{(2)})}\\\
&\leq&c||C_{\omega^{e}}||_{L^{2}(\Sigma^{(2)})}||(C_{\omega}\mathbb{I})||_{L^{2}(\Sigma^{(2)})}||\omega||_{L^{2}(\Sigma^{(2)})}\\\
&\leq&c||\omega^{e}||_{L^{2}(\Sigma^{(2)})}||\omega||^{2}_{L^{2}(\Sigma^{(2)})}\\\
&\leq&ct^{-l}.\end{array}$ (3.52)
Hence,
$|\@slowromancap i@+\@slowromancap ii@+\@slowromancap iii@+\@slowromancap
iv@|\leq ct^{-l}.$ (3.53)
Applying these estimates to equation (3.38), we can obtain equation (3.47). ∎
Let us now show that, in the sense of appropriately defined operator norms,
one may always choose to delete (or add) a portion of a contour(s) on which
the jump is $\mathbb{I}$, without altering the Riemann-Hilbert problem in the
operator sense.
Suppose that $\Sigma_{1}$ and $\Sigma_{2}$ are two oriented skeletons in
${\mathbb{C}}$ with
$\mbox{card}(\Sigma_{1}\cap\Sigma_{2})<\infty;$ (3.54)
let $u=u(\lambda)=u_{+}(\lambda)+u_{-}(\lambda)$ be a $2\times 2$ matrix-
valued function on
$\Sigma_{12}=\Sigma_{1}\cup\Sigma_{2}$ (3.55)
with entries in $L^{2}(\Sigma_{12})\cap L^{\infty}(\Sigma_{12})$ and suppose
that
$u=0\qquad\qquad\mbox{on }\Sigma_{2}.$ (3.56)
Let
$R_{\Sigma_{1}}\mbox{ denote the restriction map
}L^{2}(\Sigma_{12})\rightarrow L^{2}(\Sigma_{1}),$ (3.57)
$\mathbb{I}_{\Sigma_{1}\rightarrow\Sigma^{(12)}}\mbox{ denote the embedding
}L^{2}(\Sigma_{1})\rightarrow L^{2}(\Sigma_{12}),$ (3.58)
$C_{u}^{12}:L^{2}(\Sigma_{12})\rightarrow L^{2}(\Sigma_{12})\mbox{ denote the
operator in (\ref{BCRHPCom}) with }u\leftrightarrow\omega,$ (3.59)
$C_{u}^{1}:L^{2}(\Sigma_{1})\rightarrow L^{2}(\Sigma_{1})\mbox{ denote the
operator in (\ref{BCRHPCom}) with }u\uparrow\Sigma_{1}\leftrightarrow\omega,$
(3.60) $C_{u}^{E}:L^{2}(\Sigma_{1})\rightarrow L^{2}(\Sigma_{12})\mbox{ denote
the restriction of $C_{u}^{12}$ to }L^{2}(\Sigma_{1}).$ (3.61)
And, finally, let
$\left\\{\begin{array}[]{l}\mathbb{I}_{\Sigma_{1}}\mbox{ and
}\mathbb{I}_{\Sigma_{12}}\mbox{ denote the identity operators on}\\\
L^{2}(\Sigma_{1})\mbox{ and }L^{2}(\Sigma_{12}),\mbox{
respectively}.\end{array}\right.$ (3.62)
We then have the next lemma:
###### Lemma 3.7.
$C_{u}^{12}C_{u}^{E}=C_{u}^{E}C_{u}^{12},$ (3.63)
$(\mathbb{I}_{\Sigma_{1}}-C_{u}^{1})^{-1}=R_{\Sigma_{1}}(\mathbb{I}_{\Sigma_{12}}-C_{u}^{12})^{-1}\mathbb{I}_{\Sigma_{1}\rightarrow\Sigma_{12}},$
(3.64)
$(\mathbb{I}_{\Sigma_{12}}-C_{u}^{12})^{-1}=\mathbb{I}_{\Sigma_{12}}+C_{u}^{E}(\mathbb{I}_{\Sigma_{1}}-C_{u}^{1})^{-1}R_{\Sigma_{1}},$
(3.65)
in the sense that if the right-hand side of (3.64),resp. (3.65), exists, then
the left-hand side exists and identity (3.64),resp. (3.65), holds true.
###### Proof.
See Lemma 2.56 in [9]. ∎
We apply this lemma to the case $u=\omega^{\prime}$,
$\Sigma_{12}=\Sigma^{(2)}$ and $\Sigma_{1}=\Sigma^{(3)}$. From identity
(3.64), we get the following proposition, which is the main result of this
subsection.
###### Proposition 3.8.
$m(x,t)=-\frac{1}{2}([\sigma_{3},\int_{\Sigma^{(3)}}(\mathbb{I}-C_{\omega^{\prime}})^{-1}(\xi)\omega^{\prime}(x,t,\xi)]\frac{d\xi}{2\pi
i})_{12}.$ (3.66)
Set
$L^{\prime}=L\backslash L_{\varepsilon}$
. Then, $\Sigma^{(3)}=L^{\prime}\cup\bar{L}^{\prime}$. On $\Sigma^{(3)}$, set
$\mu^{{}^{\prime}}=(1^{\Sigma^{(3)}}-C^{\Sigma^{(3)}}_{\omega^{\prime}})^{-1}\mathbb{I}$.
Then,
$M^{(3)}(x,t,k)=\mathbb{I}+\int_{\Sigma^{(3)}}\frac{\mu^{{}^{\prime}}(\xi)\omega^{\prime}(\xi)}{\xi-k}\frac{d\xi}{2\pi
i}$ (3.67)
solves the Riemann-Hilbert problem
$\left\\{\begin{array}[]{ll}M^{(3)}_{+}(x,t,k)=M^{(3)}_{-}(x,t,k)J^{(3)}(x,t,k),&k\in\Sigma^{(3)},\\\
M^{(3)}\rightarrow\mathbb{I},&k\rightarrow\infty.\end{array}\right.$ (3.68)
where
$\displaystyle\omega^{\prime}=\omega^{\prime}_{+}+\omega^{\prime}_{-},$ (3.69)
$\displaystyle b^{\prime}_{\pm}=\mathbb{I}\pm\omega^{\prime}_{\pm},$ (3.70)
$\displaystyle J^{(3)}(x,t,k)=(b^{\prime}_{-})^{-1}b^{\prime}_{+}$ (3.71)
#### 3.2.4. The Scaling operators
In this subsection, we make a further simplification of the Riemann-Hilbert
problem on the truncated contour $\Sigma^{(3)}$ by reducing it to the one
which is stated on the four disjoint crosses,
$\Sigma_{A^{\prime}},\Sigma_{B^{\prime}},\Sigma_{C^{\prime}}$ and
$\Sigma_{D^{\prime}}$, and prove that the leading term of the asymptotic
expansion for $m(x,t)$ (proposition 3.8, (3.66)) can be written as the sum of
four terms corresponding to the solutions of four auxiliary Riemann-Hilbert
problems, each of which is set on one of the crosses; moreover, the solution
of the latter Riemann-Hilbert problem can be presented in terms of an exactly
solvable model matrix Riemann-Hilbert problem, which is studied in the next
subsection.
Let us prepare the notations which are needed for exact formulations. Write
$\Sigma^{(3)}$ as the disjoint union of the four crosses,
$\Sigma_{A^{\prime}},\Sigma_{B^{\prime}},\Sigma_{C^{\prime}}$ and
$\Sigma_{D^{\prime}}$, extend the contours
$\Sigma_{A^{\prime}},\Sigma_{B^{\prime}},\Sigma_{C^{\prime}}$ and
$\Sigma_{D^{\prime}}$ (with orientations unchanged) to the following ones,
$\begin{array}[]{c}\hat{\Sigma}_{A^{\prime}}=\\{k=k_{0}+uk_{0}e^{\pm\frac{3i\pi}{4}},u\in{\mathbb{R}}\\},\\\
\hat{\Sigma}_{B^{\prime}}=\\{k=-k_{0}+uk_{0}e^{\pm\frac{i\pi}{4}},u\in{\mathbb{R}}\\},\\\
\hat{\Sigma}_{C^{\prime}}=\\{k=ik_{0}+uk_{0}e^{-\frac{i\pi}{4}},u\in{\mathbb{R}}\\}\cup\\{k=ik_{0}+uk_{0}e^{-\frac{3i\pi}{4}},u\in{\mathbb{R}}\\},\\\
\hat{\Sigma}_{D^{\prime}}=\\{k=-ik_{0}+uk_{0}e^{\frac{i\pi}{4}},u\in{\mathbb{R}}\\}\cup\\{k=-ik_{0}+uk_{0}e^{\frac{3i\pi}{4}},u\in{\mathbb{R}}\\}.\end{array}$
and define by $\Sigma_{A},\Sigma_{B},\Sigma_{C}$ and $\Sigma_{D}$,
respectively, the contours
$\\{k=uk_{0}e^{\pm\frac{i\pi}{4}},u\in{\mathbb{R}}\\}$ oriented inward as in
$\Sigma_{A^{\prime}}$ and $\hat{\Sigma}_{A^{\prime}}$, inward as in
$\Sigma_{B^{\prime}}$ and $\hat{\Sigma}_{B^{\prime}}$, outward as in
$\Sigma_{C^{\prime}}$ and $\hat{\Sigma}_{C^{\prime}}$, and outward as in
$\Sigma_{D^{\prime}}$ and $\hat{\Sigma}_{D^{\prime}}$, respectively.
We introduce the scaling operators:
$N_{A}:f(k)\rightarrow(N_{A}f)(k)=f(\frac{k_{0}^{2}}{2\sqrt{\alpha}\beta\sqrt{t}}k+k_{0})$
(3.72a)
$N_{B}:f(k)\rightarrow(N_{B}f)(k)=f(\frac{k_{0}^{2}}{2\sqrt{\alpha}\beta\sqrt{t}}k-k_{0})$
(3.72b)
$N_{C}:f(k)\rightarrow(N_{C}f)(k)=f(\frac{-k_{0}^{2}}{2\sqrt{\alpha}\beta\sqrt{t}}k+ik_{0})$
(3.72c)
$N_{D}:f(k)\rightarrow(N_{D}f)(k)=f(\frac{-k_{0}^{2}}{2\sqrt{\alpha}\beta\sqrt{t}}k-ik_{0})$
(3.72d)
Considering the action of the operators $N_{k},k\in\\{A,B,C,D\\}$ on
$\delta(k)e^{-it\theta(k)}$, we find that,
$(N_{A}\delta e^{-it\theta})(k)=\delta_{A}^{0}(k)\delta_{A}^{1}(k)$ (3.73)
where
$\delta_{A}^{0}(k)=\frac{k_{0}^{i\nu-2i\tilde{\nu}}}{(\sqrt{\alpha
t}\beta)^{i\nu}}2^{-i\tilde{\nu}}e^{i\alpha\beta
t-i\frac{\alpha\beta^{2}}{2k_{0}^{2}}t}e^{\chi_{\pm}(k_{0})}e^{\tilde{\chi}_{\pm}^{\prime}(k_{0})}$
(3.74a)
$\begin{array}[]{rl}\delta_{A}^{1}(k)=&k^{i\nu}e^{-i\frac{k^{2}}{4}+i\frac{k^{6}_{0}k^{3}}{\zeta^{5}\sqrt{t}}}\frac{k_{0}^{2i\tilde{\nu}+i\nu}}{2^{i\nu-i\tilde{\nu}}}\frac{(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{i\nu}}{(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{2i\nu}}\\\ &((\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0}+ik_{0})(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0}-ik_{0}))^{-i\tilde{\nu}}\\\
&e^{\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\chi_{\pm}(k_{0})}e^{\tilde{\chi}_{\pm}^{\prime}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\tilde{\chi}_{\pm}^{\prime}(k_{0})}\end{array}$ (3.74b)
with
$\tilde{\chi}_{\pm}^{\prime}(k)=e^{-\frac{1}{2\pi i}\int_{\pm
k_{0}}^{0}\ln|k-ik^{\prime}|d\ln(1+|r(ik^{\prime})|^{2})}$ (3.75)
And
$(N_{B}\delta e^{-it\theta})(k)=\delta_{B}^{0}(k)\delta_{B}^{1}(k)$ (3.76)
where
$\delta_{B}^{0}(k)=\frac{k_{0}^{i\nu-2i\tilde{\nu}}}{(\sqrt{\alpha
t}\beta)^{i\nu}}2^{-i\tilde{\nu}}e^{i\alpha\beta
t-i\frac{\alpha\beta^{2}}{2k_{0}^{2}}t}e^{\chi_{\pm}(-k_{0})}e^{\tilde{\chi}_{\pm}^{\prime}(-k_{0})}$
(3.77a)
$\begin{array}[]{rl}\delta_{B}^{1}(k)=&(-k)^{i\nu}e^{-i\frac{k^{2}}{4}+i\frac{k^{6}_{0}k^{3}}{\zeta^{5}\sqrt{t}}}\frac{(-k_{0})^{2i\tilde{\nu}+i\nu}}{2^{i\nu-i\tilde{\nu}}}\frac{(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-2k_{0})^{i\nu}}{(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-k_{0})^{2i\nu}}\\\ &((\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-k_{0}+ik_{0})(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-k_{0}-ik_{0}))^{-i\tilde{\nu}}\\\
&e^{\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-k_{0})-\chi_{\pm}(-k_{0})}e^{\tilde{\chi}_{\pm}^{\prime}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-k_{0})-\tilde{\chi}_{\pm}^{\prime}(-k_{0})}\end{array}$ (3.77b)
with $\tilde{\chi}_{\pm}^{\prime}(k)$ defined by (3.75).
For $N_{C}$,
$(N_{C}\delta e^{-it\theta})(k)=\delta_{C}^{0}(k)\delta_{C}^{1}(k)$ (3.78)
where
$\delta_{C}^{0}(k)=\frac{k_{0}^{2i\nu-i\tilde{\nu}}}{(\sqrt{\alpha
t}\beta)^{-i\tilde{\nu}}}2^{i\nu}e^{i\alpha\beta
t+i\frac{\alpha\beta^{2}}{2k_{0}^{2}}t}e^{\chi^{\prime}_{\pm}(ik_{0})}e^{\tilde{\chi}_{\pm}(ik_{0})}$
(3.79a)
$\begin{array}[]{rl}\delta_{C}^{1}(k)=&(ik)^{-i\tilde{\nu}}e^{-i\frac{k^{2}}{4}+i\frac{k^{6}_{0}k^{3}}{\zeta^{5}\sqrt{t}}}\frac{(ik_{0})^{-i\tilde{\nu}}(k_{0})^{-2i\nu}}{2^{-i\nu-i\tilde{\nu}}}\frac{(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+ik_{0})^{2i\tilde{\nu}}}{(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+2ik_{0})^{i\tilde{\nu}}}\\\ &((\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+ik_{0}+k_{0})(k_{0}-(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+ik_{0})))^{i\nu}\\\
&e^{\chi^{\prime}_{\pm}(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+ik_{0})-\chi^{\prime}_{\pm}(ik_{0})}e^{\tilde{\chi}_{\pm}(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+ik_{0})-\tilde{\chi}_{\pm}(ik_{0})}\end{array}$ (3.79b)
with
$\chi_{\pm}^{\prime}(k)=e^{-\frac{1}{2\pi i}\int_{0}^{\pm
k_{0}}\ln|k-k^{\prime}|d\ln(1-|r(k^{\prime})|^{2})}$ (3.80)
For $N_{D}$
$(N_{D}\delta e^{-it\theta})(k)=\delta_{D}^{0}(k)\delta_{D}^{1}(k)$ (3.81)
where
$\delta_{D}^{0}(k)=\frac{k_{0}^{2i\nu-i\tilde{\nu}}}{(\sqrt{\alpha
t}\beta)^{-i\tilde{\nu}}}2^{i\nu}e^{i\alpha\beta
t+i\frac{\alpha\beta^{2}}{2k_{0}^{2}}t}e^{\chi^{\prime}_{\pm}(-ik_{0})}e^{\tilde{\chi}_{\pm}(-ik_{0})}$
(3.82a)
$\begin{array}[]{rl}\delta_{D}^{1}(k)=&(-ik)^{-i\tilde{\nu}}e^{-i\frac{k^{2}}{4}+i\frac{k^{6}_{0}k^{3}}{\zeta^{5}\sqrt{t}}}\frac{(-ik_{0})^{-i\tilde{\nu}}(k_{0})^{-2i\nu}}{2^{-i\nu-i\tilde{\nu}}}\frac{(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-ik_{0})^{2i\tilde{\nu}}}{(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-2ik_{0})^{i\tilde{\nu}}}\\\ &((\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-ik_{0}+k_{0})(k_{0}-(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-ik_{0})))^{i\nu}\\\
&e^{\chi^{\prime}_{\pm}(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-ik_{0})-\chi^{\prime}_{\pm}(ik_{0})}e^{\tilde{\chi}_{\pm}(\frac{-k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k-ik_{0})-\tilde{\chi}_{\pm}(ik_{0})}\end{array}$ (3.82b)
Set
$\Delta^{0}_{l}=(\delta_{l}^{0}(k))^{\sigma_{3}},\quad l\in\\{A,B,C,D\\}$
(3.83)
and let $\tilde{\Delta}^{0}_{l}$ denote right multiplication by
$\Delta^{0}_{l}$,
$\tilde{\Delta}^{0}_{l}\phi=\phi\Delta^{0}_{l}.$ (3.84)
Denote
$\begin{array}[]{lll}\omega^{l^{\prime}}=\left\\{\begin{array}[]{ll}\omega^{\prime},&k\in\Sigma_{l^{\prime}}\\\
0,&k\in\Sigma^{(3)}\backslash\Sigma_{l^{\prime}}\end{array}\right.&and&\hat{\omega}^{l^{\prime}}=\left\\{\begin{array}[]{ll}\omega^{l^{\prime}},&k\in\hat{\Sigma}_{l^{\prime}}\\\
0,&k\in\hat{\Sigma}_{l^{\prime}}\backslash\Sigma_{l^{\prime}}\end{array}\right.\end{array}$
(3.85)
According to this.
$\omega^{\prime}=\sum_{l\in\\{A,B,C,D\\}}\omega^{l^{\prime}},\quad
C^{\Sigma^{(3)}}_{\omega^{\prime}}=\sum_{l\in\\{A,B,C,D\\}}C^{\Sigma^{(3)}}_{\omega^{l^{\prime}}}=\sum_{l\in\\{A,B,C,D\\}}C^{\Sigma_{l^{\prime}}}_{\omega^{l^{\prime}}}.$
(3.86)
###### Proposition 3.9.
For $l,\iota=\\{A,B,C,D\\}$, $l\neq\iota$ we have
$||C^{\Sigma^{(3)}}_{\omega^{l^{\prime}}}C^{\Sigma^{(3)}}_{\omega^{\iota^{\prime}}}||_{L^{2}(\Sigma^{(3)})}\leq
C(k_{0})t^{-\frac{1}{2}},$ (3.87a)
$||C^{\Sigma^{(3)}}_{\omega^{l^{\prime}}}C^{\Sigma^{(3)}}_{\omega^{\iota^{\prime}}}||_{L^{\infty}(\Sigma^{(3)})\rightarrow
L^{2}(\Sigma^{(3)})}\leq C(k_{0})t^{-\frac{3}{4}}.$ (3.87b)
###### Proof.
Analogous to lemma 3.5 in [9]. ∎
Let us prove some technical results concerning the operators
$C^{\Sigma_{l^{\prime}}}_{\omega^{l^{\prime}}}$ and
$C^{\hat{\Sigma}_{l^{\prime}}}_{\hat{\omega}^{l^{\prime}}}$
###### Proposition 3.10.
For $l\in\\{A,B,C,D\\}$,
$C^{\hat{\Sigma}_{l^{\prime}}}_{\hat{\omega}^{l^{\prime}}}=(N_{l})^{-1}\tilde{(\Delta^{0}_{l})}^{-1}C^{\Sigma_{l^{\prime}}}_{\omega^{l^{\prime}}}\tilde{(\Delta^{0}_{l})}N_{l},\quad\omega^{l}=(\Delta^{0}_{l})^{-1}(N_{l}\hat{\omega}^{l^{\prime}})\Delta^{0}_{l}.$
(3.88)
where
$\left.C^{\Sigma_{l^{\prime}}}_{\omega^{l^{\prime}}}\right|_{\bar{L}_{l}}=-C_{+}(\cdot\left(\begin{array}[]{cc}0&0\\\
\overline{R(\overline{(N_{l}k)})}(\delta_{l}^{1})^{-2}&0\end{array}\right)),$
(3.89a)
$\left.C^{\Sigma_{l^{\prime}}}_{\omega^{l^{\prime}}}\right|_{L_{l}}=C_{-}(\cdot\left(\begin{array}[]{cc}0&R((N_{l}k))(\delta^{1}_{l})^{2}\\\
0&0\end{array}\right)).$ (3.89b)
here
$L_{e}=\\{k=\frac{2u\sqrt{\alpha
t}\beta}{k_{0}}e^{-\frac{i\pi}{4}},-\varepsilon<u<\infty\\},\quad e=A,B,$
(3.90a) $L_{n}=\\{k=-\frac{2u\sqrt{\alpha
t}\beta}{k_{0}}e^{\frac{i\pi}{4}},-\varepsilon<u<\infty\\},\quad n=C,D.$
(3.90b)
###### Proof.
We consider the case $l=A$, the cases $l=B,l=C$ and $l=D$ follow in an
analogous manner. Since from (3.74a), $|\delta^{0}_{A}|=1$, it follows from
the definition of the operator $\tilde{\Delta}^{0}_{A}$ in (3.83) that
$\tilde{\Delta}^{0}_{A}$ is a unitary operator. Then the equation (3.88) is a
simple change-of-variables argument.
We note that
$((\Delta^{0}_{A})^{-1}(N_{A}\hat{\omega}^{A^{\prime}})\Delta^{0}_{1})(k)=\left(\begin{array}[]{cc}0&R((N_{A}k))(\delta^{A}_{l})^{2}\\\
0&0\end{array}\right)$ (3.91)
on $L_{A}$, otherwise
$((\Delta^{0}_{A})^{-1}(N_{A}\hat{\omega}^{l^{\prime}})\Delta^{0}_{A})(k)=0$.
Similarly,
$((\Delta^{0}_{A})^{-1}(N_{A}\hat{\omega}^{A^{\prime}})\Delta^{0}_{l})(k)=\left(\begin{array}[]{cc}0&0\\\
\overline{R(\overline{(N_{A}k)})}(\delta_{A}^{1})^{-2}&0\end{array}\right)$
(3.92)
on $\bar{L}_{A}$, otherwise
$((\Delta^{0}_{A})^{-1}(N_{A}\hat{\omega}^{A^{\prime}})\Delta^{0}_{l})(k)=0$.
∎
From definitions of $R(k)$, we know that (for case $A$)
$R(k_{0}+)=\lim_{\mathrm{Re}k>k_{0}}R(k)=-\overline{r(k_{0})},$ (3.93a)
$R(k_{0}-)=\lim_{\mathrm{Re}k<k_{0}}R(k)=\frac{\overline{r(k_{0})}}{1-|r(k_{0})|^{2}}.$
(3.93b)
As $t\rightarrow\infty$,
$\bar{R}(\frac{k_{0}^{2}}{2\sqrt{\alpha}\beta\sqrt{t}}k+k_{0})(\delta_{A}^{1})^{-2}-\bar{R}(k_{0}\pm)k^{-2i\nu}e^{i\frac{k^{2}}{2}}\rightarrow
0.$ (3.94)
We obtain the following estimate on the rate of convergence:
###### Proposition 3.11.
Let $\kappa$ be a fixed small number with $0<\kappa<\frac{1}{2}$. Then, for
$k\in\bar{L}_{A}$,
$\left|\bar{R}\left(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0}\right)(\delta^{1}_{A}(k))^{-2}-\bar{R}(k_{0}\pm)k^{-2i\nu}e^{i\frac{k^{2}}{2}}\right|\leq
C(k_{0})|e^{i\frac{\kappa}{2}k^{2}}|\left(\frac{\log t}{\sqrt{t}}\right)$
(3.95)
###### Proposition 3.12.
(see Proposition 6.2 in [16]) For general operators
$C_{\omega^{l^{\prime}}}^{\Sigma^{{}^{\prime}}},l\in\\{1,2,\dots,N\\}$, if
$(1^{\prime}-C_{\omega^{l^{\prime}}}^{\Sigma^{{}^{\prime}}})^{-1}$ exist, then
$(1^{\prime}-\sum_{1\leq X\leq
N}C_{\omega^{X^{\prime}}}^{\Sigma^{{}^{\prime}}})(1^{\prime}+\sum_{1\leq Y\leq
N}C_{\omega^{Y^{\prime}}}^{\Sigma^{{}^{\prime}}}(1^{\prime}-C_{\omega^{X^{\prime}}}^{\Sigma^{{}^{\prime}}})^{-1})=1^{\prime}-\sum_{1\leq
Y\leq N}\sum_{1\leq X\leq
N}(1-\delta_{XY})C_{\omega^{Y^{\prime}}}^{\Sigma^{{}^{\prime}}}C_{\omega^{X^{\prime}}}^{\Sigma^{{}^{\prime}}}(1^{\prime}-C_{\omega^{X^{\prime}}}^{\Sigma^{{}^{\prime}}})^{-1}$
(3.96a) and $(1^{\prime}+\sum_{1\leq Y\leq
N}C_{\omega^{Y^{\prime}}}^{\Sigma^{{}^{\prime}}}(1^{\prime}-C_{\omega^{X^{\prime}}}^{\Sigma^{{}^{\prime}}})^{-1})(1^{\prime}-\sum_{1\leq
X\leq N}C_{\omega^{X^{\prime}}}^{\Sigma^{{}^{\prime}}})=1^{\prime}-\sum_{1\leq
Y\leq N}\sum_{1\leq X\leq
N}(1-\delta_{XY})(1^{\prime}-C_{\omega^{Y^{\prime}}}^{\Sigma^{{}^{\prime}}})^{-1}C_{\omega^{Y^{\prime}}}^{\Sigma^{{}^{\prime}}}C_{\omega^{X^{\prime}}}^{\Sigma^{{}^{\prime}}}$
(3.96b) where $\delta_{XY}$ is the Kronecker delta.
###### Proof.
Assumption the existence of general operators
$(1^{\prime}-C_{\omega^{l^{\prime}}}^{\Sigma^{{}^{\prime}}})^{-1},l\in\\{1,2,\dots,N\\}$,
induction, and a straightforward application of the second resolvent identity.
∎
###### Lemma 3.13.
If, for $l\in\\{A,B,C,D\\}$,
$(1_{\Sigma_{l^{\prime}}}-C_{\omega^{l^{\prime}}}^{{\Sigma_{l^{\prime}}}})^{-1}$
bounded, then as $t\rightarrow\infty$,
$m(x,t)=-\frac{1}{2}\sum_{l\in\\{A,B,C,D\\}}\left(\int_{\Sigma_{l^{\prime}}}[\sigma_{3},((1_{\Sigma_{l^{\prime}}}-C_{\omega^{l^{\prime}}}^{{\Sigma_{l^{\prime}}}})^{-1}\mathbb{I})(\xi)\omega^{l^{\prime}}(\xi)]\frac{d\xi}{2\pi
i}\right)_{12}+O(\frac{C}{t}).$ (3.97)
###### Proof.
Analogous to the proof of Lemma 6.2 in [16]. ∎
###### Lemma 3.14.
For $l\in\\{A,B,C,D\\}$,
$||(1_{\Sigma_{l^{\prime}}}-C_{\omega^{l^{\prime}}}^{{\Sigma_{l^{\prime}}}})^{-1}||_{L^{2}}\leq
C$
###### Proof.
Consider the case $l=A$, the case $l=B,C$ and $l=D$ follow in an analogous
manner. From Lemma 3.7, the boundedness of
$(1_{\Sigma_{A^{\prime}}}-C_{\omega^{A^{\prime}}}^{{\Sigma_{A^{\prime}}}})^{-1}$
follows from the boundedness of
$(1_{\hat{\Sigma}_{A^{\prime}}}-C_{\omega^{A^{\prime}}}^{\hat{\Sigma}_{A^{\prime}}})^{-1}$.
From formula (3.88) we have
$(1_{\hat{\Sigma}_{A^{\prime}}}-C_{\omega^{A^{\prime}}}^{\hat{\Sigma}_{A^{\prime}}})^{-1}=(N_{A})^{-1}\tilde{(\Delta^{0}_{A})}^{-1}(1_{\Sigma_{A^{\prime}}}-C_{\omega^{A^{\prime}}}^{{\Sigma_{A^{\prime}}}})^{-1}\tilde{(\Delta^{0}_{A})}N_{A},$
(3.98)
And then the boundedness of
$(1_{\hat{\Sigma}_{A^{\prime}}}-C_{\omega^{A^{\prime}}}^{\hat{\Sigma}_{A^{\prime}}})^{-1}$
follows from the boundedness of
$(1_{\Sigma_{A}}-C_{\omega^{A}}^{\Sigma_{A}})^{-1}$.
Set
$\omega^{A}=(\Delta^{0}_{A})^{-1}(N_{A}\hat{\omega}^{A^{\prime}})\Delta^{0}_{A},$
(3.99)
so that
$C^{\Sigma_{A}}_{\omega^{A}}=C_{+}(\cdot\omega_{-}^{A})+C_{-}(\cdot\omega_{+}^{A}).$
(3.100)
On $\Sigma_{A}$, we have the diagram in Figure 9.
Figure 9. The jump condition of cross $k_{0}$ by scaling.
Set
$J^{A^{0}}=(b_{-}^{A^{0}})^{-1}b_{+}^{A^{0}}=(\mathbb{I}-\omega_{-}^{A^{0}})^{-1}(\mathbb{I}+\omega_{+}^{A^{0}})$.
Defining as usual $\omega^{A^{0}}=\omega_{+}^{A^{0}}+\omega_{-}^{A^{0}}$, and
using Proposition 3.11, one finds that
$||\omega^{A}-\omega^{A^{0}}||_{L^{\infty}(\Sigma_{A})\cap
L^{1}(\Sigma_{A})\cap L^{2}(\Sigma_{A})}\leq C(k_{0})t^{-\frac{1}{2}}.$
(3.101)
Hence, as $t\rightarrow\infty$,
$||C^{\Sigma_{A}}_{\omega^{A}}-C^{\Sigma_{A}}_{\omega^{A^{0}}}||_{L^{2}(\Sigma_{A})}\leq
C(k_{0})t^{-\frac{1}{2}},$ (3.102)
and consequently, one sees that the boundedness of
$(1_{\Sigma_{A}}-C_{\omega^{A}}^{\Sigma_{A}})^{-1}$ follows from the
boundedness of $(1_{\Sigma_{A}}-C_{\omega^{A^{0}}}^{\Sigma_{A}})^{-1}$ as
$t\rightarrow\infty$.
Then reorient $\Sigma_{A}$ to $\Sigma_{A,r}$ as Figure 10.
Figure 10. $\Sigma_{A,r}$.
A simple computation shows that the jump matrix
$J^{A,r}=(b_{-}^{A,r})^{-1}(b_{+}^{A,r})=(\mathbb{I}-\omega_{-}^{A,r})^{-1}(\mathbb{I}+\omega_{+}^{A,r})$
on $\Sigma_{A,r}$ is determined by
$\omega_{\pm}^{A,r}(k)=-\omega_{\mp}^{A^{0}}(k),\quad for\quad\mathrm{Re}k>0,$
(3.103a) and $\omega_{\pm}^{A,r}(k)=\omega_{\pm}^{A^{0}}(k),\quad
for\quad\mathrm{Re}k<0.$ (3.103b)
The third step is that extending
$\Sigma_{A,r}\rightarrow\Sigma_{e}=\Sigma_{A,r}\cup{\mathbb{R}}$ with the
orientation on $\Sigma_{A,r}$ as Figure 10 and the orientation on
${\mathbb{R}}$ from $-\infty$ to $\infty$. And the jump
$J^{e}=(b_{-}^{e})^{-1}b_{=}^{e}=(\mathbb{I}-\omega^{e}_{-})^{-1}(\mathbb{I}+\omega^{e}_{+})$
with
$\omega^{e}(k)=\omega^{A,r}(k),\quad k\in\Sigma_{A,r},$ (3.104a)
$\omega^{e}(k)=0,\quad k\in{\mathbb{R}}.$ (3.104b)
Set ${\mathbb{C}}_{\omega^{e}}$ on $\Sigma_{e}$. Once again, by Lemma 3.7, it
is sufficient to bound $(1_{\Sigma_{e}}-C_{\omega^{e}})^{-1}$ on
$L^{2}(\Sigma_{e})$.
Figure 11. $\Sigma_{e}$.
Then define a piecewise-analytic matrix function $\phi$ as follows:
$\tilde{M}^{(k_{0})}=M^{(k_{0})}\phi,$
where
$\phi=\left\\{\begin{array}[]{ll}k^{i\nu\sigma_{3}},&k\in\Omega^{e}_{2},\Omega^{e}_{5},\\\
k^{i\nu\sigma_{3}}\left(\begin{array}[]{cc}1&0\\\
-r(k_{0})e^{-i\frac{k^{2}}{2}}&1\end{array}\right),&k\in\Omega^{e}_{1},\\\
k^{i\nu\sigma_{3}}\left(\begin{array}[]{cc}1&-\overline{r(k_{0})}e^{i\frac{k^{2}}{2}}\\\
0&1\end{array}\right),&k\in\Omega^{e}_{6},\\\
k^{i\nu\sigma_{3}}\left(\begin{array}[]{cc}1&\overline{\frac{r(k_{0})}{1-|r(k_{0})|^{2}}}e^{i\frac{k^{2}}{2}}\\\
0&1\end{array}\right),&k\in\Omega^{e}_{3},\\\
k^{i\nu\sigma_{3}}\left(\begin{array}[]{cc}1&0\\\
-\overline{\frac{r(k_{0})}{1-|r(k_{0})|^{2}}}e^{-i\frac{k^{2}}{2}}&1\end{array}\right),&k\in\Omega^{e}_{4}.\end{array}\right.$
Thus, we can get the Riemann-Hilbert problem of $\tilde{M}^{(k_{0})}$
$\begin{array}[]{ll}\tilde{M}_{+}^{(k_{0})}(x,t,k)=\tilde{M}_{-}^{(k_{0})}(x,t,k)J^{e,\phi},\\\
\\\
\tilde{M}^{(k_{0})}(x,t,k)=(\mathbb{I}+\frac{M^{A^{0}}_{1}}{k}+O(\frac{1}{k^{2}}))k^{i\nu\sigma_{3}},&k\rightarrow\infty.\end{array}$
(3.105)
where
$J^{e,\phi}=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}1-|r(k_{0})|^{2}&\overline{r(k_{0})}e^{-i\frac{k^{2}}{2}}\\\
-r(k_{0})e^{i\frac{k^{2}}{2}}&1\end{array}\right),&k\in{\mathbb{R}},\\\
\mathbb{I},&k\in\Sigma_{A,r}.\end{array}\right.$ (3.106)
On ${\mathbb{R}}$ we have
$J^{e,\phi}=(b_{-}^{e,\phi})^{-1}b_{+}^{e,\phi}=(\mathbb{I}-\omega_{-}^{e,\phi})^{-1}(\mathbb{I}+\omega_{+}^{e,\phi})=\left(\begin{array}[]{cc}1&e^{-\frac{ik^{2}}{2}}\bar{r}(k_{0})\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
-e^{\frac{ik^{2}}{2}}r(k_{0})&1\end{array}\right).$ (3.107)
Set
$C_{e,\phi}=C_{\omega^{e,\phi}}=C_{+}(\cdot\omega_{-}^{e,\phi})+C_{-}(\cdot\omega_{+}^{e,\phi})$
as thr associated operator on $\Sigma_{e}$, with
$\omega^{e,\phi}=\omega_{+}^{e,\phi}+\omega_{-}^{e,\phi}$. By Lemma 3.7, the
boundedness of $C_{e,\phi}$ follows from the boundedness of the operator
$C_{\omega^{e,\phi}|_{{\mathbb{R}}}}:L^{2}({\mathbb{R}})\rightarrow
L^{2}({\mathbb{R}})$ associated with the restriction of $\omega^{e,\phi}$ to
${\mathbb{R}}$. Howerover,
$||C_{\omega^{e,\phi}|_{{\mathbb{R}}}}||_{L^{2}({\mathbb{R}})}\leq\sup_{k\in{\mathbb{R}}}{|e^{-\frac{ik^{2}}{2}}\bar{r}(k_{0})|}\leq||r||_{L^{\infty}({\mathbb{R}})}<1$,
and hence,
$||(1_{{\mathbb{R}}}-C_{\omega^{e,\phi}|_{{\mathbb{R}}}})^{-1}||_{L^{2}({\mathbb{R}})}\leq(1-||r||_{L^{\infty}({\mathbb{R}})})^{-1}<\infty$
for all $k_{0}$, which in turn implies that $(1_{\Sigma_{e}}-C_{e,\phi})^{-1}$
is bounded. ∎
### 3.3. Model Riemann-Hilbert Problem
In this subsection, we reduce the evaluation of the integrals in Lemma 3.13 to
four Riemann-Hilbert problems on ${\mathbb{R}}$ which can be solved
explicitly.
For $l\in\\{A,B,C,D\\}$, define
$M^{l}(k)=\mathbb{I}+\int_{\Sigma_{l}}\frac{((1_{\Sigma_{l}}-C_{\omega^{l^{0}}}^{\Sigma_{l}})^{-1}\mathbb{I})(\xi)\omega^{l^{0}}(\xi)}{\xi-k}\frac{d\xi}{2\pi
i},\quad k\in{\mathbb{C}}\backslash\Sigma_{l}.$ (3.108)
Then, $M^{l}(k)$ solves the Riemann-Hilbert problem
$\left\\{\begin{array}[]{ll}M^{l}_{+}(k)=M_{-}^{l}(k)J^{l}(k)=M_{-}^{l}(k)(\mathbb{I}-\omega^{l^{0}}_{-})^{-1}(\mathbb{I}+\omega_{+}^{l^{0}}),&k\in\Sigma_{l},\\\
M^{l}(k)\rightarrow\mathbb{I},&k\rightarrow\infty.\end{array}\right.$ (3.109)
In particular we see that if
$M^{l}(k)=\mathbb{I}+\frac{M_{1}^{l}}{k}+O(k^{-2}),\quad k\rightarrow\infty,$
(3.110)
then
$M_{1}^{l}=-\int_{\Sigma_{l}}((1_{\Sigma_{l}}-C_{\omega^{l^{0}}}^{\Sigma_{l}})^{-1}\mathbb{I})(\xi)\omega^{l^{0}}(\xi)\frac{d\xi}{2\pi
i}.$ (3.111)
Substituting into (3.97), we obtain
$m(x,t)=\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}((\delta^{0}_{A})^{2}(M_{1}^{A^{0}})_{12}+(\delta_{B}^{0})^{2}(M_{1}^{B^{0}})-(\delta_{C}^{0})^{2}(M_{1}^{C^{0}})-(\delta_{D}^{0})^{2}(M_{1}^{D^{0}})_{12})+O(\frac{C}{t}).$
(3.112)
We consider in detail only case $A$. Write
$\Psi=\tilde{M}^{(k_{0})}e^{-i\frac{k^{2}}{4}\sigma_{3}}=\hat{\Psi}k^{i\nu\sigma_{3}}e^{-i\frac{k^{2}}{4}\sigma_{3}}$
(3.113)
From formula (3.105),
$\Psi_{+}(k)=\Psi_{-}(k)\tilde{J}^{(k_{0})},\quad k\in{\mathbb{R}}.$ (3.114)
where
$J^{(k_{0})}=\left(\begin{array}[]{cc}1-|r(k_{0})|^{2}&\overline{r(k_{0})}\\\
-r(k_{0})&1\end{array}\right)$
By differentiation with respect to $k$ and Liouville theorem we can get
$\frac{d\Psi}{dk}+\frac{1}{2}ik\sigma_{3}\Psi=\beta\Psi,$ (3.115)
where
$\beta=\frac{i}{2}[\sigma_{3},M^{A^{0}}_{1}]=\left(\begin{array}[]{cc}o&\beta_{12}\\\
\beta_{21}&0\end{array}\right).$
Following [9](P.350-352), we have
$\beta_{12}=-\frac{e^{-\frac{\pi}{2}\nu}}{r(k_{0})}\frac{\sqrt{2\pi}e^{i\frac{\pi}{4}}}{\Gamma(-i\nu)}.$
(3.116)
Hence,
$(M_{1}^{A^{0}})_{12}=-i\beta_{12}=i\frac{e^{-\frac{\pi}{2}\nu}}{r(k_{0})}\frac{\sqrt{2\pi}e^{i\frac{\pi}{4}}}{\Gamma(-i\nu)}.$
(3.117)
From the symmetry reduction for $M(k)$,i.e., $M(-k)=\sigma_{3}M(k)\sigma_{3}$,
we have that
$(M_{1}^{A^{0}})_{12}=(M_{1}^{B^{0}})_{12}.$ (3.118)
For $C$,
$\beta_{12}=\frac{e^{\frac{\pi}{2}\tilde{\nu}}}{r(ik_{0})}\frac{\sqrt{2\pi}e^{i\frac{\pi}{4}}}{\Gamma(i\tilde{\nu})}.$
(3.119)
And similarly $(M_{1}^{C^{0}})_{12}=(M_{1}^{D^{0}})_{12}.$
Thus, we have
###### Theorem 3.15.
As $t\rightarrow\infty$, such that $k_{0}<M$,
$\begin{array}[]{rrl}m(x,t)&=&\frac{k_{0}^{2}}{\beta}\sqrt{\frac{|\nu|}{\alpha
t}}e^{i(2\alpha\beta t+2\nu\ln{\frac{k_{0}}{\sqrt{\alpha
t}\beta}}+\frac{\pi}{4}-\frac{\alpha\beta^{2}}{k_{0}^{2}}t-2\tilde{\nu}\ln{2k_{0}^{2}}-2i\chi_{\pm}(k_{0})-2i\tilde{\chi}^{\prime}_{\pm}(k_{0})-\arg{r(k_{0})}-\arg{\Gamma(-i\nu)})}\\\
&&{}-\frac{k_{0}^{2}}{\beta}\sqrt{\frac{|\tilde{\nu}|}{\alpha
t}}e^{i(2\alpha\beta t-2\tilde{\nu}\ln{\frac{k_{0}}{\sqrt{\alpha
t}\beta}}+\frac{\pi}{4}+\frac{\alpha\beta^{2}}{k_{0}^{2}}t+2\nu\ln{2k_{0}^{2}}-2i\chi^{\prime}_{\pm}(ik_{0})-2i\tilde{\chi}_{\pm}(ik_{0})-\arg{r(ik_{0})}-\arg{\Gamma(i\tilde{\nu})})}\\\
&&{}+O(\frac{1}{t}).\end{array}$ (3.120)
###### Proposition 3.16.
$||2m||^{2}_{L^{2}({\mathbb{R}})}=\frac{2}{\pi}\left(\int_{0}^{+\infty}\frac{\log{(1+|r(i\mu)|^{2})}}{\mu}d\mu-\int_{0}^{+\infty}\frac{\log{(1-|r(\mu)|^{2})}}{\mu}d\mu\right)$
(3.121)
###### Proof.
Analogous to Proposition 8.2 in [16]. ∎
###### Lemma 3.17.
As $t\rightarrow\infty$,
$e^{4i\int_{-\infty}^{x}|m(x;,t)|^{2}dx^{\prime}}=e^{\frac{2i}{\pi}\left(\int_{k_{0}}^{+\infty}\frac{\ln(1+|r(ik^{\prime})|^{2})}{k^{\prime}}dk^{\prime}-\int_{k_{0}}^{+\infty}\frac{\ln(1-|r(k^{\prime})|^{2})}{k^{\prime}}dk^{\prime}-\tilde{\psi}\right)}+O(\frac{C}{t^{\frac{1}{2}}}).$
(3.122)
where
$\tilde{\psi}=\sqrt{\int_{0}^{k_{0}}\frac{\ln(1+|r(ik^{\prime})|^{2})}{k^{\prime}}\frac{\ln(1-|r(k^{\prime})|^{2})}{k^{\prime}}\cos{(\tilde{\xi}_{1}-\tilde{\xi}_{2})dk^{\prime}}}$
(3.123)
with
$\tilde{\xi}_{1}=2\alpha\beta t+2\nu\ln{\frac{k_{0}}{\sqrt{\alpha
t}\beta}}+\frac{\pi}{4}-\frac{\alpha\beta^{2}}{k_{0}^{2}}t-2\tilde{\nu}\ln{2k_{0}^{2}}-2i\chi_{\pm}(k_{0})-2i\tilde{\chi}^{\prime}_{\pm}(k_{0})-\arg{r(k_{0})}-\arg{\Gamma(-i\nu)}$
and
$\tilde{\xi}_{2}=2\alpha\beta t-2\tilde{\nu}\ln{\frac{k_{0}}{\sqrt{\alpha
t}\beta}}+\frac{\pi}{4}+\frac{\alpha\beta^{2}}{k_{0}^{2}}t+2\nu\ln{2k_{0}^{2}}-2i\chi^{\prime}_{\pm}(ik_{0})-2i\tilde{\chi}_{\pm}(ik_{0})-\arg{r(ik_{0})}-\arg{\Gamma(i\tilde{\nu})}$
###### Proof.
Analogous to Lemma 8.1 in [16]. ∎
###### Theorem 3.18.
As $t\rightarrow\infty$,
$\begin{array}[]{rrl}u_{x}(x,t)&=&2\frac{k_{0}^{2}}{\beta}\sqrt{\frac{|\nu|}{\alpha
t}}e^{i(2\alpha\beta t+2\nu\ln{\frac{k_{0}}{\sqrt{\alpha
t}\beta}}+\frac{3\pi}{4}-\frac{\alpha\beta^{2}}{k_{0}^{2}}t-2\tilde{\nu}\ln{2k_{0}^{2}}-2i\chi_{\pm}(k_{0})-2i\tilde{\chi}^{\prime}_{\pm}(k_{0})-\arg{r(k_{0})}-\arg{\Gamma(-i\nu)})+\tilde{\phi}}\\\
&&{}-2\frac{k_{0}^{2}}{\beta}\sqrt{\frac{|\tilde{\nu}|}{\alpha
t}}e^{i(2\alpha\beta t-2\tilde{\nu}\ln{\frac{k_{0}}{\sqrt{\alpha
t}\beta}}+\frac{3\pi}{4}+\frac{\alpha\beta^{2}}{k_{0}^{2}}t+2\nu\ln{2k_{0}^{2}}-2i\chi^{\prime}_{\pm}(ik_{0})-2i\tilde{\chi}_{\pm}(ik_{0})-\arg{r(ik_{0})}-\arg{\Gamma(i\tilde{\nu})})+\tilde{\phi}}\\\
&&{}+O(\frac{1}{t}).\end{array}$ (3.124)
where
$\tilde{\phi}=\frac{2}{\pi}\left(\int_{k_{0}}^{+\infty}\frac{\ln(1+|r(ik^{\prime})|^{2})}{k^{\prime}}dk^{\prime}-\int_{k_{0}}^{+\infty}\frac{\ln(1-|r(k^{\prime})|^{2})}{k^{\prime}}dk^{\prime}-\tilde{\psi}\right)$
Thus, the solution of the Fokas-Lenells equation $u(x,t)$ can be obtained by
integration with respect to $x$. This implies that the leading order
asymptotic of the solution to the Fokas-Lenells equation has order
$t^{-\frac{1}{2}}$.
###### Remark 3.19.
Although, Fokas-Lenells equation (2.8) is an evolution equation in $u_{x}$ and
that any solution $u(x,t)$ is undetermined up to $u(x,t)\rightarrow
u(x,t)+h(t)$ for an arbitrary function $h(t)$, the requirement that $u$ goes
to zero as $|x|\rightarrow\infty$ removes this non-uniqueness.
###### Remark 3.20.
It is not normal that we get the solution $u_{x}(x,t)$ in terms of the
solution of Rieman-Hilbert problem (2.14). And we find that if we use the
asymptotic behavior of the $M(x,t,k)$ as $k\rightarrow 0$, we can get the
solution of $u(x,t)$ from the $t-$part of Lax pair (2.10). We will use this to
deal with general initial value problem case in another paper [21].
Acknowledgements The work of Xu was partially supported by Excellent Doctor
Research Funding Project of Fudan University. The work described in this paper
was supported by grants from the National Science Foundation of China (Project
No.10971031;11271079), Doctoral Programs Foundation of the Ministry of
Education of China, and the Shanghai Shuguang Tracking Project (project
08GG01).
## Appendix A Prove Proposition 3.2 and 3.11.
Prove Proposition 3.2.
For the convenience of reader, we show the details of the procedure of the
analytic continuation.
1\. $\frac{k_{0}}{2}<|k|<k_{0},k\in{\mathbb{R}}$.
We just consider $\frac{k_{0}}{2}<k<k_{0}$, the case for
$-k_{0}<k<-\frac{k_{0}}{2}$ is similarly.
Set
$\rho(k)=\frac{-r(k)}{1-r(k)\overline{r(\bar{k})}}=\frac{-r(k)}{1-|r(k)|^{2}}.$
(A.1)
We split $\rho(k)$ into even and odd parts,
$\rho(k)=H_{e}(k^{2})+kH_{o}(k^{2})$, where $H_{e}(\cdot)$ and $H_{o}(\cdot)$
are of the Schwartz class.
For any positive integer $m$,
$H_{e}(k^{2})=\mu_{0}^{e}+\mu_{1}^{e}(k^{2}-k_{0}^{2})+\cdots+\mu_{m}^{e}(k^{2}-k_{0}^{2})^{m}+\frac{1}{m!}\int_{k_{0}^{2}}^{k^{2}}H_{e}^{(m+1)}(\gamma)(k^{2}-\gamma)^{m}d\gamma$
(A.2)
and
$H_{o}(k^{2})=\mu_{0}^{o}+\mu_{1}^{o}(k^{2}-k_{0}^{2})+\cdots+\mu_{m}^{o}(k^{2}-k_{0}^{2})^{m}+\frac{1}{m!}\int_{k_{0}^{2}}^{k^{2}}H_{o}^{(m+1)}(\gamma)(k^{2}-\gamma)^{m}d\gamma.$
(A.3)
Set
$R(k)=R_{m}(k)=\sum_{i=0}^{m}\mu_{i}^{e}(k^{2}-k_{0}^{2})^{i}+k\sum_{i=0}^{m}\mu_{i}^{o}(k^{2}-k_{0}^{2})^{i}.$
(A.4)
Assume $m=4q+1$, where $q$ is a positive integer. Write
$\rho(k)=h(k)+R(k),\quad\frac{k_{0}}{2}<k<k_{0},k\in{\mathbb{R}}.$ (A.5)
Then
$\left.\frac{d^{j}h(k)}{dk^{j}}\right|_{\pm k_{0}}=0,\quad 0\leq j\leq m.$
(A.6)
And we have
$h(k)=\frac{(k^{2}-k_{0}^{2})^{m+1}}{m!}g(k,k_{0})$ (A.7)
where
$g(k,k_{0})=\left(\int_{0}^{1}H_{e}^{(m+1)}(k_{0}^{2}+u(k^{2}-k_{0}^{2}))(1-u)^{m}du+k\int_{0}^{1}H_{o}^{(m+1)}(k_{0}^{2}+u(k^{2}-k_{0}^{2}))(1-u)^{m}du\right)$
(A.8)
and
$\left|\frac{d^{j}g(k,k_{0})}{dk^{j}}\right|\leq C,\quad\frac{k_{0}}{2}\leq
k\leq k_{0}.$ (A.9)
We will split $h$ as $h(k)=h_{\@slowromancap i@}(k)+h_{\@slowromancap
ii@}(k)$, where $h_{\@slowromancap i@}$ is small and $h_{\@slowromancap ii@}$
has an alytic continuation to $\mathrm{Im}k>0$. Thus
$\rho=h_{\@slowromancap i@}+(h_{\@slowromancap ii@}+R).$ (A.10)
Set $p(k)=(k^{2}-k_{0}^{2})^{q}$. Recall
$\begin{array}[]{rrl}\theta(k)&=&k^{2}(\frac{x}{t}+\alpha)+\frac{\alpha\beta^{2}}{4k^{2}}-\alpha\beta\\\
&=&\frac{\alpha\beta^{2}}{4k_{0}^{4}}k^{2}+\frac{\alpha\beta^{2}}{4k^{2}}-\alpha\beta.\end{array}$
(A.11)
We define
$\left\\{\begin{array}[]{rrll}\frac{h}{p}(\theta)&=&\frac{h(k(\theta))}{p(k(\theta))},&\theta(k_{0})<\theta<\theta(\frac{k_{0}}{2}),\\\
&=&0,&\theta\leq\theta(k_{0})\quad
or\quad\theta\geq\theta(\frac{k_{0}}{2}).\end{array}\right.$ (A.12)
As
$|\theta|\rightarrow\theta(k_{0})=\frac{\alpha\beta^{2}}{2k_{0}^{2}}-\alpha\beta$
and $|\theta|>\frac{\alpha\beta^{2}}{2k_{0}^{2}}-\alpha\beta$, we have
$\frac{h}{p}(\theta)=O((k^{2}(\theta)-k_{0}^{2})^{m+1-q})$ and
$\frac{d\theta}{dk}=\frac{\alpha\beta^{2}(k^{4}-k_{0}^{4})}{2k_{0}^{4}k^{3}}.$
(A.13)
We claim that $\frac{h}{p}\in H^{j}(-\infty<\theta<\infty)$ for $0\leq
j\leq\frac{3q+2}{2}$. As by Fourier inversion,
$\frac{h}{p}(k)=\int_{-\infty}^{\infty}e^{is\theta(k)}\widehat{(\frac{h}{p})}(s)\bar{d}s,\quad\frac{k_{0}}{2}<k<k_{0},$
(A.14)
where
$\widehat{(\frac{h}{p})}(s)=\int_{\theta(k_{0})}^{\theta(\frac{k_{0}}{2})}e^{-is\theta(k)}\frac{h}{p}(\theta(k))\bar{d}\theta(k),\quad
s\in{\mathbb{R}}.$ (A.15)
where $\bar{d}s=\frac{ds}{\sqrt{2\pi}}$ and
$\bar{d}\theta(k)=\frac{d\theta(k)}{\sqrt{2\pi}}$.
Thus,
$\begin{array}[]{l}\int_{\theta(k_{0})}^{\theta(\frac{k_{0}}{2})}\left|\left(\frac{d}{d\theta}\right)^{j}\frac{h}{p}(\theta(k))\right|^{2}|\bar{d}\theta(k)|\\\
=\int_{\frac{k_{0}}{2}}^{k_{0}}\left|\left(\frac{2k_{0}^{4}k^{3}}{\alpha\beta^{2}(k^{4}-k_{0}^{4})}\frac{d}{dk}\right)^{j}\frac{h}{p}(k)\right|^{2}|\frac{\alpha\beta^{2}(k^{4}-k_{0}^{4})}{2k_{0}^{4}k^{3}}|\bar{d}k\leq
C<\infty,\end{array}$ (A.16)
for $0<k_{0}<M,0\leq j\leq\frac{3q+2}{2}$. Hence,
$\int_{-\infty}^{\infty}(1+s^{2})^{j}|\widehat{(h/p)}(s)|^{2}ds\leq C<\infty$
(A.17)
for $0<k_{0}<M,0\leq j\leq\frac{3q+2}{2}$.
Split
$\begin{array}[]{rrl}h(k)&=&p(k)\int_{t}^{\infty}e^{is\theta(k)}\widehat{(h/p)}(s)\bar{d}s+p(k)\int_{-\infty}^{t}e^{is\theta(k)}\widehat{(h/p)}(s)\bar{d}s\\\
&=&h_{\@slowromancap i@}(k)+h_{\@slowromancap ii@}(k).\end{array}$ (A.18)
Thus, for $\frac{k_{0}}{2}<k<k_{0}\leq M$ and any positive integer
$n\leq\frac{3q+2}{2}$.
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
i@}(k)|&\leq&|p(k)|\int_{t}^{\infty}|\widehat{(h/p)}(s)|\bar{d}s\\\
&\leq&|p(k)|(\int_{t}^{\infty}(1+s^{2})^{-n}\bar{d}s)^{\frac{1}{2}}(\int_{t}^{\infty}(1+s^{2})^{n}|\widehat{(h/p)}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&\frac{c}{t^{n-\frac{1}{2}}}.\end{array}$ (A.19)
Consider the contour $l_{1}:k(u)=k_{0}+uk_{0}e^{i\frac{3\pi}{4}},0\leq
u\leq\frac{1}{\sqrt{2}}$. Since $\mathrm{Re}i\theta(k)$ is positive on this
contour, $h_{\@slowromancap ii@}(k)$ has an analytic continuation to contours
$l_{1}$.
On the contour $l_{1}$,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&|k+k_{0}|^{q}(k_{0}u)^{q}e^{-t\mathrm{Re}i\theta(k)}\int_{-\infty}^{t}e^{(s-t)\mathrm{Re}i\theta(k)}\widehat{(h/p)}(s)\bar{d}s\\\
&\leq&ck_{0}^{2q}u^{q}e^{-t\mathrm{Re}i\theta(k)}(\int_{-\infty}^{t}(1+s^{2})^{-1}\bar{d}s)^{\frac{1}{2}}(\int_{-\infty}^{t}(1+s^{2})|\widehat{(h/p)}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&ck_{0}^{2q}u^{q}e^{-t\mathrm{Re}i\theta(k)}.\end{array}$ (A.20)
Recall
$\theta(k)=\frac{\alpha\beta^{2}}{4k_{0}^{4}}k^{2}+\frac{\alpha\beta^{2}}{4k^{2}}-\alpha\beta$,
and set $k=k_{1}+ik_{2}$, thus
$\begin{array}[]{rrl}\mathrm{Re}i\theta(k)&=&-2\alpha\beta^{2}k_{1}k_{2}\frac{(k_{1}^{2}+k_{2}^{2})^{2}-k_{0}^{4}}{4k_{0}^{4}(k_{1}^{2}+k_{2}^{2})^{2}}\\\
&=&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{(u^{2}-\sqrt{2}u)^{2}(u^{2}-\sqrt{2}u+2)}{(u^{2}-\sqrt{2}u+1)^{2}}\\\
&\geq&\frac{\alpha\beta^{2}u^{2}}{2k_{0}^{2}},\end{array}$ (A.21)
for $0\leq u\leq\frac{1}{\sqrt{2}}$.
Thus, on the contour $l_{1}$
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&ck_{0}^{2q}u^{q}e^{-t\frac{\alpha\beta^{2}u^{2}}{2k_{0}^{2}}}\leq
ck_{0}^{2q}u^{q}e^{-t\frac{\alpha\beta^{2}u^{2}}{2M^{2}}}\\\
&\leq&\frac{c_{1}}{t^{\frac{q}{2}}},\end{array}$ (A.22)
for $k_{0}<M$.
Fix $\varepsilon$, $0<\varepsilon<\frac{1}{\sqrt{2}}$. If $k(u)$ is on the
contour $l_{1}$ , $\varepsilon<u<\frac{1}{\sqrt{2}}$, then we obtain
$|e^{-2it\theta(k)}R(k)|\leq ce^{-\frac{\alpha\beta^{2}u^{2}}{k_{0}^{2}}t}\leq
ce^{-\frac{\varepsilon^{2}\alpha\beta^{2}}{M^{2}}t}$ (A.23)
2.$0<|k|<\frac{k_{0}}{2},k\in{\mathbb{R}}$.
We consider $0<k<\frac{k_{0}}{2}$, the case for $-\frac{k_{0}}{2}<k<0$ is
similarly.
Define
$\left\\{\begin{array}[]{rrll}\rho(\theta)&=&\rho(k(\theta)),&\theta>\theta(\frac{k_{0}}{2}),\\\
&=&0,&\theta\leq\theta(\frac{k_{0}}{2}).\end{array}\right.$ (A.24)
We claim that $\rho(\theta)\in H^{j}(-\infty<\theta<\infty)$ for any
nonnegative integer $j$.
By Fourier inversion,
$\rho(\theta(k))=\int_{-\infty}^{\infty}e^{is\theta(k)}\hat{\rho}(s)\bar{d}s,\quad
0<k<\frac{k_{0}}{2},$ (A.25)
where
$\hat{\rho}(s)=\int_{\theta(\frac{k_{0}}{2})}^{\infty}e^{-is\theta(k)}\rho(\theta(k))\bar{d}\theta(k).$
(A.26)
Then,
$\begin{array}[]{l}\int_{\theta(\frac{k_{0}}{2})}^{\infty}\left|\left(\frac{d}{d\theta}\right)^{j}\rho(\theta(k))\right|^{2}|\bar{d}\theta(k)|\\\
=\int_{0}^{\frac{k_{0}}{2}}\left|\left(\frac{2k_{0}^{4}k^{3}}{\alpha\beta^{2}(k^{4}-k_{0}^{4})}\frac{d}{dk}\right)^{j}\rho(k)\right|^{2}|\frac{\alpha\beta^{2}(k^{4}-k_{0}^{4})}{2k_{0}^{4}k^{3}}|\bar{d}k\leq
C<\infty,\end{array}$ (A.27)
for any nonnegative integer $j$, $0<k_{0}<M$, since $r(k)\rightarrow 0$
rapidly, as $k\rightarrow 0$.
Hence
$\int_{-\infty}^{\infty}(1+s^{2})^{j}|\hat{\rho}(s)|^{2}\bar{d}s\leq C,$
(A.28)
for any nonnegative integer $j$.
Split
$\begin{array}[]{rrl}\rho(k)&=&\int_{t}^{\infty}e^{is\theta(k)}\hat{\rho}(s)\bar{d}s+\int_{-\infty}^{t}e^{is\theta(k)}\hat{\rho}(s)\bar{d}s\\\
&=&h_{\@slowromancap i@}(k)+h_{\@slowromancap ii@}(k).\end{array}$ (A.29)
Then, for $0<k<\frac{k_{0}}{2}$ and any positive integer $j$, we obtain,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
i@}(k)|&\leq&\int_{t}^{\infty}|\hat{\rho}|\bar{d}s\\\
&\leq&(\int_{t}^{\infty}(1+s^{2})^{-j}\bar{d}s)^{\frac{1}{2}}(\int_{t}^{\infty}(1+s^{2})^{j}|\hat{\rho}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&\frac{c}{t^{j-\frac{1}{2}}}.\end{array}$ (A.30)
Consider the contour
$l_{2}:k(u)=uk_{0}e^{i\frac{\pi}{4}},0<u<\frac{1}{\sqrt{2}}$. Since
$\mathrm{Re}i\theta(k)$ is positive on this contour, $h_{\@slowromancap ii@}$
has an analytic continuation to contour $l_{2}$.
On the contour $l_{2}$,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&e^{-t\mathrm{Re}i\theta(k)}\int_{-\infty}^{t}e^{(s-t)\mathrm{Re}i\theta(k)}|\hat{\rho}(k)|\bar{d}s\\\
&\leq&e^{-t\mathrm{Re}i\theta(k)}(\int_{-\infty}^{t}(1+s^{2})^{-1}\bar{d}s)^{\frac{1}{2}}(\int_{-\infty}^{t}(1+s^{2})|\hat{\rho}(k)|^{2}\bar{d}s)^{\frac{1}{2}},\end{array}$
(A.31)
where
$\begin{array}[]{rrl}\mathrm{Re}i\theta(k)&=&-2\alpha\beta^{2}k_{1}k_{2}\frac{(k_{1}^{2}+k_{2}^{2})^{2}-k_{0}^{4}}{4k_{0}^{4}(k_{1}^{2}+k_{2}^{2})^{2}}\\\
&=&-\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{u^{4}-1}{u^{2}}\\\
&\geq&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\end{array}$ (A.32)
for $0<u\leq\frac{1}{\sqrt{2}}$.
Thus, we obtain,
$|e^{-2it\theta(k)}h_{\@slowromancap ii@}(k)|\leq
ce^{-t\frac{\alpha\beta^{2}}{4k_{0}^{2}}}.$ (A.33)
3\. $|k|>k_{0},k\in{\mathbb{R}}$
We consider $k>k_{0}$, the case for $k<-k_{0}$ is similarly.
Set
$\rho(k)=r(k).$ (A.34)
We write
$(k-i)^{m+5}\rho(k)=\mu_{0}+\mu_{1}(k-k_{0})+\cdots+\mu_{m}(k-k_{0})^{m}+\frac{1}{m!}\int_{k_{0}}^{k}((\cdot-i)^{m+5}\rho(\cdot))^{(m+1)}(\gamma)(k-\gamma)^{m}d\gamma.$
(A.35)
Define
$R(k)=\frac{\sum_{i=0}^{m}\mu_{i}(k-k_{0})^{i}}{(k-i)^{m+5}}$ (A.36)
and write $\rho(k)=h(k)+R(k)$. We have
$\left.\frac{d^{j}h(k)}{dk^{j}}\right|_{k_{0}}=0,\quad 0\leq j\leq m.$ (A.37)
For $0<k_{0}<M$, set
$v(k)=\frac{(k-k_{0})^{q}}{(k-i)^{q+2}}.$ (A.38)
Let
$\left\\{\begin{array}[]{rrll}\frac{h}{v}(\theta)&=&\frac{h}{v}(k(\theta)),&\theta>\theta(k_{0}),\\\
&=&0,&\theta\leq\theta(k_{0}).\end{array}\right.$ (A.39)
Then
$\frac{h}{v}(\theta(k))=\int_{-\infty}^{\infty}e^{is\theta(k)}\widehat{(\frac{h}{v})}(s)\bar{d}s,\quad
k\geq k_{0},$ (A.40)
where
$\widehat{(\frac{h}{v})}(s)=\int_{\theta(k_{0})}^{\infty}e^{-is\theta(k)}\frac{h}{v}(\theta(k))\bar{d}\theta(k).$
(A.41)
Moreover, we have
$\frac{h}{v}(\theta(k))=\frac{(k-k_{0})^{3q+2}}{(k-i)^{3q+4}}g(k,k_{0}),$
(A.42)
where
$g(k,k_{0})=\frac{1}{m!}\int_{0}^{1}((\cdot-i)^{m+5}\rho(\cdot))^{(m+1)}(k_{0}+u(k-k_{0}))(1-u)^{k}du$
(A.43)
and
$\left|\frac{d^{j}g(k,k_{0})}{dk^{j}}\right|\leq C,\quad k\geq k_{0}.$ (A.44)
Using the identity $\left|\frac{k-k_{0}}{k+k_{0}}\right|\leq 1$ for $k\geq
k_{0}$, we have
$\begin{array}[]{rrl}\int_{\theta(k_{0})}^{\infty}\left|\left(\frac{d}{d\theta}\right)^{j}\left(\frac{h}{v}(\theta(k))\right)\right|^{2}\bar{d}\theta(k)&=&\int_{k_{0}}^{\infty}\left|\left(\frac{2k_{0}^{4}}{k\alpha\beta^{2}(1-\frac{k_{0}^{4}}{k^{4}})}\frac{d}{dk}\right)^{j}\frac{h}{v}(k)\right|^{2}\frac{k\alpha\beta^{2}(1-\frac{k_{0}^{4}}{k^{4}})}{2k_{0}^{4}}|\bar{d}k\\\
&\leq&c\int_{k_{0}}^{\infty}\left|\frac{(k-k_{0})^{3q+2-3j}}{(k-i)^{3q+4}}\right|^{2}k^{6j-3}(k^{4}-k_{0}^{4})\bar{d}k\leq
C_{1},\quad 0\leq j\leq\frac{3q+2}{3}.\end{array}$ (A.45)
Thus,
$\int_{-\infty}^{\infty}(1+s^{2})^{j}\left|\widehat{(\frac{h}{v})}(s)\right|^{2}\bar{d}s\leq
C<\infty.$ (A.46)
We write
$\begin{array}[]{rrl}h(k)&=&v(k)\int_{t}^{\infty}e^{is\theta(k)}\widehat{(h/v)}(s)\bar{d}s+v(k)\int_{-\infty}^{t}e^{is\theta(k)}\widehat{(h/v)}(s)\bar{d}s\\\
&=&h_{\@slowromancap i@}(k)+h_{\@slowromancap ii@}(k).\end{array}$ (A.47)
For $k\geq k_{0},0<k_{0}<M$, and any positive integer $e\leq\frac{3q+2}{3}$,
we obtain,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
i@}(k)|&\leq&\frac{|k-k_{0}|^{q}}{|k-i|^{q+2}}\int_{t}^{\infty}|\widehat{(h/v)}(s)|\bar{d}s\\\
&\leq&\frac{|k-k_{0}|^{q}}{|k-i|^{q+2}}(\int_{t}^{\infty}(1+s^{2})^{-e}\bar{d}s)^{\frac{1}{2}}(\int_{t}^{\infty}(1+s^{2})^{e}|\widehat{(h/v)}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&c\frac{1}{(1+|k|^{2})t^{e-\frac{1}{2}}}.\end{array}$ (A.48)
And $h_{\@slowromancap ii@}(k)$ has an analytic continuation to the lower
half-plane, where $\mathrm{Re}i\theta(k)$ is positive. We estimate
$e^{-2it\theta(k)}h_{\@slowromancap ii@}(k)$ on the contour
$k(u)=k_{0}+uk_{0}e^{-i\frac{\pi}{4}},u\geq 0$.
If $0<u\leq M_{1}$,
$|e^{-2it\theta(k)}h_{\@slowromancap ii@}(k)|\leq
c\frac{k_{0}^{q}u^{q}e^{-t\mathrm{Re}i\theta(k)}}{|k-i|^{q+2}},$ (A.49)
where
$\begin{array}[]{rrl}\mathrm{Re}i\theta(k)&=&-2\alpha\beta^{2}k_{1}k_{2}\frac{(k_{1}^{2}+k_{2}^{2})^{2}-k_{0}^{4}}{4k_{0}^{4}(k_{1}^{2}+k_{2}^{2})^{2}}\\\
&=&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{(u^{2}+\sqrt{2}u)^{2}(u^{2}+\sqrt{2}u+2)}{(u^{2}+\sqrt{2}u+1)^{2}}\\\
&\geq&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{4u^{2}}{(u^{2}+\sqrt{2}u+1)^{2}}.\end{array}$
(A.50)
Then
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&c\frac{k_{0}^{q}u^{q}e^{-t\mathrm{Re}i\theta(k)}}{|k-i|^{q+2}}\leq
c_{1}\frac{k_{0}^{q}u^{q}}{|k-i|^{q+2}}e^{-t\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{4u^{2}}{(u^{2}+\sqrt{2}u+1)^{2}}}\\\
&\leq&\frac{c_{2}}{(1+|k|^{2})^{q+2}t^{\frac{q}{2}}}\leq\frac{c_{2}}{(1+|k|^{2})t^{\frac{q}{2}}}\end{array}$
(A.51)
If $u>M_{1}$, then
$\mathrm{Re}i\theta(k)\geq\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{u^{6}}{(u^{2}+\sqrt{2}u+1)^{2}}$
(A.52)
and
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&c\frac{k_{0}^{q}u^{q}e^{-t\mathrm{Re}i\theta(k)}}{|k-i|^{q+2}}\leq
c_{3}\frac{k_{0}^{q}u^{q}}{|k-i|^{q+2}}e^{-t\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{u^{6}}{(u^{2}+\sqrt{2}u+1)^{2}}}\\\
&\leq&\frac{c_{4}}{(1+|k|^{2})t^{q}}\end{array}$ (A.53)
Hence, for $u>0$, we obtain
$|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|\leq\frac{c_{5}}{(1+|k|^{2})t^{\frac{q}{2}}}.$ (A.54)
4.$\frac{k_{0}}{2}<|k|<k_{0},k\in i{\mathbb{R}}$.
We just consider $\frac{k_{0}}{2}<\mathrm{Im}k<k_{0}$, the case for
$-k_{0}<\mathrm{Im}k<-\frac{k_{0}}{2}$ is similarly.
Set
$\rho(k)=\frac{-r(k)}{1+|r(k)|^{2}},\quad\frac{k_{0}}{2}<\mathrm{Im}k<k_{0},k\in
i{\mathbb{R}}.$ (A.55)
The following process is similar as the case $\frac{k_{0}}{2}<k<k_{0}$. That
is,
We split $\rho(k)$ into even and odd parts,
$\rho(k)=H_{e}(k^{2})+kH_{o}(k^{2})$, where $H_{e}(\cdot)$ and $H_{o}(\cdot)$
are of the Schwartz class.
For any positive integer $m$,
$H_{e}(k^{2})=\mu_{0}^{e}+\mu_{1}^{e}(k^{2}+k_{0}^{2})+\cdots+\mu_{m}^{e}(k^{2}+k_{0}^{2})^{m}+\frac{1}{m!}\int_{k_{0}^{2}}^{k^{2}}H_{e}^{(m+1)}(\gamma)(k^{2}-\gamma)^{m}d\gamma$
(A.56)
and
$H_{o}(k^{2})=\mu_{0}^{o}+\mu_{1}^{o}(k^{2}+k_{0}^{2})+\cdots+\mu_{m}^{o}(k^{2}+k_{0}^{2})^{m}+\frac{1}{m!}\int_{k_{0}^{2}}^{k^{2}}H_{o}^{(m+1)}(\gamma)(k^{2}-\gamma)^{m}d\gamma.$
(A.57)
Set
$R(k)=R_{m}(k)=\sum_{i=0}^{m}\mu_{i}^{e}(k^{2}+k_{0}^{2})^{i}+k\sum_{i=0}^{m}\mu_{i}^{o}(k^{2}+k_{0}^{2})^{i}.$
(A.58)
Assume $m=4q+1$, where $q$ is a positive integer. Write
$\rho(k)=h(k)+R(k),\quad\frac{k_{0}}{2}<\mathrm{Im}k<k_{0},k\in
i{\mathbb{R}}.$ (A.59)
Then
$\left.\frac{d^{j}h(k)}{dk^{j}}\right|_{\pm ik_{0}}=0,\quad 0\leq j\leq m.$
(A.60)
And we have
$h(k)=\frac{(k^{2}+k_{0}^{2})^{m+1}}{m!}g(k,k_{0})$ (A.61)
where
$g(k,ik_{0})=\left(\int_{0}^{1}H_{e}^{(m+1)}(-k_{0}^{2}+u(k^{2}+k_{0}^{2}))(1-u)^{m}du+k\int_{0}^{1}H_{o}^{(m+1)}(-k_{0}^{2}+u(k^{2}+k_{0}^{2}))(1-u)^{m}du\right)$
(A.62)
and
$\left|\frac{d^{j}g(k,ik_{0})}{dk^{j}}\right|\leq
C,\quad\frac{k_{0}}{2}\leq\mathrm{Im}k\leq k_{0}.$ (A.63)
We will split $h$ as $h(k)=h_{\@slowromancap i@}(k)+h_{\@slowromancap
ii@}(k)$, where $h_{\@slowromancap i@}$ is small and $h_{\@slowromancap ii@}$
has an analytic continuation to $\mathrm{Re}k>0$. Thus
$\rho=h_{\@slowromancap i@}+(h_{\@slowromancap ii@}+R).$ (A.64)
Set $p(k)=(k^{2}+k_{0}^{2})^{q}$. Recall
$\begin{array}[]{rrl}\theta(k)&=&k^{2}(\frac{x}{t}+\alpha)+\frac{\alpha\beta^{2}}{4k^{2}}-\alpha\beta\\\
&=&\frac{\alpha\beta^{2}}{4k_{0}^{4}}k^{2}+\frac{\alpha\beta^{2}}{4k^{2}}-\alpha\beta.\end{array}$
(A.65)
We define
$\left\\{\begin{array}[]{rrll}\frac{h}{p}(\theta)&=&\frac{h(k(\theta))}{p(k(\theta))},&\theta(ik_{0})<\theta<\theta(\frac{ik_{0}}{2}),\\\
&=&0,&\theta\leq\theta(ik_{0})\quad
or\quad\theta\geq\theta(\frac{ik_{0}}{2}).\end{array}\right.$ (A.66)
As
$|\theta|\rightarrow\theta(ik_{0})=-\frac{\alpha\beta^{2}}{2k_{0}^{2}}-\alpha\beta$
and $|\theta|>-\frac{\alpha\beta^{2}}{2k_{0}^{2}}-\alpha\beta$, we have
$\frac{h}{p}(\theta)=O((k^{2}(\theta)+k_{0}^{2})^{m+1-q})$ and
$\frac{d\theta}{dk}=\frac{\alpha\beta^{2}(k^{4}-k_{0}^{4})}{2k_{0}^{4}k^{3}}.$
(A.67)
We claim that $\frac{h}{p}\in H^{j}(-\infty<\theta<\infty)$ for $0\leq
j\leq\frac{3q+2}{2}$. As by Fourier inversion,
$\frac{h}{p}(k)=\int_{-\infty}^{\infty}e^{is\theta(k)}\widehat{(\frac{h}{p})}(s)\bar{d}s,\quad\frac{k_{0}}{2}<\mathrm{Im}k<k_{0},$
(A.68)
where
$\widehat{(\frac{h}{p})}(s)=\int_{\theta(ik_{0})}^{\theta(\frac{ik_{0}}{2})}e^{-is\theta(k)}\frac{h}{p}(\theta(k))\bar{d}\theta(k),\quad
s\in{\mathbb{R}}.$ (A.69)
where $\bar{d}s=\frac{ds}{\sqrt{2\pi}}$ and
$\bar{d}\theta(k)=\frac{d\theta(k)}{\sqrt{2\pi}}$.
Thus,
$\begin{array}[]{l}\int_{\theta(ik_{0})}^{\theta(i\frac{k_{0}}{2})}\left|\left(\frac{d}{d\theta}\right)^{j}\frac{h}{p}(\theta(k))\right|^{2}|\bar{d}\theta(k)|\\\
=\int_{i\frac{k_{0}}{2}}^{ik_{0}}\left|\left(\frac{2k_{0}^{4}k^{3}}{\alpha\beta^{2}(k^{4}-k_{0}^{4})}\frac{d}{dk}\right)^{j}\frac{h}{p}(k)\right|^{2}|\frac{\alpha\beta^{2}(k^{4}-k_{0}^{4})}{2k_{0}^{4}k^{3}}|\bar{d}k\leq
C<\infty,\end{array}$ (A.70)
for $0<k_{0}<M,0\leq j\leq\frac{3q+2}{2}$. Hence,
$\int_{-\infty}^{\infty}(1+s^{2})^{j}|\widehat{(h/p)}(s)|^{2}ds\leq C<\infty$
(A.71)
for $0<k_{0}<M,0\leq j\leq\frac{3q+2}{2}$.
Split
$\begin{array}[]{rrl}h(k)&=&p(k)\int_{t}^{\infty}e^{is\theta(k)}\widehat{(h/p)}(s)\bar{d}s+p(k)\int_{-\infty}^{t}e^{is\theta(k)}\widehat{(h/p)}(s)\bar{d}s\\\
&=&h_{\@slowromancap i@}(k)+h_{\@slowromancap ii@}(k).\end{array}$ (A.72)
Thus, for $\frac{k_{0}}{2}<\mathrm{Im}k<k_{0}\leq M$ and any positive integer
$n\leq\frac{3q+2}{2}$.
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
i@}(k)|&\leq&|p(k)|\int_{t}^{\infty}|\widehat{(h/p)}(s)|\bar{d}s\\\
&\leq&|p(k)|(\int_{t}^{\infty}(1+s^{2})^{-n}\bar{d}s)^{\frac{1}{2}}(\int_{t}^{\infty}(1+s^{2})^{n}|\widehat{(h/p)}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&\frac{c}{t^{n-\frac{1}{2}}}.\end{array}$ (A.73)
Consider the contour
$l^{\prime}_{1}:k(u)=ik_{0}+uk_{0}e^{-i\frac{\pi}{4}},0\leq
u\leq\frac{1}{\sqrt{2}}$. Since $\mathrm{Re}i\theta(k)$ is positive on this
contour, $h_{\@slowromancap ii@}(k)$ has an analytic continuation to contours
$l^{\prime}_{1}$.
On the contour $l^{\prime}_{1}$,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&|k+ik_{0}|^{q}(k_{0}u)^{q}e^{-t\mathrm{Re}i\theta(k)}\int_{-\infty}^{t}e^{(s-t)\mathrm{Re}i\theta(k)}\widehat{(h/p)}(s)\bar{d}s\\\
&\leq&ck_{0}^{2q}u^{q}e^{-t\mathrm{Re}i\theta(k)}(\int_{-\infty}^{t}(1+s^{2})^{-1}\bar{d}s)^{\frac{1}{2}}(\int_{-\infty}^{t}(1+s^{2})|\widehat{(h/p)}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&ck_{0}^{2q}u^{q}e^{-t\mathrm{Re}i\theta(k)}.\end{array}$ (A.74)
Recall
$\theta(k)=\frac{\alpha\beta^{2}}{4k_{0}^{4}}k^{2}+\frac{\alpha\beta^{2}}{4k^{2}}-\alpha\beta$,
and set $k=k_{1}+ik_{2}$, thus
$\begin{array}[]{rrl}\mathrm{Re}i\theta(k)&=&-2\alpha\beta^{2}k_{1}k_{2}\frac{(k_{1}^{2}+k_{2}^{2})^{2}-k_{0}^{4}}{4k_{0}^{4}(k_{1}^{2}+k_{2}^{2})^{2}}\\\
&=&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{(u^{2}-\sqrt{2}u)^{2}(u^{2}-\sqrt{2}u+2)}{(u^{2}-\sqrt{2}u+1)^{2}}\\\
&\geq&\frac{\alpha\beta^{2}u^{2}}{2k_{0}^{2}},\end{array}$ (A.75)
for $0\leq u\leq\frac{1}{\sqrt{2}}$.
Thus, on the contour $l^{\prime}_{1}$
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&ck_{0}^{2q}u^{q}e^{-t\frac{\alpha\beta^{2}u^{2}}{2k_{0}^{2}}}\leq
ck_{0}^{2q}u^{q}e^{-t\frac{\alpha\beta^{2}u^{2}}{2M^{2}}}\\\
&\leq&\frac{c_{1}}{t^{\frac{q}{2}}},\end{array}$ (A.76)
for $k_{0}<M$.
Fix $\varepsilon$, $0<\varepsilon<\frac{1}{\sqrt{2}}$. If $k(u)$ is on the
contour $l^{\prime}_{1}$ , $\varepsilon<u<\frac{1}{\sqrt{2}}$, then we obtain
$|e^{-2it\theta(k)}R(k)|\leq ce^{-\frac{\alpha\beta^{2}u^{2}}{k_{0}^{2}}t}\leq
ce^{-\frac{\varepsilon^{2}}{M^{2}}t}$ (A.77)
5\. $0<|k|<\frac{k_{0}}{2},k\in i{\mathbb{R}}$.
We consider $0<\mathrm{Im}k<\frac{k_{0}}{2},k\in i{\mathbb{R}}$, the case for
$-\frac{k_{0}}{2}<\mathrm{Im}k<0$ is similarly.
Define
$\left\\{\begin{array}[]{rrll}\rho(\theta)&=&\rho(k(\theta)),&\theta>\theta(i\frac{k_{0}}{2}),\\\
&=&0,&\theta\leq\theta(i\frac{k_{0}}{2}).\end{array}\right.$ (A.78)
We claim that $\rho(\theta)\in H^{j}(-\infty<\theta<\infty)$ for any
nonnegative integer $j$.
By Fourier inversion,
$\rho(\theta(k))=\int_{-\infty}^{\infty}e^{is\theta(k)}\hat{\rho}(s)\bar{d}s,\quad
0<\mathrm{Im}k<\frac{k_{0}}{2},$ (A.79)
where
$\hat{\rho}(s)=\int_{\theta(i\frac{k_{0}}{2})}^{\infty}e^{-is\theta(k)}\rho(\theta(k))\bar{d}\theta(k).$
(A.80)
Then,
$\begin{array}[]{l}\int_{\theta(i\frac{k_{0}}{2})}^{\infty}\left|\left(\frac{d}{d\theta}\right)^{j}\rho(\theta(k))\right|^{2}|\bar{d}\theta(k)|\\\
=\int_{0}^{i\frac{k_{0}}{2}}\left|\left(\frac{2k_{0}^{4}k^{3}}{\alpha\beta^{2}(k^{4}-k_{0}^{4})}\frac{d}{dk}\right)^{j}\rho(k)\right|^{2}|\frac{\alpha\beta^{2}(k^{4}-k_{0}^{4})}{2k_{0}^{4}k^{3}}|\bar{d}k\leq
C<\infty,\end{array}$ (A.81)
for any nonnegative integer $j$, $0<k_{0}<M$, since $r(k)\rightarrow 0$
rapidly, as $k\rightarrow 0$.
Hence
$\int_{-\infty}^{\infty}(1+s^{2})^{j}|\hat{\rho}(s)|^{2}\bar{d}s\leq C,$
(A.82)
for any nonnegative integer $j$.
Split
$\begin{array}[]{rrl}\rho(k)&=&\int_{t}^{\infty}e^{is\theta(k)}\hat{\rho}(s)\bar{d}s+\int_{-\infty}^{t}e^{is\theta(k)}\hat{\rho}(s)\bar{d}s\\\
&=&h_{\@slowromancap i@}(k)+h_{\@slowromancap ii@}(k).\end{array}$ (A.83)
Then, for $0<\mathrm{Im}k<i\frac{k_{0}}{2}$ and any positive integer $j$, we
obtain,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
i@}(k)|&\leq&\int_{t}^{\infty}|\hat{\rho}|\bar{d}s\\\
&\leq&(\int_{t}^{\infty}(1+s^{2})^{-j}\bar{d}s)^{\frac{1}{2}}(\int_{t}^{\infty}(1+s^{2})^{j}|\hat{\rho}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&\frac{c}{t^{j-\frac{1}{2}}}.\end{array}$ (A.84)
Consider the contour
$l^{\prime}_{2}:k(u)=uk_{0}e^{i\frac{\pi}{4}},0<u<\frac{1}{\sqrt{2}}$. Since
$\mathrm{Re}i\theta(k)$ is positive on this contour, $h_{\@slowromancap ii@}$
has an analytic continuation to contour $l^{\prime}_{2}$.
On the contour $l^{\prime}_{2}$,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&e^{-t\mathrm{Re}i\theta(k)}\int_{-\infty}^{t}e^{(s-t)\mathrm{Re}i\theta(k)}|\hat{\rho}(k)|\bar{d}s\\\
&\leq&e^{-t\mathrm{Re}i\theta(k)}(\int_{-\infty}^{t}(1+s^{2})^{-1}\bar{d}s)^{\frac{1}{2}}(\int_{-\infty}^{t}(1+s^{2})|\hat{\rho}(k)|^{2}\bar{d}s)^{\frac{1}{2}},\end{array}$
(A.85)
where
$\begin{array}[]{rrl}\mathrm{Re}i\theta(k)&=&-2\alpha\beta^{2}k_{1}k_{2}\frac{(k_{1}^{2}+k_{2}^{2})^{2}-k_{0}^{4}}{4k_{0}^{4}(k_{1}^{2}+k_{2}^{2})^{2}}\\\
&=&-\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{u^{4}-1}{u^{2}}\\\
&\geq&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\end{array}$ (A.86)
for $0<u\leq\frac{1}{\sqrt{2}}$.
Thus, we obtain,
$|e^{-2it\theta(k)}h_{\@slowromancap ii@}(k)|\leq
ce^{-t\frac{\alpha\beta^{2}}{4k_{0}^{2}}}.$ (A.87)
6\. $|k|>k_{0},k\in i{\mathbb{R}}$.
We consider $\mathrm{Im}k>k_{0},k\in{\mathbb{R}}$, the case for
$\mathrm{Im}k<-k_{0}$ is similarly.
Set
$\rho(k)=r(k).$ (A.88)
We write
$(k+1)^{m+5}\rho(k)=\mu_{0}+\mu_{1}(k-ik_{0})+\cdots+\mu_{m}(k-ik_{0})^{m}+\frac{1}{m!}\int_{k_{0}}^{k}((\cdot+1)^{m+5}\rho(\cdot))^{(m+1)}(\gamma)(k-\gamma)^{m}d\gamma.$
(A.89)
Define
$R(k)=\frac{\sum_{i=0}^{m}\mu_{i}(k-ik_{0})^{i}}{(k+1)^{m+5}}$ (A.90)
and write $\rho(k)=h(k)+R(k)$. We have
$\left.\frac{d^{j}h(k)}{dk^{j}}\right|_{ik_{0}}=0,\quad 0\leq j\leq m.$ (A.91)
For $0<k_{0}<M$, set
$v(k)=\frac{(k-ik_{0})^{q}}{(k+1)^{q+2}}.$ (A.92)
Let
$\left\\{\begin{array}[]{rrll}\frac{h}{v}(\theta)&=&\frac{h}{v}(k(\theta)),&\theta>\theta(ik_{0}),\\\
&=&0,&\theta\leq\theta(ik_{0}).\end{array}\right.$ (A.93)
Then
$\frac{h}{v}(\theta(k))=\int_{-\infty}^{\infty}e^{is\theta(k)}\widehat{(\frac{h}{v})}(s)\bar{d}s,\quad
k\geq k_{0},$ (A.94)
where
$\widehat{(\frac{h}{v})}(s)=\int_{\theta(ik_{0})}^{\infty}e^{-is\theta(k)}\frac{h}{v}(\theta(k))\bar{d}\theta(k).$
(A.95)
Moreover, we have
$\frac{h}{v}(\theta(k))=\frac{(k-ik_{0})^{3q+2}}{(k+1)^{3q+4}}g(k,ik_{0}),$
(A.96)
where
$g(k,ik_{0})=\frac{1}{m!}\int_{0}^{1}((\cdot-i)^{m+5}\rho(\cdot))^{(m+1)}(ik_{0}+u(k-ik_{0}))(1-u)^{k}du$
(A.97)
and
$\left|\frac{d^{j}g(k,ik_{0})}{dk^{j}}\right|\leq C,\quad\mathrm{Im}k\geq
k_{0}.$ (A.98)
Using the identity $\left|\frac{k-ik_{0}}{k+ik_{0}}\right|\leq 1$ for
$\mathrm{Im}k\geq k_{0}$, we have
$\begin{array}[]{rrl}\int_{\theta(ik_{0})}^{\infty}\left|\left(\frac{d}{d\theta}\right)^{j}\left(\frac{h}{v}(\theta(k))\right)\right|^{2}\bar{d}\theta(k)&=&\int_{ik_{0}}^{\infty}\left|\left(\frac{2k_{0}^{4}}{k\alpha\beta^{2}(1-\frac{k_{0}^{4}}{k^{4}})}\frac{d}{dk}\right)^{j}\frac{h}{v}(k)\right|^{2}\frac{k\alpha\beta^{2}(1-\frac{k_{0}^{4}}{k^{4}})}{2k_{0}^{4}}|\bar{d}k\\\
&\leq&c\int_{ik_{0}}^{\infty}\left|\frac{(k-ik_{0})^{3q+2-3j}}{(k+1)^{3q+4}}\right|^{2}k^{6j-3}(k^{4}-k_{0}^{4})\bar{d}k\leq
C_{1},\quad 0\leq j\leq\frac{3q+2}{3}.\end{array}$ (A.99)
Thus,
$\int_{-\infty}^{\infty}(1+s^{2})^{j}\left|\widehat{(\frac{h}{v})}(s)\right|^{2}\bar{d}s\leq
C<\infty.$ (A.100)
We write
$\begin{array}[]{rrl}h(k)&=&v(k)\int_{t}^{\infty}e^{is\theta(k)}\widehat{(h/v)}(s)\bar{d}s+v(k)\int_{-\infty}^{t}e^{is\theta(k)}\widehat{(h/v)}(s)\bar{d}s\\\
&=&h_{\@slowromancap i@}(k)+h_{\@slowromancap ii@}(k).\end{array}$ (A.101)
For $\mathrm{Im}k\geq k_{0},k\in i{\mathbb{R}},0<k_{0}<M$, and any positive
integer $e\leq\frac{3q+2}{3}$, we obtain,
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
i@}(k)|&\leq&\frac{|k-ik_{0}|^{q}}{|k+1|^{q+2}}\int_{t}^{\infty}|\widehat{(h/v)}(s)|\bar{d}s\\\
&\leq&\frac{|k-ik_{0}|^{q}}{|k+1|^{q+2}}(\int_{t}^{\infty}(1+s^{2})^{-e}\bar{d}s)^{\frac{1}{2}}(\int_{t}^{\infty}(1+s^{2})^{e}|\widehat{(h/v)}(s)|^{2}\bar{d}s)^{\frac{1}{2}}\\\
&\leq&c\frac{1}{(1+|k|^{2})t^{e-\frac{1}{2}}}.\end{array}$ (A.102)
And $h_{\@slowromancap ii@}(k)$ has an analytic continuation to the left half-
plane, where $\mathrm{Re}i\theta(k)$ is positive. We estimate
$e^{-2it\theta(k)}h_{\@slowromancap ii@}(k)$ on the contour
$k(u)=ik_{0}+uk_{0}e^{i\frac{3\pi}{4}},u\geq 0$.
If $0<u\leq M_{1}$,
$|e^{-2it\theta(k)}h_{\@slowromancap ii@}(k)|\leq
c\frac{k_{0}^{q}u^{q}e^{-t\mathrm{Re}i\theta(k)}}{|k-i|^{q+2}},$ (A.103)
where
$\begin{array}[]{rrl}\mathrm{Re}i\theta(k)&=&-2\alpha\beta^{2}k_{1}k_{2}\frac{(k_{1}^{2}+k_{2}^{2})^{2}-k_{0}^{4}}{4k_{0}^{4}(k_{1}^{2}+k_{2}^{2})^{2}}\\\
&=&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{(u^{2}+\sqrt{2}u)^{2}(u^{2}+\sqrt{2}u+2)}{(u^{2}+\sqrt{2}u+1)^{2}}\\\
&\geq&\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{4u^{2}}{(u^{2}+\sqrt{2}u+1)^{2}}.\end{array}$
(A.104)
Then
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&c\frac{k_{0}^{q}u^{q}e^{-t\mathrm{Re}i\theta(k)}}{|k+1|^{q+2}}\leq
c_{1}\frac{k_{0}^{q}u^{q}}{|k+1|^{q+2}}e^{-t\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{4u^{2}}{(u^{2}+\sqrt{2}u+1)^{2}}}\\\
&\leq&\frac{c_{2}}{(1+|k|^{2})^{q+2}t^{\frac{q}{2}}}\leq\frac{c_{2}}{(1+|k|^{2})t^{\frac{q}{2}}}\end{array}$
(A.105)
If $u>M_{1}$, then
$\mathrm{Re}i\theta(k)\geq\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{u^{6}}{(u^{2}+\sqrt{2}u+1)^{2}}$
(A.106)
and
$\begin{array}[]{rrl}|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|&\leq&c\frac{k_{0}^{q}u^{q}e^{-t\mathrm{Re}i\theta(k)}}{|k+1|^{q+2}}\leq
c_{3}\frac{k_{0}^{q}u^{q}}{|k-i|^{q+2}}e^{-t\frac{\alpha\beta^{2}}{4k_{0}^{2}}\frac{u^{6}}{(u^{2}+\sqrt{2}u+1)^{2}}}\\\
&\leq&\frac{c_{4}}{(1+|k|^{2})t^{q}}\end{array}$ (A.107)
Hence, for $u>0$, we obtain
$|e^{-2it\theta(k)}h_{\@slowromancap
ii@}(k)|\leq\frac{c_{5}}{(1+|k|^{2})t^{\frac{q}{2}}}.$ (A.108)
Note that if $l$ is an arbitrary positive integer, we can choose $m$ large
enough such that $\frac{3q+2}{2}-\frac{1}{2}>q-\frac{1}{2}>\frac{q}{2}>l$ and
Proposition 3.2 holds.
Prove Proposition 3.11.
###### Proof.
Write
$\begin{array}[]{l}\bar{R}\left(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0}\right)(\delta^{1}_{A}(k))^{-2}-\bar{R}(k_{0}\pm)k^{-2i\nu}e^{i\frac{k^{2}}{2}}=\\\
e^{i\frac{\kappa}{2}k^{2}}e^{i\frac{\kappa}{2}k^{2}}\bar{R}\left(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0}\right)k^{-2i\nu}e^{i(1-2\kappa)\frac{k^{2}}{2}(1-\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}\\\
{}\frac{k_{0}^{-4i\tilde{\nu}-2i\nu}}{2^{-2i\nu+2i\tilde{\nu}}}\frac{(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{-2i\nu}}{(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{-4i\nu}}((\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0}+ik_{0})(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0}-ik_{0}))^{2i\tilde{\nu}}\\\
{}e^{-2\left(\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\chi_{\pm}(k_{0})\right)}e^{-2\left(\tilde{\chi}_{\pm}^{\prime}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\tilde{\chi}_{\pm}^{\prime}(k_{0})\right)}\\\
{}-e^{i\frac{\kappa}{2}k^{2}}e^{i\frac{\kappa}{2}k^{2}}\bar{R}(k_{0}\pm)k^{-2i\nu}e^{i(1-2\kappa)\frac{k^{2}}{2}}\end{array}$
(A.109)
and also divided it into six terms
$\bar{R}\left(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0}\right)(\delta^{1}_{A}(k))^{-2}-\bar{R}(k_{0}\pm)k^{-2i\nu}e^{i\frac{k^{2}}{2}}=e^{i\kappa\frac{k^{2}}{2}}(\@slowromancap
i@+\@slowromancap ii@+\@slowromancap iii@+\@slowromancap iv@+\@slowromancap
v@+\@slowromancap vi@)$ (A.110)
where
$\begin{array}[]{l}\@slowromancap
i@=e^{i\kappa\frac{k^{2}}{2}}k^{-2i\nu}[\bar{R}(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\bar{R}(k_{0}\pm)]\\\ \@slowromancap
ii@=e^{i\kappa\frac{k^{2}}{2}}k^{-2i\nu}\bar{R}(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})\left(e^{i(1-2\kappa)\frac{k^{2}}{2}(1-\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}-e^{i(1-2\kappa)\frac{k^{2}}{2}}\right)\\\ \@slowromancap
iii@=e^{i\kappa\frac{k^{2}}{2}}k^{-2i\nu}\bar{R}(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})e^{i(1-2\kappa)\frac{k^{2}}{2}(1-\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}\left(\frac{2^{2i\nu}}{k_{0}^{2i\nu}}\frac{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{4i\nu}}{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{2i\nu}}-1\right)\\\ \@slowromancap
iv@=e^{i\kappa\frac{k^{2}}{2}}k^{-2i\nu}\bar{R}(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})e^{i(1-2\kappa)\frac{k^{2}}{2}(1-\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}\\\
{}\frac{2^{2i\nu}}{k_{0}^{2i\nu}}\frac{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{4i\nu}}{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{2i\nu}}\left(\frac{((\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{2}+k_{0}^{2})^{2i\tilde{\nu}}}{2^{2i\tilde{\nu}}k_{0}^{4i\tilde{\nu}}}-1\right)\\\
\@slowromancap
v@=e^{i\kappa\frac{k^{2}}{2}}k^{-2i\nu}\bar{R}(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})e^{i(1-2\kappa)\frac{k^{2}}{2}(1-\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}\frac{2^{2i\nu}}{k_{0}^{2i\nu}}\frac{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{4i\nu}}{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{2i\nu}}\\\ {}\frac{((\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{2}+k_{0}^{2})^{2i\tilde{\nu}}}{2^{2i\tilde{\nu}}k_{0}^{4i\tilde{\nu}}}\left(e^{-2\left(\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\chi_{\pm}(k_{0})\right)}-1\right)\\\ \@slowromancap
vi@=e^{i\kappa\frac{k^{2}}{2}}k^{-2i\nu}\bar{R}(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})e^{i(1-2\kappa)\frac{k^{2}}{2}(1-\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}\frac{2^{2i\nu}}{k_{0}^{2i\nu}}\frac{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{4i\nu}}{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{2i\nu}}\\\ {}\frac{((\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{2}+k_{0}^{2})^{2i\tilde{\nu}}}{2^{2i\tilde{\nu}}k_{0}^{4i\tilde{\nu}}}e^{-2\left(\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\chi_{\pm}(k_{0})\right)}\left(e^{-2\left(\tilde{\chi}_{\pm}^{\prime}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\tilde{\chi}_{\pm}^{\prime}(k_{0})\right)}-1\right)\\\
\end{array}$
Note that $|e^{i\kappa\frac{k^{2}}{2}}|=e^{-\kappa u^{2}\frac{2\alpha
t\beta^{2}}{k_{0}^{2}}}$. The terms $\@slowromancap i@,\@slowromancap
ii@,\@slowromancap iii@,\@slowromancap iv@,\@slowromancap v@$ and
$\@slowromancap vi@$ can be estimated as follows.
$\begin{array}[]{rrl}|\@slowromancap
i@|&\leq&|k^{-2i\nu}||e^{i\kappa\frac{k^{2}}{2}}||\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k|||\partial_{k}\bar{R}(k)||_{L^{\infty}(\bar{L}_{A})}\\\
&\leq&\frac{C}{\sqrt{t}},\end{array}$
where $C$ is independent of $k$.
$\begin{array}[]{rrl}|\@slowromancap
ii@|&\leq&|k^{-2i\nu}||e^{i\kappa\frac{k^{2}}{2}}|||\bar{R}||_{L^{\infty}(\bar{L}_{A})}|\frac{d}{ds}e^{i(1-2\kappa)\frac{k^{2}}{2}(1-s\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}|,\quad 0<s<1\\\ &\leq&\frac{C}{\sqrt{t}}\end{array}$
To estimate $\@slowromancap iii@$, we write
$\begin{array}[]{rrl}|\@slowromancap
iii@|&\leq&|k^{-2i\nu}||e^{i\kappa\frac{k^{2}}{2}}|||\bar{R}||_{L^{\infty}(\bar{L}_{A})}|e^{i(1-2\kappa)\frac{k^{2}}{2}(1-\frac{4k_{0}^{6}k}{(1-2\kappa)2\sqrt{\alpha
t}\beta\eta^{5}})}|(\@slowromancap iii@_{1}+\@slowromancap iii@_{2})\\\
\end{array}$
where
$\begin{array}[]{l}\@slowromancap
iii@_{1}=\frac{2^{2i\nu}}{k_{0}^{2i\nu}}\frac{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{4i\nu}}{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{2i\nu}}\left[1-\frac{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{2i\nu}}{(2k_{0})^{2i\nu}}\right]\\\ \@slowromancap
iii@_{2}=\left(\frac{\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0}}{k_{0}}\right)^{4i\nu}-1\end{array}$
The estimate of $\@slowromancap iii@_{2}$ is as follows,
$\begin{array}[]{rrl}|\@slowromancap
iii@_{2}|&=&|\int_{1}^{1+\frac{k_{0}}{2\sqrt{\alpha
t}\beta}k}4i\nu\xi^{4i\nu-1}d\xi|\\\ &\leq&\frac{C}{\sqrt{t}},\end{array}$
as $|\xi^{4i\nu-1}|\leq ce^{-4\nu arg\xi}\leq c$ for
$\xi=1+sk\frac{k_{0}}{2\sqrt{\alpha t}\beta}=1+sue^{i\frac{\pi}{4}},0\leq
s\leq 1,-\varepsilon<u<\infty$. Since the first term on the right-hand side of
the equation for $\@slowromancap iii@_{1}$ is bounded, namely,
$\left|\frac{2^{2i\nu}}{k_{0}^{2i\nu}}\frac{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+k_{0})^{4i\nu}}{(\frac{k_{0}^{2}}{2\sqrt{\alpha
t}\beta}k+2k_{0})^{2i\nu}}\right|\leq e^{\frac{\pi\nu}{2}}$
one obtains an analogous estimate for $\@slowromancap iii@_{1}$. And the
estimate for $\@slowromancap iv@$ is similar as $\@slowromancap iii@$.
$\begin{array}[]{rrl}|\@slowromancap v@|&\leq&C\sup_{0\leq s\leq
1}|e^{-2se^{-2\left(\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\chi_{\pm}(k_{0})\right)}}|\left|2e^{i\kappa\frac{k^{2}}{2}}(\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\chi_{\pm}(k_{0}))\right|\end{array}$
using the Lipschitz property of the function
$\log\left(\frac{1-|r(\xi)|^{2}}{1-|r(k_{0})|^{2}}\right),|\xi|\leq k_{0}$,
integrating by parts shows that
$\begin{array}[]{rrl}\left|2e^{i\kappa\frac{k^{2}}{2}}(\chi_{\pm}(\frac{k^{2}_{0}}{2\sqrt{\alpha
t}\beta}k+k_{0})-\chi_{\pm}(k_{0}))\right|&\leq&C\frac{\log
t}{\sqrt{t}},\end{array}$
The analogous estimates for $\@slowromancap vi@$ can be also obtained. ∎
## References
* [1] A. S. Fokas, On a class of physically important integrable equations, Physica D 87(1995), 145-150.
* [2] J. Lenells, Exactly solvable model for nonlinear pulse propagation in optical fibers, Stud. Appl. Math.123 (2009), 215-232.
* [3] J. Lenells, Dressing for a novel integrable generalization of the nonlinear Schrödinger equation, J. Nonlinear Sci.20 (2010), 709-722.
* [4] J. Lenells and A. S. Fokas, On a novel integrable generalization of the nonlinear Schrödinger equation, Nonlinearity22 (2009), 11-27.
* [5] J. Lenells and A. S. Fokas, An integrable generalization of the nonlinear Schr odinger equation on the half-line and solitons, Inverse Problems25 (2009), 115006 (32pp).
* [6] V. E. Vekslerchik, Lattice representation and dark solitons of the Fokas–Lenells equation, Nonlinearity 24 (2011), 1165.
* [7] Y. Matsuno, A direct method of solution for the Fokas Lenells derivative nonlinear Schrödinger equation: I. Bright soliton solutions, J. Phys. A: Math. Theor. 45 (2012) 235202 (19pp).
* [8] Y. Matsuno, A direct method of solution for the Fokas Lenells derivative nonlinear Schrödinger equation: I. Dark soliton solutions, J. Phys. A: Math. Theor. 45 (2012) 475202 (31pp).
* [9] P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems, Ann. of Math. (2) 137(1993), 295-368.
* [10] P. A. Deift, A. R. Its, and X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, in “Important developments in soliton theory”, 181-204, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993.
* [11] R. Beals and R. Coifman, Scattering and inverse scattering for first order systems, Comm. in Pure and Applied Math. 37(1984), 39–90.
* [12] Schuur P C 1986 Asymptotic Analysis of Soliton Problems (Lecture Notes in Mathematics 1232) (Berlin: Springer)
Bikbaev R F 1988 Theor. Math. Phys. 77 1117 23
Fokas A S and Its A R 1992 Phys. Rev. Lett. 68 3117 20
Rybin A and Timonen J 1993 J. Phys. A: Math. Gen. 26 3869 82
* [13] Manakov S V 1974 Sov. Phys. JETP 38 693 6
Zakharov V E and Manakov S V 1976 Sov. Phys. JETP 44 106 12
Its A R 1981 Sov. Math. Dokl. 24 452 6
* [14] Zakharov V E and Shabat A B 1979 Funct. Anal. Appl. 13 166 74
* [15] Po-Jen Cheng, S. Venakides and X. Zhou, Long-time asymptotics for the pure radiation solution of the Sine–Gordon equation, Comm. Part. Diff. Equ. 24(1999), 1195-1262.
* [16] A. V. Kitaev and A. H. Vartanian, Leading-order temporal asymptotics of the modified nonlinear Schrödinger equation:solitonless sector, Inverse Problems 13(1997),1311-1339.
* [17] A.V.Kitaev and A.H.Vartanian, Asymptotics of solutions to the modified nonlinear Schrödinger equation: Solution on a nonvanishing continuous background, SIAM Journal of Mathematical Analysis. 30,no.4(1999),787-832.
* [18] A.V.Kitaev and A.H.Vartanian, Higher order asymptotics of the modified non-linear schrödinger equation, Comm. Part. Diff. Equ. 25(2000), 1043-1098.
* [19] K. Grunert and G. Teschl, Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent, Math. Phys. Anal. Geom. 12(2009), 287-324.
* [20] A. Boutet de Monvel, A. Kostenko, D. Shepelsky, and G. Teschl, Long-Time Asymptotics for the Camassa-Holm Equation, SIAM J. Math. Anal. 41(4)(2009), 1559-1588.
* [21] J. Xu and E. Fan, Long-Time Asymptotics for the Fokas-Lenells Equation with Shock Problem, in preparation.
|
arxiv-papers
| 2013-08-03T22:41:16 |
2024-09-04T02:49:48.953249
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian Xu and Engui Fan",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1308.0755"
}
|
1308.0809
|
# Improved Nucleon Properties in the Extended Quark Sigma Model
M. Abu-Shady Department of Mathematics, Faculty of Science, Menoufia
University, Egypt
([; date; date; date; date)
###### Abstract
The quark sigma model describes the quarks interacting via exchange the pions
and sigma meson fields. A new version of mesonic potential is suggested in the
frame of some aspects of the quantum chromodynamics (QCD). The field equations
have been solved in the mean-field approximation for the hedgehog baryon
state. The obtained results are compared with previous works and other models.
We conclude that the suggested mesonic potential successfully calculates
nucleon properties..
Quark models, Chiral symmetry, Nucleon properties
###### pacs:
11.10 Wx, 12.39 Fe
††preprint: HEP/123-qed
year number number identifier Date text]date
LABEL:FirstPage101 LABEL:LastPage#1102
###### Contents
1. I $\mathbf{Introduction}$
2. II The Chiral-Quark Sigma Model
3. III The Chiral Higher-Order Quark Sigma Model
4. IV Numerical Calculations and Discussions
1. IV.1 The scalar field $\sigma^{\prime}$
2. IV.2 The pion field $\mathbf{\pi}$
3. IV.3 The Properties of the Nucleon
4. IV.4 Discussion of the Results
5. V Comparison with Other Models
6. VI Conclusion
7. VII References
## I $\mathbf{Introduction}$
The description of the processes involving strong interactions is very
difficult in the frame of the quantum chromodynamics (QCD) due to its non-
abelian color and flavor structure and strong coupling constants. These
effective models, like quark sigma model, which are constructed in such a way
as to respect general properties from the more fundamental theory (QCD), such
as the chiral symmetry and its spontaneous breaking [1]. It is known that the
linear sigma model of Gell-Mann and Levy [2] does not always give the correct
phenomenology such as the value of the isoscalar pion-nucleon scattering
length is too large as in Refs. [3-5]. Birse and Banerjee [3] constructed
equations of motion treating both $\sigma$ and $\mathbf{\pi}$ fields as time-
independence classical fields and the quarks in hedgehog spinor state. This
work is reexamined by Broniowski and Banerjee [4] with corrected numerical
errors in Ref. [3]. Birse [5] generalized this mean-field approximation to
include angular momentum and isospin projection.
Recently, the mesons play an important role for improving the nucleon
properties in the chiral quark models. In the framework of the perturbative
chiral quark model [6, 7] which extended to include the kaon and eta mesons
cloud contributions to analyze the electromagnetic structure of nucleon.
Horvat et al. [8] applied Tamm-Dancoff method to the chiral quark model which
extended to include additional degrees of freedom as a pseudoscalar isoscalar
field and a triplet of scalar isovector to provide a better description of
nucleon properties. In Refs. [9-11], the authors analyzed a particular
extension of the linear sigma model coupled to valence quarks in which
contained an additional term with gradients of the chiral fields and
investigated the dynamically consequence of this term and its relevant to the
phenomenology. In addition, Rashdan et al. [12, 13] 1and Abu-shady [14]
increased the order mesonic interactions in the chiral quark sigma model using
mean-field approximation to improve nucleon properties.
The aim of the paper is to introduce the suggested mesonic potential to
improve nucleon properties and avoid the difficulty which found in the
previous works. The paper is organized as follow: In the following Section, we
review briefly the linear sigma model. The higher-order mesonic interactions
are studied in details in Sec. 3. The numerical calculations and the
discussion of results are presented in Secs. 4 and 5, respectively.
## II The Chiral-Quark Sigma Model
Brise and Banerjee [3] described the interactions of quarks via the exchange
of $\sigma$ and $\mathbf{\pi}$ \- meson fields. The Lagrangian density is
$L\left(r\right)=i\overline{\Psi}\partial_{\mu}\gamma^{\mu}\Psi+\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma+\partial_{\mu}\mathbf{\pi}.\partial^{\mu}\mathbf{\pi}\right)+g\overline{\Psi}\left(\sigma+i\gamma_{5}\mathbf{\tau}.\mathbf{\pi}\right)\Psi-
U_{1}\left(\sigma,\mathbf{\pi}\right),$ (1)
with
$U_{1}\left(\sigma,\mathbf{\pi}\right)=\frac{\lambda^{2}}{4}\left(\sigma^{2}+\mathbf{\pi}^{2}-\nu^{2}\right)^{2}+m_{\pi}^{2}f_{\pi}\sigma,$
(2)
is the meson-meson interaction potential where the $\Psi,\sigma$ and
$\mathbf{\pi}$ are the quark, sigma, and pion fields, respectively. In the
mean-field approximation, the meson fields treat as time-independent classical
fields. This means that we replace the power and the products of the meson
fields by the corresponding powers and the products of their expectation
values. In Eq. (2), the meson-meson interactions leads to the hidden chiral
symmetry $SU(2)\times SU(2)$ with $\sigma\left(r\right)$ taking on a vacuum
expectation value
$\ \ \ \ \ \ \left\langle\sigma\right\rangle=-f_{\pi},$ (3)
where $f_{\pi}=92.4$ MeV is the pion decay constant. The final term in Eq. (2)
is included to break the chiral symmetry explicitly. It leads to the partial
conservation of axial-vector current (PCAC). The parameters
$\lambda^{2},\nu^{2}$ can be expressed in terms of$\ f_{\pi}$ and the masses
of mesons as,
$\lambda^{2}=\frac{m_{\sigma}^{2}-m_{\pi}^{2}}{2f_{\pi}^{2}},$ (4)
$\nu^{2}=f_{\pi}^{2}-\frac{m_{\pi}^{2}}{\lambda^{2}}.$ (5)
## III The Chiral Higher-Order Quark Sigma Model
The Lagrangian density of the extended linear sigma model which describes the
interactions between quarks via the $\sigma$ and $\mathbf{\pi}$ mesons
$\left[14\right]$
$L\left(r\right)=i\overline{\Psi}\gamma_{\mu}\partial^{\mu}\Psi+\frac{1}{2}\left(\partial_{\mu}\sigma\partial^{\mu}\sigma+\partial_{\mu}\mathbf{\pi}.\partial^{\mu}\mathbf{\pi}\right)+g\overline{\Psi}\left(\sigma+i\gamma_{5}\mathbf{\tau}.\mathbf{\pi}\right)\Psi-
U_{2}\left(\sigma,\mathbf{\pi}\right),$ (6)
with
$\displaystyle U_{2}\left(\sigma,\mathbf{\pi}\right)$
$\displaystyle=\frac{\lambda_{1}^{2}}{4}\left(\sigma^{2}+\mathbf{\pi}^{2}-\nu_{1}^{2}\right)^{2}+\frac{\lambda_{2}^{2}}{4}\left(\left(\sigma^{2}+\mathbf{\pi}^{2}\right)^{2}-\nu_{2}^{2}\right)^{2}$
(7) $\displaystyle+m_{\pi}^{2}f_{\pi}\sigma\text{.}$
It is clear that potential satisfies the chiral symmetry when
$m_{\pi}\rightarrow 0$. In the original model [3], the higher-order term in
Eq. 7 is excluded by the requirement of renormalizability. Since we are going
to use Eq. (7) as an approximating effective model. The model did not need and
should not be renormalizable as in Ref. [9]. By using the PCAC and the
minimization conditions of mesonic potential $\left[14\right]$, we obtain
$\lambda_{1}^{2}=\frac{m_{\sigma}^{2}-m_{\pi}^{2}}{4f_{\pi}^{2}},\ \ \ \ \ \ \
\ \nu_{1}^{2}=f_{\pi}^{2}-\frac{m_{\pi}^{2}}{\lambda_{1}^{2}},$ (8)
$\lambda_{2}^{2}=\frac{m_{\sigma}^{2}-3m_{\pi}^{2}}{16f_{\pi}^{6}},\ \ \
\nu_{2}^{2}=f_{\pi}^{4}-\frac{m_{\pi}^{2}}{2\lambda_{2}^{2}f_{\pi}^{2}}.$ (9)
Now we can expand the extremum with the shifted field defined as
$\sigma=\sigma^{\prime}-f_{\pi},$ (10)
substituting Eq. (10) into Eq. (6), we get
$\displaystyle L\left(r\right)$
$\displaystyle=i\overline{\Psi}\gamma_{\mu}\partial^{\mu}\Psi+\frac{1}{2}\left(\partial_{\mu}\sigma^{\prime}\partial^{\mu}\sigma^{\prime}+\partial_{\mu}\mathbf{\pi}.\partial^{\mu}\mathbf{\pi}\right)-g\overline{\Psi}f_{\pi}\Psi+g\overline{\Psi}\sigma^{\prime}\Psi+ig\overline{\Psi}\mathbf{\gamma}_{5}.\mathbf{\pi}\Psi$
$\displaystyle-U_{2}\left(\sigma^{\prime},\mathbf{\pi}\right),$ (11)
with
$\displaystyle U_{2}\left(\sigma^{\prime},\mathbf{\pi}\right)$
$\displaystyle=\frac{\lambda_{1}^{2}}{4}(\left(\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}-\nu_{1}^{2}\right)^{2}+\frac{\lambda_{2}^{2}}{4}\left(\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)^{2}-\nu_{2}^{2}\right)^{2}$
$\displaystyle+m_{\pi}^{2}f_{\pi}(\sigma^{\prime}-f_{\pi}).$ (12)
The time-independent fields $\sigma^{{}^{\prime}}\left(r\right)\,\,$and
$\mathbf{\pi}\left(r\right)$ satisfy the Euler$-$Lagrange equations, and the
quark wave function satisfies the Dirac eigenvalue equation. Substituting Eq.
(11) in Euler$-$Lagrange equation, we get
$\displaystyle\square\sigma^{\prime}$
$\displaystyle=g\overline{\Psi}\Psi-\lambda_{1}^{2}(f_{\pi}-\sigma^{\prime})((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}-\nu_{1}^{2})-$
$\displaystyle
2\lambda_{2}^{2}(f_{\pi}-\sigma^{\prime})\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)(\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)^{2}-\nu_{2}^{2})-m_{\pi}^{2}f_{\pi},$
(13) $\displaystyle\square\mathbf{\pi}$
$\displaystyle=ig\overline{\Psi}\gamma_{5\cdot}\mathbf{\tau}\Psi-\lambda_{1}^{2}((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}-\nu_{1}^{2}))\mathbf{\pi}-$
$\displaystyle
2\lambda_{2}^{2}\mathbf{\pi}\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)(\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)^{2}-\nu_{2}^{2}),$
(14)
where $\mathbf{\tau}$ refers to Pauli isospin matrices,
$\gamma_{5}=\left(\begin{array}[c]{cc}0&1\\\ 1&0\end{array}\right)$. Including
the color degree of freedom, one has $g\overline{\Psi}\Psi\rightarrow
N_{c}g\overline{\Psi}\Psi$ where $N_{c}=3$ colors. Thus
$\Psi\left(r\right)=\frac{1}{\sqrt{4\pi}}\left[\begin{array}[c]{c}u\left(r\right)\\\
iw\left(r\right)\end{array}\right]\qquad\text{and}\qquad\bar{\Psi}\left(r\right)=\frac{1}{\sqrt{4\pi}}\left[\begin{array}[c]{cc}u\left(r\right)&iw\left(r\right)\end{array}\right],$
(15)
then
$\displaystyle\rho_{s}$
$\displaystyle=N_{c}\overline{\Psi}\Psi=\frac{3g}{4\pi}\left(u^{2}-w^{2}\right),$
(16) $\displaystyle\rho_{p}$
$\displaystyle=iN_{c}\overline{\Psi}\gamma_{5}\mathbf{\tau}\Psi=\frac{3g}{2\pi}(uw),$
(17) $\displaystyle\rho_{v}$
$\displaystyle=\frac{3g}{4\pi}\left(u^{2}+w^{2}\right),$ (18)
where $\rho_{s},$ $\rho_{p}$ and $\rho_{v}$ are sigma, pion and vector
densities, respectively. These equations are subject to the boundary
conditions as follows,
$\sigma\left(r\right){\sim}-f_{\pi},\ \ \ \ \pi\left(r\right){\sim}0\text{ \ \
\ \ at }r\rightarrow\infty\text{.}$ (19)
By using hedgehog ansatz [12], where
$\mathbf{\pi}\left(r\right)=\pi\left(r\right)\overset{\char
94\relax}{\mathbf{r}}.$ (20)
The chiral Dirac equation for the quarks is [12]
$\frac{du}{dr}=-P\left(r\right)u+\left(W+m_{q}-S(r)\right)w,$ (21)
where the scalar potential $S(r)=g\left\langle\sigma^{\prime}\right\rangle$,
the pseudoscalar potential
$P(r)=\left\langle\mathbf{\pi}\cdot\mathbf{\hat{r}}\right\rangle$, and $W$ is
the eigenvalue of the quarks spinor $\Psi$
$\frac{dw}{dr}=-\left(W-m_{q}+S(r)\right)u-\left(\frac{2}{r}-P\left(r\right)\right)w.$
(22)
## IV Numerical Calculations and Discussions
### IV.1 The scalar field $\sigma^{\prime}$
To solve Eq. (13), we integrate a suitable Green’s function over the source
fields as in Refs. $\left[12,13\right].$ Thus
$\displaystyle\sigma^{\prime}\left(\mathbf{r}\right)$ $\displaystyle=\int
d^{3}\mathbf{r}^{\prime}D_{\sigma}(\mathbf{r-\grave{r}})[g\rho_{s}(\mathbf{\grave{r}})-\lambda_{1}^{2}(f_{\pi}-\sigma^{\prime})((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}-\nu_{1}^{2})-$
$\displaystyle
2\lambda_{2}^{2}(f_{\pi}-\sigma^{\prime})\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)(\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)^{2}-\nu_{2}^{2})-m_{\pi}^{2}f_{\pi}],\;\;\;\;\;\;\;\;\;\;\;\;\
$ (23)
where
$D_{\sigma}(\mathbf{r-\grave{r}})=\frac{1}{4\pi\left|\mathbf{r-\grave{r}}\right|}\exp(-m_{\sigma}\left|\mathbf{r-\grave{r}}\right|),\;$
the scalar field is spherical in this model so we only need the $l=0$ term
$D_{\sigma}\left(\mathbf{r-\grave{r}}\right)=\frac{1}{4\pi}\sinh\left(m_{\sigma}r_{<}\right)\frac{\exp\left(-m_{\sigma}r_{>}\right)}{r_{>}},\;\;$
(24)
therefore
$\displaystyle\sigma^{\prime}\left(\mathbf{r}\right)$
$\displaystyle=m_{\sigma}\int\limits_{0}^{\infty}r^{\prime
2}dr^{\prime}(\frac{\sinh\left(m_{\sigma}r_{>}\right)\exp\left(-m_{\sigma}r_{>}\right)}{m_{\sigma}r_{>}})[g\rho_{s}(\mathbf{\grave{r}})-$
(25)
$\displaystyle\lambda_{1}^{2}(f_{\pi}-\sigma^{\prime})((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}-\nu_{1}^{2})-2\lambda_{2}^{2}(f_{\pi}-\sigma^{\prime})\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)\times$
$\displaystyle\times(\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)^{2}-\nu_{2}^{2})-m_{\pi}^{2}f_{\pi}]\text{.}$
Note that this form is implicit in the solution of $\sigma^{\prime}$involves
integrals over the unknown $\sigma^{\prime}$ itself. We will solve this
implicit integral equation by iterating to self-consistency.
### IV.2 The pion field $\mathbf{\pi}$
To solve Eq. (14), we integrate a suitable Green’s function over the source
fields. We use the $l=1$ component of the pion Green’s function. Thus
$\displaystyle\;\;\;\ \ \mathbf{\pi}\left(r\right)$
$\displaystyle=m_{\pi}\int_{0}^{\infty}r^{\prime
2}dr^{\prime}\frac{[-\sinh\left(m_{\pi}r_{<}\right)+m_{\pi}r_{<}\cosh\left(m_{\pi}r_{<}\right)]}{\left(m_{\pi}r_{>}\right)^{2}}\times\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(26)
$\displaystyle[(1+\frac{1}{m_{\pi}r_{>}})\frac{\exp\left(-m_{\pi}r_{>}\right)}{m_{\pi}r_{>}})(g\rho_{p}-\lambda_{1}^{2}((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}-\nu_{1}^{2}))\mathbf{\pi-}$
$\displaystyle
2\lambda_{2}^{2}\mathbf{\pi}\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)(\left((\sigma^{\prime}-f_{\pi})^{2}+\mathbf{\pi}^{2}\right)^{2}-\nu_{2}^{2})].$
We have solved Dirac Eqs. (21), (22) using fourth-order Rung Kutta method. Due
to the implicit nonlinearly of these Eqs. (13), (14) it is necessary to
iterate the solution until self-consistency is achieved. To start this
iteration process, we could use the chiral circle form for the meson fields
[12, 13]:
$S(r)=m_{q}(1-\cos\theta),\text{ }P(r)=-m_{q}\sin\theta,$ (27)
where $\theta=\tanh r$.
### IV.3 The Properties of the Nucleon
The proton and neutron magnetic moments are given by [3]
$\mu_{p,n}=<P\uparrow\left|\int\frac{1}{2}\mathbf{r}\times\mathbf{j}_{\varepsilon
M}(\mathbf{r})d^{3}\mathbf{r}\right|P\uparrow>,$ (28)
where, the electromagnetic current is
$j_{\epsilon
M}(\mathbf{r})=\bar{\Psi}\left(\mathbf{r}\right)\mathbf{\gamma}\left(\frac{1}{6}+\frac{\tau_{3}}{2}\right)\Psi(\mathbf{r})-\varepsilon_{\alpha\beta_{3}}\pi_{\alpha}\left(\mathbf{r}\right)\mathbf{\nabla}\pi_{\beta}\left(\mathbf{r}\right),$
(29)
such that
$\left(\mathbf{j}_{\epsilon
M}(\mathbf{r})\right)_{nucleon}=\bar{\Psi}\left(\mathbf{r}\right)\mathbf{\gamma}\left(\frac{1}{6}+\frac{\tau_{3}}{2}\right)\Psi\left(\mathbf{r}\right),$
(30) $\left(\mathbf{j}_{\epsilon
M}(\mathbf{r})\right)_{meson}=-\epsilon_{\alpha\beta
3}\pi_{\alpha}\left(\mathbf{r}\right)\mathbf{\nabla}\pi_{\beta}\left(\mathbf{r}\right).$
(31)
The nucleon axial-vector coupling constant is found from
$\frac{1}{2}g_{A}(0)=\left\langle P\uparrow\left|\int
d^{3}rA_{3}^{z}(\mathbf{r})\right|P\uparrow\right\rangle,$ (32)
where the z-component of the axial vector current is given by
$A_{3}^{z}(\mathbf{r})=\bar{\Psi}\left(\mathbf{r}\right)\frac{1}{2}\gamma_{5}\gamma^{3}\tau_{3}\Psi\left(\mathbf{r}\right)-\sigma\left(\mathbf{r}\right)\frac{\partial}{\partial
z}\pi_{3}\left(\mathbf{r}\right)+\pi_{3}\left(\mathbf{r}\right)\frac{\partial}{\partial
z}\sigma\left(\mathbf{r}\right).$ (33)
The pion-nucleus $\sigma$ commutator is defined
$\sigma(\pi N)=\left\langle
P\uparrow\left|\int\sigma^{\prime}(\mathbf{r})d^{3}r\right|P\uparrow\right\rangle.$
(34)
In calculation of $\sigma(\pi N),$ we replace $\sigma^{\prime}(\mathbf{r})$ by
$\frac{j_{\sigma}(\mathbf{r})}{m_{\sigma}^{2}}$ where $j_{\sigma}(\mathbf{r})$
is the source current defined by
$(\square+m_{\sigma}^{2})\sigma^{\prime}=j_{\sigma}(\mathbf{r}).$
The hedgehog mass is calculated in details in Refs. $\left[12,13\right]$.
### IV.4 Discussion of the Results
The set of equations (13-22) are numerically solved by the iteration method as
Refs. [12-14] for different values of the sigma and quark masses. The
dependence of the nucleon properties on the sigma and the quark masses are
listed in the tables (1), (2), (3), and (4). In Table (1), we note that the
hedgehog mass, the magnetic moments of the proton and neutron, and the sigma
commutator increase by increasing sigma mass. We obtain a good value of the
hedgehog mass equals to 1090 MeV which closed to experimental data 1086 MeV.
In Table (2), we examine the effect of quark mass on the nucleon properties.
We note that the hedgehog mass decreases with increasing quark mass. This
interpreted that an increase in the quark mass leads to increase in the
coupling constant $(g=\frac{m_{q}}{f_{\pi}})$. Therefore, the coupling between
meson and the quark more tight, leading the decrease in the hedgehog mass as
in Refs. [3, 12, 13]. Also, we note that the magnetic moments of proton and
neutron increase by increasing quark mass. A similar effect occurred respect
to sigma commutator $\sigma(\pi N).$ In comparison between the results in the
tables 1 and 2. We note that quark mass is more affected on nucleon properties
that the strong change of sigma mass leads to the change of nucleon properties
as in the table 1. In Table (3), we compare between the original quark model
and the higher-order quark model. We fixed all parameters in the two models to
show the effect of the higher-order mesonic interactions on the nucleon
properties. We note that the dynamic of kinetic energy of quark increases by
increasing mesonic contributions in the original quark model. In addition, the
meson-quark interaction energy decreases by increasing higher-order
interactions. We note that meson-meson interaction decreases by increasing
mesonic contributions in the original sigma model. We obtain the excellent
value of hedgehog mass $M_{H}$ $\cong 1090$ MeV while we obtain $M_{H}$ $\cong
1068$ MeV in the original sigma model at the same free parameters. Therefore,
an increase of the mesonic interactions improved the hedgehog mass which
closed to experimental data ($M_{H}$ $\cong 1086$ MeV). The magnetic moments
of proton and neutron are improved in comparison with the original model.
Sigma commutator $\sigma(\pi N)$ is one of problems in the original sigma
model that is a largest value in comparison with data. By increasing mesonic
contributions in the original sigma model. This value reduced from 126 MeV to
78 MeV. Therefore, the value improved about 38 % and it is acceptable
agreement with experimental data. The quantity $g_{A}(0)$ is improve in
comparsion with the original model but still a large value in comparing with
experimental data $\left(1.25\right)$. Since the $g_{A}(0)$ depends on the
meson fields only not on the coupling of higher-order term in the extended
sigma model. Therefore, we need to add a vector meson to our model to improve
this quantity, which will be a future paper.
Table (1). Values of magnetic moments of proton and neutron, the hedgehog mass
$M_{B}$, and $\sigma(\pi N)$ for $m_{\pi}=139.6$ MeV$,m_{q}=500$ MeV,
$f_{\pi}=92.4\,$MeV. All quantities in MeV.
$m_{\sigma}\left(\text{MeV}\right)$ | 600 | 700 | 800 | 900
---|---|---|---|---
Hedgehog mass $M_{B}$ | 1090.92 | 1108.98 | 1125.54 | 1139.27
Total moment proton $\mu_{p}\left(N\right)$ | 2.8456 | 2.8641 | 2.8643 | 2.8646
Total moment neutron $\mu_{n}\left(N\right)$ | -2.2076 | -2.2374 | -2.2494 | -2.259
$\sigma(\pi N)$ | 77.025 | 78.158 | 78.440 | 78.770
Table (2). Values of magnetic moments of proton and neutron, the hedgehog mass
$M_{B}$, and $\sigma(\pi N)$ for $m_{\pi}=139.6$ MeV$,m_{\sigma}=600$ MeV,
$f_{\pi}=92.4\,$MeV. All quantities in MeV.
$m_{q}\left(\text{MeV}\right)$ | 400 | 420 | 440 | 460 | 480 | 500
---|---|---|---|---|---|---
Hedgehog mass $M_{B}$ | 1230 | 1210 | 1185 | 1157 | 1124 | 1089
Total moment proton $\mu_{p}\left(N\right)$ | 2.574 | 2.653 | 2.719 | 2.775 | 2.823 | 2.845
Total moment neutron $\mu_{n}\left(N\right)$ | -1.899 | -1.985 | -2.05 | -2.121 | -2.175 | -2.207
$\sigma(\pi N)$ | 49.19 | 57.57 | 64.28 | 69.79 | 74.32 | 77.02
Table(3). Details of energy calculations of the hedgehog mass, the magnetic
moments of proton and neutron, and the sigma commutator $\sigma(\pi N)$ for
$m_{q}=500$ MeV$,m_{\pi}=139.6$ MeV$,m_{\sigma}=600$ MeV, and
$f_{\pi}=92.4\,$MeV. All quantities in MeV.
Quantity | Original Sigma Model | Higher-order Sigma Model
---|---|---
Quark kinetic energy | 1166.38 | 1171.068
Sigma kinetic energy | 353.15 | 375.038
Pion kinetic energy | 461.85 | 451.827
Sigma interaction energy | -165.84 | -165.975
Pion interaction energy | -860.87 | -854.098
Meson interaction energy | 114.0 | 113.069
Hedgehog mass baryon | 1068.67 | 1090.92
Total moment of proton $\mu_{p}$ | 2.89 | 2.84
Total moment of neutron $\mu_{n}$ | -2.24 | 2.20
$g_{A}(0)$ | 1.80 | 1.78
$\sigma(\pi N)$ | 126.99 | 77
## V Comparison with Other Models
It is interesting to compare the nucleon properties in the present approach
with the previous works and other models. The higher-order mesonic potential
was suggested in Refs. [12-14]. In Ref. [12], the sigma commutator $\sigma(\pi
N)$ is not calculated in this work. It is an essential property of nucleon
properties. In addition, the mesonic potential has a weakness point at
$m_{\pi}=0$ so the model did not satisfy the chiral limit case. We note that
the hedgehog mass improved in comparison with result of Ref. [12 ]. In Ref.
[13], the authors suggested another form of mesonic potential to avoid the
difficulty which came from $m_{\pi}=0.$ We have two advantages in comparison
with Ref. [13]. The first, our results in the present work are improved, in
particular the hedgehog mass and the $\sigma(\pi N)$. The second, the mesonic
potential in Eq. 7, has the similar form when the coupling constant of higher-
order $\lambda_{2}^{2}$ is vanished as in Eq. 2. This advantage is not found
in Ref. [13]. In Ref. [14], the author studied the effect of large pion masses
on the magnetic moments of proton and neutron only.
It is important to compare present model with other models such as the
perturbative chiral quark Model [6, 7] and the extended Skyrme model [15]. The
perturbative chiral quark model is an effective model of baryons based on
chiral symmetry. The baryon is described as a state of three localized
relativistic quarks supplemented by a pseudoscalar meson cloud as dictated by
chiral symmetry requirements. In this model, the effect of the meson cloud is
evaluated perturbatively in a systematic fashion. The model has been
successfully applied to the nucleon properties (see Table 4). We obtain
reasonable results in comparison with this model for the $\sigma(\pi N)$ which
backs to perturbative chiral quark model based on non-linear $\sigma-$ model
Lagrangian. In particular, nucleon magnetic moments are improved in comparison
with this model. Moreover, Hedgehog mass $M_{B}$ is not calculated in this
model. The original Skyrme model [16] consists of the non-linear sigma term
and the fourth-order derivative term, which guarantees the stabilization of
the soliton so that the degree of freedom of the sigma field may be replaced
by a variable chiral radius, which becomes the new dynamical degree of freedom
and plays an important role in the modified Skyrmion Lagrangian density [15],
leading to a better description of nucleon properties. In comparison with the
extended Skyrme model [15], the results obtained for the hedgehog mass have
been improved and the other properties are in agreement with this model (see
Table 4).
Table (4). Values of the observables calculated from the extended linear sigma
model [12, 13], the perturbative chiral quark model [6, 7], and the extended
Skyrme model [15] in comparison with the present work.
Quantity | Present work | [ 13 ] | [6, 7] | [ 12 ] | [15] | Expt.
---|---|---|---|---|---|---
Hedgehog mass $M_{B}$ | 1090 | 1200 | - | 1081 | 1157 | 1086
$\mu_{p}\left(N\right)$ | 2.84 | 2.76 | 2.62$\pm 0.02$ | 2.768 | 2.77 | 2.79
$\mu_{n}\left(N\right)$ | -2.20 | -1.91 | -2.02$\pm 0.02$ | -1.909 | -2.11 | -1.91
$\sigma(\pi N)$ | 77 | 88 | 54.7 | - | 70 | 50$\pm 20$
## VI Conclusion
The present calculations have shown the importance of mesonic corrections of
higher-order than that normally used in most soliton models. The obtained
results are improved in comparison with previous calculations. In addition, we
avoid the difficulty that found in the previous works. The advantage of the
present work that hedgehog mass is corrected and closed with data. The
magnetic moments of proton and neutron and sigma commutator $\sigma(\pi N)$
are improved in comparison with other models.
## VII References
1. 1.
S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. 41, 531 (1969).
2. 2.
M. Gell-Mann, M. Levy, Nuovo Cimento 16, 705 (1960).
3. 3.
M. Birse and M. Banerjee, Phys. Rev. D 31, 118 (1985).
4. 4.
W. Broniowski and M. K. Banerjee, Phys. Lett. B 158, 335 (1985).
5. 5.
M. Birse, Phys. Rev. D 33, 1934 (1986).
6. 6.
V. E. Lyubovitskij, T. Gutsche and A. Faessler, Phys. Rev. C 64, 065203
(2001).
7. 7.
T. Inoue, V. E. Lyubovitskij, T. Gutsche, A. Faessler, Phys. Rev. C 69, 035207
(2004).
8. 8.
D. Horvat, D. Horvatic, B. Podobnik and D. Tadic, FIZIKA B 9, 181 (2000).
9. 9.
W. Broniowski and B. Golli, Nucl. Phys. A 714, 575 (2003).
10. 10.
M. Abu-Shady, Acta Phys. Polo. B 40, 8 (2009).
11. 11.
M. Abu-Shady, Int. J. Theor. Phys. 48, 1110 (2009).
12. 12.
M. Rashdan, M. Abu-Shady, and T.S.T Ali, Inter. J. Mod. Phys. A 22, 2673
(2007)
13. 13.
M. Rashdan, M. Abu-shady, and T.S.T. Ali, Int. J. Mod. Phys. E 15, 143 (
2006).
14. 14.
M. Abu-Shady, Phys. Atom. Nucl. 73, 978 (2010).
15. 15.
F. L. Braghin and I. P. Cavalcante, Phys. Rev. C 67, 065207 (2003).
16. 16.
T. H. R. Skyrme, Proc. R. Soc. London, Ser. A 260, 127 (1961).
|
arxiv-papers
| 2013-08-04T13:24:59 |
2024-09-04T02:49:48.967638
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Abu-Shady",
"submitter": "Mohamed Mohamed",
"url": "https://arxiv.org/abs/1308.0809"
}
|
1308.0822
|
# Ondas sonoras estacionárias em um tubo: análise de problemas & sugestões
(Standing Sound Waves in a tube: Approach analysis & sugestions )
L. P. Vieira†, D. F. Amaral ‡ and V. O. M. Lara⋆ † Instituto de Física -
Universidade Federal do Rio de Janeiro, Rio de Janeiro - Rio de Janeiro Brasil
‡ Consórcio de Ensino à distância do Rio de Janeiro (CEDERJ), pólo São Gonçalo
- Rio de Janeiro Brasil
⋆ Instituto de Física - Universidade Federal Fluminense, Niterói - Rio de
Janeiro Brasil
###### Abstract
No presente trabalho temos como objetivo apresentar alguns questionamentos com
respeito à abordagem utilizada em alguns livros didáticos de nível médio sobre
o tema de ondas sonoras estacionárias em tubos. Além de classificar os livros
didáticos dentro de um conjunto de critérios estabelecidos, apresentamos
também algumas sugestões para uma discussão mais aprofundada deste tema.
Sugerimos o uso de gifs e animações e a utilização de dois experimentos
simples, que permitem a visualização dos perfis de variação de pressão e
deslocamento de ar para os modos harmônicos de vibração.
Palavras-chave: Ondas sonoras estacionárias, Análise de livros didáticos,
Tablets e smartphones, Uso de Tecnologias no Ensino de Ciências.
In this paper we attempt to present some questions with respect to the
approach used in some brazilian mid-level textbooks on the topic of stationary
sound waves in tubes. In addition to ranking the textbooks within a set of
criteria, we also present some suggestions for further discussions of this
topic. We suggest the use of gifs and animations and the use of two
experiments that allow you to view the profiles of variation of pressure and
air displacement for the harmonic modes of vibration.
Keywords: Standing sound waves, Textbook analysis, Tablets and smartphones,
technology support for science classes.
###### pacs:
## I Introdução
Diversos estudos mostram que a utilização de experimentos em sala de aula são
de grande importância na aprendizagem dos conteúdos apresentados nas áreas das
Ciências da Natureza bybee ; krasilchik ; longhini . Infelizmente a
inexistência de um espaço físico e/ou a falta de infraestrutura prejudicam a
prática destas atividades experimentais. Este cenário é bastante comum nas
escolas públicas e em uma boa parcela das escolas particulares borges .
Alguns trabalhos, tais como previous_mag ; accelerometer ; artigo_gota , já
demonstraram a gama de aplicativos que tais dispositivos apresentam e que
podem ser explorados para fins didáticos. Em particular na área da Física,
muitas leis podem ser confirmadas e visualizadas, o que torna alguns conceitos
físicos mais concretos e palatáveis.
Tendo isto em mente, apresentamos neste trabalho dois experimentos que podem
ser utilizados para se discutir ondas sonoras estacionárias em um tubo semi
aberto. Embora tenhamos nos restringido ao caso de tubos semiabertos por
brevidade, nada impede o uso dos experimentos apresentados aqui em um tubo
aberto, enquanto que o caso de um tubo fechado exigiria uma sofisticação um
pouco maior. Além disto, realizamos também uma avaliação da maneira com que
este assunto é discutido em diversos livros de Física de nível médio. Conforme
veremos, o tratamento realizado em boa parte dos livros peca em diversos
aspectos, tais como a falta de clareza à respeito das quantidades físicas
medidas, a natureza das ondas sonoras no ar (boa parte dos livros apresenta
perfis relacionados à ondas transversais sem maiores preocupações), que,
conforme sabemos, são longitudinais, e o comportamento oscilatório deste tipo
de onda.
## II Discussão Teórica
O estudo de ondas sonoras em um tubo é assunto delicado devido à uma série de
fatores. O problema surge quando desejamos representar ondas sonoras
estacionárias em um livro site_d_russell . Conforme sabemos, as ondas sonoras
no ar são longitudinais. Entretanto, ao se discutir o caso de ondas sonoras
estacionárias em um tubo, praticamente todos os livros apresentam imagens
estáticas cujo formato é basicamente o de uma onda em uma corda esticada (como
a corda de um violão), sendo associada portanto à uma onda transversal. Isto
pode gerar uma série de dúvidas conceituais, fazendo com que o leitor incauto
acredite que as imagens são a onda sonora em si, e não a representação
esquemática da variação de uma grandeza física específica associada à esta
onda sonora.
Outro problema deve-se ao fato de as ondas sonoras estacionárias em um tubo
apresentarem oscilações temporais. As imagens ilustram apenas a amplitude de
uma grandeza física em específico, no caso o deslocamento de ar. Conforme
veremos posteriormente, podemos visualizar a oscilação temporal tanto para o
deslocamento de ar (utilizando bolinhas de isopor) quanto para a variação de
pressão (que está relacionada à intensidade da onda sonora) utilizando-se uma
montagem experimental simples, ou fazendo uso de gifs e animações
site_d_russell .
Outro ponto bastante importante, e que não recebe a devida atenção em nenhum
dos livros relacionados neste trabalho relaciona-se à uma aproximação
fundamental que deve ser feita. Todo o tratamento feito à esse respeito deve
supor que as ondas sonoras emitidas pela fonte sonora são ondas planas. Se a
fonte sonora puder ser considerada uma fonte pontual, por exemplo, a discussão
feita só será válida no regime em que estivermos suficientemente afastados da
fonte, de modo que possamos considerar que as ondas sonoras no interior do
tubo são essencialmente planas. Se uma fonte sonora pontual for posta na
extremidade aberta de um tubo, por exemplo, o fenômeno físico seria
consideravelmente mais complicado, uma vez que teríamos sucessivas reflexões
nas paredes do tubo, dado o caráter esférico da onda emitida pela fonte.
## III Avaliação dos livros
A fim de classificar adequadamente os livros, elaboramos cinco critérios,
enumerando-os de $1$ a $5$:
1. 1.
Quando apresenta as ondas sonoras estacionárias em um tubo, o livro salienta a
natureza longitudinal deste tipo de onda?
2. 2.
O livro deixa claro qual é a quantidade física que está sendo representada
(deslocamento de ar, ou mesmo ”vibração da coluna de ar”)?
3. 3.
O livro levanta a possibilidade de se medir outras quantidades (variação de
pressão, ou intensidade da onda sonora)?
4. 4.
O livro discute o caráter temporal oscilatório da onda sonora estacionária,
deixando claro que a imagem representa a amplitude do deslocamento de ar?
5. 5.
Apresenta a condição fundamental de que as ondas sonoras no tubo devem ser
ondas planas?
Tendo em vista os critérios listados àcima, elaboramos a Tabela I.
Livros | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
Máximo & Alvarenga maximo_alvarenga | ✓ | ✓ | $\times$ | $\times$ | $\times$
Guimarães & Fonte Boa guimaraes_boa | ✓ | ✓ | $\times$ | $\times$ | $\times$
Ramalho et al ramalho | $\times$ | ✓ | $\times$ | $\times$ | $\times$
Gaspar gaspar | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
Helou et al helou | ✓ | ✓ | ✓ | $\times$ | $\times$
Hewitt * hewitt | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
Gref * gref | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
Tabela I: Tabela que classifica os livros selecionados de acordo com os
critérios apontados no texto. Os livros cujos autores aparecem com * indicam
que o livro em questão não discute especificamente o caso de ondas sonoras
estacionárias em um tubo.
De todos os livros avaliados nesta pesquisa bibliográfica, apenas a coleção de
Helou et al diferencia o deslocamento de ar e a variação de pressão, apontando
inclusive a defasagem de $90^{\circ}$ existente entre ambos os perfis.
Conforme pode-se ver na Tabela I, boa parte dos livros salienta a natureza
longitudinal das ondas sonoras quando discutem as ondas estacionárias no tubo,
com exceção das coleções de Ramalho et al e Gaspar. A discussão realizada
neste último é a menos cuidadosa. Além de não salientar a natureza
longitudinal das ondas sonoras na seção onde as ondas sonoras estacionárias
são discutidas, há ainda uma confusão à respeito das quantidades físicas em
questão. São empregados os termos ”rarefação” e ”compressão” para a imagem que
representa o deslocamento de ar. Entretanto, sem a separação entre
deslocamento de ar e variação de pressão, estes termos podem confundir mais do
que explicar.
Deve-se salientar que nenhum dos livros avaliados discute o comportamento
oscilatório das quantidades medidas, nem a necessidade de se considerar ondas
sonoras planas no interior do tubo.
## IV Sugestões
Para tornar a discussão sobre o assunto mais clara, apresentamos algumas
sugestões:
* •
Elaboração de experimentos;
* •
O uso de aplicativos, applets, gifs, etc.;
* •
Tópicos interessantes que podem ser discutidos.
Sugerimos a elaboração de dois experimentos, onde é possível evidenciar
grandezas físicas importantes que não são devidamente discutidas nos livros
didáticos (veja a seção III). O primeiro trata da visualização do perfil da
pressão do ar e o segundo se destina à visualização do perfil do deslocamento
do ar ao longo de um tubo com a uma extremidade aberta e outra fechada.
Tendo isto em mente, elaboramos um arranjo experimental simples que permite
observar o perfil de ondas sonoras estacionárias formadas em um tubo semi
aberto. Uma grande vantagem desta montagem é que ela pode ser reproduzida não
só em um laboratório, mas também em sala de aula e outros ambientes.
### IV.1 Experimentos
Para reproduzir os experimentos que discutimos neste trabalho o leitor deverá
dispor da seguinte relação de materiais:
1. (i)
Dois Tablets (ou smartphones);
2. (ii)
Um tubo de vidro aberto em uma extremidade e fechado na outra;
3. (iii)
uma vareta de madeira (de tamanho compatível com o do tubo de vidro);
4. (iv)
um alto-falante;
5. (v)
trena (ou régua);
6. (vi)
fita adesiva;
7. (vii)
um microfone;
8. (viii)
pequenas bolas de isopor.
Além desta estrutura física também faz-se necessário que os tablets tenham
alguns aplicativos previamente instalados.
Para a reprodução do experimento utilizamos dois tablets. Entretanto, nada
impede que se utilizem dois smartphones, ou um smartphone e um tablet. O
importante é que os aplicativos necessários estejam instalados e funcionando
devidamente [veja a Figura (1) onde mostramos o experimento proposto sendo
realizado].
Figure 1: Imagem que mostra a realização do experimento proposto neste
trabalho.
Um dos tablets servirá como uma fonte sonora, em conjunto com o alto-falante.
Neste caso utilizamos o aplicativo SGenerator Lite sgenerator , que embora
seja gratuito e possua menos recursos que a versão paga, já nos permite
escolher a intensidade e a frequência da onda sonora a ser gerada. Conectando
o alto-falante ao tablet já com o SGenerator Lite aberto, você terá em mãos um
gerador de sinal [veja a Figura (2)].
Figure 2: Imagem capturada da tela do aplicativo SGenerator Lite, a versão
gratuita do SGenerator.
Já o outro tablet será responsável por realizar medidas da onda sonora formada
no interior do tubo. Para isto instalamos no mesmo o aplicativo oScope
Liteoscope , que é gratuito, e conectamos o microfone, que está preso à vareta
por meio da fita adesiva. Uma vez que o aplicativo oScope Lite está aberto,
podemos passear com a vareta no interior do tubo de vidro e observar padrões
de máximos e mínimos.
Deste modo, é importante ressaltar que o perfil do harmônico que será
visualizado pelo aplicativo oScope Lite é o perfil da variação da pressão do
ar com relacão ao eixo perpendicular da secção reta transversal do tubo, ou
seja, uma excitação mecânica que é interpretada pelo aplicativo como a
intensidade, ou grosso modo volume, da onda sonora [veja a figura (3)]. Também
seria possível reproduzir o mesmo experimento em um tubo aberto em ambas as
extremidades, o que não foi feito aqui.
Figure 3: Imagem capturada da tela do aplicativo oScope Lite, versão gratuita
do oScope.
Antes de mais nada, podemos facilmente estimar com boa precisão quais são as
frequências dos harmônicos formados em um tubo semiaberto. Para isto, basta
medir o comprimento do tubo com o auxílio de uma trena ou régua, e tendo em
vista que a velocidade do som no ar é de aproximadamente $340$ $m/s$ obtemos a
frequência do harmônico fundamental através da relação $f_{0}=v_{som}/4L$.
Para os demais harmônicos, basta utilizar a expressão harmonicos
$\displaystyle f_{n}=\Big{(}\frac{n}{4L}\Big{)}v_{som},$ $\displaystyle
n=1,3,5,...$ (1)
Primeiramente ajustamos a frequência da onda sonora emitida pelo alto-falante
para o valor de $f_{0}=404$ Hz, estimado a partir do valor do comprimento do
tubo. Em seguida colocamos o tubo cilindríco em frente ao auto falante. À
medida que passeamos com o microfone acoplado à vareta pelo interior do tubo
visualizamos no aplicativo oScope Lite o perfil do harmônico fundamental, que
apresenta um único mínimo de intesidade, localizado na extremidade aberta e um
único máximo na extremidade fechada.
Para visualizar o deslocamento de ar podemos espaçar pequenas bolas de isopor
pela extensão do tubo e apontar o alto-falante para a extremidade aberta do
tubo. Para que possamos visualizar esta grandeza, é fundamental que o alto-
falante tenha uma potência razoável em torno de 10 W e que as bolinhas estejam
suficientemente espaçadas, de modo que a inércia das mesmas não atrapalhe a
sua movimentação.
Para reforçar o caráter longitudinal da onda sonora no ar sugerimos a animação
encontrada no site de D. Russel site_d_russell .
Reproduzimos também o segundo harmônico possível para o tubo semiaberto. Para
isto, utilizamos a equação (1), obtendo $f_{1}=1212$ Hz. Os padrões
encontrados podem ser vistos na figura (4).
Figure 4: Imagens obtidas com o aplicativo Oscilloscope oscilloscope . À
esquerda temos o perfil da variação de pressão num tubo semi aberto para
$f=404\,Hz$ ao longo do eixo do tubo. À direita, o mesmo para $f=1212\,Hz$.
A reprodução de alguns dos demais harmônicos pode ficar um pouco comprometida
a medida que as estimativas das frequências associadas à cada um deles
necessitam de uma precisão cada vez maior. Nós conseguimos obter com cuidado
suficiente ao menos os próximos dois harmônicos, além dos dois primeiros
discutidos aqui.
## V Conclusões e Perspectivas
Neste trabalho apresentamos uma montagem experimental bastante simples e
interessante, que permite ao professor e/ou estudante reproduzir ondas sonoras
estacionárias em um tubo semiaberto. Além de propiciar a visualização de um
fenômeno que normalmente desperta muitas dificuldades em uma abordagem
tradicional, podemos discutir a natureza longitudinal de uma onda sonora no ar
e apontar os problemas que normalmente são encontrados em livros texto de
Ensino Médio e Superior.
Pretendemos futuramente estender o experimento apresentado aqui para o caso de
outras geometrias possíveis, tais como a formação de harmônicos sonoros em uma
sala de aula ou corredor, por exemplo.
## Saiba mais
O leitor interessado pode assistir à uma série de vídeos produzidos pelos
autores deste trabalho, onde apresentamos o experimento mostrado neste
trabalho e muitos outros.
http://www.youtube.com/channel/UC7E_sQiahyAzwd4FiUDhJxg
## Agradecimentos
Os autores são gratos à Agência de Fomento CAPES e ao professor Anderson
Ribeiro de Souza do colégio Pedro Segundo, Niterói, RJ.
## References
* [1] R. W. Bybee, G. E. Deboer, “Research on goals for the science curriculum”, Handbook of Research on Science Teaching and Learning, p. 357-387, McMillan, 1996.
* [2] M. Krasilchik, “Reformas e Realidade - o caso do ensino de Ciências. ”, São Paulo em Perspectiva, v. 14, n 1, p. 85-93, 2000.
* [3] M. D. Longhini, “O Uno e o Diverso na Educação”, EDUFU, 2011.
* [4] A. T. Borges, “Novos rumos para o laboratório escolar de ciências”, Caderno Brasileiro de Ensino de Física, capa, v. 19, n. 3 (2002)
* [5] N. Silva, “Magnetic field sensor”, The Physics Teacher, v. 50, p.372-373 (2009).
* [6] Página do MagnetMeter na Apple Store https://itunes.apple.com/us/app/magnetmeter-3d-vector-magnetometer/id346516607?mt=8. Acesso em 03/08/2013.
* [7] L. P. Vieira, V. O. M. Lara, “Macrofotografia com um tablet: aplicações ao Ensino de Ciências ”, versão arxiv: http://arxiv.org/abs/1307.4345.
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|
arxiv-papers
| 2013-08-04T16:10:58 |
2024-09-04T02:49:48.974052
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. P. Vieira, D. F. Amaral, V. O. M. Lara",
"submitter": "Vitor de Oliveira Moraes Lara",
"url": "https://arxiv.org/abs/1308.0822"
}
|
1308.0961
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-138 LHCb-PAPER-2013-038 August 5, 2013
First evidence for the two-body charmless baryonic decay $B^{0}\\!\rightarrow
p\overline{}p$
The LHCb collaboration†††Authors are listed on the following pages.
The results of a search for the rare two-body charmless baryonic decays
$B^{0}\\!\rightarrow p\overline{}p$ and $B^{0}_{s}\\!\rightarrow
p\overline{}p$ are reported. The analysis uses a data sample, corresponding to
an integrated luminosity of 0.9 $\mbox{\,fb}^{-1}$, of $pp$ collision data
collected by the LHCb experiment at a centre-of-mass energy of 7
$\mathrm{\,Te\kern-1.00006ptV}$. An excess of $B^{0}\\!\rightarrow
p\overline{}p$ candidates with respect to background expectations is seen with
a statistical significance of 3.3 standard deviations. This is the first
evidence for a two-body charmless baryonic $B^{0}$ decay. No significant
$B^{0}_{s}\\!\rightarrow p\overline{}p$ signal is observed, leading to an
improvement of three orders of magnitude over previous bounds. If the excess
events are interpreted as signal, the 68.3% confidence level intervals on the
branching fractions are
$\displaystyle{\cal B}(B^{0}\\!\rightarrow p\overline{}p)$ $\displaystyle=$
$\displaystyle(1.47\,^{+0.62}_{-0.51}\,{}^{+0.35}_{-0.14})\times 10^{-8}\,,$
$\displaystyle\vspace*{0.3cm}{\cal B}(B^{0}_{s}\\!\rightarrow p\overline{}p)$
$\displaystyle=$
$\displaystyle(2.84\,^{+2.03}_{-1.68}\,{}^{+0.85}_{-0.18})\times 10^{-8}\,,$
where the first uncertainty is statistical and the second is systematic.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, E.
Cowie45, D.C. Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8,
P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O.
Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M.
Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F.
Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M.
Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra
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T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C.
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John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
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Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The observation of $B$ meson decays into two charmless mesons has been
reported in several decay modes [1]. Despite various searches at $e^{+}e^{-}$
colliders [2, 3, 4, 5], it is only recently that the LHCb collaboration
reported the first observation of a two-body charmless baryonic $B$ decay, the
$B^{+}\\!\rightarrow p\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}(1520)$ mode [6]. This situation is in contrast with the
observation of a multitude of three-body charmless baryonic $B$ decays whose
branching fractions are known to be larger than those of the two-body modes;
the former exhibit a so-called threshold enhancement, with the baryon-
antibaryon pair being preferentially produced at low invariant mass, while the
suppression of the latter may be related to the same effect [7].
In this paper, a search for the $B^{0}\\!\rightarrow p\overline{}p$ and
$B^{0}_{s}\\!\rightarrow p\overline{}p$ rare decay modes at LHCb is presented.
Both branching fractions are measured with respect to that of the
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ decay mode. The inclusion of charge-
conjugate processes is implied throughout this paper.
In the Standard Model (SM), the $B^{0}\\!\rightarrow p\overline{}p$ mode
decays via the $b\rightarrow u$ tree-level process whereas the penguin-
dominated decay $B^{0}_{s}\\!\rightarrow p\overline{}p$ is expected to be
further suppressed. Theoretical predictions of the branching fractions for
two-body charmless baryonic $B^{0}$ decays within the SM vary depending on the
method of calculation used, e.g. quantum chromodynamics sum rules, diquark
model and pole model. The predicted branching fractions are typically of order
$10^{-7}\\!-\\!10^{-6}$ [8, 9, 10, 11, 12]. No theoretical predictions have
been published for the branching fraction of two-body charmless baryonic
decays of the $B^{0}_{s}$ meson.
The experimental 90$\%$ confidence level (CL) upper limit on the
$B^{0}\\!\rightarrow p\overline{}p$ branching fraction, ${\cal
B}(B^{0}\\!\rightarrow p\overline{}p)<1.1\times 10^{-7}$, is dominated by the
latest search by the Belle experiment [5] and has already ruled out most
theoretical predictions. A single experimental search exists for the
corresponding $B^{0}_{s}\\!\rightarrow p\overline{}p$ mode, performed by
ALEPH, yielding the upper limit ${\cal B}(B^{0}_{s}\\!\rightarrow
p\overline{}p)<5.9\times 10^{-5}$ at 90% CL [2].
## 2 Detector and trigger
The LHCb detector [13] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors [14]. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [15]. The trigger [16] consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage, which applies a full event reconstruction.
Events are triggered and subsequently selected in a similar way for both
$B^{0}_{(s)}\\!\rightarrow p\overline{}p$ signal modes and the normalisation
channel $B^{0}\\!\rightarrow K^{+}\pi^{-}$. The software trigger requires a
two-track secondary vertex with a large sum of track $p_{\rm T}$ and
significant displacement from the primary $pp$ interaction vertices (PVs). At
least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any primary interaction greater than 16, where $\chi^{2}_{\rm IP}$
is defined as the difference in $\chi^{2}$ from the fit of a given PV
reconstructed with and without the considered track. A multivariate algorithm
[17] is used for the identification of secondary vertices consistent with the
decay of a $b$ hadron.
Simulated data samples are used for determining the relative detector and
selection efficiencies between the signal and the normalisation modes: $pp$
collisions are generated using Pythia 6.4 [18] with a specific LHCb
configuration [19]; decays of hadronic particles are described by EvtGen [20],
in which final state radiation is generated using Photos [21]; and the
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [22, *Agostinelli:2002hh] as described in
Ref. [24].
## 3 Candidate selection
The selection requirements of both signal modes and the normalisation channel
exploit the characteristic topology of two-body decays and their kinematics.
All daughter tracks tend to have larger $p_{\rm T}$ compared to generic tracks
from light-quark background owing to the high $B$ mass, therefore a minimum
$p_{\rm T}$ requirement is imposed for all daughter candidates. Furthermore,
the two daughters form a secondary vertex (SV) displaced from the PV due to
the relatively long $B$ lifetime. The reconstructed $B$ momentum vector points
to its production vertex, the PV, which results in the $B$ meson having a
small IP with respect to the PV. This is in contrast with the daughters, which
tend to have a large IP with respect to the PV as they originate from the SV,
therefore a minimum $\chi^{2}_{\rm IP}$ with respect to the PVs is imposed on
the daughters. The condition that the $B$ candidate comes from the PV is
further reinforced by requiring that the angle between the $B$ candidate
momentum vector and the line joining the associated PV and the $B$ decay
vertex ($B$ direction angle) is close to zero.
To avoid potential biases, $p\overline{}p$ candidates with invariant mass
within $\pm 50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ ($\approx 3\sigma$)
around the known $B^{0}$ and $B^{0}_{s}$ masses, specifically the region
$[5230,5417]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, are not examined until
all analysis choices are finalised. The final selection of $p\overline{}p$
candidates relies on a boosted decision tree (BDT) algorithm [25] as a
multivariate classifier to separate signal from background. Additional
preselection criteria are applied prior to the BDT training.
The BDT is trained with simulated signal samples and data from the sidebands
of the $p\overline{}p$ mass distribution as background. Of the
$1.0\mbox{\,fb}^{-1}$ of data recorded in 2011, 10% of the sample is exploited
for the training of the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ selection,
and 90% for the actual search. The BDT training relies on an accurate
description of the distributions of the selection variables in simulated
events. The agreement between simulation and data is checked on the
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ proxy decay with distributions obtained
from data using the sPlot technique [26]. No significant deviations are found,
giving confidence that the inputs to the BDT yield a nearly optimal selection.
The variables used in the BDT classifier are properties of the $B$ candidate
and of the $B$ daughters, i.e. the proton and the antiproton. The $B$
candidate variables are: the vertex $\chi^{2}$ per number of degrees of
freedom; the vertex $\chi^{2}_{\rm IP}$; the direction angle; the distance in
$z$ (the direction of the interacting proton beams) between its decay vertex
and the related PV; and the $p_{\rm T}$ asymmetry within a cone around the $B$
direction defined by $A_{\mbox{$p_{\rm T}$}}=(\mbox{$p_{\rm
T}$}^{B}-\mbox{$p_{\rm T}$}^{\text{cone}})/(\mbox{$p_{\rm
T}$}^{B}+\mbox{$p_{\rm T}$}^{\text{cone}})$, with $\mbox{$p_{\rm
T}$}^{\text{cone}}$ being the $p_{\rm T}$ of the vector sum of the momenta of
all tracks measured within the cone radius $R=0.6$ around the $B$ direction,
except for the $B$-daughter particles. The cone radius is defined in
pseudorapidity and azimuthal angle $(\eta,\phi)$ as
$R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. The BDT selection variables on
the daughters are: their distance of closest approach; the minimum of their
$p_{\rm T}$; the sum of their $p_{\rm T}$; the minimum of their $\chi^{2}_{\rm
IP}$; the maximum of their $\chi^{2}_{\rm IP}$; and the minimum of their cone
multiplicities within the cone of radius $R=0.6$ around them, the daughter
cone multiplicity being calculated as the number of charged particles within
the cone around each $B$ daughter.
The cone-related discriminators are motivated as isolation variables. The cone
multiplicity requirement ensures that the $B$ daughters are reasonably
isolated in space. The $A_{\mbox{$p_{\rm T}$}}$ requirement further exploits
the isolation of signal daughters in comparison to random combinations of
particles.
The figure of merit suggested in Ref. [27] is used to determine the optimal
selection point of the BDT classifier
$\text{FoM}=\frac{\epsilon^{\rm BDT}}{a/2+\sqrt{B_{\rm BDT}}}\,,$ (1)
where $\epsilon^{\rm BDT}$ is the efficiency of the BDT selection on the
$B^{0}_{(s)}\\!\rightarrow p\overline{}p$ signal candidates, which is
determined from simulation, $B_{\rm BDT}$ is the expected number of background
events within the (initially excluded) signal region, estimated from the data
sidebands, and the term $a=3$ quantifies the target level of significance in
units of standard deviation. With this optimisation the BDT classifier is
found to retain 44% of the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ signals
while reducing the combinatorial background level by 99.6%.
The kinematic selection of the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ decay is
performed using individual requirements on a set of variables similar to that
used for the BDT selection of the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$
decays, except that the cone variables are not used. This selection differs
from the selection used for signal modes and follows from the synergy with
ongoing LHCb analyses on two-body charmless $B$ decays, e.g. Ref. [28].
The particle identification (PID) criteria applied in addition to the
$B^{0}_{(s)}\\!\rightarrow p\overline{}p$ BDT classifier are also optimised
via Eq. 1. In this instance, the signal efficiencies are determined from data
control samples owing to known discrepancies between data and simulation for
the PID variables. Proton PID efficiencies are tabulated in bins of $p$,
$p_{\rm T}$ and the number of tracks in the event from data control samples of
$\mathchar 28931\relax\\!\rightarrow p\pi^{-}$ decays that are selected solely
using kinematic criteria. Pion and kaon efficiencies are likewise tabulated
from data control samples of $D^{*+}\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+})\,\pi^{+}$ decays. The kinematic distributions of the simulated
decay modes are then used to determine an average PID efficiency.
Specific PID criteria are separately defined for the two signal modes and the
normalisation channel. The PID efficiencies are found to be approximately 56%
for the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ signals and 42% for
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ decays.
The ratio of efficiencies of $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ with
respect to $B^{0}\\!\rightarrow K^{+}\pi^{-}$,
$\epsilon_{B^{0}_{(s)}\\!\rightarrow
p\overline{}p}/\epsilon_{B^{0}\\!\rightarrow K^{+}\pi^{-}}$, including
contributions from the detector acceptance, trigger, selection and PID, is
0.60 (0.61). After all selection criteria are applied, 45 and 58009 candidates
remain in the invariant mass ranges
$[5080,5480]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[5000,5800]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $p\overline{}p$
and $K^{+}\pi^{-}$ spectra, respectively.
Possible sources of background to the $p\overline{}p$ and $K^{+}\pi^{-}$
spectra are investigated using simulation samples. These include partially
reconstructed backgrounds with one or more particles from the decay of the $b$
hadron escaping detection, and two-body $b$-hadron decays where one or both
daughters are misidentified.
## 4 Signal yield determination
The signal and background candidates, in both the signal and normalisation
channels, are separated, after full selection, using unbinned maximum
likelihood fits to the invariant mass spectra.
The $K^{+}\pi^{-}$ mass spectrum of the normalisation mode is described with a
series of probability density functions (PDFs) for the various components,
similarly to Ref. [29]: the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ signal, the
$B^{0}_{s}\\!\rightarrow\pi^{+}K^{-}$ signal, the $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$, $B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and the $\mathchar
28931\relax_{b}^{0}\\!\rightarrow p\pi^{-}$ misidentified backgrounds,
partially reconstructed backgrounds, and combinatorial background. Any
contamination from other decays is treated as a source of systematic
uncertainty.
Both signal distributions are modelled by the sum of two Crystal Ball (CB)
functions [30] describing the high and low-mass asymmetric tails. The peak
values and the widths of the two CB components are constrained to be the same.
All CB tail parameters and the relative normalisation of the two CB functions
are fixed to the values obtained from simulation whereas the signal peak value
and width are free to vary in the fit to the $K^{+}\pi^{-}$ spectrum. The
$B^{0}_{s}\\!\rightarrow\pi^{+}K^{-}$ signal width is constrained to the
fitted $B^{0}\\!\rightarrow K^{+}\pi^{-}$ width such that the ratio of the
widths is identical to that obtained in simulation.
The invariant mass distributions of the misidentified $B^{0}_{s}\\!\rightarrow
K^{+}K^{-}$, $B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and $\mathchar
28931\relax_{b}^{0}\\!\rightarrow p\pi^{-}$ backgrounds are determined from
simulation and modelled with non-parametric PDFs. The fractions of these
misidentified backgrounds are related to the fraction of the
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ signal in the data via scaling factors that
take into account the relative branching fractions [1, 31], $b$-hadron
production fractions $f_{q}$ [32, 33], and relevant misidentification rates.
The latter are determined from calibration data samples.
Partially reconstructed backgrounds represent decay modes that can populate
the spectrum when misreconstructed as signal with one or more undetected
final-state particles, possibly in conjunction with misidentifications. The
shape of this distribution is determined from simulation, where each
contributing mode is assigned a weight dependent on its relative branching
fraction, $f_{q}$ and selection efficiency. The weighted sum of these
partially-reconstructed backgrounds is shown to be well modelled with the sum
of two exponentially-modified Gaussian (EMG) functions
$\mbox{EMG}(x;\mu,\sigma,\lambda)=\frac{\lambda}{2}e^{\frac{\lambda}{2}(2x+\lambda\sigma^{2}-2\mu)}\cdot{\rm
erfc}\Big{(}\frac{x+\lambda\sigma^{2}-\mu}{\sqrt{2}\sigma}\Big{)}\,,$ (2)
where $\rm{erfc}(x)=1-\rm{erf}(x)$ is the complementary error function. The
signs of the variable $x$ and parameter $\mu$ are reversed compared to the
standard definition of an EMG function. The parameters defining the shape of
the two EMG functions and their relative weight are determined from
simulation. The component fraction of the partially-reconstructed backgrounds
is obtained from the fit to the data, all other parameters being fixed from
simulation. The mass distribution of the combinatorial background is found to
be well described by a linear function whose gradient is determined by the
fit.
The fit to the $K^{+}\pi^{-}$ spectrum, presented in Fig. 1, determines seven
parameters, and yields $N(B^{0}\\!\rightarrow K^{+}\pi^{-})=24\,968\pm 198$
signal events, where the uncertainty is statistical only.
Figure 1: Invariant mass distribution of $K^{+}\pi^{-}$ candidates after full
selection. The fit result (blue, solid) is superposed together with each fit
model component as described in the legend. The normalised fit residual
distribution is shown at the bottom. Figure 2: Invariant mass distribution of
$p\overline{}p$ candidates after full selection. The fit result (blue, solid)
is superposed with each fit model component: the $B^{0}\\!\rightarrow
p\overline{}p$ signal (red, dashed), the $B^{0}_{s}\\!\rightarrow
p\overline{}p$ signal (grey, dotted) and the combinatorial background (green,
dot-dashed).
The $p\overline{}p$ spectrum is described by PDFs for the three components:
the $B^{0}\\!\rightarrow p\overline{}p$ and $B^{0}_{s}\\!\rightarrow
p\overline{}p$ signals, and the combinatorial background. In particular, any
contamination from partially reconstructed backgrounds, with or without
misidentified particles, is treated as a source of systematic uncertainty.
Potential sources of non-combinatorial background to the $p\overline{}p$
spectrum are two- and three-body decays of $b$ hadrons into protons, pions and
kaons, and many-body $b$-baryon modes partially reconstructed, with one or
multiple misidentifications. It is verified from extensive simulation studies
that the ensemble of specific backgrounds do not peak in the signal region but
rather contribute to a smooth mass spectrum, which can be accommodated by the
dominant combinatorial background contribution. The most relevant backgrounds
are found to be $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax^{+}_{c}(\rightarrow p\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0})\pi^{-}$, $\mathchar
28931\relax^{0}_{b}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0}p\pi^{-}$, $B^{0}\\!\rightarrow
K^{+}K^{-}\pi^{0}$ and $B^{0}\\!\rightarrow\pi^{+}\pi^{-}\pi^{0}$ decays.
Calibration data samples are exploited to determine the PID efficiencies of
these decay modes, thereby confirming the suppression with respect to the
combinatorial background by typically one or two orders of magnitude.
Henceforth physics-specific backgrounds are neglected in the fit to the
$p\overline{}p$ mass spectrum.
The $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ signal mass shapes are verified
in simulation to be well described by a single Gaussian function. The widths
of both Gaussian functions are assumed to be the same for $B^{0}\\!\rightarrow
p\overline{}p$ and $B^{0}_{s}\\!\rightarrow p\overline{}p$; a systematic
uncertainty associated to this assumption is evaluated. They are determined
from simulation with a scaling factor to account for differences in the
resolution between data and simulation; the scaling factor is determined from
the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ data and simulation samples. The mean
of the $B^{0}_{s}\\!\rightarrow p\overline{}p$ Gaussian function is
constrained according to the $B^{0}_{s}$–$B^{0}$ mass difference [1]. The mass
distribution of the combinatorial background is described by a linear
function.
The fit to the $p\overline{}p$ mass spectrum is presented in Fig. 2. The
yields for the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ signals in the full
mass range are $N(B^{0}\\!\rightarrow p\overline{}p)=11.4^{+4.3}_{-4.1}$ and
$N(B^{0}_{s}\\!\rightarrow p\overline{}p)=5.7^{+3.5}_{-3.2}$, where the
uncertainties are statistical only.
Figure 3: Negative logarithm of the profile likelihoods as a function of
(left) the $B^{0}\\!\rightarrow p\overline{}p$ signal yield and (right) the
$B^{0}_{s}\\!\rightarrow p\overline{}p$ signal yield. The orange solid curves
correspond to the statistical-only profiles whereas the blue dashed curves
include systematic uncertainties.
The statistical significances of the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$
signals are computed, using Wilks’ theorem [34], from the change in the mass
fit likelihood profiles when omitting the signal under scrutiny, namely
$\sqrt{2\ln(L_{\rm S+B}/L_{\rm B})}$, where $L_{\rm S+B}$ and $L_{\rm B}$ are
the likelihoods from the baseline fit and from the fit without the signal
component, respectively. The statistical significances are $3.5\,\sigma$ and
$1.9\,\sigma$ for the $B^{0}\\!\rightarrow p\overline{}p$ and
$B^{0}_{s}\\!\rightarrow p\overline{}p$ decay modes, respectively. Each
statistical-only likelihood curve is convolved with a Gaussian resolution
function of width equal to the systematic uncertainty (discussed below) on the
signal yield. The resulting likelihood profiles are presented in Fig. 3. The
total signal significances are $3.3\,\sigma$ and $1.9\,\sigma$ for the
$B^{0}\\!\rightarrow p\overline{}p$ and $B^{0}_{s}\\!\rightarrow
p\overline{}p$ modes, respectively. We observe an excess of
$B^{0}\\!\rightarrow p\overline{}p$ candidates with respect to background
expectations; the $B^{0}_{s}\\!\rightarrow p\overline{}p$ signal is not
considered to be statistically significant.
## 5 Systematic uncertainties
The sources of systematic uncertainty are minimised by performing the
branching fraction measurement relative to a decay mode topologically
identical to the decays of interest. They are summarised in Table 1.
Table 1: Relative systematic uncertainties contributing to the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ branching fractions. The total corresponds to the sum of all contributions added in quadrature. Source | Value (%)
---|---
| $B^{0}\\!\rightarrow p\overline{}p$ | $B^{0}_{s}\\!\rightarrow p\overline{}p$ | $B^{0}\\!\rightarrow K^{+}\pi^{-}$
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ branching fraction | – | – | 2.8
Trigger efficiency relative to $B^{0}\\!\rightarrow K^{+}\pi^{-}$ | 2.0 | 2.0 | –
Selection efficiency relative to $B^{0}\\!\rightarrow K^{+}\pi^{-}$ | 8.0 | 8.0 | –
PID efficiency | 10.6 | 10.7 | 1.0
Yield from mass fit | 6.8 | 4.6 | 1.6
$f_{s}/f_{d}$ | – | 7.8 | –
Total | 15.1 | 16.3 | 3.4
The branching fraction of the normalisation channel $B^{0}\\!\rightarrow
K^{+}\pi^{-}$, ${\cal B}(B^{0}\\!\rightarrow K^{+}\pi^{-})=(19.55\pm
0.54)\times 10^{-6}$ [31], is known to a precision of 2.8%, which is taken as
a systematic uncertainty. For the measurement of the $B^{0}_{s}\\!\rightarrow
p\overline{}p$ branching fraction, an extra uncertainty arises from the 7.8%
uncertainty on the ratio of fragmentation fractions $f_{s}/f_{d}=0.256\pm
0.020$ [33].
The trigger efficiencies are assessed from simulation for all decay modes. The
simulation describes well the ratio of efficiencies of the relevant modes that
comprise the same number of tracks in the final state. Neglecting small $p$
and $p_{\rm T}$ differences between the $B^{0}\\!\rightarrow p\overline{}p$
and $B^{0}_{s}\\!\rightarrow p\overline{}p$ modes, the ratios of
$B^{0}\\!\rightarrow K^{+}\pi^{-}/B^{0}_{(s)}\\!\rightarrow p\overline{}p$
trigger efficiencies should be consistent within uncertainties. The difference
of about 2% observed in simulation is taken as systematic uncertainty.
The $B^{0}\\!\rightarrow K^{+}\pi^{-}$ mode is used as a proxy for the
assessment of the systematic uncertainties related to the selection;
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ signal distributions are obtained from
data, using the sPlot technique, for a variety of selection variables. From
the level of agreement between simulation and data, a systematic uncertainty
of 8% is derived for the $B^{0}_{(s)}\\!\rightarrow p\overline{}p$ selection
efficiencies relative to $B^{0}\\!\rightarrow K^{+}\pi^{-}$.
The PID efficiencies are determined from data control samples. The associated
systematic uncertainties are estimated by repeating the procedure with
simulated control samples, the uncertainties being equal to the differences
observed betweeen data and simulation, scaled by the PID efficiencies
estimated with the data control samples. The systematic uncertainties on the
PID efficiencies are found to be 10.6%, 10.7% and 1.0% for the
$B^{0}\\!\rightarrow p\overline{}p$, $B^{0}_{s}\\!\rightarrow p\overline{}p$
and $B^{0}\\!\rightarrow K^{+}\pi^{-}$ decay modes, respectively. The large
uncertainties on the proton PID efficiencies arise from limited coverage of
the proton control samples in the kinematic region of interest for the signal.
Systematic uncertainties on the fit yields arise from the limited knowledge or
the choice of the mass fit models, and from the uncertainties on the values of
the parameters fixed in the fits. They are investigated by studying a large
number of simulated datasets, with parameters varying within their estimated
uncertainties. Combining all sources of uncertainty in quadrature, the
uncertainties on the $B^{0}\\!\rightarrow p\overline{}p$,
$B^{0}_{s}\\!\rightarrow p\overline{}p$ and $B^{0}\\!\rightarrow K^{+}\pi^{-}$
yields are 6.8%, 4.6% and 1.6%, respectively.
## 6 Results and conclusion
The branching fractions are determined relative to the $B^{0}\\!\rightarrow
K^{+}\pi^{-}$ normalisation channel according to
$\displaystyle{\cal B}(B^{0}_{(s)}\\!\rightarrow p\overline{}p)$
$\displaystyle=$ $\displaystyle\frac{N(B^{0}_{(s)}\\!\rightarrow
p\overline{}p)}{N(B^{0}\\!\rightarrow
K^{+}\pi^{-})}\cdot\frac{\epsilon_{B^{0}\\!\rightarrow
K^{+}\pi^{-}}}{\epsilon_{B^{0}_{(s)}\\!\rightarrow p\overline{}p}}\cdot
f_{d}/f_{d(s)}\cdot{\cal B}(B^{0}\\!\rightarrow K^{+}\pi^{-})$ (3)
$\displaystyle=$ $\displaystyle\alpha_{d(s)}\cdot N(B^{0}_{(s)}\\!\rightarrow
p\overline{}p)\,,$
where $\alpha_{d(s)}$ are the single-event sensitivities equal to $(1.31\pm
0.18)\times 10^{-9}$ and $(5.04\pm 0.81)\times 10^{-9}$ for the
$B^{0}\\!\rightarrow p\overline{}p$ and $B^{0}_{s}\\!\rightarrow
p\overline{}p$ decay modes, respectively; their uncertainties amount to 14%
and 16%, respectively.
The Feldman-Cousins (FC) frequentist method [35] is chosen for the calculation
of the branching fractions. The determination of the 68.3% and 90% CL bands is
performed with simulation studies relating the measured signal yields to
branching fractions, and accounting for systematic uncertainties. The 68.3%
and 90% CL intervals are
$\begin{array}[]{rcrc}{\cal B}(B^{0}\\!\rightarrow
p\overline{}p)=(1.47\,^{+0.62}_{-0.51}\,{}^{+0.35}_{-0.14})\times
10^{-8}&\mbox{at}&68.3\%&\mbox{CL}\,,\vspace*{0.2cm}\\\ {\cal
B}(B^{0}\\!\rightarrow
p\overline{}p)=(1.47\,^{+1.09}_{-0.81}\,{}^{+0.69}_{-0.18})\times
10^{-8}&\mbox{at}&90\%&\mbox{CL}\,,\vspace*{0.2cm}\\\ {\cal
B}(B^{0}_{s}\\!\rightarrow
p\overline{}p)=(2.84\,^{+2.03}_{-1.68}\,{}^{+0.85}_{-0.18})\times
10^{-8}&\mbox{at}&68.3\%&\mbox{CL}\,,\vspace*{0.2cm}\\\ {\cal
B}(B^{0}_{s}\\!\rightarrow
p\overline{}p)=(2.84\,^{+3.57}_{-2.12}\,{}^{+2.00}_{-0.21})\times
10^{-8}&\mbox{at}&90\%&\mbox{CL}\,,\vspace*{0.2cm}\\\ \end{array}$
where the first uncertainties are statistical and the second are systematic.
In summary, a search has been performed for the rare two-body charmless
baryonic decays $B^{0}\\!\rightarrow p\overline{}p$ and
$B^{0}_{s}\\!\rightarrow p\overline{}p$ using a data sample, corresponding to
an integrated luminosity of 0.9 $\mbox{\,fb}^{-1}$, of $pp$ collisions
collected at a centre-of-mass energy of 7 $\mathrm{\,Te\kern-1.00006ptV}$ by
the LHCb experiment. The results allow two-sided confidence limits to be
placed on the branching fractions of both $B^{0}\\!\rightarrow p\overline{}p$
and $B^{0}_{s}\\!\rightarrow p\overline{}p$ for the first time. We observe an
excess of $B^{0}\\!\rightarrow p\overline{}p$ candidates with respect to
background expectations with a statistical significance of $3.3\,\sigma$. This
is the first evidence for a two-body charmless baryonic $B^{0}$ decay. No
significant $B^{0}_{s}\\!\rightarrow p\overline{}p$ signal is observed and the
present result improves the previous bound by three orders of magnitude.
The measured $B^{0}\\!\rightarrow p\overline{}p$ branching fraction is
incompatible with all published theoretical predictions by one to two orders
of magnitude and motivates new and more precise theoretical calculations of
two-body charmless baryonic $B$ decays. An improved experimental search for
these decay modes at LHCb with the full 2011 and 2012 dataset will help to
clarify the situation, in particular for the $B^{0}_{s}\\!\rightarrow
p\overline{}p$ mode.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
## References
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arxiv-papers
| 2013-08-05T12:50:40 |
2024-09-04T02:49:48.982969
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, E. Cowie, D.C. Craik, S.\n Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P.\n Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin,\n U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez,\n P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Eduardo Rodrigues",
"url": "https://arxiv.org/abs/1308.0961"
}
|
1308.1005
|
# The profinite dimensional manifold structure of formal solution spaces of
formally integrable PDE’s
Batu Güneysu and Markus J. Pflaum
Batu Güneysu, [email protected]
Institut für Mathematik, Humboldt-Universität zu Berlin,
Rudower Chaussee 25, 12489 Berlin, Germany
Markus J. Pflaum, [email protected]
Department of Mathematics, University of Colorado,
Boulder CO 80309, USA
###### Abstract.
In this paper, we study the formal solution space of a nonlinear PDE in a
fiber bundle. To this end, we start with foundational material and introduce
the notion of a pfd structure to build up a new concept of profinite
dimensional manifolds. We show that the infinite jet space of the fiber bundle
is a profinite dimensional manifold in a natural way. The formal solution
space of the nonlinear PDE then is a subspace of this jet space, and inherits
from it the structure of a profinite dimensional manifold, if the PDE is
formally integrable. We apply our concept to scalar PDE’s and prove a new
criterion for formal integrability of such PDE’s. In particular, this result
entails that the Euler-Lagrange Equation of a relativistic scalar field with a
polynomial self-interaction is formally integrable.
###### Contents
1. 1 Some notation
2. 2 Profinite dimensional manifolds
1. 2.1 The category of profinite dimensional manifolds
2. 2.2 Tangent bundles and vector fields
3. 2.3 Differential forms
3. 3 Jet bundles and formal solutions of nonlinear PDE’s
1. 3.1 Finite order jet bundles
2. 3.2 Partial differential equations
1. 3.2.1 General facts
2. 3.2.2 Linear partial differential equations
3. 3.3 The manifold of $\infty$-jets and formally integrable PDE’s
4. 3.4 Scalar PDE’s and interacting relativistic scalar fields
1. 3.4.1 A criterion for formal integrability of scalar PDE’s
2. 3.4.2 Interacting relativistic scalar fields
4. A Two results on completed projective tensor products
## Introduction
Even though it appears to be unsolvable in general, the problem to describe
the moduli space of solutions of a particular nonlinear PDE has lead to
powerful new results in geometric analysis and mathematical physics. Notably
this can be seen, for example, by the fundamental work on the structure of the
moduli space of Yang–Mills equations [5, 12, 34]. Among the many challenging
problems which arise when studying moduli spaces of solutions of nonlinear
PDE’s is that the space under consideration does in general not have a
manifold structure, usually not even one modelled on an infinite dimensional
Hilbert or Banach space. Moreover, the solution space can possess
singularities. A way out of this dilemma is to study compactifications of the
moduli space like the completion of the moduli space with respect to a certain
Sobolev metric, cf. [15]. Another way, and that is the one we are advocating
in this article, is to consider a “coarse” moduli space consisting of so-
called formal solutions of a PDE, i.e. the space of those smooth functions
whose power series expansion at each point solves the PDE. In case the PDE is
formally integrable in a sense defined in this article, the formal solution
space turns out to be a profinite dimensional manifold. These possibly
infinite dimensional spaces are ringed spaces which can be regarded as
projective limits of projective systems of finite dimensional manifolds.
Profinite dimensional manifolds appear naturally in several areas of
mathematics, in particular in deformation quantization, see for example [27],
the structure theory of Lie-projective groups [6, 21], in connection with
functional integration on spaces of connections [4], and in the secondary
calculus invented by Vinogradov [36, 22] which inspired the approach in this
paper.
The paper consists of two main parts. The first, Section 2, lays out the
foundations of the theory of profinite dimensional manifolds. Besides the work
[1], which is taylored towards explaining the differential calculus by
Ashtekar and Lewandowski [4], literature on profinite dimensional manifolds is
scarce. Moreover, our approach to profinite dimensional manifolds is novel in
the sense that we define them as ringed spaces together with a so-called pfd
structure, which consists not only of one but a whole equivalence class of
representations by projective systems of finite dimensional manifolds. The
major point hereby is that all the projective systems appearing in the pfd
structure induce the same structure sheaf, which allows to define differential
geometric concepts depending only on the pfd structure and not a particular
representative. One way to construct differential geometric objects is by
dualizing projective limits of manifolds to injective limits of, for example,
differential forms, and then sheafify the thus obtain presheaves of “local”
objects. Again, it is crucial to observe that the thus obtained sheaves are
independant of the particular choice of a representative within the pfd
structure, whereas the “local” objects obtain a filtration which depends on
the choice of a particular representative. Using variants of this approach or
directly the structure sheaf of smooth functions, we introduce in Section 2
tangent bundles of profinite dimensional manifolds and their higher tensor
powers, vector fields, and differential forms.
The second main part is Section 3, where we introduce the formal solution
space of a nonlinear PDE. We first explain the necessary concepts from jet
bundle theory and on prolongations of PDE’s in fiber bundles, following
essentially Goldschmidt [19], cf. also [28, 36, 37, 22]. In Section 3.2.2 we
introduce in the jet bundle setting a notion of an operator symbol of a
nonlinear PDE such that, in the linear case, it coincides with the well-known
(principal) symbol of a partial differential operator up to canonical
isomorphisms. The corresponding result, Theorem 3.17, appears to be new.
Afterwards, we show that the bundle of infinite jets is a profinite
dimensional manifold. This result immediately entails that the formal solution
space of a formally integrable PDE is a profinite dimensional submanifold of
the infinite jet bundle. Finally, in Section 3.4, we consider scalar PDE’s. We
prove there a widely applicable criterion for the formal integrability of
scalar PDE’s, which to our knowledge has not appeared in the mathematical
literature yet. Moreover, we conclude from our criterion that the Euler-
Lagrange Equation of a scalar field with a polynomial self-interaction on an
arbitrary Lorentzian manifold is formally integrable, so its formal solution
space is a profinite dimensional manifold. We expect that this observation
will have fundamental consequences for a mathematically rigorous formulation
of the quantization theory of such scalar fields.
Acknowledgements: The first named author (B.G.) is indebted to Werner Seiler
for many discussions on jet bundles, and would also like to thank B. Kruglikov
and A.D. Lewis for helpful discussions. B.G. has been financially supported by
the SFB 647: Raum–Zeit–Materie, and would like to thank the University of
Colorado at Boulder for its hospitality. The second named author (M.P.) has
been partially supported by NSF grant DMS 1105670 and would like to thank
Humboldt-University, Berlin for its hospitality.
## 1\. Some notation
Let us introduce some notation and conventions which will be used throughout
the paper.
If nothing else is said, all manifolds and corresponding concepts, such as
submersions, bundles etc., are understood to be smooth and finite dimensional.
The symbol $\mathrm{T}^{k,l}$ stands for the functor of $k$-times
contravariant and $l$-times covariant tensors, where as usual
$\mathrm{T}:=\mathrm{T}^{1,0}$ and $\mathrm{T}^{*}:=\mathrm{T}^{0,1}$. If $X$
is a manifold, then the corresponding tensor bundles will be denoted by
$\pi_{\mathrm{T}^{k,l}X}:\>\mathrm{T}^{k,l}X\to X$. Moreover, we write
$\mathscr{X}^{\infty}$ and $\Omega^{k}$ for the sheaves of smooth vector
fields and of smooth $k$-forms, respectively.
Given a fibered manifold, i.e. a surjective submersion $\pi:\>E\to X$, we
write $\Gamma^{\infty}(\pi)$ for the sheaf of smooth sections of $\pi$. Its
space of sections over an open $U\subset X$ will be denoted by
$\Gamma^{\infty}(U;\pi)$. The set of _local smooth sections of_ $\pi$ _around
a point_ $p\in M$ is the set of smooth sections defined on some open
neighborhood of $p$ and will be denoted by $\Gamma^{\infty}(p;\pi)$. The stalk
at $p$ then is a quotient space of $\Gamma^{\infty}(p;\pi)$ and is written as
$\Gamma^{\infty}_{p}(\pi)$.
The _vertical vector bundle_ corresponding to the fibered manifold $\pi$ is
defined as the subvector bundle
(1.1)
$\pi^{\mathsf{V}}:\>\mathsf{V}(\pi):=\operatorname{ker}(\mathrm{T}\pi)\longrightarrow
E$
of $\pi_{\mathrm{T}E}:\mathrm{T}E\rightarrow E$. If
$\pi^{\prime}:\>E^{\prime}\to X$ is a second fibered manifold, the _vertical
morphism_ corresponding to a morphism $h:E\to E^{\prime}$ of fibered manifolds
over $X$ is given by
$h^{\mathsf{V}}:\>\mathsf{V}(\pi)\longrightarrow\mathsf{V}(\pi^{\prime}),\>v\longmapsto\mathrm{T}h(v).$
If $\pi:\>E\to X$ is a vector bundle, then the fibers of $\pi$ are
$\mathbb{R}$-vector spaces, hence one can apply tensor functors fiberwise to
obtain the corresponding tensor bundles. In particular,
$\pi^{\odot^{k}}:\>\operatorname{Sym}^{k}(\pi)\to X$ will stand for the _$k$
-fold symmetric tensor product bundle of_ $\pi$.
Finally, unless otherwise stated, the notions “projective system” and
“projective limit” will always be understood in the category of topological
spaces, where they of course exist; see [14, Chap. VIII, Sec. 3]. In fact,
given such a projective system
$\big{(}M_{i},\mu_{ij}\big{)}_{i,j\in\mathbb{N},i\leq j}$, a distinguished
projective limit is given as follows. Define
$M:=\Big{\\{}(p_{i})_{i\in\mathbb{N}}\in\prod_{i\in\mathbb{N}}M_{i}\mid\mu_{ij}(p_{j})=p_{i}\text{
for all $i,j\in\mathbb{N}$ with $i\leq j$}\Big{\\}}$
to be the subspace of all threads in the product, and the continuous maps
$\mu_{i}:M\rightarrow M_{i}$ as the restrictions of the canonical projections
$\prod_{i\in\mathbb{N}}M_{i}\rightarrow M_{i}$ to $M$. Then one obviously has
$\mu_{ij}\circ\mu_{j}=\mu_{i}$ for all $i,j\in\mathbb{N}$ with $i\leq j$. Note
that a basis of the topology of $M$ is given by the set of all open sets of
the form $\mu_{i}^{-1}(U)$, where $i\in\mathbb{N}$ and $U\subset M_{i}$ is
open. In the following, we will refer to the thus defined $M$ together with
the maps $(\mu_{i})_{i\in\mathbb{N}}$ as _the canonical projective limit_ of
$\big{(}M_{i},\mu_{ij}\big{)}_{i,j\in\mathbb{N},i\leq j}$, and denote it by
$M=\lim\limits_{\longleftarrow\atop i\in\mathbb{N}}M_{i}$.
## 2\. Profinite dimensional manifolds
In this section, we introduce the concept of profinite dimensional manifolds
and establish the differential geometric foundations of this new category.
### 2.1. The category of profinite dimensional manifolds
The following definition lies in the center of the paper:
###### Definition 2.1.
1. a)
By a _smooth projective system_ we understand a family
$\big{(}M_{i},\mu_{ij}\big{)}_{i,j\in\mathbb{N},\>i\leq j}$ of smooth
manifolds $M_{i}$ and surjective submersions $\mu_{ij}:M_{j}\rightarrow M_{i}$
for $i\leq j$ such that the following conditions hold true:
1. (SPS1)
$\mu_{ii}=\operatorname{id}_{M_{i}}$ for all $i\in\mathbb{N}$.
2. (SPS2)
$\mu_{ij}\circ\mu_{jk}=\mu_{ik}$ for all $i,j,k\in\mathbb{N}$ such that $i\leq
j\leq k$.
2. b)
If $\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}_{a,b\in\mathbb{N},\>a\leq
b}$ denotes a second smooth projective system, a _morphism of smooth
projective systems_ between
$\big{(}M_{i},\mu_{ij}\big{)}_{i,j\in\mathbb{N},\>i\leq j}$ and
$\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}_{a,b\in\mathbb{N},\>a\leq b}$
is a pair $\big{(}\varphi,(F_{a})_{a\in\mathbb{N}}\big{)}$ consisting of a
strictly increasing map $\varphi:\mathbb{N}\rightarrow\mathbb{N}$ and a family
of smooth maps $F_{a}:M_{\varphi(a)}\rightarrow M_{a}^{\prime}$,
$a\in\mathbb{N}$ such that for each pair $a,b\in\mathbb{N}$ with $a\leq b$ the
diagram
$\textstyle{M_{\varphi(a)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{a}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M_{\varphi(b)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\varphi(a)\varphi(b)}}$$\scriptstyle{F_{b}}$$\textstyle{M_{a}^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M_{b}^{\prime}}$$\scriptstyle{\mu_{ab}^{\prime}}$
commutes. We usually denote a smooth projective system shortly by
$\big{(}M_{i},\mu_{ij}\big{)}$ and write
$\big{(}\varphi,F_{a}\big{)}:\big{(}M_{i},\mu_{ij}\big{)}\longrightarrow\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}$
to indicate that $\big{(}\varphi,(F_{a})_{a\in\mathbb{N}}\big{)}$ is a
morphism of smooth projective systems. If each of the maps $F_{a}$ is a
submersion (resp. immersion), we call the morphism
$\big{(}\varphi,F_{a}\big{)}$ a _submersion_ (resp. _immersion_).
3. c)
Two smooth projective systems $\big{(}M_{i},\mu_{ij}\big{)}$ and
$\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}$ are called _equivalent_ , if
there are surjective submersions
$\big{(}\varphi,F_{a}\big{)}:\big{(}M_{i},\mu_{ij}\big{)}\longrightarrow\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)},\>\big{(}\psi,G_{i}\big{)}:\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}\longrightarrow\big{(}M_{i},\mu_{ij}\big{)}$
such that the diagrams
(2.1)
---
$\textstyle{M_{i}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M_{\varphi(\psi(i))}}$$\scriptstyle{\mu_{i\,\varphi(\psi(i))}}$$\scriptstyle{F_{\psi(i)}}$$\textstyle{N_{\psi(i)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G_{i}}$
and
---
$\textstyle{M_{a}^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
N_{\psi(\varphi(a))}}$$\scriptstyle{\mu_{a\,\psi(\varphi(a))}^{\prime}}$$\scriptstyle{G_{\varphi(a)}}$$\textstyle{M_{\varphi(a)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{a}}$
commute for all $i,a\in\mathbb{N}$. A pair of such surjective submersions will
be called an _equivalence transformation of smooth projective systems_.
###### Remark 2.2.
In the definition of smooth projective systems and later in the one of smooth
projective representations we use the partially ordered set $\mathbb{N}$ as
index set. Obviously, $\mathbb{N}$ can be replaced there by any partially
ordered set canonically isomorphic to $\mathbb{N}$ such as an infinite subset
of $\mathbb{Z}$ bounded from below. We will silently use this observation in
later applications for convenience of notation.
###### Example 2.3.
1. a)
Let $M$ be a manifold. Then $\big{(}M_{i},\mu_{ij}\big{)}$ with $M_{i}:=M$ and
$\mu_{ij}:=\operatorname{id}_{M}$ for $i\leq j$ is a smooth projective system
which we call _trivial_ and which we denote shortly by
$\big{(}M,\operatorname{id}_{M}\big{)}$.
2. b)
Assume that for $i\leq j$ one has given surjective linear maps
$\lambda_{ij}:V_{j}\rightarrow V_{i}$ between real finite dimensional vector
spaces such that a)(SPS1) and a)(SPS2) are satisfied. Then
$\big{(}V_{i},\lambda_{ij}\big{)}$ is a smooth projective system. For example,
this situation arises in deformation quantization of symplectic manifolds when
constructing the completed symmetric tensor algebra of a finite dimensional
real vector space; see [27] for details. Of course, a simpler example is given
by the canonical projections
$\pi_{ij}:\mathbb{R}^{j}\rightarrow\mathbb{R}^{i}$ onto the first $i$
coordinates, hence $\big{(}\mathbb{R}^{i},\pi_{ij}\big{)}$ is a (non-trivial)
smooth projective system.
3. c)
In the structure theory of topological groups [21, 6] one considers smooth
projective systems $(\mathsf{G}_{i},\eta_{ij})$ such that each
$\mathsf{G}_{j}$ is a Lie Group and the
$\eta_{ij}:\mathsf{G}_{j}\to\mathsf{G}_{i}$ are continuous group
homomorphisms. See Example 2.8 c) below for a precise description of the
projective limits of such projective systems of Lie groups.
4. d)
The tower of $k$-jets over a fiber bundle together with their canonical
projections forms a smooth projective system (see Section 3.1).
Within the category of (smooth finite dimensional) manifolds, a projective
limit of a smooth projective system obviously does in general not exist. In
the following, we will enlarge the category of manifolds by the so-called
profinite dimensional manifolds (and appropriate morphisms). The thus obtained
category will contain projective limits of smooth projective systems.
###### Definition 2.4.
1. a)
By a _smooth projective representation_ of a commutative locally
$\mathbb{R}$-ringed space $(M,\mathscr{C}^{\infty}_{M})$ we understand a
smooth projective system $\big{(}M_{i},\mu_{ij}\big{)}$ together with a family
of continuous maps $\mu_{i}:M\rightarrow M_{i}$, $i\in\mathbb{N}$, such that
the following conditions hold true:
1. (PFM1)
As a topological space, $M$ together with the family of maps $\mu_{i}$,
$i\in\mathbb{N}$, is a projective limit of $\big{(}M_{i},\mu_{ij}\big{)}$.
2. (PFM2)
The section space $\mathscr{C}^{\infty}_{M}(U)$ of the structure sheaf over an
open subset $U\subset M$ is given by the set of all $f\in\mathscr{C}(U)$ such
that for every $x\in U$ there exists an $i\in\mathbb{N}$, an open
$U_{i}\subset M_{i}$ and an $f_{i}\in\mathscr{C}^{\infty}(U_{i})$ such that
$p\in\mu_{i}^{-1}(U_{i})\subset U$ and
$f_{|\mu_{i}^{-1}(U_{i})}=f_{i}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})}$
hold true.
We usually denote a smooth projective representation briefly as a family
$\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$.
2. b)
A smooth projective representation $\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$ of
$(M,\mathscr{C}^{\infty}_{M})$ is said to be _regular_ , if each of the maps
$\mu_{ij}:M_{j}\rightarrow M_{i}$ is a fiber bundle.
3. c)
Two smooth projective representations $\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$
and $\big{(}M_{a}^{\prime},\mu_{ab}^{\prime},\mu_{a}^{\prime}\big{)}$ of
$(M,\mathscr{C}^{\infty}_{M})$ are called _equivalent_ , if there is an
equivalence transformation of smooth projective systems
$\big{(}\varphi,F_{a}\big{)}:\big{(}M_{i},\mu_{ij}\big{)}\longrightarrow\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)},\>\big{(}\psi,G_{i}\big{)}:\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}\longrightarrow\big{(}M_{i},\mu_{ij}\big{)}$
such that
(2.2)
$\mu_{i}=G_{i}\circ\mu_{\psi(i)}^{\prime}\quad\text{and}\quad\mu_{a}^{\prime}=F_{a}\circ\mu_{\varphi(a)}\quad\text{for
all $i,a\in\mathbb{N}$}.$
In the following, we will sometimes call such a pair of surjective submersions
an _equivalence transformation of smooth projective representations_. The
equivalence class of a smooth projective system
$\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$ will be simply denoted by
$[\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}]$ and called a _pfd structure_ on
$(M,\mathscr{C}^{\infty}_{M})$.
###### Proposition 2.5.
Let $(M,\mathscr{C}^{\infty}_{M})$ be a commutative locally
$\mathbb{R}$-ringed space with a smooth projective representation
$\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$. Assume further that
$\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}$ is a smooth projective system
which is equivalent to $\big{(}M_{i},\mu_{ij}\big{)}$. Then there are
continuous maps $\mu_{a}^{\prime}:M\rightarrow M_{a}^{\prime}$,
$a\in\mathbb{N}$, such that
$\big{(}M_{a}^{\prime},\mu_{ab}^{\prime},\mu_{a}^{\prime}\big{)}$ becomes a
smooth projective representation of $(M,\mathscr{C}^{\infty}_{M})$ which is
eqivalent to $\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$.
###### Proof.
Choose an equivalence transformation of smooth projective systems
$\big{(}\varphi,F_{a}\big{)}:\big{(}M_{i},\mu_{ij}\big{)}\longrightarrow\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)},\>\big{(}\psi,G_{i}\big{)}:\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}\longrightarrow\big{(}M_{i},\mu_{ij}\big{)}.$
Put $\mu_{a}^{\prime}:=F_{a}\circ\mu_{\varphi(a)}$. Let us show first that $M$
together with the family of continuous maps $\mu_{a}^{\prime}$,
$a\in\mathbb{N}$ is a projective limit of
$\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}$. So assume that $X$ is a
topological space, and $h_{a}:X\rightarrow M_{a}^{\prime}$, $a\in\mathbb{N}$ a
family of continuous maps such that $h_{a}=\mu_{ab}^{\prime}\circ h_{b}$ for
$a\leq b$. Since $M$ is a projective limit of $\big{(}M_{i},\mu_{ij}\big{)}$,
there exists a uniquely determined $h:X\rightarrow M$ such that $\mu_{i}\circ
h=G_{i}\circ h_{\psi(i)}$ for all $i\in\mathbb{N}$. But then
$\mu_{a}^{\prime}\circ h=F_{a}\circ\mu_{\varphi(a)}\circ h=F_{a}\circ
G_{\varphi(a)}\circ h_{\psi(\varphi(a))}=\mu_{a\psi(\varphi(a))}^{\prime}\circ
h_{\psi(\varphi(a))}=h_{a}.$
Moreover, if $\widetilde{h}:X\rightarrow M$ is a continuous function such that
$\mu_{a}^{\prime}\circ\widetilde{h}=h_{a}$ for all $a\in\mathbb{N}$, one
computes
$\displaystyle\mu_{i}\circ\widetilde{h}=\mu_{i\varphi(\psi(i))}\circ\mu_{\varphi(\psi(i))}\circ\widetilde{h}=G_{i}\circ
F_{\psi(i)}\circ\mu_{\varphi(\psi(i))}\circ\widetilde{h}=G_{i}\circ\mu_{\psi(i)}^{\prime}\circ\widetilde{h}$
$\displaystyle=G_{i}\circ h_{\psi(i)}.$
Since $M$ is a projective limit of $\big{(}M_{i},\mu_{ij}\big{)}$, this
entails $\widetilde{h}=h$. This proves that $M$ is a projective limit of
$\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}$.
Next let us show that a)(PFM2) holds true with the $\mu_{i}$ replaced by the
$\mu_{a}^{\prime}$. So let $U\subset M$ be open,
$f\in\mathscr{C}^{\infty}_{M}(M)$, and $p\in U$. Choose $i\in\mathbb{N}$ such
that there is an open $U_{i}\subset M_{I}$ and a smooth
$f_{i}:U_{i}\rightarrow\mathbb{R}$ with $p\in\mu_{i}^{-1}(U_{i})\subset U$ and
$f_{|\mu_{i}^{-1}(U_{i})}=f_{i}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})}$. Put
$a:=\psi(i)$, $V_{a}:=G_{i}^{-1}(U_{i})$, and define
$\widetilde{f}_{a}:V_{a}\rightarrow\mathbb{R}$ by
$\widetilde{f}_{a}:=f_{i}\circ{G_{i}}_{|V_{a}}$. Then $\widetilde{f}_{a}$ is
smooth, and
$\begin{split}\widetilde{f}_{a}\circ{\mu_{a}^{\prime}}_{|{\mu_{a}^{\prime}}^{-1}(V_{a})}&=f_{i}\circ
G_{i}\circ
F_{\psi(i)}\circ{\mu_{\varphi(\psi(i))}}_{|{\mu_{a}^{\prime}}^{-1}(V_{a})}\\\
&=f_{i}\circ\mu_{i\varphi(\psi(i))}\circ{\mu_{\varphi(\psi(i))}}_{|{\mu_{a}^{\prime}}^{-1}(V_{a})}=f_{i}\circ{\mu_{i}}_{|{\mu_{a}^{\prime}}^{-1}(V_{a})}\\\
&=f_{|{\mu_{a}^{\prime}}^{-1}(V_{a})},\end{split}$
where we have used that ${\mu_{a}^{\prime}}^{-1}(V_{a})=\mu_{i}^{-1}(U_{i})$.
Similarly one shows that a continuous $\widetilde{f}:U\rightarrow\mathbb{R}$
is an element of $\mathscr{C}^{\infty}_{M}(U)$, if for every $p\in U$ there is
an $a\in\mathbb{N}$, an open $V_{a}\subset M_{a}^{\prime}$, and a smooth
function $\widetilde{f}_{a}:V_{a}\rightarrow\mathbb{R}$ such that
$p\in{\mu_{a}^{\prime}}^{-1}(V_{a})\subset U$ and
$\widetilde{f}_{a}\circ{\mu_{a}^{\prime}}_{|{\mu_{a}^{\prime}}^{-1}(V_{a})}=f_{|{\mu_{a}^{\prime}}^{-1}(V_{a})}$.
Finally, it remains to prove that $\mu_{i}=G_{i}\circ\mu_{\psi(i)}^{\prime}$
for all $i\in\mathbb{N}$, but this follows from
$G_{i}\circ\mu_{\psi(i)}^{\prime}=G_{i}\circ
F_{\psi(i)}\circ\mu_{\varphi(\psi(i)))}=\mu_{i\varphi(\psi(i)))}\circ\mu_{\varphi(\psi(i)))}=\mu_{i}.$
This finishes the proof. ∎
###### Remark 2.6.
The preceding proposition entails that the structure sheaf of a commutative
locally $\mathbb{R}$-ringed space $\big{(}M,\mathscr{C}^{\infty}_{M}\big{)}$
for which a smooth projective representation
$\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$ exists depends only on the equivalence
class $[\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}]$.
The latter remark justifies the following definition:
###### Definition 2.7.
1. a)
By a _profinite dimensional manifold_ we understand a commutative locally
$\mathbb{R}$-ringed space $(M,\mathscr{C}^{\infty}_{M})$ together with a pfd
structure defined on it. The profinite dimensional manifold
$(M,\mathscr{C}^{\infty}_{M})$ is called _regular_ , if there exists a regular
smooth representation within the pfd structure on
$(M,\mathscr{C}^{\infty}_{M})$.
2. b)
Assume that $(M,\mathscr{C}^{\infty}_{M})$ and $(N,\mathscr{C}^{\infty}_{N})$
are profinite dimensional manifolds. Then a continuous map $f:M\rightarrow N$
is said to be _smooth_ , if the following condition holds true:
1. (PFM1)
For every open $U\subset N$, and $g\in\mathscr{C}^{\infty}_{N}(U)$ one has
$g\circ f_{|f^{-1}(U)}\in\mathscr{C}^{\infty}_{M}\big{(}f^{-1}(U)\big{)}\>.$
By definition, it is clear that the composition of smooth maps between
profinite dimensional manifolds is smooth, hence profinite dimensional
manifolds and the smooth maps between them as morphisms form a category, the
isomorphisms of which can be safely called _diffeomorphisms_. All of this
terminology is justified by the simple observation Example 2.8 a) below.
###### Example 2.8.
1. a)
Given a manifold $M$, the trivial smooth projective system
$\big{(}M,\operatorname{id}_{M}\big{)}$ defines a smooth projective
representation for the ringed space $(M,\mathscr{C}^{\infty}_{M})$. Hence,
every manifold is a profinite dimensional manifold in a natural way, and the
category of manifolds a full subcategory of the category of profinite
dimensional manifolds.
2. b)
Assume that $\big{(}M_{i},\mu_{ij}\big{)}$ is a smooth projective system. Let
$M:=\lim\limits_{\longleftarrow\atop i\in\mathbb{N}}M_{i}$
together with the natural projections $\mu_{i}:M\rightarrow M_{i}$ denote the
canonical projective limit of $\big{(}M_{i},\mu_{ij}\big{)}$. Then, a)(PFM1)
is fulfilled by assumption, and it is immediate that $M$ carries a uniquely
determined structure sheaf $\mathscr{C}^{\infty}_{M}$ which satisfies
a)(PFM2). The locally ringed space $(M,\mathscr{C}^{\infty}_{M})$ together
with the pfd structure $[\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}]$ then is a
profinite dimensional manifold. This profinite dimensioal manifold is even a
projective limit of the projective system $\big{(}M_{i},\mu_{ij}\big{)}$
within the category of profinite dimensional manifolds. We therefore write in
this situation
$(M,\mathscr{C}^{\infty}_{M})=\lim\limits_{\longleftarrow\atop
i\in\mathbb{N}}(M_{i},\mathscr{C}^{\infty}_{M_{i}})$
and call $(M,\mathscr{C}^{\infty}_{M})$ (together with
$[\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}]$) _the canonical smooth projective
limit_ of $\big{(}M_{i},\mu_{ij}\big{)}$.
3. c)
A locally compact Hausdorff topological group $\mathsf{G}$ is called _Lie
projective_ , if every neighbourhood of the identity contains a compact Lie
normal subgroup, i.e. a normal subgroup $N\subset G$ such that $G/N$ is a Lie
group. One has the following structure theorem [6, Thm. 4.4], [21]. A locally
compact metrizable group $\mathsf{G}$ is Lie projective, if and only if there
is a smooth projective system $(\mathsf{G}_{i},\eta_{ij})$ as in Example 2.3
c) together with continuous group homomorphisms
$\eta_{i}:\mathsf{G}\to\mathsf{G}_{i}$, $i\in\mathbb{N}$ such that
$(\mathsf{G},\eta_{i})$ is a projective limit of $(\mathsf{G}_{i},\eta_{ij})$.
Again, it follows that $\mathsf{G}$ carries a uniquely determined structure
sheaf $\mathscr{C}^{\infty}_{\mathsf{G}}$ satisfying a)(PFM2). The locally
ringed space $(\mathsf{G},\mathscr{C}^{\infty}_{\mathsf{G}})$ together with
the pfd structure $[\big{(}\mathsf{G}_{i},\eta_{ij},\eta_{i}\big{)}]$ becomes
a regular profinite dimensional manifold with a group structure such that all
of its structure maps are smooth.
4. d)
The space of infinite jets over a fiber bundle canonically is a profinite
dimensional manifold (see Section 3.3).
###### Remark 2.9.
In the sequel, $(M,\mathscr{C}^{\infty}_{M})$ or briefly $M$ will always
denote a profinite dimensional manifold. Moreover,
$\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$ always stands for a smooth projective
representation defining the pfd structure on $M$. The sheaf of smooth
functions on a profinite dimensional manifold will often briefly be denoted by
$\mathscr{C}^{\infty}$, if no confusion can arise.
Let $N\subset M$ be a subset, and assume further that for some smooth
projective representation $\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$ of the pfd
structure on $M$ the following holds true:
1. (PFSM1)
There is a stricly increasing sequence $(l_{i})_{i\in\mathbb{N}}$ such that
for every $i\in\mathbb{N}$ the set $N_{i}:=\mu_{l_{i}}(N)$ is a submanifold of
$M_{l_{i}}$.
2. (PFSM2)
One has $N=\bigcap\limits_{i\in\mathbb{N}}\mu_{l_{i}}^{-1}(N_{i})$.
3. (PFSM3)
The induced map
$\nu_{ij}:={\mu_{l_{i}l_{j}}}_{|N_{j}}:N_{j}\longrightarrow N_{i}$
is a submersion for all $i,j\in\mathbb{N}$ with $j\geq i$.
Observe that the $\nu_{ij}$ are surjective by definition of the manifolds
$N_{i}$ and by $\nu_{i}=\nu_{ij}\circ\nu_{j}$, where we have put
$\nu_{i}:={\mu_{l_{i}}^{\prime}}_{|N}$. In particular, $(N_{i},\nu_{ij})$
becomes a smooth projective system.
###### Proposition and Definition 2.10.
Let $N\subset M$ be a subset such that for some smooth projective
representation $(M_{i},\mu_{ij},\mu_{i})$ of the pfd structure on $M$ the
axioms (PFSM1) to (PFSM3) are fulfilled. Then $N$ carries in a natural way the
structure of a profinite dimensional manifold such that its sheaf of smooth
functions coincides with the sheaf $\mathscr{C}^{\infty}_{|N}$ of continuous
functions on open subset of $N$ which are locally restrictions of smooth
functions on $M$. A smooth projective representation of $N$ defining its
natural pfd structure is given by the family $(N_{i},\nu_{ij},\nu_{i})$. ¿From
now on, such a subset $N\subset M$ will be called a _profinite dimensional
submanifold of_ $M$, and $(M_{i},\mu_{ij},\mu_{i})$ a smooth projective
representation of $M$ _inducing the submanifold structure on_ $N$.
###### Proof.
We first show that $N$ together with the maps $\nu_{i}$ is a (topological)
projective limit of the projective system $(N_{i},\nu_{ij})$. Let $p_{i}\in
N_{i}$, $i\in\mathbb{N}$ such that $\nu_{ij}(p_{j})=p_{i}$ for all $j\geq i$.
Since $M$ together with the $\mu_{i}$ is a projective limit of
$(M_{i},\mu_{ij})$, there exists an $p\in M$ such that $\mu_{l_{i}}(p)=p_{i}$
for all $i\in\mathbb{N}$. By axiom (PFSM2), $p\in N$, hence one concludes that
$N$ is a projective limit of the manifolds $N_{i}$.
Next, we show that $\mathscr{C}^{\infty}_{|N}$ coincides with the uniquely
determined sheaf $\mathscr{C}^{\infty}_{N}$ satisfying axiom a)(PFM2). Since
the canonical embeddings $N_{i}\hookrightarrow M_{l_{i}}$ are smooth by
(PFSM1), the embedding $N\hookrightarrow M$ is smooth as well, and
$\mathscr{C}^{\infty}_{|N}$ is a subsheaf of the sheaf
$\mathscr{C}^{\infty}_{N}$. It remains to prove that for every open $V\subset
N$ a function $f\in\mathscr{C}^{\infty}_{N}(V)$ is locally the restriction of
a smooth function on $M$. To show this let $p\in V$ and $V_{i}$ an open subset
of some $N_{i}$ such that $p\in\nu_{i}^{-1}(V_{i})\subset V$, and such that
there is an $f_{i}\in\mathscr{C}^{\infty}(V_{i})$ with
$f_{|\nu_{i}^{-1}(V_{i})}=f_{i}\circ{\nu_{i}}_{|\nu_{i}^{-1}(V_{i})}$. Since
$N_{i}$ is locally closed in $M_{l_{i}}$, we can assume after possibly
shrinking $V_{i}$ that there is an open $U_{i}\subset M_{l_{i}}$ with
$V_{i}=N_{i}\cap U_{i}$ and such that $N_{i}\cap U_{i}$ is closed in $U_{i}$.
Then there exists ${F_{i}}\in\mathscr{C}^{\infty}(U_{i})$ such that
${F_{i}}_{|V_{i}}=f_{i}$. Put
$F:=F_{i}\circ{\mu_{l_{i}}}_{|\mu_{l_{i}}^{-1}(U_{i})}$. Then
$F\in\mathscr{C}^{\infty}(\mu_{l_{i}}^{-1}(U_{i}))$, and
$f_{|\nu_{i}^{-1}(V_{i})}=F_{|\nu_{i}^{-1}(V_{i})},$
which proves that $f\in\mathscr{C}^{\infty}_{|N}(V)$. The claim follows. ∎
###### Example 2.11.
1. a)
Every open subset $U$ of $M$ is naturally a profinite dimensional submanifold
since for each $i\in\mathbb{N}$ the set $U_{i}:=\mu_{i}(U)$ is an open
submanifold of $M_{i}$.
2. b)
Consider the profinite dimensional manifold
$\big{(}\mathbb{R}^{\infty},\mathscr{C}^{\infty}_{\mathbb{R}^{\infty}}\big{)}:=\lim\limits_{\longleftarrow\atop
n\in\mathbb{N}}\big{(}\mathbb{R}^{n},\mathscr{C}^{\infty}_{\mathbb{R}^{n}}\big{)},$
and let $\mathrm{B}^{n}(0)$ be the open unit ball in $\mathbb{R}^{n}$. The
projective limit
$\big{(}\mathrm{B}^{\infty}(0),\mathscr{C}^{\infty}_{\mathrm{B}^{\infty}(0)}\big{)}:=\lim\limits_{\longleftarrow\atop
n\in\mathbb{N}}\big{(}\mathrm{B}^{n}(0),\mathscr{C}^{\infty}_{\mathrm{B}^{n}(0)}\big{)}$
then becomes a profinite dimensional submanifold of $\mathbb{R}^{\infty}$.
Note that it is not locally closed in $\mathbb{R}^{\infty}$.
3. c)
The space of formal solutions of a formally integrable partial differential
equation is a profinite dimensional submanifold of the space of infinite jets
over the underlying fiber bundle (see Section 3.3).
We continue with:
###### Definition 2.12.
Let $U\subset M$ be open. A smooth function $f\in\mathscr{C}^{\infty}(U)$ then
is called _local_ , if there is an open $U_{i}\subset M_{i}$ for some
$i\in\mathbb{N}$ and a function $f_{i}\in\mathscr{C}^{\infty}(U_{i})$ such
that $U\subset\mu_{i}^{-1}(U_{i})$ and $f=f_{i}\circ{\mu_{i}}_{|U}$. We denote
the space of local functions over $U$ by
$\mathscr{C}_{\textup{loc}}^{\infty}(U)$.
###### Remark 2.13.
1. a)
Observe that $\mathscr{C}_{\textup{loc}}^{\infty}$ forms a presheaf on $M$,
which depends only on the pfd structure
$[\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}]$. Moreover, it is clear by
construction that for every open $U\subset M$ and every representative
$(M_{i},\mu_{ij},\mu_{i})$ of the pfd structure,
$\mathscr{C}_{\textup{loc}}^{\infty}(U)$ together with the family of pull-back
maps
$\mu_{i}^{*}:\mathscr{C}^{\infty}(\mu_{i}(U))\rightarrow\mathscr{C}_{\textup{loc}}^{\infty}(U)$
is an injective limit of the injective system of linear spaces
$\big{(}\mathscr{C}^{\infty}(\mu_{i}(U)),\mu_{ij}^{*}\big{)}_{i\in\mathbb{N}}$.
2. b)
$\mathscr{C}_{\textup{loc}}^{\infty}$ is in general not a sheaf unless $M$ is
a finite dimensional manifold. The sheaf associated to
$\mathscr{C}_{\textup{loc}}^{\infty}$ naturally coincides with
$\mathscr{C}^{\infty}$ since locally, every smooth function is local.
3. c)
By naming sections of $\mathscr{C}_{\textup{loc}}^{\infty}$ local functions we
essentially follow Stasheff [30, Def. 1.1] and Barnich [10, Def. 1.1], where
the authors consider jet bundles. Note that in [1], local functions are called
cylindrical functions.
4. d)
The representative $\mathcal{M}:=(M_{i},\mu_{ij},\mu_{j})$ leads to a
particular filtration $\mathcal{F}^{\mathcal{M}}_{\bullet}$ of the presheaf of
local functions by putting, for $l\in\mathbb{N}$,
$\mathcal{F}^{\mathcal{M}}_{l}\big{(}\mathscr{C}_{\textup{loc}}^{\infty}\big{)}:=\mu_{l}^{*}\mathscr{C}^{\infty}_{M_{l}}\>.$
Observe that this filtration has the property that
$\mathscr{C}_{\textup{loc}}^{\infty}=\bigcup_{l\in\mathbb{N}}\mathcal{F}^{\mathcal{M}}_{l}\big{(}\mathscr{C}_{\textup{loc}}^{\infty}\big{)}\>.$
### 2.2. Tangent bundles and vector fields
The tangent space at a point of a finite dimensional manifold can be defined
as a set of equivalence classes of germs of smooth paths at that point or as
the space of derivations on the stalk of the sheaf of smooth functions at that
point. The definition via paths can not be immediately carried over to the
profinite dimensional case, so we use the derivation approach.
###### Definition 2.14.
Given a point $p$ of the profinite dimensional manifold $M$, the _tangent
space_ of $M$ at $p$ is defined as the space of derivations on
$\mathscr{C}^{\infty}_{p}$, the stalk of smooth functions at $p$, i.e. as the
space
$\mathrm{T}_{p}M:=\operatorname{Der}\big{(}\mathscr{C}^{\infty}_{p},\mathbb{R}\big{)}\>.$
Elements of $\mathrm{T}_{p}M$ will be called _tangent vectors_ of $M$ at $p$.
The _tangent bundle_ of $M$ is the disjoint union
$\mathrm{T}M:=\bigcup_{p\in M}\mathrm{T}_{p}M,$
and
$\pi_{\mathrm{T}M}:\mathrm{T}M\longrightarrow M,\mathrm{T}_{p}M\ni
Y\longmapsto p$
the _canonical projection_.
Note that for every $i\in\mathbb{N}$ there is a canonical map
$\mathrm{T}\mu_{i}:\mathrm{T}M\rightarrow\mathrm{T}M_{i}$ which maps a tangent
vector $Y\in\mathrm{T}_{p}M$ to the tangent vector
$Y_{i}:\>\mathscr{C}^{\infty}_{M_{i},p_{i}}\rightarrow\mathbb{R},\quad[f_{i}]_{p_{i}}\mapsto
Y\big{(}[f_{i}\circ\mu_{i}]_{p}\big{)},\quad\text{where $p_{i}:=\mu_{i}(p)$}.$
By construction, one has
$\mathrm{T}\mu_{ij}\circ\mathrm{T}\mu_{j}=\mathrm{T}\mu_{i}$ for $i\leq j$. We
give $\mathrm{T}M$ the coarsest topology such that all the maps
$\mathrm{T}\mu_{i}$, $i\in\mathbb{N}$ are continuous. Now we record the
following observation:
###### Lemma 2.15.
The topological space $\mathrm{T}M$ together with the maps $\mathrm{T}\mu_{i}$
is a projective limit of the projective system
$\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij}\big{)}$.
###### Proof.
Assume that $X$ is a topological space, and
$\big{(}\Phi_{i}\big{)}_{i\in\mathbb{N}}$ a family of continuous maps
$\Phi_{i}:X\rightarrow\mathrm{T}M_{i}$ such that
$\mathrm{T}\mu_{ij}\circ\Phi_{j}=\Phi_{i}$ for all $i\leq j$. Since $M$ is a
projective limit of the projective system $\big{(}M_{i},\mu_{ij}\big{)}$,
there exists a uniquely determined continuous map $\varphi:X\rightarrow M$
such that $\mu_{i}\circ\Phi_{i}=\mu_{i}\circ\varphi$ for all $i\in\mathbb{N}$.
Now let $x\in X$, and put $p:=\varphi(x)$ and $p_{i}:=\mu_{i}(p)$. Then, for
every $i\in\mathbb{N}$, $\Phi_{i}(x)$ is a tangent vector of $M_{i}$ with
footpoint $p_{i}$. We now construct a derivation
$\Phi(x)\in\operatorname{Der}\big{(}\mathscr{C}^{\infty}_{p},\mathbb{R}\big{)}$.
Let $[f]_{p}\in\mathscr{C}^{\infty}_{p}$, i.e. let $f$ be a smooth function
defined on a neighborhood $U$ of $p$, and $[f]_{p}$ its germ at $p$. Then
there exists $i\in\mathbb{N}$, an open neighborhood $U_{i}\subset M_{i}$ of
$p_{i}$ and a smooth function $f_{i}:U_{i}\rightarrow\mathbb{R}$ such that
$\mu_{i}^{-1}(U_{i})\subset U\quad\text{and}\quad
f_{|\mu_{i}^{-1}(U_{i})}=f_{i}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})}.$
We now put
(2.3)
$\Phi(x)\big{(}[f]_{p}\big{)}:=\Phi_{i}(x)\big{(}[f_{i}]_{p_{i}}\big{)},\quad\text{where
$p_{i}:=\mu_{i}(p)$}.$
We have to show that $\Phi(x)$ is independant of the choices made, and that it
is a derivation indeed. So let $f^{\prime}:U^{\prime}\rightarrow\mathbb{R}$ be
another smooth function defining the germ $[f]_{p}$. Choose $j\in\mathbb{N}$,
an open neighborhood $U_{j}^{\prime}\subset M_{j}$ of $p_{j}$, and a smooth
function $f_{j}^{\prime}:U_{j}^{\prime}\rightarrow\mathbb{R}$ such that
$\mu_{j}^{-1}(U_{j}^{\prime})\subset U\quad\text{and}\quad
f_{|\mu_{j}^{-1}(U_{j}^{\prime})}=f_{j}^{\prime}\circ{\mu_{j}}_{|\mu_{j}^{-1}(U_{j}^{\prime})}.$
Without loss of generality, we can assume $i\leq j$. By assumption
$[f]_{p}=[f^{\prime}]_{p}$, hence one concludes that
$f_{i}\circ{\mu_{ij}}_{|V_{j}}={f_{j}^{\prime}}_{|V_{j}}$
for some open neighborhood $V_{j}\subset M_{j}$ of $p_{j}:=\mu_{j}(p)$. But
this implies, using the assumption on the $\Phi_{i}$ that
$\Phi_{j}(x)\big{(}[f_{j}^{\prime}]_{p_{j}}\big{)}=\mathrm{T}\mu_{ij}\Phi_{j}(x)\big{(}[f_{i}]_{p_{i}}\big{)}=\Phi_{i}(x)\big{(}[f_{i}]_{p_{i}}\big{)}.$
Hence, $\Phi(x)$ is well-defined, indeed.
Next, we show that $\Phi(x)$ is derivation. So let
$[f]_{p},[g]_{p}\in\mathscr{C}^{\infty}_{p}$ be two germs of smooth functions
at $p$. Then, after possibly shrinking the domains of $f$ and $g$, one can
find an $i\in\mathbb{N}$, an open neighborhood $U_{i}\subset M_{i}$ of
$p_{i}$, and $f_{i},g_{i}\in\mathscr{C}^{\infty}(U_{i})$ such that
$f_{|\mu_{i}^{-1}(U_{i})}=f_{i}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})}\quad\text{and}\quad
g_{|\mu_{i}^{-1}(U_{i})}=g_{i}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})}\>.$
Since $\Phi_{i}(x)$ acts as a derviation on $\mathscr{C}^{\infty}_{p_{i}}$,
one checks
$\begin{split}\Phi(x)\big{(}[f]_{p}[g]_{p}\big{)}\,&=\Phi_{i}(x)\big{(}[f_{i}]_{p_{i}}[g_{i}]_{p_{i}}\big{)}=\\\
&=f_{i}(p_{i})\Phi_{i}(x)\big{(}[g_{i}]_{p_{i}}\big{)}+g_{i}(p_{i})\Phi_{i}(x)\big{(}[f_{i}]_{p_{i}}\big{)}=\\\
&=f(p)\Phi(x)\big{(}[g]_{p}\big{)}+g(p)\Phi(x)\big{(}[f]_{p}\big{)},\end{split}$
which means that $\Phi(x)$ is a derivation.
By construction, it is clear that
$\mathrm{T}\mu_{i}\Phi(x)=\Phi_{i}(x)\quad\text{for all $i\in\mathbb{N}$}.$
Let us verify that $\Phi(x)$ is uniquely determined by this property. So
assume that $\Phi^{\prime}(x)$ is another element of $\mathrm{T}_{p}M$ such
that $\mathrm{T}\mu_{i}\Phi^{\prime}(x)=\Phi_{i}(x)$ for all $i\in\mathbb{N}$.
For $[f]_{p}\in\mathscr{C}^{\infty}_{p}$ of the form
$f=f_{i}\circ{\mu_{i}}_{|U_{i}}$ with $U_{i}\subset M_{i}$ an open
neighborhood of $p_{i}$ and $f_{i}\in\mathscr{C}^{\infty}(U_{i})$ this
assumption entails
$\Phi(x)\big{(}[f]_{p}\big{)}=\Phi_{i}(x)\big{(}[f_{i}]_{p_{i}}\big{)}=\Phi^{\prime}(x)\big{(}[f]_{p}\big{)}.$
Since every germ $[f]_{p}$ is locally of the form
$f_{i}\circ{\mu_{i}}_{|U_{i}}$, we obtain $\Phi(x)=\Phi^{\prime}(x)$.
Finally, we observe that $\Phi:X\rightarrow\mathrm{T}M$ is continuous, since
all maps $\Phi_{i}=\mathrm{T}\mu_{i}\Phi$ are continuous, and $\mathrm{T}M$
carries the initial topology with respect to the maps $\mathrm{T}\mu_{i}$.
This concludes the proof that $\mathrm{T}M$ together with the maps
$\mathrm{T}\mu_{i}$ is a projective limit of the projective system
$\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij}\big{)}$. ∎
###### Remark 2.16.
1. a)
If $p\in M$, $Y_{p},Z_{p}\in\mathrm{T}_{p}M$, and $\lambda\in\mathbb{R}$, then
the maps $Y_{p}+Z_{p}:\mathscr{C}^{\infty}_{p}\rightarrow\mathbb{R}$ and
$\lambda Y_{p}:\mathscr{C}^{\infty}_{p}\rightarrow\mathbb{R}$ are derivations
again. Hence $\mathrm{T}_{p}M$ becomes a topological vector space in a natural
way and one has $\mathrm{T}_{p}M\cong\lim\limits_{\longleftarrow\atop
i\in\mathbb{N}}\mathrm{T}_{\mu_{i}(p)}M_{i}$ canonically as topological vector
spaces. In particular, this implies that
$\pi_{\mathrm{T}M}:\mathrm{T}M\rightarrow M$ is a continuous family of vector
spaces. Note that this family need not be locally trivial, in general.
2. b)
Denote by $\mathscr{P}^{\infty}_{M,p}$ the set of germs of smooth paths
$\gamma:(\mathbb{R},0)\rightarrow(M,p)$. There is a canonical map
$\mathscr{P}^{\infty}_{M,p}\rightarrow\mathrm{T}_{p}M$ which associates to
each germ of a smooth path $\gamma:(\mathbb{R},0)\rightarrow(M,p)$ the
derivation
$\dot{\gamma}:\mathscr{C}^{\infty}_{p}\longrightarrow\mathbb{R},\quad[f]_{p}\longmapsto(f\circ\gamma){\dot{\mbox{\hskip
2.84526pt}}\,}(0).$
Unlike in the finite dimensional case, this map need not be surjective, in
general. But note the following result.
###### Proposition 2.17.
In case the profinite dimensional manifold $M$ is regular, the “dot map”
$\mathscr{P}^{\infty}_{M,p}\longrightarrow\mathrm{T}_{p}M,\>[\gamma]_{0}\longmapsto\dot{\gamma}(0)$
is surjective for every $p\in M$.
###### Proof.
We start with an auxiliary construction. Choose a smooth projective
representation $(M_{i},\mu_{ij},\mu_{i})$ within the pdf structure on $M$ such
that all $\mu_{ij}$ are fiber bundles. Put $p_{i}:=\mu(p_{i})$ for every
$i\in\mathbb{N}$. Then choose a relatively compact open neighborhood
$U_{0}\subset M_{0}$ of $p_{0}$ which is diffeoemorphic to an open ball in
some $\mathbb{R}^{n}$. In particular, $U_{0}$ is contractible, hence the fiber
bundle ${\mu_{01}}_{|\mu_{01}^{-1}(U_{0})}:\mu_{01}^{-1}(U_{0})\rightarrow
U_{0}$ is trivial with typical fiber $F_{1}:=\mu_{01}^{-1}(p_{0})$. Let
$\Psi_{0}:\mu_{01}^{-1}(U_{0})\rightarrow U_{0}\times F_{1}$ be a
trivialization of that fiber bundle, and $D_{1}\subset F_{1}$ an open
neighborhood of $p_{1}$ which is diffeomeorphic to an open ball in some
euclidean space. Put $U_{1}:=\Psi_{0}^{-1}(U_{0}\times D_{1})$. Then, $U_{1}$
is diffeomeorphic to a ball in some euclidean space, and
${\mu_{01}}_{|U_{1}}:U_{1}\rightarrow U_{0}$ is a trivial fiber bundle with
fiber $D_{1}$. Assume now that we have constructed $U_{0}\subset
M_{0},\ldots,U_{j}\subset M_{j}$ such that for all $i\leq j$ the following
holds true:
1. (1)
the set $U_{i}$ is a relatively compact open neighborhood of $p_{i}$
diffeomorphic to an open ball in some euclidean space,
2. (2)
for $i>0$, the identity $\mu_{i-1i}(U_{i})=U_{i-1}$ holds true,
3. (3)
for $i>0$, the restricted map ${\mu_{i-1i}}_{|U_{i}}:U_{i}\rightarrow U_{i-1}$
is a trivial fiber bundle with fiber $D_{i}$ diffeomorphic to an open ball in
some euclidean space.
Let us now construct $U_{j+1}$ and $D_{j+1}$. To this end note first that
${\mu_{jj+1}}_{|\mu_{jj+1}^{-1}(U_{j})}:\mu_{jj+1}^{-1}(U_{j})\rightarrow
U_{j}$ is a trivial fiber bundle with typical fiber
$F_{j}:=\mu_{jj+1}^{-1}(p_{j})$, since $U_{j}$ is contractible. Choose a
trivialization $\Psi_{j+1}:{\mu_{jj+1}}_{|\mu_{jj+1}^{-1}(U_{j})}\rightarrow
U_{j}\times F_{j}$, and an open neighborhood $D_{j+1}\subset F_{j+1}$ of
$p_{j+1}$ which is diffeomorphic to an open ball in some euclidean space. Put
$U_{j+1}:=\Psi_{j+1}^{-1}\big{(}U_{j}\times D_{j+1}\big{)}$. Then, $U_{j}$ is
diffeomeorphic to a ball in some euclidean space, and
${\mu_{jj+1}}_{|U_{j+1}}:U_{j+1}\rightarrow U_{j}$ is a trivial fiber bundle
with fiber $D_{j+1}$. This finishes the induction step, and we obtain
$U_{i}\subset M_{i}$ and $D_{i}$ such that the three conditions above are
satisfied.
After these preliminaries, assume that $Z\in T_{p}M$ is a tangent vector. Let
$Z_{i}:=\mathrm{T}\mu_{i}(Z)$ for $i\in\mathbb{N}$. We now inductively
construct smooth paths $\gamma_{i}:\mathbb{R}\rightarrow U_{i}$ such that
(2.4) $\gamma_{i}(0)=p_{i},\quad\dot{\gamma}_{i}(0)=Z_{i},\quad\text{and, if
$i>0$,}\quad\mu_{i-1i}\circ\gamma_{i}=\gamma_{i-1}.$
To start, choose a smooth path $\gamma_{0}:\mathbb{R}\rightarrow U_{0}$ such
that $\gamma_{0}(0)=p_{0}$, and $\dot{\gamma}_{0}(0)=Z_{0}$. Assume that we
have constructed $\gamma_{0},\ldots,\gamma_{j}$ such that (2.4) is satisfied
for all $i\leq j$. Consider the trivial fiber bundle
${\mu_{jj+1}}_{|U_{j+1}}:U_{j+1}\rightarrow U_{j}$, and let
$\Psi_{j+1}:U_{j+1}\rightarrow U_{j}\times D_{j+1}$ be a trivialization. Then,
$\mathrm{T}\Psi_{j+1}(Z_{j+1})=\big{(}Z_{j},Y_{j+1}\big{)}$ for some tangent
vector $Y_{j+1}\in\mathrm{T}_{p_{j+1}}D_{j+1}$. Choose a smooth path
$\varrho_{j+1}:\mathbb{R}\rightarrow D_{j+1}$ such that
$\varrho_{j+1}(0)=p_{j+1}$, and $\dot{\varrho}_{j+1}(0)=Y_{j+1}$. Put
$\gamma_{j+1}(t)=\Psi_{j+1}^{-1}\big{(}\gamma_{i}(t),\varrho_{j+1}(t)\big{)}\quad\text{for
all $t\in\mathbb{R}$}.$
By construction, $\gamma_{j+1}$ is a smooth path in $U_{j+1}$ such that (2.4)
is fulfilled for $i=j+1$. This finishes the induction step, and we obtain a
family of smooth paths $\gamma_{i}$ with the desired properties.
Since $M$ is the smooth projective limit of the $M_{i}$, there exists a
uniquely determined smooth path $\gamma:\mathbb{R}\rightarrow M$ such that
$\mu_{i}\circ\gamma=\gamma_{i}$ for all $i\in\mathbb{N}$. In particular, this
entails $\gamma(0)=p$, and $\dot{\gamma}(0)=Z$, or in other words that $Z$ is
in the image of the map
$\mathscr{P}^{\infty}_{M,p}\rightarrow\mathrm{T}_{p}M$. ∎
Let us define a structure sheaf $\mathscr{C}^{\infty}_{\mathrm{T}M}$ on
$\mathrm{T}M$. To this end call a continuous map $f\in\mathscr{C}(U)$ defined
on an open set $U\subset\mathrm{T}M$ _smooth_ , if for every tangent vector
$Z\in U$ there is an $i\in\mathbb{N}$, an open neighborhood
$U_{i}\subset\mathrm{T}M_{i}$ of $Z_{i}:=\mathrm{T}\mu_{i}(Z)$, and a smooth
map $f_{i}\in\mathscr{C}^{\infty}(U_{i})$ such that
$(\mathrm{T}\mu_{i})^{-1}(U_{i})\subset U$ and
$f_{|(\mathrm{T}\mu_{i})^{-1}(U_{i})}=f_{i}\circ(\mathrm{T}\mu_{i})_{|(\mathrm{T}\mu_{i})^{-1}(U_{i})}$.
The spaces
$\mathscr{C}^{\infty}_{\mathrm{T}M}(U):=\big{\\{}f\in\mathscr{C}(U)\mid\text{$f$
is smooth}\big{\\}}$
for $U\subset\mathrm{T}M$ open then form the section spaces of a sheaf
$\mathscr{C}^{\infty}_{\mathrm{T}M}$ which we call the _sheaf of smooth
functions_ on $\mathrm{T}M$. By construction, the family
$\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij},\mathrm{T}\mu_{i}\big{)}$ now is a
smooth projective representation of the locally ringed space
$\big{(}\mathrm{T}M,\mathscr{C}^{\infty}_{\mathrm{T}M}\big{)}$, hence
$\big{(}\mathrm{T}M,\mathscr{C}^{\infty}_{\mathrm{T}M}\big{)}$ becomes a
profinite dimensional manifold. Since
$\mu_{i}\circ\pi_{\mathrm{T}M}=\pi_{\mathrm{T}M_{i}}\circ\mathrm{T}\mu_{i}$
for all $i\in\mathbb{N}$, one immediatly checks that the canonical map
$\pi_{\mathrm{T}M}:\mathrm{T}M\rightarrow M$ is even a smooth map between
profinite dimensional manifolds. With these preparations we can state:
###### Proposition and Definition 2.18.
The profinite dimensional manifold given by
$\big{(}\mathrm{T}M,\mathscr{C}^{\infty}_{\mathrm{T}M}\big{)}$ and the pfd
structure
$[\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij},\mathrm{T}\mu_{i}\big{)}]$ is
called the _tangent bundle_ of $M$, and
$\pi_{\mathrm{T}M}:\mathrm{T}M\rightarrow M$ its _canonical projection_. The
pfd structure
$[\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij},\mathrm{T}\mu_{i}\big{)}]$ depends
only on the equivalence class $[\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}]$.
###### Proof.
In order to check the last statement, consider a smooth projective
representation
$\big{(}M_{a}^{\prime},\mu_{ab}^{\prime},\mu_{a}^{\prime}\big{)}$ which is
equivalent to $\big{(}M_{i},\mu_{ij},\mu_{i}\big{)}$. Choose an equivalence
transformation of smooth projective representations
$\big{(}\varphi,F_{a}\big{)}:\big{(}M_{i},\mu_{ij}\big{)}\longrightarrow\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)},\>\big{(}\psi,G_{i}\big{)}:\big{(}M_{a}^{\prime},\mu_{ab}^{\prime}\big{)}\longrightarrow\big{(}M_{i},\mu_{ij}\big{)}\>.$
Then one obtains surjective submersions
$\displaystyle\big{(}\varphi,\mathrm{T}F_{a}\big{)}:\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij}\big{)}\longrightarrow\big{(}\mathrm{T}M_{a}^{\prime},\mathrm{T}\mu_{ab}^{\prime}\big{)},$
$\displaystyle\big{(}\psi,\mathrm{T}G_{i}\big{)}:\big{(}\mathrm{T}M_{a}^{\prime},\mathrm{T}\mu_{ab}^{\prime}\big{)}\longrightarrow\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij}\big{)}$
such that the following diagrams commute for all $i,a\in\mathbb{N}$:
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.23903pt\raise-27.1972pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.14166pt\hbox{$\scriptstyle{\mathrm{T}G_{\varphi(a)}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
70.05026pt\raise-31.27777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern-3.0pt\raise-41.11108pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
37.48698pt\raise-41.11108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{T}M_{\varphi(a)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.19684pt\raise-26.37776pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.9611pt\hbox{$\scriptstyle{\mathrm{T}F_{a}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
5.62245pt\raise-4.20555pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces\end{split}$
Hence, $\big{(}\mathrm{T}M_{a}^{\prime},\mathrm{T}\mu_{ab}^{\prime}\big{)}$ is
a smooth projective system which is equivalent to
$\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij}\big{)}$. Now recall that the map
$\mathrm{T}\mu_{a}^{\prime}:\mathrm{T}M\rightarrow M$ is defined by
$\mathrm{T}\mu_{a}^{\prime}\big{(}Z_{p}\big{)}=Z_{p}\circ(\mu_{a}^{\prime})^{*}$,
where $Z_{p}\in\mathrm{T}_{p}M$, $p\in M$, and $(\mu_{a}^{\prime})^{*}$
denotes the pullback by $\mu_{a}^{\prime}$. One concludes that for all
$i\in\mathbb{N}$
$\begin{split}\mathrm{T}G_{i}\circ\mathrm{T}\mu_{\psi(i)}^{\prime}(Z_{p})&=\mathrm{T}G_{i}\big{(}Z_{p}\circ(\mu_{\psi(i)}^{\prime})^{*}\big{)}=Z_{p}\circ(\mu_{\psi(i)}^{\prime})^{*}\circ
G_{i}^{*}=\\\ &=Z_{p}\circ\mu_{i}^{*}=\mathrm{T}\mu_{i}(Z_{p})\>,\end{split}$
and likewise that
$\mathrm{T}F_{a}\circ\mathrm{T}\mu_{\varphi(a)}(Y_{p})=\mathrm{T}\mu_{a}^{\prime}(Y_{p})$
for all $a\in\mathbb{N}$. This entails that the smooth projective
representations
$\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij},\mathrm{T}\mu_{i}\big{)}$ and
$\big{(}\mathrm{T}M_{a}^{\prime},\mathrm{T}\mu_{ab}^{\prime},\mathrm{T}\mu_{a}^{\prime}\big{)}$
of the tangent bundle $(\mathrm{T}M,\mathscr{C}^{\infty}_{\mathrm{T}M})$ are
equivalent, and the proof is finished. ∎
###### Remark 2.19.
1. a)
By Example 2.8 b), the induced smooth projective system
$\big{(}\mathrm{T}M_{i},\mathrm{T}\mu_{ij}\big{)}$ has the canonical smooth
projective limit
$\big{(}\widetilde{\mathrm{T}}M,\mathscr{C}^{\infty}_{\widetilde{\mathrm{T}}M}\big{)}:=\lim\limits_{\longleftarrow\atop
i\in\mathbb{N}}(\mathrm{T}M_{i},\mathscr{C}^{\infty}_{\mathrm{T}M_{i}}).$
Denote its canonical maps by
$\widetilde{\mathrm{T}}\mu_{i}:\widetilde{\mathrm{T}}M\rightarrow\mathrm{T}M_{i}$.
By the universal property of projective limits there exists a unique smooth
map
$\tau:\mathrm{T}M\longrightarrow\widetilde{\mathrm{T}}M$
such that $\widetilde{\mathrm{T}}\mu_{i}\circ\tau=\mathrm{T}\mu_{i}$ for all
$i\in\mathbb{N}$. By construction of the profinite dimensional manifold
structure on the tangent bundle $\mathrm{T}M$, the map $\tau$ is even a linear
diffeomorphism, and is in fact given by
$\mathrm{T}M\ni
Y\longmapsto\big{(}\mathrm{T}\mu_{i}(Y)\big{)}_{i\in\mathbb{N}}\in\widetilde{\mathrm{T}}M.$
2. b)
As a generalization of the tangent bundle, one can define for every
$k\in\mathbb{N}^{*}$ the tensor bundle $\mathrm{T}^{k,0}M$ of $M$. First, one
puts for every $p\in M$
$\mathrm{T}^{k,0}_{p}M:=\widehat{\bigotimes}^{k}\mathrm{T}_{p}M,$
where $\widehat{\otimes}$ denotes the completed projective tensor product
LABEL: . The canonical maps
$\mathrm{T}\mu_{p,i}:={\mathrm{T}\mu_{i}}_{|\mathrm{T}_{p}M}:\mathrm{T}_{p}M\rightarrow\mathrm{T}_{p_{i}}M_{i}$,
$p_{i}:=\mu_{i}(p)$ induce continuous linear maps
$\mathrm{T}^{k,0}\mu_{p,i}:=\widehat{\bigotimes}^{k}\mathrm{T}\mu_{p,i}:\mathrm{T}^{k,0}_{p}M\longrightarrow\mathrm{T}^{k,0}_{p_{i}}M_{i}$
by the universal property of the completed projective tensor product.
Likewise, one constructs for $i\leq j$ the continuous linear maps
$\mathrm{T}^{k,0}\mu_{p_{j},ij}:\mathrm{T}^{k,0}_{p_{j}}M_{j}\longrightarrow\mathrm{T}^{k,0}_{p_{i}}M_{i}$
which turn
$\big{(}\mathrm{T}^{k,0}_{p_{i}}M_{i},\mathrm{T}^{k,0}\mu_{p_{j},ij}\big{)}$
into a projective system of (finite dimensional) real vector spaces. By
Theorem A.4, its projective limit within the category of locally convex
topological Hausdorff spaces is given by $\mathrm{T}^{k,0}_{p}M$ together with
the continuous linear maps $\mathrm{T}^{k,0}\mu_{p,i}$, that means we have
(2.5) $\mathrm{T}^{k,0}_{p}M=\lim\limits_{\longleftarrow\atop
i\in\mathbb{N}}\mathrm{T}^{k,0}_{p_{i}}M_{i}.$
Now define
$\mathrm{T}^{k,0}M:=\bigcup_{p\in M}\mathrm{T}^{k,0}_{p}M,$
and give $\mathrm{T}^{k,0}M$ the coarsest topology such that all the canonical
maps
$\displaystyle\mathrm{T}^{k,0}\mu_{i}:\>$
$\displaystyle\mathrm{T}^{k,0}M\longrightarrow\mathrm{T}^{k,0}M_{i},$
$\displaystyle Z_{1}\otimes\ldots\otimes
Z_{k}\longmapsto\mathrm{T}\mu_{i}(Z_{1})\otimes\ldots\otimes\mathrm{T}\mu_{i}(Z_{k})$
are continuous. By construction, $\mathrm{T}^{k,0}M$ together with the maps
$\mathrm{T}^{k,0}\mu_{i}$ has to be a projective limit of the projective
system $\big{(}\mathrm{T}^{k,0}M_{i},\mathrm{T}^{k,0}\mu_{ij}\big{)}$. The
sheaf of smooth functions $\mathscr{C}^{\infty}_{\mathrm{T}^{k,0}M}$ is
uniquely determined by requiring axiom a)(PFM2) to hold true. One thus obtains
a profinite dimensional manifold which depends only on the equivalence class
of the smooth projective representation and which will be denoted by
$\mathrm{T}^{k,0}M$ in the following. Moreover, $\mathrm{T}^{k,0}$ even
becomes a functor on the category of profinite dimensional manifolds. If
$(N,\mathscr{C}^{\infty}_{N})$ is another profinite dimensional manifold and
$f:M\to N$ a smooth map, then one naturally obtains the smooth map
$\begin{split}\mathrm{T}^{k,0}f:\>&\mathrm{T}^{k,0}M\longrightarrow\mathrm{T}^{k,0}N,\\\
&Z_{1}\otimes\ldots\otimes
Z_{k}\longmapsto\mathrm{T}f(Z_{1})\otimes\ldots\otimes\mathrm{T}f(Z_{k})\end{split}$
which satisfies
$\pi_{\mathrm{T}^{k,0}N}\circ\mathrm{T}^{k,0}f=f\circ\pi_{\mathrm{T}^{k,0}M}$.
We continue with:
###### Definition 2.20.
Let $U\subset M$ be open. Then a smooth section $V:U\rightarrow\mathrm{T}M$ of
$\pi_{\mathrm{T}M}:\mathrm{T}M\rightarrow M$ is called a _smooth vector field_
on $M$ over $U$. The space of smooth vector fields over $U$ will be denoted by
$\mathscr{X}^{\infty}(U)$.
Assume that for $U\subset M$ open we are given a smooth vector field
$V:U\rightarrow\mathrm{T}M$ and a smooth function $f:U\rightarrow\mathbb{R}$.
We then define a function $Vf$ over $U$ by putting for $p\in U$
(2.6) $Vf\,(p):=V(p)\big{(}[f]_{p}\big{)}\>.$
###### Lemma 2.21.
For every $V\in\mathscr{X}^{\infty}(U)$ and $f\in\mathscr{C}^{\infty}(U)$, the
function $Vf$ is smooth.
###### Proof.
Choose a point $p\in U$, and then an open $U_{i}\subset M_{i}$ and a function
$f_{i}\in\mathscr{C}^{\infty}(U_{i})$ for some appropriate $i\in\mathbb{N}$
such that $p\in\mu_{i}^{-1}(U_{i})\subset U$ and
(2.7) $f_{|\mu_{i}^{-1}(U_{i})}=f_{i}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})}.$
Consider $V_{i}:M\rightarrow\mathrm{T}M_{i}$, $V_{i}:=\mathrm{T}\mu_{i}\circ
V$. Since $V_{i}$ takes values in a finite dimensional smooth manifold, there
exists an integer $j_{p}\geq i$ (which we briefly denote by $j$, if no
confusion can arise), an open $U_{pj}\subset M_{j}$ and a smooth vector field
$V_{p}:U_{pj}\rightarrow\mathrm{T}M_{i}$ along $\mu_{ij}$ such that
$p\in\mu_{j}^{-1}(U_{pj})$, $U_{pj}\subset\mu_{ij}^{-1}(U_{i})$ and
(2.8) ${\mathrm{T}\mu_{i}\circ
V}_{|\mu_{j}^{-1}(U_{pj})}=V_{p}\circ{\mu_{j}}_{|\mu_{j}^{-1}(U_{pj})}.$
Now define $g_{pj}:U_{pj}\rightarrow\mathbb{R}$ by
$g_{pj}(q_{j}):=V_{p}(q_{j})\big{(}[f_{i}]_{\mu_{ij}(q_{j})}\big{)}\quad\text{for
all $q_{j}\in U_{pj}$}.$
Then $g_{pj}$ is smooth, hence
$g_{p}:=g_{pj}\circ{\mu_{j}}_{|\mu_{j}^{-1}(U_{pj})}$ is an element of
$\mathscr{C}^{\infty}\big{(}\mu_{j}^{-1}(U_{pj})\big{)}$. Now one checks for
$q\in\mu_{j}^{-1}(U_{pj})$ that
(2.9)
$g_{p}(q)=V_{p}(\mu_{j}(q))\big{(}[f_{i}]_{\mu_{i}(q)}\big{)}=V_{i}(q)\big{(}[f_{i}]_{\mu_{i}(q)}\big{)}=V(q)\big{(}[f]_{q}\big{)}$
by Eq. (2.7). Hence
$g_{p}=(Vf)_{|\mu_{j}^{-1}(U_{pj})},$
and $Vf$ is smooth indeed. ∎
###### Proposition 2.22.
Every vector field $V\in\mathscr{X}^{\infty}(U)$ defined over an open subset
$U\subset M$ induces a derivation
$\delta_{V}:\mathscr{C}^{\infty}(U)\longrightarrow\mathscr{C}^{\infty}(U),\>f\longmapsto
Vf$
###### Proof.
By construction, it is clear that the map
$\mathscr{C}^{\infty}(U)\ni f\longmapsto Vf\in\mathscr{C}^{\infty}(U)$
is $\mathbb{R}$-linear. It remains to check that $\delta_{V}$ is a derivation,
or in other words that it satisfies Leibniz’ rule. But this follows
immediately by the definition of the action of $V$ on
$\mathscr{C}^{\infty}(U)$ and the fact that
$V(p)\in\operatorname{Der}\big{(}\mathscr{C}^{\infty}_{p},\mathbb{R}\big{)}$
for all $p\in U$. More precisely, one has, for $p\in U$ and
$f,g\in\mathscr{C}^{\infty}(U)$,
$\begin{split}V(fg)\,(p)&=V(p)\big{(}[fg]_{p}\big{)}=f(p)\,V(p)\big{(}[g]_{p}\big{)}+g(p)\,V(p)\big{(}[f]_{p}\big{)}=\\\
&=\big{(}f\,V(g)+g\,V(f)\big{)}\,(p).\end{split}$
This finishes the proof. ∎
###### Definition 2.23.
Let $U\subset M$ be open. A smooth vector field $V\in\mathscr{X}^{\infty}(U)$
is called _local_ , if for every $i\in\mathbb{N}$ there is an integer
$m_{i}\geq i$ and a smooth vector field
$V_{im_{i}}:\mu_{m_{i}}(U)\rightarrow\mathrm{T}M_{i}$ along $\mu_{im_{i}}$
such that
(2.10) $\displaystyle\mathrm{T}\mu_{i}\circ
V=V_{im_{i}}\circ{p_{m_{i}}}_{|U}.$
The space of local vector fields over $U$ will be denoted by
$\mathscr{X}_{\textup{loc}}^{\infty}(U)$.
###### Remark 2.24.
1. a)
Obviously, $\mathscr{X}^{\infty}$ is a sheaf of $\mathscr{C}^{\infty}$-modules
on $M$, and $\mathscr{X}_{\textup{loc}}^{\infty}$ a presheaf of
$\mathscr{C}_{\textup{loc}}^{\infty}$-modules. Note that
$\mathscr{X}_{\textup{loc}}^{\infty}$ depends only on the pfd structure
$[(M_{i},\mu_{ij},\mu_{i})]$.
2. b)
Let $V\in\mathscr{X}_{\textup{loc}}^{\infty}(U)$, and pick a representative
$(M_{i},\mu_{ij},\mu_{i})$ of the underlying pfd structure. If
$(m_{i})_{i\in\mathbb{N}}$ is a sequence of integers such that (2.10) holds
true, we sometimes say that $V$ is of _type $(m_{0},m_{1},m_{2},\ldots)$ with
respect to the smooth projective representation $(M_{i},\mu_{ij},\mu_{i})$_.
The notion of the type of a local vector field is known from jet bundle
literature [2], where it makes perfekt sense, since the profinite dimensional
manifold of infinite jets has a distinguished representative of the underlying
pfd structure, see Section 3.3.
Now we are in the position to prove the following structure theorem:
###### Theorem 2.25.
The map
$\delta:\mathscr{X}^{\infty}(M)\longrightarrow\operatorname{Der}\big{(}\mathscr{C}^{\infty}(M),\mathscr{C}^{\infty}(M)\big{)},\>V\longmapsto\delta_{V}$
is a bijection. Moreover, for every $V\in\mathscr{X}^{\infty}(M)$, the
derivation $\delta_{V}$ leaves the algebra
$\mathscr{C}_{\textup{loc}}^{\infty}(M)$ of local functions on $M$ invariant,
if and only if one has $V\in\mathscr{X}_{\textup{loc}}^{\infty}(M)$.
###### Proof.
_Surjectivity_ : Assume that
$D:\mathscr{C}^{\infty}(M)\rightarrow\mathscr{C}^{\infty}(M)$ is a derivation.
Then one obtains for each $i\in\mathbb{N}$ and point $p\in M$ a linear map
$D_{pi}:\mathscr{C}^{\infty}(M_{i})\longrightarrow\mathbb{R},\>f\longmapsto
D(f\circ\mu_{i})(p)\>.$
Note that for $f,f^{\prime}\in\mathscr{C}^{\infty}(M_{i})$
$\begin{split}D_{pi}&(ff^{\prime})=D\big{(}(ff^{\prime})\circ\mu_{i}\big{)}(p)=\\\
&=f\circ\mu_{i}(p)D(f^{\prime}\circ\mu_{i})(p)+f^{\prime}\circ\mu_{i}(p)D(f\circ\mu_{i})(p)=\\\
&=f\circ\mu_{i}(p)D_{pi}(f^{\prime})+f^{\prime}\circ\mu_{i}(p)D_{pi}(f),\end{split}$
which entails that there is a tangent vector
$V_{pi}\in\mathrm{T}_{\mu_{i}(p)}M_{i}$ such that $D_{pi}=V_{pi}$. Observe
that for $j\geq i$ the relation
$D_{pi}(f)=D(f\circ\mu_{i})(p)=D(f\circ\mu_{ij}\circ\mu_{j})(p)=D_{pj}(f\circ\mu_{ij})$
holds true, which entails that $V_{pi}=\mathrm{T}\mu_{ij}\circ V_{pj}$. Hence,
the sequence of tangent vectors $(V_{pi})_{i\in\mathbb{N}}$ defines an element
$V_{p}$ in
$\mathrm{T}_{p}M\cong\lim\limits_{\longleftarrow\atop
i\in\mathbb{N}}\mathrm{T}_{\mu_{i}(p)}M_{i}.$
We thus obtain a section $V:M\to\mathrm{T}M,p\mapsto V_{p}$. Let us show that
$V$ is smooth. To this end, consider the composition
$V_{i}:=\mathrm{T}\mu_{i}\circ V:M\longrightarrow\mathrm{T}M_{i}.$
By construction $V_{i}(p)=V_{pi}$ for all $p\in M$. It suffices to show that
each of the maps $V_{i}$ is smooth. To show this, choose a coordinate
neighborhood $U_{i}\subset M_{i}$ of $\mu_{i}(p)$, and coordinates
$(x^{1},\cdots,x^{k}):U_{i}\longrightarrow\mathbb{R}^{k}.$
Then
$(x^{1}\circ\mu_{i}\circ\pi_{\mathrm{T}U_{i}},\cdots,x^{k}\circ\mu_{i}\circ\pi_{\mathrm{T}U_{i}},\mathrm{d}x^{1},\cdots,\mathrm{d}x^{k}):\mathrm{T}U_{i}\longrightarrow\mathbb{R}^{2k}$
is a local coordinate system of $\mathrm{T}M_{i}$. The map $V_{i}$ now is
proven to be smooth, if $\mathrm{d}x^{l}\circ V_{i}$ is smooth for $1\leq
l\leq k$. But
$\mathrm{d}x^{l}\circ{V_{i}}_{|\mu_{i}^{-1}(U_{i})}=D(x^{l}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})}),$
since for $q\in\mu_{i}^{-1}(U_{i})$
$\mathrm{d}x^{l}\circ{V_{i}}_{|\mu_{i}^{-1}(U_{i})}(q)=V_{qi}(q)\big{(}[x^{l}]_{\mu_{i}(q)}\big{)}=D_{qi}(x^{l})=D(x^{l}\circ{\mu_{i}}_{|\mu_{i}^{-1}(U_{i})})(q).$
Hence each $V_{i}$ is smooth, and $V$ is a smooth vector field on $M$ which
satisfies $\delta_{V}=D$. This proves surjectivity.
_Injectivity_ : Assume that $V$ is a smooth vector field on $M$ such that
$\delta_{V}=0$. This means that $\delta_{V}f(p)=0$ for all
$f\in\mathscr{C}^{\infty}(M)$ and $p\in M$. Choose now a $i\in\mathbb{N}$ and
let $f_{i}$ be a smooth function on $M_{i}$. Put $f:=f_{i}\circ\mu_{i}$ and
$V_{i}=\mathrm{T}\mu_{i}\circ V$. Then, we have for all $p\in M$
$V_{i}(p)\big{(}[f_{i}]_{\mu_{i}(p)}\big{)}=\delta_{V}f(p)=0,$
which implies that $V_{i}(p)=0$ for all $p\in M$. Since $V(p)$ is the
projective limit of the $V_{i}(p)$, we obtain $V(p)=0$ for all $p\in M$, hence
$V=0$. This finishes the proof that $\delta$ is bijective.
_Local vector fields_ : Next, let us show that for a local vector field
$V:M\rightarrow\mathrm{T}M$ the derivation $\delta_{V}$ maps local functions
to local ones. To this end choose for every $i\in\mathbb{N}$ an integer
$m_{i}\geq i$ such that there exists a smooth vector field
$V_{im_{i}}:M_{m_{i}}\rightarrow\mathrm{T}M_{i}$ along $\mu_{im_{i}}$ which
satisfies
$\mathrm{T}\mu_{i}\circ V=V_{im_{i}}\circ\mu_{m_{i}}.$
Now let $f$ be a local function on $M$, which means that $f=f_{i}\circ\mu_{i}$
for some $i\in\mathbb{N}$ and $f_{i}\in\mathscr{C}^{\infty}(M_{i})$. Define
$g_{m_{i}}\in\mathscr{C}^{\infty}(M_{m_{i}})$ by
$g_{m_{i}}(q)=V_{im_{i}}(q)\big{(}[f_{i}]_{\mu_{im_{i}}(q)}\big{)}$ for all
$q\in M_{m_{i}}$. Then, one obtains for $p\in M$
$\delta_{V}f(p)=V_{im_{i}}(\mu_{m_{i}}(p))\big{(}[f_{i}]_{\mu_{i}(p)}\big{)}=g_{m_{i}}(\mu_{m_{i}}(p)),$
which means that $\delta_{V}f=g_{m_{i}}\circ\mu_{m_{i}}$ is local.
_Invariance of $\mathscr{C}_{\textup{loc}}^{\infty}(M)$_: Finally, we have to
show that if $\delta_{V}$ for $V\in\mathscr{X}^{\infty}(M)$ leaves the space
$\mathscr{C}_{\textup{loc}}^{\infty}(M)$ invariant, the vector field $V$ has
to be local. To this end fix $i\in\mathbb{N}$ and choose a proper embedding
$\chi=(\chi_{1},\ldots,\chi_{N}):M_{i}\lhook\joinrel\relbar\joinrel\rightarrow\mathbb{R}^{N}.$
Then $\chi_{l}\circ\mu_{i}\in\mathscr{C}_{\textup{loc}}^{\infty}(M)$ for
$l=1,\ldots,N$, hence there exist by assumption
$j_{1},\ldots,j_{N}\in\mathbb{N}$ and
$g_{il}\in\mathscr{C}^{\infty}(M_{j_{l}})$ such that
$\delta_{V}\big{(}\chi_{l}\circ\mu_{i}\big{)}=g_{il}\circ\mu_{j_{l}}.$
After possibly increasing the $j_{l}$, we can assume that
$m_{i}:=j_{1}=\ldots=j_{N}\geq i$. Denote by
$z_{l}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ the canonical projection onto the
$l$-th coordinate, and define the vector field
$\widetilde{V}_{im_{i}}:M_{m_{i}}\rightarrow\mathrm{T}\mathbb{R}^{N}$ along
$\chi\circ\mu_{im_{i}}$ by
$\widetilde{V}_{im_{i}}(q):=\sum_{l=1}^{N}\,g_{lj_{l}}(q)\dfrac{\partial}{\partial
z_{l}}_{\mid\chi(\mu_{im_{i}}(q))}\quad\text{for $q\in M_{m_{i}}$}.$
Since by construction
$\widetilde{V}_{im_{i}}(\mu_{m_{i}}(p))\big{(}[z_{l}]_{\chi(\mu_{i}(p))}\big{)}=g_{il}(\mu_{m_{i}}(p))=\mathrm{T}\mu_{i}\circ
V(p)\big{(}[\chi_{l}]_{\mu_{i}(p)}\big{)}$
for all $p\in M$, $\widetilde{V}_{im_{i}}(y)$ is in the image of
$\mathrm{T}_{q}\chi$ for every $q\in M_{m_{i}}$, hence
$V_{im_{i}}:M_{m_{i}}\longrightarrow\mathrm{T}M_{i},\>q\longmapsto(\mathrm{T}_{y}\chi)^{-1}\big{(}\widetilde{V}_{im_{i}}(q)\big{)}$
is well-defined and satisfies $\mathrm{T}\mu_{i}\circ
V=V_{im_{i}}\circ\mu_{m_{i}}$. Therefore, $V$ is a local vector field. ∎
The following result is an immediate consequence of Theorem 2.25.
###### Corollary 2.26.
For all $V,W\in\mathscr{X}^{\infty}(M)$, the map
$[V,W]:\,\mathscr{C}^{\infty}(M)\longrightarrow\mathscr{C}^{\infty}(M),\>f\longmapsto
V(Wf)-W(Vf)$
is a derivation on $\mathscr{C}^{\infty}(M)$. Its corresponding underlying
vector field will be denoted by $[V,W]$ as well, and will be called the _Lie
bracket_ of $V$ and $W$. The Lie bracket of vector fields turns
$\mathscr{X}^{\infty}(M)$ into a Lie algebra.
### 2.3. Differential forms
###### Definition 2.27.
Let $k\in\mathbb{N}$ and $U\subset M$ open.
1. a)
A continuous map
$\omega:(\pi_{\mathrm{T}^{k,0}M})^{-1}(U)\longrightarrow\mathbb{R}$
is called a _differential form of order $k$_ or a _$k$ -form_ on $M$ over $U$,
if for every point $p\in U$ there is some $i\in\mathbb{N}$, an open subset
$U_{i}\subset M_{i}$ with $p\in\mu_{i}^{-1}(U_{i})\subset U$ and a $k$-form
$\omega_{i}\in\Omega^{k}(U_{i})$ such that
$\omega_{|\left(\mu_{i}\circ\pi_{\mathrm{T}^{k,0}M}\right)^{-1}(U_{i})}=\omega_{i}\circ{\mathrm{T}^{k,0}\mu_{i}}_{|\left(\mu_{i}\circ\pi_{\mathrm{T}^{k,0}M}\right)^{-1}(U_{i})}\>.$
More precisely, this means that for all $y\in\mu_{i}^{-1}(U_{i})$, and
$V_{1},\ldots,V_{k}\in\pi^{-1}(y)\subset\mathrm{T}M$ the relation
$\omega(V_{1}\otimes\ldots\otimes
V_{k})=\omega_{i}\big{(}\mathrm{T}\mu_{i}(V_{1})\otimes\ldots\otimes\mathrm{T}\mu_{i}(V_{k})\big{)}$
holds true. In particular, a $k$-form $\omega$ over $U$ is antisymmetric and
$k$-multilinear in its arguments. The space of $k$-forms over $U$ will be
denoted by $\Omega^{k}(U)$.
2. b)
A $k$-form $\omega\in\Omega^{k}(U)$ is called _local_ , if there is an open
$U_{i}\subset M_{i}$ for some $i\in\mathbb{N}$ and a $k$-form
$\omega_{i}\in\Omega^{k}(U_{i})$ such that $U\subset\mu_{i}^{-1}(U_{i})$ and
$\omega=(\mu_{i}^{*}\omega_{i})_{|U}$, where here and from now on we use the
notation $\mu_{i}^{*}\omega_{i}$ for the form
$\omega_{i}\circ{\mathrm{T}^{k,0}\mu_{i}}$. The space of local $k$-forms over
$U$ will be denoted by $\Omega^{k}_{\textup{loc}}(U)$.
###### Remark 2.28.
1. a)
By a straightforward argument one checks that the spaces $\Omega^{k}(U)$ and
$\Omega^{k}_{\textup{loc}}(U)$ only depend on the pfd structure
$[(M_{i},\mu_{ij},\mu_{i})]$. Moreover, for every representative
$(M_{i},\mu_{ij},\mu_{i})$ of the pfd structure,
$\Omega^{k}_{\textup{loc}}(U)$ together with the family of pull-back maps
$\mu_{i}^{*}:\Omega^{k}(\mu_{i}(U))\rightarrow\Omega^{k}_{\textup{loc}}(U)$ is
an injective limit of the injective system of linear spaces
$\big{(}\Omega^{k}(\mu_{i}(U)),\mu_{ij}^{*}\big{)}_{i\in\mathbb{N}}$.
2. b)
By construction, it is clear that $\Omega^{k}$ forms a sheaf of
$\mathscr{C}^{\infty}$-modules on $M$ and $\Omega^{k}_{\textup{loc}}$ a
presheaf of $\mathscr{C}^{\infty}_{\mathrm{loc}}$-modules. Moreover,
$\Omega^{k}$ coincides with the sheaf associated to
$\Omega^{k}_{\textup{loc}}$.
3. c)
The representative $\mathcal{M}:=(M_{i},\mu_{ij},{\mu}_{j})$ of the pfd
structure on $M$ leads to the particular filtration
$\mathcal{F}^{\mathcal{M}}_{\bullet}$ of the presheaf
$\Omega^{k}_{\textup{loc}}$ of local $k$-forms on $M$ by putting, for
$l\in\mathbb{N}$,
$\mathcal{F}^{\mathcal{M}}_{l}\big{(}\Omega^{k}_{\textup{loc}}\big{)}:=\mu_{l}^{*}\Omega^{k}_{M_{l}}\>.$
Observe that this filtration has the property that
$\Omega^{k}_{\textup{loc}}=\bigcup_{l\in\mathbb{N}}\mathcal{F}^{\mathcal{M}}_{l}\big{(}\Omega^{k}_{\textup{loc}}\big{)}.$
###### Proposition and Definition 2.29.
1. a)
There exists a uniquely determined morphism of sheaves
$\mathrm{d}:\Omega^{k}\rightarrow\Omega^{k+1}$ such that
$\mathrm{d}(\mu_{i}^{*}\omega_{i})=\mu_{i}^{*}(\mathrm{d}\omega_{i})\>\text{
for all $i\in\mathbb{N}$, $U_{i}\subset M_{i}$ open,
$\omega_{i}\in\Omega^{k}(U_{i})$.}$
The morphism $\mathrm{d}$ is called the _exterior derivative_ , fulfills
$\mathrm{d}\circ\mathrm{d}=0$, and maps $\Omega^{k}_{\textup{loc}}$ to
$\Omega^{k+1}_{\textup{loc}}$.
2. b)
There exists a uniquely determined morphism of sheaves
$\wedge:\Omega^{k}\times\Omega^{l}\longrightarrow\Omega^{k+l},$
called the _wedge product_ , such that for all $i\in\mathbb{N}$, $U_{i}\subset
M_{i}$ open, $\omega_{i}\in\Omega^{k}(U_{i})$, and
$\mu_{i}\in\Omega^{l}(U_{i})$ one has
$\mu_{i}^{*}\omega_{i}\wedge\mu_{i}^{*}\mu_{i}=\mu_{i}^{*}(\omega_{i}\wedge\mu_{i}).$
The wedge product also leaves $\Omega^{\bullet}_{\textup{loc}}$ invariant.
3. c)
Given a vector field $V\in\mathscr{X}^{\infty}(M)$, there exists the
_contraction with $V$_ that means the sheaf morphism
$i_{V}:\Omega^{k}\longrightarrow\Omega^{k-1},$
which is uniquely determined by the requirement that for all
$\omega\in\Omega^{k}(U)$ with $U\subset M$ open, $p\in U$, and
$W_{1},\ldots,W_{k-1}\in\mathrm{T}_{p}M$ the relation
$i_{V}(\omega)(W_{1}\otimes\dots\otimes W_{k-1})=\omega\big{(}V(p)\otimes
W_{1}\otimes\dots\otimes W_{k-1}\big{)}$
holds true. If $V$ is a local vector field, contraction with $V$ leaves
$\Omega^{\bullet}_{\textup{loc}}$ invariant.
###### Proof.
Using the sheaf property of $\Omega^{k}$ one can reduce the claims to local
statements which are immediately proved. ∎
## 3\. Jet bundles and formal solutions of nonlinear PDE’s
The aim of this section is to develop a precise geometric notion of formally
integrable (systems of) partial differential equations, and to show that the
formal solution spaces of these equations canonically become a profinite
dimensional manifold in the sense of Section 2. Finally, we are going to give
a criterion for the formal integrability of nonlinear scalar partial
differential equations, and apply this result to a class of interacting
relativistic scalar field theories that arise in theoretical physics.
We refer the reader to [31, 18, 22] and also to [28, 32] for introductionary
texts on jet bundles, where the latter two references have a strong focus on
the highly nontrivial algorithmic aspects of this theory. A nice short
overview is also included in the introduction of [37].
### 3.1. Finite order jet bundles
For the rest of the paper, we fix a fiber bundle $\pi:E\to X$. Moreover, $F$
will denote the typical fiber of $\pi$ and we set $m:=\dim X$, $n:=\dim F$.
Then one has $\dim E=m+n$ and the fibers $\pi^{-1}(p)\subset E$ become
$n$-dimensional submanifolds, which are diffeomorphic to $F$. There are
distinguished charts for $E$:
###### Definition 3.1.
A manifold chart $(x,u):W\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ of $E$ defined
over some open $W\subset E$ is called a _fibered_ _chart of $\pi$_, if for all
$e,e^{\prime}\in W$ with $\pi(e)=\pi(e^{\prime})$ the equality
$x(e)=x(e^{\prime})$ holds true.
###### Remark 3.2.
1. a)
Sometimes, fibered charts are called _adapted_ _charts_.
2. b)
Note that a fibered chart $(x,u):W\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ for
$\pi$ canonically gives rise to a well-defined manifold chart on $X$. It is
given by
(3.1)
$\displaystyle\tilde{x}:\pi(W)\longrightarrow\mathbb{R}^{m},\>\>p\longmapsto
x(e),$
where $e\in W\cap\pi^{-1}(p)$ is arbitrary.
3. c)
On the other hand, a manifold atlas for $E$ that consists of fibered charts
for $\pi$ can be constructed from manifold charts for $X$ and from the local
triviality of $E$ as follows: For an arbitrary $e\in E$, take a bundle chart
$\phi:\pi^{-1}(U)\to U\times F$ around $\pi(e)$, that is, $U$ is an open
neighbourhood of $\pi(e)$ and $\phi:\pi^{-1}(U)\to U\times F$ is a
diffeomorphism such that
(3.6)
commutes. Let $\tilde{x}:U\to\mathbb{R}^{m}$ be a manifold chart of $X$ (here
we assume that $U$ is small enough), and let $\tilde{u}:B\to\mathbb{R}^{n}$ be
a manifold chart of $F$. Then
$(\tilde{x}\circ\pi,\tilde{u}\circ\mathrm{pr}_{2}\circ\phi)=(\tilde{x}\circ\mathrm{pr}_{1}\circ\phi,\tilde{u}\circ\mathrm{pr}_{2}\circ\phi):\pi^{-1}(U)\longrightarrow\mathbb{R}^{m}\times\mathbb{R}^{n}$
is a fibered chart of $\pi$. Note here that by the commutativity of (3.6), the
notation “$\tilde{x}$” is consistent with (3.1).
Let us introduce the following notation for multi-indices, which will be
convenient in the following: For any $k_{1},k_{2}\in\mathbb{N}$ with
$k_{1}\leq k_{2}$ let $\mathbb{N}^{m}_{k_{1},k_{2}}$ denote the set of all
multi-indices $I\in\mathbb{N}^{m}$ such that
$k_{1}\leq|I|:=\sum^{m}_{j=1}I_{j}\leq k_{2}$
and let $\mathsf{F}(m,k_{1},k_{2})$ denote the linear space of all maps
$\mathbb{N}^{m}_{k_{1},k_{2}}\to\mathbb{R}$. For any $l\leq m$,
$i_{1},\dots,i_{l}\in\\{1,\dots,m\\}$, the symbol $1_{i_{1}\dots
i_{l}}\in\mathbb{N}^{m}$ will denote the multi-index which has a $1$ in its
$i_{j}$’s slot for $j=1,\dots,l$, and a $0$ elsewhere.
Any $\psi\in\Gamma^{\infty}(p;\pi)$ allows the following local description:
Choose a fibered chart $(x,u):W\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ of $\pi$
with $W\cap\pi^{-1}(p)\neq\emptyset$. Then one has $x\circ\psi=\tilde{x}$ near
$p$, so that $\psi$ is determined by the coordinates
$(u^{1}\circ\psi,\dots,u^{n}\circ\psi)=u\circ\psi$ near $p$. The special form
of the following definition is motivated by the latter fact:
###### Definition 3.3.
Let $p\in X$, $k\in\mathbb{N}$. Any two
$\psi,\varphi\in\Gamma^{\infty}(p;\pi)$ are called _$k$ -equivalent at $p$_,
if for every fibered chart $(x,u):W\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ of
$\pi$ with $W\cap\pi^{-1}(p)\neq\emptyset$ one has
(3.7)
$\displaystyle\frac{\partial^{|I|}\left(u^{\alpha}\circ\psi\right)}{\partial\tilde{x}^{I}}(p)=\frac{\partial^{|I|}\left(u^{\alpha}\circ\varphi\right)}{\partial\tilde{x}^{I}}(p)$
for all $\alpha=1,\dots,n$ and all $I\in\mathbb{N}^{m}_{0,k}$. The
corresponding equivalence class $\mathsf{j}^{k}_{p}\psi$ of $\psi$ is called
the _$k$ -jet of $\psi$ at $p$. _
###### Remark 3.4.
In fact, it is enough to check (3.7) in _some_ fibered chart. This can be
proved by induction on $k$, using the multivariate version of Faa di Bruno’s
formula [11, Theorem 2.1] (details can be found in Lemma 6.2.1 in [31]).
Let us now come to several structures that can be defined via jets. Denoting
by
$\mathsf{J}^{k}(\pi):=\bigcup_{p\in
X}\left\\{\mathsf{j}^{k}_{p}\psi\mid\psi\in\Gamma^{\infty}(p;\pi)\right\\}$
the collection of all $k$-jets in $\pi$, we obtain the surjective maps
(3.8) $\displaystyle\pi_{k}:\>$
$\displaystyle\mathsf{J}^{k}(\pi)\longrightarrow
X,\>\mathsf{j}^{k}_{p}\psi\longmapsto p,$ (3.9) $\displaystyle\pi_{0,k}:\>$
$\displaystyle\mathsf{J}^{k}(\pi)\longrightarrow
E,\>\mathsf{j}^{k}_{p}\psi\longmapsto\psi(p).$
Using these maps, one can give $\mathsf{J}^{k}(\pi)$ the structure of a finite
dimensional manifold in a canonical way: For every fibered chart
$(x,u):W\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ of $\pi$ and every
$I\in\mathbb{N}^{m}_{0,k}$, one defines the map
(3.10)
$\begin{split}(x_{k},u_{k,I})\\!:\>&\pi_{0,k}^{-1}(W)\longrightarrow\mathbb{R}^{m}\times\mathbb{R}^{n\dim\mathsf{F}(m,0,k)}\\\
&\mathsf{j}^{k}_{p}\psi\longmapsto\left(\tilde{x}(p),\frac{\partial^{|I|}\left(u^{1}\circ\psi\right)}{\partial\tilde{x}^{I}}(p),\dots,\frac{\partial^{|I|}\left(u^{n}\circ\psi\right)}{\partial\tilde{x}^{I}}(p)\right).\end{split}$
The following result is checked straightforwardly (cf. [31]).
###### Proposition and Definition 3.5.
The maps $(\ref{mf1})$ define an $m+n\dim\mathsf{F}(m,0,k)$-dimensional
manifold structure on $\mathsf{J}^{k}(\pi)$. In view of this fact,
$\mathsf{J}^{k}(\pi)$ is called the _$k$ -jet manifold_ corresponding to
$\pi$.
For convenience, we set $\mathsf{J}^{0}(\pi):=E$ and
$\mathsf{j}^{0}_{p}\psi:=\psi(p)$ for any $\psi\in\Gamma^{\infty}(p;\pi)$, and
$\pi_{0}:=\pi$. More generally, we have for any $k_{1}\leq k_{2}$ the smooth
surjective maps
$\pi_{k_{1},k_{2}}:\mathsf{J}^{k_{2}}(\pi)\longrightarrow\mathsf{J}^{k_{1}}(\pi),\>\mathsf{j}^{k_{2}}_{p}\psi\longmapsto\mathsf{j}^{k_{1}}_{p}\psi,$
which satisfy $\pi_{k,k}=\mathrm{id}_{\mathsf{J}^{k}(\pi)}$, and if one also
has $k_{2}\leq k_{3}$, then the following diagram commutes:
(3.17)
Let us collect all structures underlying the above maps. Let
$(x,u):W\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ be a fibered chart of $\pi$.
Then we set
(3.18)
$\begin{split}\left(\pi_{k},u_{k}\right):\>&\pi^{-1}_{k,0}(W)\longrightarrow\pi(W)\times\mathsf{F}(m,0,k)^{n}\\\
&\mathsf{j}^{k}_{p}\psi\longmapsto\left(p,\left\\{\frac{\partial^{|I|}\left(u\circ\psi\right)}{\partial\tilde{x}^{I}}(p)\right\\}_{I\in\mathbb{N}^{m}_{0,k}}\right)\\\
&=\left(p,\left\\{\frac{\partial^{|I|}\left(u^{1}\circ\psi\right)}{\partial\tilde{x}^{I}}(p)\right\\}_{I\in\mathbb{N}^{m}_{0,k}},\dots,\left\\{\frac{\partial^{|I|}\left(u^{n}\circ\psi\right)}{\partial\tilde{x}^{I}}(p)\right\\}_{I\in\mathbb{N}^{m}_{0,k}}\right),\end{split}$
(3.19)
$\begin{split}\left(\pi_{k_{1},k_{2}},u_{k_{1},k_{2}}\right):\>&\pi^{-1}_{k_{2},0}(W)\longrightarrow\pi^{-1}_{k_{1},0}(W)\times\mathsf{F}(m,k_{1}+1,k_{2})^{n}\\\
&\mathsf{j}^{k_{2}}_{p}\psi\longmapsto\left(\mathsf{j}^{k_{1}}_{p}\psi,\left\\{\frac{\partial^{|I|}\left(u\circ\psi\right)}{\partial\tilde{x}^{I}}(p)\right\\}_{I\in\mathbb{N}^{m}_{k_{1}+1,k_{2}}}\right).\end{split}$
If $\pi$ is a vector bundle, then, for every $p\in X$, the fiber
$\pi_{k}^{-1}(p)$ canonically becomes a linear space through
$c_{1}(\mathsf{j}^{k}_{p}\psi)+c_{2}(\mathsf{j}^{k}_{p}\varphi):=\mathsf{j}^{k}_{p}(c_{1}\psi+c_{2}\varphi),\>\>c_{j}\in\mathbb{R}.$
Furthermore, if $k_{2}=k$, $k_{1}=k-1$ and if $a\in\mathsf{J}^{k-1}(\pi)$,
then the fiber $\pi_{k-1,k}^{-1}(a)$ carries a canonical affine structure
which is modelled on the linear space
(3.20)
$\displaystyle\mathrm{Sym}^{k}\left(\mathrm{T}^{*}_{\pi_{k-1}(a)}X\right)\otimes\operatorname{ker}\big{(}{\mathrm{T}\pi}_{|\pi_{0,k-1}(a)}\big{)}.$
To see the latter fact, assume that $\pi_{0,k-1}(a)\in W$, let
$\mathsf{j}^{k}_{\pi_{k-1}(a)}\psi\in\pi_{k-1,k}^{-1}(a)$ and let $v$ be an
element of (3.20). Then $v$ can be uniquely expanded as
$v=\sum_{I\in\mathbb{N}^{m}_{k,k}}\sum^{n}_{\alpha=1}v^{\alpha}_{I}{\mathrm{d}\tilde{x}_{I}}_{|\pi_{k-1}(a)}\otimes\frac{\partial}{\partial{u}^{\alpha}}_{\mid\pi_{0,k-1}(a)},\>\>v^{\alpha}_{I}\in\mathbb{R},$
where we have used the abbreviation
$\displaystyle\mathrm{d}\tilde{x}_{I}:=(\mathrm{d}\tilde{x}_{1})^{\otimes
I_{1}}\odot\dots\odot(\mathrm{d}\tilde{x}_{m})^{\otimes I_{m}},$
so that one can define
$\mathsf{j}^{k}_{\pi_{k-1}(a)}\psi+v\in\pi_{k-1,k}^{-1}(a)$ to be the uniquely
determined element whose image under (3.19) is given by
$\displaystyle\left(\mathsf{j}^{k-1}_{\pi_{k-1}(a)}\psi,\left\\{\frac{\partial^{|I|}\left(u\circ\psi\right)}{\partial\tilde{x}^{I}}_{\mid\pi_{k-1}(a)}+v_{I}\right\\}_{I\in\mathbb{N}^{m}_{k,k}}\right).$
With these preparations, one has:
###### Lemma 3.6.
Let $k,k_{1},k_{2}\in\mathbb{N}$ with $k_{1}\leq k_{2}$. Then the following
assertions hold.
1. a)
The maps (3.18) turn $\pi_{k}:\mathsf{J}^{k}(\pi)\to X$ into a fiber bundle
with typical fiber $\mathsf{F}(m,0,k)^{n}$. If $\pi$ is a vector bundle, then
so is $\pi_{k}$.
2. b)
The maps (3.19) turn
$\pi_{k_{1},k_{2}}:\mathsf{J}^{k_{2}}(\pi)\to\mathsf{J}^{k_{1}}(\pi)$ into a
fiber bundle with typical fiber $\mathsf{F}(m,l+1,k)^{n}$, and
$\pi_{k-1,k}:\mathsf{J}^{k}(\pi)\to\mathsf{J}^{k-1}(\pi)$ becomes an affine
bundle, modelled on the vector bundle
$\pi^{*}_{k-1}\mathrm{Sym}^{k}\big{(}\pi_{\mathrm{T}^{*}X}\big{)}\otimes\pi^{*}_{k-1,0}\mathsf{V}(\pi)\longrightarrow\mathsf{J}^{k-1}(\pi).$
###### Proof.
The reader can find a detailed proof of Lemma 3.6 in Chapter 6 of [31].
∎
We close this section with a simple observation about distinguished elements
of $\Gamma^{\infty}(\pi_{k})$. Let $U\subset X$ be an open subset for the
moment. Then for any $\psi\in\Gamma^{\infty}(U;\pi)$, the map
$U\to\mathsf{J}^{k}(\pi)$, $p\mapsto\mathsf{j}^{k}_{p}\psi$, defines an
element of $\Gamma^{\infty}(U;\pi_{k})$, called the _$k$ -jet prolongation of
$\psi$._ In fact, this construction induces a morphism of sheaves
$\mathsf{j}^{k}:\Gamma^{\infty}(\pi)\rightarrow\Gamma^{\infty}(\pi_{k})$ (with
values in the category of sets) such that
(3.21) $\pi_{k_{1},k_{2}}\circ\mathsf{j}^{k_{2}}=\mathsf{j}^{k_{1}}\quad\text{
for $k_{1}\leq k_{2}$}.$
It should be noted that it is not possible to write an arbitrary element of
$\Gamma^{\infty}(U;\pi_{k})$ as $\mathsf{j}^{k}_{U}\psi$ for some
$\psi\in\Gamma^{\infty}(U;\pi)$. The elements of $\Gamma^{\infty}(U;\pi_{k})$
having the latter property are called _projectable_. This notion is motivated
by the following simple observation which follows readily from (3.21).
###### Lemma 3.7.
Let $U\subset X$ be an open subset and $\Psi\in\Gamma^{\infty}(U;\pi_{k})$.
Then the map $p\mapsto\pi_{0,k}(\Psi(p))$ defines an element of
$\Gamma^{\infty}(U;\pi)$, and $\Psi$ is projectable, if and only if one has
$\mathsf{j}^{k}\big{(}\pi_{0,k}\circ\Psi\big{)}=\Psi$.
### 3.2. Partial differential equations
The aim of this section is to give a precise global definition of partial
differential equations and the solutions thereof in the setting of arbitrary
fiber bundles. We shall first consider the general (possibly nonlinear)
situation in Section 3.2.1. Then, in Section 3.2.2, we are going to relate
everything with the corresponding classical linear concepts.
Throughout this section, let $\underline{\pi}:\underline{E}\to X$ be a second
fiber bundle, with typical fiber $\underline{F}$ and fiber dimension
$\underline{n}$.
#### 3.2.1. General facts
We start with:
###### Definition 3.8.
1. a)
A subset $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ is called a _partial
differential equation on $\pi$ of order $\leq k$_, if
${\pi_{k}}_{|\mathsf{E}}:\mathsf{E}\to X$ is a fibered submanifold of
$\pi_{k}$.
2. b)
Let $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ be a partial differential equation
on $\pi$ of order $\leq k$. Some $\psi\in\Gamma^{\infty}(p;\pi)$ is called a
_solution of $\mathsf{E}$ in $p$_, if $\mathsf{j}^{k}_{p}\psi\in\mathsf{E}$.
For an open $U\subset X$, a section $\psi\in\Gamma^{\infty}(U;\pi)$ will
simply be called a _solution_ of $\mathsf{E}$, if $\psi$ is a solution of
$\mathsf{E}$ in $p$ for every $p\in U$, that is, if
$\mathrm{im}(\mathsf{j}^{k}\psi)\subset\mathsf{E}$.
The point of this definition is that one has seperated and globalized the
notions “partial differential equation”, “solution of a partial differential
equation” and “partial differential operator”. We shall first clarify how the
latter concept fits into definition 3.8 a).
###### Definition 3.9.
A morphism $h:\mathsf{J}^{k}(\pi)\to\underline{E}$ of fibered manifolds over
$X$ is called a _partial differential operator of order $\leq k$ from $\pi$ to
$\underline{\pi}$._
Of course, the notion “operator” in definition 3.9 is justified by the fact
that as a morphism of fibered manifolds, any $h$ as in Definition 3.9 induces
the morphism of set theoretic sheaves
$P^{h}:=h\circ\mathsf{j}^{k}:\Gamma^{\infty}(\pi)\longrightarrow\Gamma^{\infty}(\underline{\pi}).$
We define $\mathrm{D}^{k}(\pi,\underline{\pi})$ to be the set of all partial
differential operators of order $\leq k$ from $\pi$ to $\underline{\pi}$, and
remark that the assignment $h\mapsto P^{h}$ induces an injection $P^{\bullet}$
of $\mathrm{D}^{k}(\pi,\underline{\pi})$ into the set theoretic sheaf
morphisms $\Gamma^{\infty}(\pi)\to\Gamma^{\infty}(\underline{\pi})$. The
connection between partial differential operators and partial differential
equations is given in this abstract setting as follows: For every
$h\in\mathrm{D}^{k}(\pi,\underline{\pi})$ and
$O\in\Gamma^{\infty}(X;\underline{\pi})$, the _$O$ -kernel $\ker_{O}(h)$ of
$h$_ is defined by
$\ker_{O}(h):=h^{-1}(\mathrm{im}(O))\subset\mathsf{J}^{k}(\pi)$, with the
convention $\ker(h):=\ker_{0}(h)$, if $\pi$ is a vector bundle. Observe that
one has by definition
$\operatorname{ker}_{O}(h)=\left\\{a\in\mathsf{J}^{k}(\pi)\mid
h(a)=O(\pi_{k}(a))\right\\}.$
The following fact is well-known:
###### Proposition 3.10.
If $h\in\mathrm{D}^{k}(\pi,\underline{\pi})$ has constant rank, and if
$O\in\Gamma^{\infty}(X;\underline{\pi})$ fulfills
$\mathrm{im}(O)\subset\mathrm{im}(h)$, then
$\ker_{O}(h)\subset\mathsf{J}^{k}(\pi)$ is a partial differential equation.
It is clear that with an open subset $U\subset X$, a section
$\psi\in\Gamma^{\infty}(U;\pi)$ is a solution of $\ker_{O}(h)$ (in $p\in U$),
if and only if one has $P^{h}_{U}(\psi)=O$ (in $p$).
Next, we explain how the affine structure of $\pi_{k-1,k}$ can be used to
introduce the notion of “operator symbols of (possibly nonlinear) partial
differential operators”. To avoid any confusion, we remark that with “symbol”
we will exclusively mean “principal symbol” in this paper.
To this end, note that the assignment
$\displaystyle\mu^{\pi}_{k}:\>$
$\displaystyle\pi^{*}_{k}\mathrm{Sym}^{k}\big{(}\pi_{\mathrm{T}^{*}X}\big{)}\otimes\pi^{*}_{k,0}\mathsf{V}(\pi)\longrightarrow\mathsf{V}(\pi_{k}),$
$\displaystyle{\mu^{\pi}_{k}}_{|a}(v):=\frac{\mathrm{d}}{\mathrm{d}t}\Big{[}a+tv\Big{]}_{\mid
t=0},$
for $a\in\mathsf{J}^{k}(\pi)$,
$v\in\mathrm{Sym}^{k}\big{(}\mathrm{T}^{*}_{\pi_{k}(a)}X\big{)}\otimes\ker\big{(}{\pi}_{|\pi_{0,k}(a)}\big{)}$,
is a (mono)morphism of vector bundles over $\mathsf{J}^{k}(\pi)$. Note here
that $\mu^{\pi}_{k}$ essentially extracts the pure $k$-th order part of
vertical $k$-jets. Using the map $\mu^{\pi}_{k}$, we can provide the following
definition (see also [8]):
###### Definition 3.11.
For every $h\in\mathrm{D}^{k}(\pi,\underline{\pi})$, the morphism $\sigma(h)$
of vector bundles over $h$ given by the composition
(3.22)
---
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\pi^{*}_{k}\mathrm{Sym}^{k}\big{(}\pi_{\mathrm{T}^{*}X}\big{)}\otimes\pi^{*}_{k,0}\mathsf{V}(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\sigma(h)}$$\scriptstyle{\mu^{\pi}_{k}}$$\textstyle{\mathsf{V}(\underline{\pi})}$$\textstyle{\mathsf{V}(\pi_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{\mathsf{V}}}$
is called the _operator symbol_ of $h$.
Given a partial differential operator, one can use its symbol to check whether
it defines a partial differential equation in the sense of Proposition 3.10
(see also Theorem 3.28 below):
###### Proposition 3.12.
Let $h\in\mathrm{D}^{k}(\pi,\underline{\pi})$. If $\sigma(h)$ is surjective,
then so is $h$. If $\sigma(h)$ is a submersion, then $h$ is a submersion, too,
which in particularly means that for every
$O\in\Gamma^{\infty}(X;\underline{\pi})$ with
$\mathrm{im}(O)\subset\mathrm{im}(h)$ the set
$\ker_{O}(h)\subset\mathsf{J}^{k}(\pi)$ is a partial differential equation.
###### Proof.
We have the following commuting diagrams,
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.64297pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\\\&&\crcr}}}\ignorespaces{\hbox{\kern-17.25703pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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37.74048pt\raise 6.11389pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{h_{\mathsf{V}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 71.25703pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-20.64297pt\raise-35.31943pt\hbox{{}\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43333pt\hbox{$\scriptstyle{(\pi_{k})^{\mathsf{V}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-58.86111pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.25703pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 71.25703pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathsf{V}(\underline{\pi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
84.39595pt\raise-35.31943pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.52222pt\hbox{$\scriptstyle{\underline{\pi}^{\mathsf{V}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
84.39595pt\raise-61.9611pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-35.0pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
41.25703pt\raise-35.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
81.39595pt\raise-35.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-13.85431pt\raise-70.63887pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\kern
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74.86815pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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90.84035pt\raise-35.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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where the maps for the second diagram are given by the tangential maps
corresponding to the first one. If
$\sigma(h)=h_{\mathsf{V}}\circ\mu^{\pi}_{k}$ is surjective, then so is
$h_{\mathsf{V}}$ and $\underline{\pi}^{\mathsf{V}}\circ h_{\mathsf{V}}$, so
that the first assertion follows from the first diagramm. If $\sigma(h)$ is a
submersion, then one can use the analogous argument for the second diagram to
deduce that $\mathrm{T}h$ has full rank everywhere. ∎
#### 3.2.2. Linear partial differential equations
We are now going to explain how the classical concepts of linear partial
differential equations and partial differential operators fit into the general
setting of Section 3.2.1. In fact, it will turn out that the notions “linear
partial differential equation” and “linear partial differential operator” are
equivalent under natural assumptions (this is only locally true in the
nonlinear case [16]), and that the space of linear partial differential
operators coincides with the space of classical linear partial differential
operators (see Theorem 3.16 below). We begin with:
###### Definition 3.13.
Let $\pi$ and $\underline{\pi}$ be vector bundles.
1. a)
A subset $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ is called a _linear partial
differential equation on $\pi$ of order $\leq k$_, if
${\pi_{k}}_{|\mathsf{E}}:\mathsf{E}\to X$ is a sub-vector-bundle of $\pi_{k}$.
2. b)
A morphism $h:\mathsf{J}^{k}(\pi)\to\underline{E}$ of vector bundles over $X$
is called a _linear partial differential operator of order $\leq k$ from $\pi$
to $\underline{\pi}$_.
###### Remark 3.14.
Let $\pi$ and $\underline{\pi}$ be vector bundles. Then
$h\in\mathrm{D}^{k}(\pi,\underline{\pi})$ is linear, if and only if
$P^{h}_{X}:\Gamma^{\infty}(X;\pi)\to\Gamma^{\infty}(X;\underline{\pi})$ is
linear. We denote the linear space of linear partial differential operators by
$\mathrm{D}^{k}_{\mathrm{lin}}(\pi,\underline{\pi})\subset\mathrm{D}^{k}(\pi,\underline{\pi})$
and remark that if $h\in\mathrm{D}^{k}_{\mathrm{lin}}(\pi,\underline{\pi})$,
then one has
$h^{(l)}\in\mathrm{D}^{k+l}_{\mathrm{lin}}(\pi,\underline{\pi}_{l})$ for all
$l\in\mathbb{N}$.
Let us recall the definition of “classical” linear partial differential
operators:
###### Definition 3.15.
Let $\pi$ and $\underline{\pi}$ be vector bundles. A _classical linear partial
differential operator of order $\leq k$_ from $\pi$ to $\underline{\pi}$ is a
morphism of sheaves
$D:\Gamma^{\infty}(\pi)\longrightarrow\Gamma^{\infty}(\underline{\pi})$
with the following property: For every manifold chart
$\tilde{x}:U\to\mathbb{R}^{m}$ of $X$ for which there are frames
$e_{1},\dots,e_{n}\in\Gamma^{\infty}(U;\pi)$ and
$\underline{e}_{1},\dots,\underline{e}_{\underline{n}}\in\Gamma^{\infty}(U;\underline{\pi})$
there exist (necessarily unique) functions
$D^{\alpha,\beta}_{I}\in\mathscr{C}^{\infty}(U)$ for $\alpha=1,\dots,n$,
$\beta=1,\dots,\underline{n}$, and $I\in\mathbb{N}^{m}_{0,k}$ such that one
has for all $\psi^{1},\dots,\psi^{n}\in\mathscr{C}^{\infty}(U)$
(3.23) $\displaystyle
D_{U}\left(\sum^{n}_{j=1}\psi^{\alpha}e_{\alpha}\right)=\sum^{\underline{n}}_{\beta=1}\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{0,k}}D^{\alpha,\beta}_{I}\frac{\partial^{|I|}\psi^{\alpha}}{\partial\tilde{x}^{I}}\underline{e}_{\beta}.$
The linear space of classical partial differential operators will be denoted
by $\mathrm{D}^{k}_{\mathrm{cl},\mathrm{lin}}(\pi,\underline{\pi})$.
Now one has:
###### Theorem 3.16.
Let $\pi$ and $\underline{\pi}$ be vector bundles.
1. a)
$P^{\bullet}$ induces the isomorphism of linear spaces
$P^{\bullet}_{\mathrm{lin}}:\mathrm{D}^{k}_{\mathrm{lin}}(\pi,\underline{\pi})\longrightarrow\mathrm{D}^{k}_{\mathrm{cl},\mathrm{lin}}(\pi,\underline{\pi}),\>h\longmapsto
P^{h}.$
2. b)
If $h\in\mathrm{D}^{k}_{\mathrm{lin}}(\pi,\underline{\pi})$ has constant rank,
then $\ker(h)\subset\mathsf{J}(\pi_{k})$ is a linear partial differential
equation. Conversely, if $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ is a linear
partial differential equation, then there is a vector bundle
$\underline{\underline{\pi}}:\underline{\underline{E}}\to X$ and an
$\underline{\underline{h}}\in\mathrm{D}^{k}_{\mathrm{lin}}\left(\pi,\underline{\underline{\pi}}\right)$
with constant rank such that
$\mathsf{E}=\ker\left(\underline{\underline{h}}\right)$.
###### Proof.
a) We first have to show that $P^{\bullet}_{\mathrm{lin}}$ is well-defined,
which means that for any
$h\in\mathrm{D}^{k}_{\mathrm{lin}}(\pi,\underline{\pi})$, $P^{h}$ is in
$\mathrm{D}^{k}_{\mathrm{cl},\mathrm{lin}}(\pi,\underline{\pi})$. It is then
clear that $P^{\bullet}_{\mathrm{lin}}$ is a linear monomorphism. To this end,
let $\tilde{x}$, $e_{\alpha}$, $\underline{e}_{\beta}$, $\psi^{\alpha}$ be as
in Definition 3.15 and let $a_{\alpha}$ be a basis for $F$. Then we have the
vector bundle chart
$\phi:\pi^{-1}(U)\to U\times
F,\>\>\sum^{n}_{\alpha=1}v^{\alpha}e_{\alpha}(p)\longmapsto\left(p,\sum^{n}_{\alpha=1}v^{\alpha}a_{\alpha}\right),\>\>p\in
U,$
so that we get the fibered chart
$(x,u):\pi^{-1}(U)\longrightarrow\mathbb{R}^{m}\times\mathbb{R}^{n},\>\>\sum^{n}_{\alpha=1}v^{\alpha}e_{\alpha}(p)\longmapsto\Big{(}\tilde{x}(\pi(p)),(v^{1},\dots,v^{n})\Big{)}$
of $\pi$ as in Remark 3.2.3. Then
$\left(\pi_{k},u_{k}\right):\pi_{k}^{-1}(U)\longrightarrow
U\times\mathsf{F}(m,0,k)^{n}$
is a vector bundle chart of $\pi_{k}$ by Lemma 3.6, and we get the frame
$e_{I,\alpha}\in\Gamma(U;\pi_{k})$, $\alpha=1,\dots n$,
$I\in\mathbb{N}^{m}_{0,k}$, given by
$e_{I,\alpha}:=\left(\pi_{k},u_{k}\right)^{-1}(\bullet,\delta_{I,\alpha}).$
Hereby, $\delta_{I,\alpha}:\mathbb{N}^{m}_{0,k}\to\mathbb{R}^{n}$ is defined
by $\delta_{I,\alpha}(J):=1_{\alpha}$, if $I=J$, and to be $0$ elsewhere.
Since $h$ is a homomorphism of linear bundles over $X$, there are uniquely
determined $h^{\alpha,\beta}_{I}\in\mathscr{C}^{\infty}(U)$ such that one has
for all $\alpha=1,\dots,n$, $I\in\mathbb{N}^{m}_{0,k}$ and
$\psi^{\alpha,I}\in\mathscr{C}^{\infty}(U)$
$h\left(\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{0,k}}\psi^{\alpha,I}e_{\alpha,I}\right)=\sum^{\underline{n}}_{\beta=1}\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{0,k}}h^{\alpha,\beta}_{I}\psi^{\alpha,I}\underline{e}_{\beta}.$
The proof of the asserted well-definedness of $P^{\bullet}_{\mathrm{lin}}$ is
completed by observing that by the above construction of the frame
$e_{I,\alpha}$ for $\Gamma^{\infty}(U;\pi_{k})$ the following equality holds
true:
$\displaystyle\mathsf{j}^{k}\left(\sum^{n}_{\alpha=1}\psi^{\alpha}e_{\alpha}\right)=\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{0,k}}\frac{\partial^{|I|}\psi^{\alpha}}{\partial\tilde{x}^{I}}e_{\alpha,I}.$
In order to prove surjectivity of $P^{\bullet}_{\mathrm{lin}}$, let
$D\in\mathrm{D}^{k}_{\mathrm{cl},\mathrm{lin}}(\pi,\underline{\pi})$, and let
$\mathsf{j}^{k}_{p}\psi\in\mathsf{J}(\pi_{k})$, with $p$ from an open subset
$U\subset X$. Then $h(\mathsf{j}^{k}_{p}\psi):=D_{U}\psi(p)$ gives rise to a
well-defined element $h\in\mathrm{D}^{k}_{\mathrm{lin}}(\pi,\underline{\pi})$,
which of course satisfies $P^{h}_{\mathrm{lin}}=D$.
b) The first fact is well-known. For the second assertion, we can simply take
$\underline{\underline{E}}\to X$ to be given by the quotient bundle
$\mathsf{J}^{k}(\pi)/\mathsf{E}\to X$, and $\underline{\underline{h}}$ to be
given by the canonical projection
$\mathsf{J}^{k}(\pi)\to\mathsf{J}^{k}(\pi)/\mathsf{E}$ (see [25, Prop. 3.10]
for a more general statement). ∎
Finally, we explain in which sense the classical concept of linear operator
symbols fits into the general setting of Section 3.2.1.
Let $\pi$ and $\underline{\pi}$ be vector bundles for the moment. From the
canonical identification of $\operatorname{ker}(\mathrm{T}\pi_{\mid e})$ with
$\pi^{-1}(\pi(e))$, for $e\in E$, (and analogous ones for $\underline{\pi}$),
we obtain canonical morphisms of vector bundles over the base map $\pi$ resp.
$\underline{\pi}$
(3.24) $\begin{split}\sigma^{\pi}:\>&\mathsf{V}(\pi)\longrightarrow E\\\
&{\sigma^{\pi}}_{|\ker({\mathrm{T}\pi}_{|e})}:\ker({\mathrm{T}\pi}_{|e})\longrightarrow\pi^{-1}(\pi(e)),\>\>\>\>e\in
E,\text{ and}\\\ \end{split}$ (3.25)
$\begin{split}\sigma^{\underline{\pi}}:\>&\mathsf{V}(\underline{\pi})\longrightarrow\underline{E}\\\
&{\sigma^{\underline{\pi}}}_{|\ker(\mathrm{T}\underline{\pi}_{|\underline{e}})}:\ker(\mathrm{T}\underline{\pi}_{|\underline{e}})\longrightarrow\underline{\pi}^{-1}(\underline{\pi}(\underline{e})),\>\>\>\>\underline{e}\in\underline{E},\end{split}$
which both are fiberwise isomorphisms. It follows that for each
$k\in\mathbb{N}^{*}$ the map
(3.26)
$\begin{split}\sigma^{\pi}_{k}:\>&\pi^{*}_{k}\mathrm{Sym}^{k}\big{(}\pi_{\mathrm{T}^{*}X}\big{)}\otimes\pi^{*}_{k,0}\mathsf{V}(\pi)\longrightarrow\mathrm{Sym}^{k}\big{(}\pi_{\mathrm{T}^{*}X}\big{)}\otimes
E,\\\ &{\sigma^{\pi}_{k}}(v\otimes w):=v\otimes\sigma^{\pi}(w),\\\
&\text{where $v\otimes
w\in\operatorname{Sym}^{k}\left(\mathrm{T}^{*}_{\pi_{k}(a)}X\right)\otimes\operatorname{ker}\left({\mathrm{T}\pi}_{|\pi_{0,k}(a)}\right)$
for $a\in\mathsf{J}^{k}(\pi)$, }\end{split}$
is a morphism of vector bundles over the base map $\pi_{k}$ and also acts by
ismorphisms, fiberwise. Furthermore, for later reference, we record that there
is a canonical (mono)morphism of vector bundles over $X$ which is defined by
$\displaystyle\mu^{\pi}_{k,\mathrm{lin}}\\!:\>$
$\displaystyle\mathrm{Sym}^{k}\big{(}\pi_{\mathrm{T}^{*}X}\big{)}\otimes
E\longrightarrow\mathsf{J}^{k}(\pi)$
$\displaystyle\mathrm{d}f_{1}(p)\odot\cdots\odot\mathrm{d}f_{k}(p)\otimes\psi(p)\longmapsto\mathsf{j}^{k}_{p}(f_{1}\cdots
f_{k}\psi).$
Here, each $f_{j}$ denotes a smooth function defined in a neighbourhood of
$p\in X$ and satisfies $f_{j}(p)=0$, and $\psi\in\Gamma^{\infty}_{p}(\pi)$.
Analogously to $\mu^{\pi}_{k}$, the map $\mu^{\pi}_{k,\mathrm{lin}}$ also
extracts the pure $k$-th order part of $k$-jets in an appropriate sense
(taking into account the canonical isomorphisms (3.25) and (3.26)).
The following result recalls the classical definition of linear operator
symbols and shows the naturality of Definition 3.11, in the sense that in the
linear case, the linear operator symbol coincides with the operator symbol up
to the canonical isomorphisms (3.25) and (3.26):
###### Proposition and Definition 3.17.
Let $\pi$ and $\underline{\pi}$ be vector bundles and let
$h\in\mathrm{D}^{k}_{\mathrm{lin}}(\pi,\underline{\pi})$.
1. a)
There is a unique morphism of vector bundles over $X$
(3.27)
$\displaystyle\sigma_{\mathrm{lin}}(h):\mathrm{Sym}^{k}(\pi_{\mathrm{T}^{*}X})\otimes
E\longrightarrow\underline{E}$
with the following property: For every manifold chart
$\tilde{x}:U\to\mathbb{R}^{m}$ of $X$ for which there are frames
$e_{1},\dots,e_{n}\in\Gamma^{\infty}(U;\pi)$ and
$\underline{e}_{1},\dots,\underline{e}_{\underline{n}}\in\Gamma^{\infty}(U;\underline{\pi})$,
one has
(3.28)
$\displaystyle\sigma_{\mathrm{lin}}(h)\left(\sum_{I\in\mathbb{N}^{m}_{k,k}}\sum^{n}_{\alpha=1}v^{\alpha}_{I}\mathrm{d}\tilde{x}_{I}\otimes
e_{\alpha}\right)=\sum^{\underline{n}}_{\beta=1}\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{k,k}}(P^{h}_{\mathrm{lin}})^{\alpha,\beta}_{I}v^{\alpha}_{I}\underline{e}_{\beta},$
where $v^{\alpha}_{I}\in\mathscr{C}^{\infty}(U)$, and where we have used the
notation from Definition 3.15 and Theorem 3.16. The morphism
$\sigma_{\mathrm{lin}}(h)$ is called the _linear operator symbol of $h$_.
2. b)
The following diagram commutes,
$\textstyle{\pi^{*}_{k}\mathrm{Sym}^{k}(\pi_{\mathrm{T}^{*}X})\otimes\pi^{*}_{k,0}\mathsf{V}(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma(h)}$$\scriptstyle{\sigma^{\pi}_{k}}$$\textstyle{\mathsf{V}(\underline{\pi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{\underline{\pi}}}$$\textstyle{\mathrm{Sym}^{k}(\pi_{\mathrm{T}^{*}X})\otimes
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\sigma_{\mathrm{lin}}(h)}$$\textstyle{\underline{E}}$
###### Proof.
a) Here, one only has to prove that the representation (3.28) does not depend
on a particular choice of local data. In fact, the easiest way to see this, is
to note that one can simply define $\sigma_{\mathrm{lin}}(h)$ by the diagram
(3.29)
---
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{Sym}^{k}(\pi_{\mathrm{T}^{*}X})\otimes
E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\sigma_{\mathrm{lin}}(h)}$$\scriptstyle{\mu^{\pi}_{k,\mathrm{lin}}}$$\textstyle{\underline{E}}$$\textstyle{\mathsf{J}^{k}(\pi).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$
To see that $\sigma_{\mathrm{lin}}(h)$ defined like this satsfies (3.28), let
$\tilde{x}:U\to\mathbb{R}^{m}$ be a manifold chart of $X$ such that there are
frames $e_{1},\dots,e_{n}\in\Gamma^{\infty}(U;\pi)$,
$\underline{e}_{1},\dots,\underline{e}_{\underline{n}}\in\Gamma^{\infty}(U;\underline{\pi})$.
Then, as in the proof of Theorem 3.16 a), picking a basis $a_{\alpha}$ for
$F$, we get the corresponding frame
$e_{I,\alpha}\in\Gamma^{\infty}(U;\pi_{k})$, $\alpha=1,\dots n$,
$I\in\mathbb{N}^{m}_{0,k}$, and we denote the representation of $h$ with
respect to $e_{I,\alpha}$ and $\underline{e}_{\beta}$ by
$h^{\alpha,\beta}_{I}$. Furthermore, by the proof of Theorem 3.16 a), we have
$h^{\alpha,\beta}_{I}=(P^{h}_{\mathrm{lin}})^{\alpha,\beta}_{I}$. Now one has
$\mu^{\pi}_{k,\mathrm{lin}}\left(\sum_{I\in\mathbb{N}^{m}_{k,k}}\sum^{n}_{\alpha=1}v^{\alpha}_{I}\mathrm{d}\tilde{x}_{I}\otimes
e_{\alpha}\right)=\sum_{I\in\mathbb{N}^{m}_{k,k}}\sum^{n}_{\alpha=1}v^{\alpha}_{I}e_{I,\alpha},$
so that
$h\circ\mu^{\pi}_{k,\mathrm{lin}}\left(\sum_{I\in\mathbb{N}^{m}_{k,k}}\sum^{n}_{\alpha=1}v^{\alpha}_{I}\mathrm{d}\tilde{x}_{I}\otimes
e_{\alpha}\right)=\sum^{\underline{n}}_{\beta=1}\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{k,k}}h^{\alpha,\beta}_{I}v^{\alpha}_{I}\underline{e}_{\beta},$
which completes the proof of part a).
b) Let $a\in\mathsf{J}^{k}(\pi)$ be arbitrary, and let
$\tilde{x}:U\to\mathbb{R}^{m}$ be a manifold chart of $X$ around $\pi_{k}(a)$
such that there are frames $e_{1},\dots,e_{n}\in\Gamma^{\infty}(U;\pi)$,
$\underline{e}_{1},\dots,\underline{e}_{\underline{n}}\in\Gamma^{\infty}(U;\underline{\pi})$.
Then, again as in the proof of Theorem 3.16 a), picking a basis $a_{\alpha}$
for $F$ and a basis $\underline{a}_{\beta}$ for $\underline{F}$, we get the
corresponding adapted coordinates
$(x,u):\pi^{-1}(U)\longrightarrow\mathbb{R}^{m}\times\mathbb{R}^{n},\>\>(\underline{x},\underline{u}):\underline{\pi}^{-1}(U)\longrightarrow\mathbb{R}^{m}\times\mathbb{R}^{\underline{n}}.$
We can expand an arbitrary
$v\in\mathrm{Sym}^{k}\left(\mathrm{T}^{*}_{\pi_{k}(a)}X\right)\otimes\ker(\mathrm{T\pi}_{\mid\pi_{0,k}(a)})$
uniquely as
$v=\sum_{I\in\mathbb{N}^{m}_{k,k}}\sum^{n}_{\alpha=1}v^{\alpha}_{I}\mathrm{d}\tilde{x}_{I\mid\pi_{k}(a)}\otimes\frac{\partial}{\partial
u^{\alpha}}_{\mid\pi_{0,k}(a)},\>\>v^{\alpha}_{I}\in\mathbb{R},$
so that
$\sigma^{\pi_{k}}(v)=\sum_{I\in\mathbb{N}^{m}_{k,k}}\sum^{n}_{\alpha=1}v^{\alpha}_{I}\mathrm{d}\tilde{x}_{I}\otimes
e_{\alpha\mid\pi_{k}(a)},$
and we arrive at
(3.30) $\displaystyle\sigma_{\mathrm{lin}}(h)\circ\sigma^{\pi_{k}}_{\mid
v}\>=\sum^{\underline{n}}_{\beta=1}\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{k,k}}(P^{h}_{\mathrm{lin}})^{\alpha,\beta}_{I}v^{\alpha}_{I}\underline{e}_{\beta\mid\pi_{k}(a)}.$
Let us now evaluate
$\sigma^{\underline{\pi}}\circ\sigma(h)=\sigma^{\underline{\pi}}\circ
h_{\mathsf{V}}\circ\mu^{\pi}_{k}$ in $v$: By Proposition 3.5 and Lemma 3.6 we
have the frame
$\frac{\partial}{\partial
u^{k}_{I,\alpha}}\in\Gamma^{\infty}\left(\pi^{-1}_{k}(U);(\pi_{k})^{\mathsf{V}}\right),\>\>I\in\mathbb{N}^{m}_{0,k},\>\alpha=1,\dots,n\>.$
As $h$ is a linear morphism, the linear morphism $h_{\mathsf{V}}$ is
represented with respect to the frames $\frac{\partial}{\partial
u^{\alpha}_{k,I}}$ and $\frac{\partial}{\partial\underline{u}^{\beta}}$
precisely by the functions
$h^{\alpha,\beta}_{I}\circ\left(\pi_{k\mid\pi^{-1}_{k}(U)}\right)\in\mathscr{C}^{\infty}(\pi^{-1}_{k}(U)),$
where $h^{\alpha,\beta}_{I}\in\mathscr{C}^{\infty}(U)$ is the representation
of $h$ with respect to $(x,u)$ and $(\underline{x},\underline{u})$ (cf. the
proof of Theorem 3.16 a)). Thus, in view of
$\mu^{\pi}_{k}(v)=\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{k,k}}v^{\alpha}_{I}\frac{\partial}{\partial
u^{\alpha}_{k,I}}_{\mid a},$
we have
$\sigma(h)_{\mid_{v}}=\sum^{\underline{n}}_{\beta=1}\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{k,k}}v^{\alpha}_{I}h^{\alpha,\beta}_{I\mid\pi_{k}(a)}\frac{\partial}{\partial\underline{u}^{\beta}}_{\mid\pi_{0,k}(a)},$
so that
$\sigma^{\underline{\pi}}\circ\sigma(h)_{\mid
v}=\sum^{\underline{n}}_{\beta=1}\sum^{n}_{\alpha=1}\sum_{I\in\mathbb{N}^{m}_{k,k}}h^{\alpha,\beta}_{I}v^{\alpha}_{I}\underline{e}_{\beta\mid\pi_{k}(a)}\>.$
But this is equal to (3.30), in view of
$(P^{h}_{\mathrm{lin}})^{\alpha,\beta}_{I}=h^{\alpha,\beta}_{I}$. The claim
follows. ∎
### 3.3. The manifold of $\infty$-jets and formally integrable PDE’s
Throughout Section 3.3, $\pi$ will again be an arbitrary fiber bundle.
Finally, in this section we are going to make contact with the abstract theory
on profinite dimensional manifolds from Section 2: We are going to prove that
the space of “$\infty$-jets” in $\pi$ canonically becomes a profinite
dimensional manifold (see Proposition 3.20), and that the space of “formal
solutions” of a “formally integrable” partial differential equation on $\pi$
canonically is a profinite dimensional submanifold of the latter (see
Proposition 3.27).
We start by introducing the space of $\infty$-jets. In analogy to Definition
3.3, we have:
###### Definition 3.18.
Let $p\in X$. Any two $\psi,\varphi\in\Gamma^{\infty}(p;\pi)$ are called
_$\infty$ -equivalent at $p$_, if $\psi(p)=\varphi(p)$ and if for every
fibered chart $(x,u):W\to\mathbb{R}^{m}\times\mathbb{R}^{n}$ of $\pi$ with
$W\cap\pi^{-1}(p)\neq\emptyset$ one has
(3.31)
$\displaystyle\frac{\partial^{|I|}\left(u^{\alpha}\circ\psi\right)}{\partial\tilde{x}^{I}}(p)=\frac{\partial^{|I|}\left(u^{\alpha}\circ\varphi\right)}{\partial\tilde{x}^{I}}(p)$
for all $\alpha=1,\dots,n$ and all $I\in\mathbb{N}^{m}$ with
$1\leq|I|<\infty$. The corresponding equivalence class
$\mathsf{j}^{\infty}_{p}\psi$ of $\psi$ is called the _$\infty$ -jet of $\psi$
at $p$_.
###### Remark 3.19.
In view of Remark 3.4, $\infty$-equivalence also only has to be checked in
_some_ fibered chart.
It will be convenient in what follows to set $\mathsf{J}^{-1}(\pi):=X$,
$\mathsf{j}^{-1}_{p}\psi_{p}:=p$, and $\pi_{-1,0}:=\pi$. We define
$\mathsf{J}^{\infty}(\pi):=\bigcup_{p\in
X}\left\\{\mathsf{j}^{\infty}_{p}\psi\mid\psi\in\Gamma^{\infty}(p;\pi)\right\\},$
and obtain for every $i\in\mathbb{Z}_{\geq-1}$ a surjective map
(3.32)
$\displaystyle\pi_{i,\infty}:\mathsf{J}^{\infty}(\pi)\longrightarrow\mathsf{J}^{i}(\pi),\>\>\mathsf{j}^{\infty}_{p}\psi\longmapsto\mathsf{j}^{i}_{p}\psi\>.$
We equip $\mathsf{J}^{\infty}(\pi)$ with the initial topology with respect to
the maps $\pi_{i,\infty}$, $i\in\mathbb{Z}_{\geq-1}$. Furthermore, we define
$\mathscr{C}^{\infty}_{\pi}$ to be the sheaf on $\mathsf{J}^{\infty}(\pi)$,
whose section space $\mathscr{C}^{\infty}_{\pi}(U)$ over an open
$U\subset\mathsf{J}^{\infty}(\pi)$ is given by the set of all
$f\in\mathscr{C}(U)$ such that for every $x\in U$ there is an
$i\in\mathbb{Z}_{\geq-1}$, an open $U_{i}\subset\mathsf{J}^{i}(\pi)$ and an
$f_{i}\in\mathscr{C}^{\infty}(U_{i})$ with
$x\in\pi_{i,\infty}^{-1}(U_{i})\subset U$ and
$f_{|\pi_{i,\infty}^{-1}(U_{i})}=f_{i}\circ{\pi_{i,\infty}}_{|\pi_{i,\infty}^{-1}(U_{i})}\>.$
In particular, $(\mathsf{J}^{\infty}(\pi),\mathscr{C}^{\infty}_{\pi})$ becomes
a locally $\mathbb{R}$-ringed space. Now observe that we have, in view of
(3.17), a smooth projective system
$\big{(}\mathsf{J}^{i}(\pi),\pi_{i,j}\big{)}$, which graphically can be
depicted by
(3.33)
$\begin{split}\mathsf{J}^{-1}(\pi)\xleftarrow{\pi_{-1,0}}\mathsf{J}^{0}(\pi)\xleftarrow{\pi_{0,1}}\dots\longleftarrow\mathsf{J}^{i}(\pi)\xleftarrow{\pi_{i,i+1}}\mathsf{J}^{i+1}(\pi)\longleftarrow\dots,\end{split}$
together with a family of continuous maps
$\pi_{i,\infty}:\mathsf{J}^{\infty}(\pi)\longrightarrow\mathsf{J}^{i}(\pi),\>i\in\mathbb{Z}_{\geq-1}.$
These data have the following crucial property.
###### Proposition and Definition 3.20.
The family $\big{(}\mathsf{J}^{i}(\pi),\pi_{i,j},\pi_{i,\infty}\big{)}$ is a
smooth projective representation of
$(\mathsf{J}^{\infty}(\pi),\mathscr{C}^{\infty}_{\pi})$. In particular, when
equipped with the corresponding pfd structure,
$(\mathsf{J}^{\infty}(\pi),\mathscr{C}^{\infty}_{\pi})$ canonically becomes a
smooth profinite dimensional manifold, called the _manifold of $\infty$-jets
given by $\pi$._
###### Proof.
Let $\mathsf{J}^{\infty}(\pi)^{\prime}{}:=\lim\limits_{\longleftarrow\atop
i\in\mathbb{Z}_{\geq-1}}\mathsf{J}^{i}(\pi)$ denote the canonical projective
limit of (3.33), that means let
$\begin{split}\mathsf{J}^{\infty}&(\pi)^{\prime}{}=\\\
&=\Big{\\{}b=(b_{-1},b_{0},b_{1},\dots)\in\prod_{i\in\mathbb{Z}_{\geq-1}}\mathsf{J}^{i}(\pi)\mid
b_{i}=\pi_{i,j}(b_{j})\text{ for all $i\leq j$}\Big{\\}},\end{split}$
and let
$\pi_{i,\infty}^{\prime}{}:\mathsf{J}^{\infty}(\pi)^{\prime}{}\to\mathsf{J}^{i}(\pi)$
denote the canonical projections. We are going to prove the existence of a
homeomorphism $\Xi$ such that the diagrams
(3.38)
commute for all $i\in\mathbb{Z}_{\geq-1}$. Then the universal property of
$(\mathsf{J}^{\infty}(\pi)^{\prime}{},\pi_{i,\infty}^{\prime}{})$ will
directly imply the same property for
$(\mathsf{J}^{\infty}(\pi),\pi_{i,\infty})$, which is precisely (PFM1). As a
consequence, (PFM2) is trivially satisfied by the definition of the structure
sheaf $\mathscr{C}^{\infty}_{\pi}$.
We now simply define $\Xi(a)_{j}:=\pi_{j,\infty}(a)$ for
$a\in\mathsf{J}^{\infty}(\pi)$ and $j\in\mathbb{Z}_{\geq-1}$. Then it is
obvious that $\Xi$ is a well-defined injective map, and that the $\Xi$-diagram
in (3.38) commutes. In particular, the continuity of $\Xi$ is directly implied
by that of the maps $\pi_{i,\infty}$. In order to see that $\Xi$ is surjective
and that $\Xi^{-1}$ is continuous, let us recall that Borel’s Theorem states
that for any map
$t:\mathbb{N}^{m}=\bigcup_{j\in\mathbb{N}}\mathbb{N}^{m}_{0,j}\longrightarrow\mathbb{R}^{n}$
there is a smooth function $\tilde{\psi}:\mathbb{R}^{m}\to\mathbb{R}^{n}$ such
that $t_{I}=\partial_{I}\tilde{\psi}(0)/I!$ for all $I\in\mathbb{N}^{m}$. Let
$b\in\mathsf{J}^{\infty}(\pi)^{\prime}{}$ and let
$\tilde{x}:U\to\mathbb{R}^{m}$ be a manifold chart of $X$ around $b_{-1}$ with
$\tilde{x}(b_{-1})=0$. Choosing furthermore a bundle chart
$\phi:\pi^{-1}(U)\to U\times F$ and a manifold chart
$\tilde{u}:B\to\mathbb{R}^{n}$ of $F$, we get the fibered chart
$(x,u):=(\tilde{x}\circ\pi,\tilde{u}\circ\mathrm{pr}_{2}\circ\phi):\pi^{-1}(U)\longrightarrow\mathbb{R}^{m}\times\mathbb{R}^{n}$
of $\pi$ by Remark 3.2.3. Moreover, the function
$t:\mathbb{N}^{m}\to\mathbb{R}^{n}$, $t_{I}:=u_{j,I}(b_{j})/I!$, if
$I\in\mathbb{N}^{m}_{0,j}$, is well-defined. Borel’s Theorem then produces a
function $\tilde{\psi}:\mathbb{R}^{m}\to\mathbb{R}^{n}$ such that
$t_{I}=\partial_{I}\tilde{\psi}(0)/I!$. It is clear that the section
$\psi\in\Gamma^{\infty}_{b_{-1}}(\pi)$ defined by
$\psi:=\phi^{-1}\left(\bullet,\tilde{u}^{-1}\circ\tilde{\psi}\circ\tilde{x}\right)$
satisfies $x_{j}(\mathsf{j}^{j}_{b_{-1}}\psi)=0$, and
$u_{j,I}(\mathsf{j}^{j}_{b_{-1}}\psi)=u_{j,I}(b_{j})$ for all
$j\in\mathbb{Z}_{\geq-1}$ and $I\in\mathbb{N}^{m}_{0,j}$, thus
$\Xi(\mathsf{j}^{\infty}_{b_{-1}}\psi)=b$, and $\Xi$ is surjective, indeed.
Furthermore, by the construction of $\Xi^{-1}(b)$, it is also clear that the
$\Xi^{-1}$-diagram in (3.38) commutes, so that the continuity of $\Xi^{-1}$
trivially follows from that of the $\pi_{i,\infty}^{\prime}{}$. This completes
the proof. ∎
Next, we will prepare the introduction of formal integrability. Let us first
note the following simple result:
###### Proposition and Definition 3.21.
1. a)
For every $h\in\mathrm{D}^{k}(\pi,\underline{\pi})$ there exists a unique
$h^{(l)}\in\mathrm{D}^{k+l}(\pi,\underline{\pi}_{l})$ such that the following
diagram of set theoretic sheaf morphisms
(3.45)
commutes. The partial differential operator $h^{(l)}$ is called the _$l$ -jet
prolongation of $h$_.
2. b)
The partial differential operator
$\displaystyle\iota^{\pi}_{l,k}\>\left(=\mathrm{id}_{\mathsf{J}^{k}(\pi)}^{(l)}\right)\>:\mathsf{J}^{k+l}(\pi)$
$\displaystyle\longrightarrow\mathsf{J}^{l}(\pi_{k}),\>\mathsf{j}^{k+l}_{p}\psi\longmapsto\mathsf{j}^{l}_{p}(\mathsf{j}^{k}\psi)$
is an embedding of manifolds.
Let $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ be an arbitrary partial
differential equation for the moment. Since, by definition, the map
${\pi_{k}}_{|\mathsf{E}}:\mathsf{J}^{k}(\pi)\supset\mathsf{E}\longrightarrow
X$
is again a fibered manifold, there exists for every $l\in\mathbb{N}$ an
obvious well-defined map
$\iota_{l,\mathsf{E}}:\mathsf{J}^{l}({\pi_{k}}_{|\mathsf{E}})\longrightarrow\mathsf{J}^{l}(\pi_{k}),$
which comes from considering a locally defined section in
${\pi_{k}}_{|\mathsf{E}}$ as taking values in $\mathsf{J}^{k}(\pi)$.
###### Definition 3.22.
Let $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ be a partial differential equation.
Then the set
(3.46) $\displaystyle\mathsf{E}^{(l)}:=\begin{cases}\mathsf{E},&\text{for
$l=0$},\\\
\iota_{l,k}^{\pi,-1}\Big{(}\iota_{l,\mathsf{E}}(\mathsf{J}^{l}(\pi_{k\mid\mathsf{E}}))\Big{)}\subset\mathsf{J}^{k+l}(\pi),&\text{for
$l\in\mathbb{N}^{*}$},\end{cases}$
is called the _$l$ -jet prolongation_ of $\mathsf{E}$.
If the underlying partial differential equation is actually given by a partial
differential operator, then there is an explicit description of the
corresponding $l$-jet prolongation ([19], p. 294):
###### Proposition 3.23.
Let $h\in\mathrm{D}^{k}(\pi,\underline{\pi})$ with constant rank and let
$O\in\Gamma^{\infty}(X;\underline{\pi})$ with
$\mathrm{im}(O)\subset\mathrm{im}(h)$. Then one has, for every
$l\in\mathbb{N}$,
$\ker_{O}(h)^{(l)}=\ker_{\mathsf{j}^{l}O}(h^{(l)})\subset\mathsf{J}^{k+l}(\pi).$
Let us note the simple fact that the following diagramm commutes, for every
$r\in\mathbb{N}$,
$\textstyle{\mathsf{J}^{k+l+r}(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\pi}_{l+r,k}}$$\scriptstyle{\pi_{k+l,k+l+r}}$$\textstyle{\mathsf{J}^{l+r}(\pi_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\pi_{k})_{l+r,l}}$$\textstyle{\mathsf{J}^{k+l}(\pi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\pi}_{l,k}}$$\textstyle{\mathsf{J}^{l}(\pi_{k})\>.}$
Applying this in the case $r=1$ implies
$\pi_{k+l,k+l+1}(\mathsf{E}^{(l+1)})\subset\mathsf{E}^{(l)}$ for any partial
differential equation $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ and every
$l\in\mathbb{N}$, so that we obtain the maps
(3.47)
$\displaystyle\mathsf{E}^{(l+1)}\longrightarrow\mathsf{E}^{(l)},\>a\longmapsto\pi_{k+l,k+l+1}(a).$
Now we have the tools to give
###### Definition 3.24.
A partial differential equation $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ is
called _formally integrable_ , if $\mathsf{E}^{(l)}$ is a submanifold of
$\mathsf{J}^{k+l}(\pi)$ and if (3.47) is a fibered manifold for every
$l\in\mathbb{N}$.
###### Remark 3.25.
1. a)
Here, it should be noted that $E$ itself can always be considered as a trivial
formally integrable partial differential equation on $\pi$ of order $0$, where
in this case one has $E^{(l)}=\mathsf{J}^{l}(\pi)$ for all $l\in\mathbb{N}$.
2. b)
Furthermore, there are abstract cohomological tests for partial differential
equations to be formally integrable [19]. In fact, we will use such a test in
the proof of Theorem 3.28 below; we refer the reader to [28] and particularly
to [32] for the algorithmic aspects of these tests. Although it can become
very involved to verify these test properties in particular examples, it is
widely believed that most partial differential equations that arise naturally
from geometry and physics are formally integrable. In accordance with the
latter statement, Theorem 3.28 below states that all reasonable (possibly
nonlinear) scalar partial differential equations are formally integrable. See
for example [18] for a full treatement of the Yang–Mills–Higgs equations, and
[23] for a treatement of Einstein’s field equations under the viewpoint of
formal integrability.
An important purely analytic consequence of formal integrability is given by
the highly nontrivial Theorem 3.26 below, which essentially states that if all
underlying data are real analytic, then formal integrability implies the
existence of local analytic solutions with prescribed finite order Taylor
expansions. Theorem 3.26 goes back to Goldschmidt [19] and heavily relies on
(cohomological) results by Spencer [33] and Ehrenpreis–Guillemin–Sternberg
[13]. This result can also be regarded as a variant of Michael Artin’s
Approximation Theorem [3].
###### Theorem 3.26.
Assume that $X$ is real analytic, that $\pi$ is a real analytic fiber bundle
(then so is $\pi_{k}$), and that $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ is
formally integrable such that in fact $\pi_{k,\mid\mathsf{E}}:\mathsf{E}\to X$
is a real analytic fibered submanifold of $\pi_{k}$. Then, for every
$l\in\mathbb{N}$ and $a\in\mathsf{E}^{(l)}$ there exists an open neighborhood
$U\subset X$ of $\pi_{k+l}(a)$ and a real analytic solution
$\psi\in\Gamma^{\infty}(U;\pi)$ of $\mathsf{E}$ such that
$\mathsf{j}^{k+l}_{\pi_{k+l}(a)}\psi=a$.
###### Proof.
This result follows directly from Theorem 9.1 in [19] (in combination with
Proposition 7.1 therein). ∎
Now let $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ be a formally integrable
partial differential equation. Then we can define a subset
$\mathsf{E}^{(\infty)}\subset\mathsf{J}^{\infty}(\pi)$ by
$\mathsf{E}^{(\infty)}:=\pi^{-1}_{\infty,k}(\mathsf{E})$. Inductively, one
checks that the maps (3.32) restrict to surjective maps
(3.48)
$\displaystyle\mathsf{E}^{(\infty)}\longrightarrow\mathsf{E}^{(i)},\>a\longmapsto\pi_{k+i,\infty}(a),\>\>i\in\mathbb{N},$
(3.49) $\displaystyle\mathsf{E}^{(\infty)}\longrightarrow
X,\>a\longmapsto\pi_{k-1,\infty}(a),$
so that
$\mathsf{E}^{(\infty)}=\bigcap_{i\in\mathbb{N}}\pi^{-1}_{\infty,k+i}(\mathsf{E}).$
In other words, this means that axioms (PFSM1) to (PFSM3) are satisfied for
the subset $\mathsf{E}^{(\infty)}\subset\mathsf{J}^{\infty}(\pi)$ and the
smooth projective representation
$\big{(}\mathsf{J}^{i}(\pi),\pi_{i,j},\pi_{i,\infty}\big{)}$. Hence, one
readily obtains
###### Proposition and Definition 3.27.
Let $\mathsf{E}\subset\mathsf{J}^{k}(\pi)$ be a formally integrable partial
differential equation. Then
$\big{(}\mathsf{J}^{i}(\pi),\pi_{i,j},\pi_{i,\infty}\big{)}$ induces on
$\mathsf{E}^{(\infty)}$ the structure of a profinite dimensional submanifold
of $(\mathsf{J}^{\infty}(\pi),\mathscr{C}^{\infty}_{\pi})$. In view of this
fact, $\mathsf{E}^{(\infty)}$ will be called the _manifold of formal solutions
of $\mathsf{E}$._
### 3.4. Scalar PDE’s and interacting relativistic scalar fields
Let us first clarify that throughout Section 3.4, $\pi:X\times\mathbb{R}\to X$
will denote the canonical ine bundle.
#### 3.4.1. A criterion for formal integrability of scalar PDE’s
We now come to the aforementioned result on formal integrability of scalar
PDE’s.
In the scalar situation, the sheaf of sections of $\pi$ can be canonically
identified with the sheaf of smooth functions on $X$. The smooth functions
which are defined near $p\in X$ will be denoted with
$\mathscr{C}^{\infty}(p;X)$. For any $h\in\mathrm{D}^{k}(\pi,\pi)$, the space
of vector bundle morphisms
$\pi^{*}_{k}\mathrm{Sym}^{k}(\pi_{\mathrm{T}^{*}X})\otimes\pi^{*}_{k,0}\mathsf{V}(\pi)\longrightarrow\mathsf{V}(\pi_{k})$
over $h$ can be identified canonically as a linear space (remember here the
maps (3.25) and (3.26)) with
$\Gamma^{\infty}\left(\mathsf{J}^{k}(\pi);\left[\pi^{*}_{k}\pi^{\odot^{k}}_{\mathrm{T}^{*}X}\right]^{*}\right).$
It follows that the symbol $\sigma(h)$ of an $h$ as above can be identified
with an element of
$\Gamma^{\infty}\left(\mathsf{J}^{k}(\pi);\left[\pi^{*}_{k}\pi^{\odot^{k}}_{\mathrm{T}^{*}X}\right]^{*}\right)$,
implying that
${\sigma(h)}_{|a}:\mathrm{Sym}^{k}\left(\mathrm{T}^{*}_{\pi_{k}(a)}X\right)\longrightarrow\mathbb{R}$
is a linear map for every $a\in\mathsf{J}^{k}(\pi)$. Thus, $h$ induces the
globally defined section $\sigma(h)^{(1)}$ of the vector bundle
$\mathrm{Hom}\Big{(}\pi_{k}^{*}\mathrm{Sym}^{k+1}(\pi_{\mathrm{T}^{*}X}),\pi_{k}^{*}\mathrm{T}^{*}X\Big{)}\longrightarrow\mathsf{J}^{k}(\pi),$
which, for every $a\in\mathsf{J}^{k}(\pi)$, is given by
$\displaystyle{\sigma(h)^{(1)}}_{|a}:\>$
$\displaystyle\mathrm{Sym}^{k+1}\left(\mathrm{T}^{*}_{\pi_{k}(a)}X\right)\longrightarrow\mathrm{T}^{*}_{\pi_{k}(a)}X,$
$\displaystyle v_{1}\odot\cdots\odot
v_{k+1}\longmapsto\underbrace{{\sigma(h)}_{|a}(v_{2}\odot\cdots\odot
v_{k+1})}_{\in\mathbb{R}}v_{1}.$
Finally, we note that the space of $k$-th order partial differential operators
$\mathrm{D}^{k}(\pi,\pi)$ can be canonically identified as a linear space with
$\mathscr{C}^{\infty}(\mathsf{J}^{k}(\pi))$.
With these preparations, we have:
###### Theorem 3.28.
Let $h\in\mathscr{C}^{\infty}(\mathsf{J}^{k}(\pi))$ and assume that the
following assumptions are satisfied:
1. _(1)_
One has ${\sigma(h)}_{|a}\neq 0$ for all $a\in\mathsf{J}^{k}(\pi)$.
2. _(2)_
With $\iota_{h}:\ker(h)\hookrightarrow\mathsf{J}^{k}(\pi)$ denoting the
inclusion, the pull-back $\iota^{*}_{h}[\sigma(h)^{(1)}]$ has constant rank.
3. _(3)_
The map $\ker(h)^{(1)}\to\ker(h)$, $a\mapsto\pi_{k,k+1}(a)$ is surjective.
Then $\ker(h)\subset\mathsf{J}^{k}(\pi)$ is a formally integrable partial
differential equation on $\pi$.
###### Remark 3.29.
Note that assumption (1) together with Proposition 3.12 imply that $\ker(h)$
indeed is a partial differential equation, in particular, assumption (2) makes
sense.
Proof of Theorem 3.28. The seemingly short proof that we are going to give
actually combines two heavy machineries: The already mentioned abstract
cohomological criterion for formal integrability of partial differential
equations from [19], with a highly nontrivial reduction result for the
cohomology of Cohen-Macaulay symbolic systems [23]. There seems to be no
reasonable elementary proof of Theorem 3.28.
To prove our claim, let an arbitrary $a\in\mathsf{J}^{k}(\pi)$ be given. By
assumption (1), we can pick some $v\in\mathrm{T}^{*}_{\pi_{k}(a)}X$ with
${\sigma(h)}_{|a}(v^{\otimes k})\neq 0$. Then, in the terminology of [23],
$V^{*}:=\mathbb{C}v\subset\left(\mathrm{T}^{*}_{\pi_{k}(a)}X\right)_{\mathbb{C}}$
is a one-dimensional noncharacteristic subspace corresponding to the Cohen-
Macaulay symbolic system $\mathrm{g}(h;a)$ given by $\ker(h)$ over $a$. Thus
we may apply Theorem A from [23] to deduce that all Spencer cohomology groups
$\mathrm{H}^{i,j}(\mathrm{g}(h;a))$ except possibly
$\mathrm{H}^{0,0}(\mathrm{g}(h;a))$ and $\mathrm{H}^{1,1}(\mathrm{g}(h;a))$
vanish. But now the result follows from combining (3), [19, Theorem 8.1] and
[19, Proposition 7.1], noting that by assumption (2), the first prolongation
$\bigcup_{a\in\ker(h)}\mathrm{g}(h;a)^{(1)}\longrightarrow\ker(h)$
becomes a vector bundle. $\blacksquare$
The assumptions (2) and (3) from Theorem 3.28 are technical regularity
assumptions (which can become tedious to check in applications), whereas the
reader should notice that assumption (1) therein is essentially trivial and
means nothing but that the underlying differential operator globally is a
“genuine” $k$-th order operator.
#### 3.4.2. Interacting relativistic scalar fields
As an application of Theorem 3.28, we will now consider evolution equations
that correspond to (possibly nonlinearly!) interacting relativistic scalar
fields on semi-riemannian manifolds. To this end, let $(X,\mathsf{g})$ be a
smooth semi-riemannian $m$-manifold with an arbitrary signature. The
corresponding d’Alembert operator will be written as
$\Box_{\mathsf{g}}:\mathscr{C}^{\infty}(X)\longrightarrow\mathscr{C}^{\infty}(X).$
With functions $F_{1},F_{2}\in\mathscr{C}^{\infty}(X)$,
$K\in\mathscr{C}^{\infty}(\mathbb{R})$, we consider the partial differential
operator
$h_{\mathsf{g},F_{1},F_{2},K}\in\mathscr{C}^{\infty}(\mathsf{J}^{2}(\pi))$
given for $p\in X$, $\varphi\in\mathscr{C}^{\infty}(p;X)$ by
$h_{\mathsf{g},F_{1},F_{2},K}\left(\mathsf{j}^{2}_{p}\varphi\right):=\Box_{\mathsf{g}}\varphi(p)+F_{1}(p)\varphi(p)+F_{2}(p)K(\varphi(p)).$
What we have in mind here is:
###### Example 3.30.
Let us assume that $m=4$, that $(X,\mathsf{g})$ has a Lorentz signature, and
that
$F_{1}=\alpha_{1}\,\mathrm{scal}_{\mathsf{g}}+\alpha_{2}^{2},\>F_{2}=0,\>K=\alpha_{3}\underline{K},$
where $\alpha_{1},\alpha_{3}\in\mathbb{R}$, $\alpha_{2}\geq 0$,
$\mathrm{scal}_{\mathsf{g}}\in\mathscr{C}^{\infty}(X)$ denotes the scalar
curvature of $\mathsf{g}$ and
$\underline{K}\in\mathscr{C}^{\infty}(\mathbb{R})$. Then
$\ker(h_{\mathsf{g},F_{1},0,K})\subset\mathsf{J}^{2}(\pi)$ describes the on-
shell dynamics of a relativistic (real) scalar field with mass $\alpha_{2}$,
where $\underline{K}$ is the field self-interaction with coupling strength
$\alpha_{3}$, and where the number $\alpha_{1}$ is an additional parameter,
which is sometimes set equal to zero. For example, $\underline{K}(z)=z^{3}$
corresponds to what is called a _$\varphi^{4}$ -perturbation_ in the physics
literature (since the corresponding potential in the Lagrange densitiy which
has $\operatorname{ker}(h_{\mathsf{g},F_{1},0,K})$ as its Euler-Lagrange
equation is given by $V(\varphi)=\varphi^{4}$). We refer the reader to [9] for
the perturbative aspects of this equation in the flat $\varphi^{4}$ case.
Returning to the general situation, we can now prove the following result on
scalar partial differential equations on semi-riemannian manifolds:
###### Proposition 3.31.
In the above situation, the assumptions _(1), (2) and (3)_ from Theorem 3.28
are satisfied by $h_{\mathsf{g},F_{1},F_{2},K}$. In particular,
$\ker(h_{\mathsf{g},F_{1},F_{2},K})\subset\mathsf{J}^{2}(\pi)$ is formally
integrable, and the corresponding space of formal solutions canonically
becomes a profinite dimensional manifold via Proposition 3.27. Moreover, if in
addition $(X,\mathsf{g})$, $F_{1}$, $F_{2}$ and $K$ are real analytic, then
there exists for every $l\in\mathbb{N}$ and
$a\in\ker(h_{\mathsf{g},F_{1},F_{2},K})^{(l)}$ an open neighborhood $U\subset
X$ of $\pi_{k+l}(a)$ and a real analytic solution
$\varphi\in\mathscr{C}^{\infty}(U)$ of $\ker(h_{\mathsf{g},F_{1},F_{2},K})$
such that $\mathsf{j}^{k+l}_{\pi_{k+l}(a)}\varphi=a$.
###### Proof.
In view of Theorem 3.26, we only have to prove that the assumptions (1), (2),
(3) from 3.28 are satisfied. To this end, we set
$h:=h_{\mathsf{g},F_{1},F_{2},K}$ and assume $F_{2}=0$. Firstly, in view of
${\sigma(h)}_{|a}(v\odot v)=\mathsf{g}^{*}_{\pi_{2}(a)}(v,v)\>\text{ for all
$a\in\mathsf{J}^{2}(\pi)$, $v\in\mathrm{T}^{*}_{\pi_{2}(a)}X$},$
assumption (1) is obviously satisfied and $\ker(h)$ indeed is a partial
differential equation. Analogously, to see that assumption (2) is satisfied,
one just has to note that
${\sigma(h)^{(1)}}_{|a}(v_{1}\odot v_{2}\odot
v_{3})=\mathsf{g}^{*}_{\pi_{2}(a)}(v_{2},v_{3})v_{1}\>\text{ for all
$a\in\ker(h)$, $v\in\mathrm{T}^{*}_{\pi_{2}(a)}X$}.$
Thus, ${\sigma(h)^{(1)}}_{|a}$ is surjective for fixed $a$. As a consequence
of this and of being a vector bundle morphism, $\sigma(h)^{(1)}$ has constant
rank.
It remains to prove that the map $\ker(h)^{(1)}\to\ker(h)$,
$a\mapsto\pi_{k,k+1}(a)$ is surjective. To see this, assume to be given
$b\in\ker(h)$ and consider a $\mathsf{g}$-exponential manifold chart
$\tilde{x}:U\to\mathbb{R}^{m}$ of $X$ centered at $\pi_{2}(b)$. Then one gets
the trivial fibered chart
$(x,u):=(\tilde{x},\mathrm{id}_{\mathbb{R}}):U\times\mathbb{R}\longrightarrow\mathbb{R}^{m}\times\mathbb{R}$
of $\pi$, and $b\in\ker(h)$ means nothing but $b\in\mathsf{J}^{2}(\pi)$ and
$\begin{split}\sum^{m}_{i,j=1}\mathsf{g}^{ij}(\pi_{2}(b))\,u_{2,1_{ij}}(b)-\sum^{m}_{i,j,k=1}\mathsf{g}^{ij}(\pi_{2}(b))\Gamma^{k}_{ij}(\pi_{2}(b))u_{2,1_{k}}(b)\,+&\\\
+F_{1}(\pi_{2}(b))u_{2,(0,\dots,0)}(b)+K(u_{2,(0,\dots,0)}(b))\,&=0,\end{split}$
where $\mathsf{g}^{ij},\Gamma^{k}_{ij}\in\mathscr{C}^{\infty}(U)$ denote the
components of the metric tensor and the Christoffel symbols of $g$ with
respect to $\tilde{x}$, respectively. Noting that Proposition 3.23 implies
$\ker(h)^{(1)}=\ker(h^{(1)})$, one easily finds that some
$a\in\mathsf{J}^{3}(\pi)$ is in $\ker(h)^{(1)}$, if and only if
$\begin{split}\sum^{m}_{i,j=1}\mathsf{g}^{ij}(\pi_{3}(a))u_{3,1_{ij}}(a)&-\sum^{m}_{i,j,k=1}\mathsf{g}^{ij}(\pi_{3}(a))\Gamma^{k}_{ij}(\pi_{3}(a))u_{1_{k}}(a)\,+\\\
&+F_{1}(\pi_{3}(a))u_{3,(0,\dots,0)}(a)+K\Big{(}u_{3,(0,\dots,0)}(a)\Big{)}=0,\end{split}$
and, for all $l=1,\dots,m$,
$\begin{split}&\sum^{m}_{i,j=1}\Big{(}\partial_{l}\mathsf{g}^{ij}(\pi_{3}(a))u_{3,1_{ij}}(a)+\mathsf{g}^{ij}(\pi_{3}(a))u_{3,1_{ijl}}(a)\Big{)}-\\\
&-\sum^{m}_{i,j,k=1}\\!\\!\Big{(}\partial_{l}\mathsf{g}^{ij}(\pi_{3}(a))\Gamma^{k}_{ij}(\pi_{3}(a))u_{3,1_{k}}(a)-\mathsf{g}^{ij}(\pi_{3}(a))\partial_{l}\Gamma^{k}_{ij}(\pi_{3}(a))u_{3,1_{k}}(a)\Big{)}-\\\
&-\sum^{m}_{i,j,k=1}\mathsf{g}^{ij}(\pi_{3}(a))\Gamma^{k}_{ij}(\pi_{3}(a))u_{3,1_{lk}}(a)+\partial_{l}F_{1}(\pi_{3}(a))u_{3,(0,\dots,0)}(a)\,+\\\
&+F_{1}(\pi_{3}(a))u_{3,1_{l}}(a)+K^{\prime}{}\Big{(}u_{3,(0,\dots,0)}(a)\Big{)}u_{3,1_{l}}(a)=0.\end{split}$
Here, we have used $\partial_{l}:=\frac{\partial}{\partial\tilde{x}^{l}}$. Let
us now assume that the signature of $\mathsf{g}$ is given by
$(\varepsilon_{1},\dots,\varepsilon_{m})=(1,-1,\dots,-1)$. The general case
can be treated with the same method. We define some $a\in\mathsf{J}^{3}(\pi)$
by requiring $\tilde{x}_{3}(a):=\tilde{x}(\pi_{2}(b))$, and, for
$I\in\mathbb{N}^{m}_{0,3}$,
$\begin{split}u_{3,I}&(a):=\\\ &\begin{cases}u_{2,I}(b),&\text{ if
$I\in\mathbb{N}^{m}_{0,2}$,}\\\
-\sum^{m}_{i,j=1}\partial_{l}\mathsf{g}^{ij}(\pi_{2}(b))u_{2,1_{ij}}(b)+\\\
+\sum^{m}_{i,j,k=1}\partial_{l}\mathsf{g}^{ij}(\pi_{2}(b))\Gamma^{k}_{ij}(\pi_{2}(b))u_{2,1_{k}}(b)+\\\
+\sum^{m}_{i,j,k=1}\mathsf{g}^{ij}(\pi_{2}(b))\partial_{l}\Gamma^{k}_{ij}(\pi_{2}(b))u_{2,1_{k}}(b)+\\\
+\sum^{m}_{i,j,k=1}\mathsf{g}^{ij}(\pi_{2}(b))\Gamma^{k}_{ij}(\pi_{2}(b))u_{2,1_{lk}}(b)-\\\
-\partial_{l}F_{1}(\pi_{2}(b))u_{2,(0,\dots,0)}(b)-F_{1}(\pi_{2}(b))u_{2,1_{l}}(b)-\\\
-K^{\prime}{}\Big{(}u_{2,(0,\dots,0)}(b)\Big{)}u_{2,1_{l}}(b),&\text{ if
$I=1_{11l}$ for some $l=1,\dots,m$,}\\\ 0,&\text{
else}.\end{cases}\end{split}$
Now we are almost done: Indeed, our construction of $a$ directly gives
$\pi_{2,3}(a)=b$, so $\pi_{3}(a)=\pi_{2}(b)$. Since we have
(3.50)
$\displaystyle\mathsf{g}^{ij}(\pi_{3}(a))=\mathsf{g}^{ij}(\pi_{2}(b))=\begin{cases}\varepsilon_{j},&\text{if
$i=j$},\\\ 0,&\text{else},\end{cases}$
it follows immediately that $a\in\ker(h)^{(1)}$, and the proof is complete,
noting that $F_{2}$ has not played a role in the above argument. ∎
## Appendix A Two results on completed projective tensor products
Assume to be given two locally convex topological vector spaces $V$ and $W$,
and consider their algebraic tensor product $V\otimes W$. A topology $\tau$ on
$V\otimes W$ is called _compatible_ (in the sense of Grothendieck [20]) or a
_tensor product topology_ , if the following axioms hold true:
1. (TPT1)
$V\otimes W$ equipped with $\tau$ is a locally convex topological vector space
which will be denoted by $V\otimes_{\tau}W$.
2. (TPT2)
The canonical map $V\times W\rightarrow V\otimes_{\tau}W$ is seperately
continuous.
3. (TPT3)
For every equicontinuous subset $A$ of the topological dual $V^{\prime}$ and
every equicontinuous subset $B$ of the topological dual $W^{\prime}$, the set
$A\otimes B:=\\{\lambda\otimes\mu\mid\lambda\in A,\>\mu\in B\\}$ is an
equicontinuous subset of $\big{(}V\otimes_{\tau}W\big{)}^{\prime}$.
If $\tau$ is a tensor product topology on $V\otimes W$, we denote by
$V\widehat{\otimes}_{\tau}W$ the completion of $V\otimes_{\tau}W$.
###### Example A.1.
1. a)
The _projective tensor product topology_ is the finest locally convex vector
space topology on $V\otimes W$ such that the canonical map $V\times
W\rightarrow V\otimes W$ is continuous, cf. [20, 35]. The projective tensor
product topology is denoted by $\pi$. It is generated by seminorms
$p_{A}\otimes_{\pi}q_{B}$, where $p_{A}$, $A\in\mathcal{A}$ and $q_{B}$,
$B\in\mathcal{B}$ each run through a family of seminorms generating the
locally convex topology on $V$ respectively $W$, and $p_{A}\otimes_{\pi}q_{B}$
is defined by
$p_{A}\otimes_{\pi}q_{B}(z):=\inf\left\\{\sum_{l=1}^{n}p_{A}(v_{l})\,q_{B}(w_{l})\mid
z=\sum_{l=1}^{n}v_{l}\otimes w_{l}\right\\}\>.$
The seminorm $p_{A}\otimes_{\pi}q_{B}$ is in particular a _cross seminorm_ ,
i.e. it satisfies the relation
$p_{A}\otimes_{\pi}q_{B}(v\otimes w)=p_{A}(v)\,q_{B}(w)\quad\text{for all
$v\in V$ and $w\in W$}.$
2. b)
The _injective tensor product topology_ on $V\otimes W$, denoted by
$\varepsilon$, is the locally convex topology inherited from the canonical
embedding $V\otimes W\hookrightarrow\mathcal{B}_{s}(V_{s}^{\prime}\otimes
W_{s}^{\prime})$, where $\mathcal{B}_{s}(V^{\prime},W^{\prime})$ denotes the
space of seperately continuous bilinear forms on the product $V^{\prime}\times
W^{\prime}$ of the weak topological duals $V^{\prime}$ and $W^{\prime}$
endowed with the topology of uniform convergence on products of equicontinuous
subsets of $V^{\prime}$ and $W^{\prime}$. See [20] and [35, Sec. 43] for
details.
###### Remark A.2.
1. a)
By definition, the $\varepsilon$-topology on $V\otimes W$ is coarser than the
$\pi$-topology. If $V$ (or $W$) is a nuclear locally convex topological vector
space, then these two topologies coincide, cf. [20, 35]. Since finite
dimensional vector spaces over $\mathbb{R}$ are nuclear, this entails in
particular that for finite dimensional $V$ and $W$ the natural vector space
topology on $V\otimes W$ coincides with the (completed) $\pi$\- and
$\varepsilon$-topology.
2. b)
The projective tensor product, the injective tensor product, and their
completed versions are in fact functors, so it is clear what is meant by
$f\otimes_{\varepsilon}g$, $f\widehat{\otimes}_{\pi}g$, and so on, where $f$
and $g$ denote continuous linear maps.
###### Theorem A.3.
Let $\big{(}V_{i}\big{)}_{i\in\mathbb{N}}$ and
$\big{(}W_{i}\big{)}_{i\in\mathbb{N}}$ be two families of finite dimensional
real vector spaces. Denote by $V$ and $W$ their respective product (within the
category of locally convex topological vector spaces), i.e. let
$V:=\prod\limits_{i\in\mathbb{N}}V_{i}\quad\text{and}\quad
W:=\prod\limits_{i\in\mathbb{N}}W_{i}\>.$
Then $V$, $W$, and the completed projective tensor product
$V\widehat{\otimes}_{\pi}W$ are nuclear Fréchet spaces. Moreover, one has the
canonical isomorphism
(A.1)
$V\widehat{\otimes}_{\pi}W\cong\prod_{(k,l)\in\mathbb{N}\times\mathbb{N}}V_{k}\otimes
W_{l}\>.$
###### Proof.
Since each of the vector spaces $V_{i}$ and $W_{i}$ is a nuclear Fréchet
space, and countable products of nuclear Fréchet are again nuclear Fréchet
spaces by [35], the spaces $V$ and $W$ are nuclear Fréchet. Moreover, the same
argument shows that $V\widehat{\otimes}_{\pi}W$ is nuclear Fréchet, if Eq. A.1
holds true. So let us show Eq. A.1. To this end recall first [7, §3.7] that
there is a canonical injection
$\begin{split}\iota:\>&V\otimes
W\lhook\joinrel\relbar\joinrel\rightarrow\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}}V_{i}\otimes
W_{j},\\\
&(v_{i})_{i\in\mathbb{N}}\otimes(w_{j})_{j\in\mathbb{N}}\longmapsto(v_{i}\otimes
w_{j})_{(i,j)\in\mathbb{N}\times\mathbb{N}}\>.\end{split}$
Choose norms $p_{i}:V_{i}\rightarrow\mathbb{R}$ and
$q_{i}:W_{i}\rightarrow\mathbb{R}$. The product topology on
$\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}}V_{i}\otimes W_{j}$ then is defined
by the sequence of seminorms
$r_{k,l}:\>\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}}V_{i}\otimes
W_{j}\longrightarrow\mathbb{R},\>\big{(}z_{i,j}\big{)}_{(i,j)\in\mathbb{N}\times\mathbb{N}}\longmapsto(p_{k}\otimes_{\pi}q_{l})(z_{k,l}).$
The product topology on $V$ is generated by the seminorms
$p_{k}^{V}:V\rightarrow\mathbb{R}$,
$\big{(}v_{i}\big{)}_{i\in\mathbb{N}}\mapsto p_{k}(v_{k})$, the topology on
$W$ by the seminorms $q_{l}^{W}:W\rightarrow\mathbb{R}$,
$\big{(}w_{i}\big{)}_{i\in\mathbb{N}}\mapsto q_{l}(w_{l})$. Hence, the
$\pi$-topology on $V\otimes W$ is generated by the seminorms
$p_{k}^{V}\otimes_{\pi}q_{l}^{W}$. But since these are cross seminorms, one
obtains for $(v_{i})_{i\in\mathbb{N}}\in V$ and $(w_{i})_{i\in\mathbb{N}}\in
W$ the equality
$\begin{split}p_{k}^{V}\otimes_{\pi}q_{l}^{W}\,&\big{(}(v_{i})_{i\in\mathbb{N}}\otimes(w_{i})_{i\in\mathbb{N}}\big{)}=p_{k}^{V}\big{(}(v_{i})_{i\in\mathbb{N}}\big{)}\,q_{l}^{W}\big{(}(w_{i})_{i\in\mathbb{N}}\big{)}=\\\
&=p_{k}(v_{k})\,q_{l}(w_{l})=p_{k}\otimes_{\pi}q_{l}(v_{k}\otimes
w_{l})=r_{k,l}\big{(}(v_{i}\otimes
w_{j})_{(i,j)\in\mathbb{N}\times\mathbb{N}}\big{)}.\end{split}$
This entails $p_{k}^{V}\otimes_{\pi}q_{l}^{W}=r_{k,l}\circ\iota$, or in other
words that the $\pi$-topology on $V\otimes W$ coincides with the pull-back of
the product topology on
$\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}}V_{i}\otimes W_{j}$ by the
embedding $\iota$. The claim now follows, if we can yet show that the image of
$\iota$ is dense in its range. To prove this let
$z=\big{(}z_{i,j}\big{)}_{(i,j)\in\mathbb{N}\times\mathbb{N}}$ be an element
of the product $\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}}V_{i}\otimes W_{j}$.
Choose representations
$z_{i,j}=\sum_{l=1}^{n_{i,j}}v_{i,j,l}\otimes w_{i,j,l},\quad\text{where
}v_{i,j,l}\in V_{i},\>w_{i,j,l}\in V_{j}\>.$
Put $v_{i,j,l}=0$ and $w_{i,j,l}=0$, if $l>n_{i,j}$. Let
$\iota_{i}^{V}:V_{i}\rightarrow V$ the embedding of the $i$-th factor in $V$,
i.e. the map which associates to $v_{i}\in V_{i}$ the family
$(v_{j})_{j\in\mathbb{N}}$, where $v_{j}:=0$, if $j\neq i$. Likewise, denote
by $\iota_{i}^{W}:W_{i}\hookrightarrow W$ the embedding of the $i$-th factor
in $W$. Then define for $n\in\mathbb{N}$
$z_{n}:=\sum_{i,j\leq
n}\sum_{l\in\mathbb{N}}\iota_{i}^{V}(v_{i,j,l})\otimes\iota_{j}^{W}(w_{i,j,l}),$
and note that by construction the sum on the right side is finite. The
sequence $(z_{n})_{n\in\mathbb{N}}$ then is a family in $V\otimes_{\pi}W$. By
construction, it is clear that
$\lim\limits_{n\rightarrow\infty}\iota(z_{n})=z$. The proof is finished.
∎
###### Theorem A.4.
Assume that $V$ and $W$ are projective limits of projective systems of finite
dimensional real vector spaces $\big{(}V_{i},\lambda_{ij}\big{)}$ and
$\big{(}W_{i},\mu_{ij}\big{)}$, respectively. Denote by
$\lambda_{i}:V\longrightarrow V_{i},\>\text{ respectively by
}\>\mu_{i}:W\longrightarrow W_{i},$
the corresponding canonical maps. The completed $\pi$-tensor product
$V\widehat{\otimes}_{\pi}W$ together with the family of canonical maps
$\lambda_{i}\widehat{\otimes}_{\pi}\mu_{i}:V\widehat{\otimes}_{\pi}W\longrightarrow
V_{i}\otimes W_{i}$
then is a projective limit of the projective system $\big{(}V_{i}\otimes
W_{i},\lambda_{ij}\otimes\mu_{ij}\big{)}$ within the category of locally
convex topological vector spaces. Moreover, both $V$ and $W$ are nuclear,
hence $V\widehat{\otimes}_{\pi}W=V\widehat{\otimes}_{\varepsilon}W$.
###### Proof.
First observe that $\big{(}V_{i}\otimes
W_{i},\lambda_{ij}\otimes\mu_{ij}\big{)}$ is a projective systems of finite
dimensional real vector spaces, indeed. Next recall that projective limits of
nuclear Fréchet spaces are nuclear by [35]. This proves the second claim. It
remains to show the first one. To this end put $\widetilde{V}_{0}:=V_{0}$,
$\widetilde{W}_{0}:=W_{0}$, and denote for every $i\in\mathbb{N}^{*}$ by
$\widetilde{V}_{i}$ be the kernel of the map $\lambda_{i-1i}$ and by
$\widetilde{W}_{i}$ the kernel of $\mu_{i-1i}$. Morever, choose for every
$i\in\mathbb{N}^{*}$ a splitting $f_{i}:V_{i-1}\rightarrow V_{i}$ of
$\lambda_{i-1i}$, and a splitting $g_{i}:W_{i-1}\rightarrow W_{i}$ of
$\mu_{i-1i}$. Put
$\widetilde{V}:=\prod\limits_{i\in\mathbb{N}}\widetilde{V}_{i}\quad\text{and}\quad\widetilde{W}:=\prod\limits_{i\in\mathbb{N}}\widetilde{W}_{i}\>.$
Let $\pi_{i}^{\widetilde{V}}:\widetilde{V}\rightarrow\widetilde{V}_{i}$ be the
projection onto the $i$-th factor of $\widetilde{V}$, and
$\pi_{j}^{\widetilde{W}}:\widetilde{W}\rightarrow\widetilde{W}_{j}$ the
projection on the $j$-th factor of $\widetilde{W}$.
Now we inductively construct $\widetilde{\lambda}_{i}:\widetilde{V}\rightarrow
V_{i}$ and $\widetilde{\mu}_{i}:\widetilde{W}\rightarrow W_{i}$. First, put
$\widetilde{\lambda}_{0}:=\pi_{0}^{\widetilde{V}}$ and
$\widetilde{\mu}_{0}:=\pi_{0}^{\widetilde{W}}$. Next, assume that we have
constructed $\widetilde{\lambda}_{0},\ldots,\widetilde{\lambda}_{j}$ and
$\widetilde{\mu}_{0},\ldots,\widetilde{\mu}_{j}$ such that for $i\leq k\leq j$
(A.2)
$\widetilde{\lambda}_{i}=\lambda_{ik}\circ\widetilde{\lambda}_{k}\quad\text{and}\quad\widetilde{\mu}_{i}=\mu_{ik}\circ\widetilde{\mu}_{k}\>.$
Then we define $\widetilde{\lambda}_{j+1}:\widetilde{V}\rightarrow V_{j+1}$
and $\widetilde{\mu}_{j+1}:\widetilde{W}\rightarrow W_{j+1}$ by
$\widetilde{\lambda}_{j+1}(v)=\pi_{j+1}^{\widetilde{V}}(v)+f_{j+1}\widetilde{\lambda}_{j}(v)\quad\text{and}\quad\widetilde{\mu}_{j+1}(w)=\pi_{j+1}^{\widetilde{W}}(w)+g_{j+1}\widetilde{\lambda}_{j}(w),$
where $v\in\widetilde{V}$, and $w\in\widetilde{W}$. By assumption on $f_{j+1}$
and $g_{j+1}$ one concludes that
$\widetilde{\lambda}_{j}=\lambda_{j+1j}\circ\widetilde{\lambda}_{j+1}\quad\text{and}\quad\widetilde{\mu}_{j}=\mu_{j+1j}\circ\widetilde{\mu}_{j+1},$
which entails that Eq. (A.2) holds true for $i\leq k\leq j+1$. We now claim
that $\widetilde{V}$ together with the family
$\big{(}\widetilde{\lambda}_{i}\big{)}$ is a projective limit of
$\big{(}V_{i},\lambda_{ij}\big{)}$, and likewise for $\widetilde{W}$. We only
need to prove the claim for $\widetilde{V}$. Let $Z$ be a locally convex
topological vector space, and $\nu_{i}:Z\rightarrow V_{i}$ a family of
continuous linear maps such that $\nu_{i}=\lambda_{ij}\circ\nu_{j}$ for $i\leq
j$. Put for every $z\in Z$
$\widetilde{\nu}_{0}(z):=\nu_{0}(z)\quad\text{and}\quad\widetilde{\nu}_{i}(z):=\nu_{i}(z)-f_{i}\big{(}\nu_{i-1}(z))\big{)}\text{
for $i\in\mathbb{N}^{*}$}.$
Then $\widetilde{\nu}_{i}(z)\in\widetilde{V_{i}}$ for all $i\in\mathbb{N}$,
and
$\nu:Z\longrightarrow\widetilde{V},\>z\longmapsto\big{(}\widetilde{\nu}_{i}(z)\big{)}_{i\in\mathbb{N}}$
is well-defined, linear, and continuous. Moreover, it follows by induction on
$i\in\mathbb{N}$ that
$\widetilde{\lambda}_{i}\nu=\nu_{i}.$
For $i=0$ this is clear, so assume that we have shown this for some
$i\in\mathbb{N}$. Then, for $z\in Z$,
$\widetilde{\lambda}_{i+1}\nu(z)=\nu_{i+1}(z)-f_{i+1}\big{(}\nu_{i}(z)\big{)}+f_{i+1}\widetilde{\lambda}_{i}\big{(}\nu(z)\big{)}=\nu_{i+1}(z),$
which finishes the inductive argument. Assume that
$\nu^{\prime}:Z\rightarrow\widetilde{V}$ is another continuous linear map such
that $\widetilde{\lambda}_{i}\nu^{\prime}=\nu_{i}$ for all $i\in\mathbb{N}$.
First, this entails that
$\pi_{0}^{\widetilde{V}}\nu^{\prime}=\widetilde{\lambda}_{0}\nu^{\prime}=\nu_{0}=\widetilde{\nu}_{0}.$
Assume that $\pi_{i}^{\widetilde{V}}\nu^{\prime}=\widetilde{\nu}_{i}$ for some
$i\in\mathbb{N}$. Then
$\pi_{i+1}^{\widetilde{V}}\nu^{\prime}=\widetilde{\lambda}_{i+1}\nu^{\prime}-f_{i+1}\widetilde{\lambda}_{i}\nu^{\prime}=\nu_{i+1}-f_{i+1}\nu_{i}=\widetilde{\nu}_{i+1}.$
Hence, one obtains, for all $i\in\mathbb{N}$,
$\pi_{i}^{\widetilde{V}}\nu^{\prime}=\widetilde{\nu}_{i}=\pi_{i}^{\widetilde{V}}\nu,$
which proves $\nu^{\prime}=\nu$. So $\widetilde{V}$ is a projective limit of
$\big{(}V_{i},\lambda_{ij}\big{)}$, and $\widetilde{W}$ a projective limit of
$\big{(}W_{i},\mu_{ij}\big{)}$. Moreover, $\widetilde{V}$ is canonically
isomorphic to $V$, and $\widetilde{W}$ to $W$.
The theorem is now proved, if we can show that
$\widetilde{V}\otimes_{\pi}\widetilde{W}$ together with the family of
canonical maps
$\widetilde{\lambda}_{i}\widehat{\otimes}_{\pi}\widetilde{\mu}_{i}:\widetilde{V}\otimes_{\pi}\widetilde{W}\rightarrow
V_{i}\otimes W_{i}$ is a projective limit of the projective system
$\big{(}V_{i}\otimes W_{i},\lambda_{ij}\otimes\mu_{ij}\big{)}$. But this is
clear, since by the preceeding theorem,
$\widetilde{V}\otimes_{\pi}\widetilde{W}\cong\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}}\widetilde{V}_{i}\otimes\widetilde{W}_{j}\cong\lim_{\longleftarrow\atop
k\in\mathbb{N}}\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}\atop i,j\leq
k}\widetilde{V}_{i}\otimes\widetilde{W}_{j}$
and, for $k\in\mathbb{N}$,
$\prod_{(i,j)\in\mathbb{N}\times\mathbb{N}\atop i,j\leq
k}\widetilde{V}_{i}\otimes\widetilde{W}_{j}\cong\prod_{i\leq
k}\widetilde{V}_{i}\otimes\prod_{j\leq k}\widetilde{W}_{j}\cong V_{k}\otimes
W_{k}.$
∎
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|
arxiv-papers
| 2013-08-05T15:18:45 |
2024-09-04T02:49:48.996552
|
{
"license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/",
"authors": "Batu G\\\"uneysu and Markus J. Pflaum",
"submitter": "Markus J. Pflaum",
"url": "https://arxiv.org/abs/1308.1005"
}
|
1308.1048
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-141 LHCb-PAPER-2013-033 October 25, 2013
The LHCb collaboration†††Authors are listed on the following pages.
The $C\\!P$-violating asymmetry $a_{\rm sl}^{\rm s}$ is studied using
semileptonic decays of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons produced in $pp$
collisions at a centre-of-mass energy of 7 TeV at the LHC, exploiting a data
sample corresponding to an integrated luminosity of 1.0 fb-1. The
reconstructed final states are $D^{\pm}_{s}\mu^{\mp}$, with the $D^{\pm}_{s}$
particle decaying in the $\phi\pi^{\pm}$ mode. The $D^{\pm}_{s}\mu^{\mp}$
yields are summed over $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
and $B^{0}_{s}$ initial states, and integrated with respect to decay time.
Data-driven methods are used to measure efficiency ratios. We obtain $a_{\rm
sl}^{\rm s}$ = $(-0.06\pm 0.50\pm 0.36)$%, where the first uncertainty is
statistical and the second systematic.
# Measurement of the flavour-specific $C\\!P$-violating asymmetry $a_{\rm
sl}^{s}$ in $B^{0}_{s}$ decays
Published in Phys. Lett. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
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Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
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H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
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Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
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David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
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Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
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Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S. Farry51, D.
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Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
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A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
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J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
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Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
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Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
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J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-
March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, J. Maratas5, U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11,
G. Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D.
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Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, A. Mordà6, M.J.
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B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1,
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Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M.
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Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L.
Pescatore44, E. Pesen61, K. Petridis52, A. Petrolini19,i, A. Phan58, E.
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Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A.
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Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
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V. Salustino Guimaraes2, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
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Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
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N. Skidmore45, T. Skwarnicki58, N.A. Smith51, E. Smith54,48, J. Smith46, M.
Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro18, D. Souza45, B. Souza De
Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B. Storaci39, M.
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M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D. Tonelli37, S. Topp-
Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M.
Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37,
A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G.
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Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S.
Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
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Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, S.A. Wotton46, S.
Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, R. Young49,
X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The $C\\!P$ asymmetry in $B^{0}_{s}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing is a sensitive probe of
new physics. In the neutral $B$ system ($B^{0}$ or $B^{0}_{s}$), the mixing of
the flavour eigenstates (the neutral $B$ and its antiparticle $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$) is governed by a $2\times 2$ complex
effective Hamiltonian matrix [1]
$\begin{pmatrix}\noindent
M_{11}-\frac{i}{2}\Gamma_{11}&M_{12}-\frac{i}{2}\Gamma_{12}\\\
M_{12}^{*}-\frac{i}{2}\Gamma_{12}^{*}&M_{22}-\frac{i}{2}\Gamma_{22}\end{pmatrix},$
(1)
which operates on the neutral $B$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ flavour eigenstates. The mass
eigenstates have eigenvalues $M_{\rm H}$ and $M_{\rm L}$. Other measurable
quantities are the mass difference $\Delta M$, the width difference
$\Delta\Gamma$, and the semileptonic (or flavour-specific) asymmetry $a_{\rm
sl}$. These quantities are related to the off-diagonal matrix elements and the
phase $\phi_{12}\equiv\arg\left(-M_{12}/\Gamma_{12}\right)$ by
$\displaystyle\Delta M$ $\displaystyle\equiv$ $\displaystyle M_{\rm H}-M_{\rm
L}=2|M_{12}|\left(1-\frac{1}{8}\frac{|\Gamma_{12}|^{2}}{|M_{12}|^{2}}\sin^{2}\phi_{12}+....\right),$
$\displaystyle\Delta\Gamma$ $\displaystyle\equiv$ $\displaystyle\Gamma_{\rm
L}-\Gamma_{\rm
H}=2|\Gamma_{12}|\cos\phi_{12}\left(1+\frac{1}{8}\frac{|\Gamma_{12}|^{2}}{|M_{12}|^{2}}\sin^{2}\phi_{12}+....\right),$
$\displaystyle a_{\rm sl}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}(t)\rightarrow
f\right)-\Gamma\left(B(t)\rightarrow\bar{f}\right)}{\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}(t)\rightarrow
f\right)+\Gamma\left(B(t)\rightarrow\bar{f}\right)}\simeq\frac{\Delta\Gamma}{\Delta
M}\tan{\phi_{12}}\,,$ (2)
where $B(t)$ is the state into which a produced $B$ meson has evolved after a
proper time $t$ measured in the meson rest frame, and $f$ indicates a flavour-
specific final state. The term flavour-specific means that the final state is
only reachable by the decay of the $B$ meson, and consequently reachable by a
meson originally produced as a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$
only through mixing. We use the semileptonic flavour specific final state and
thus refer to this quantity as $a_{\rm sl}$. Note that $a_{\rm sl}$ is decay
time independent. Throughout the paper, mention of a specific channel implies
the inclusion of the charge-conjugate mode, except in reference to
asymmetries.
The phase $\phi_{12}$ is very small in the Standard Model (SM), in particular,
for $B^{0}_{s}$ mixing, $\phi_{12}^{s}$ is approximately $0.2^{\circ}$
[2].111This phase should not be confused with the $C\\!P$ violation phase
measured in $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$ and $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays, sometimes called $\phi_{s}$ [4]. New physics can
affect this phase [3, 4] and therefore $a_{\rm sl}^{s}$. The D0 collaboration
has reported evidence for a decay asymmetry $A_{\rm sl}^{b}=(-0.787\pm
0.172\pm 0.093)\%$ in a mixture of $B^{0}$ and $B^{0}_{s}$ semileptonic
decays, where the first uncertainty is statistical and the second systematic
[5, *Abazov:2010hv, *Abazov:2010hj]. This asymmetry is much larger in
magnitude than the SM predictions for semileptonic asymmetries in $B^{0}_{s}$
and $B^{0}$ decays, namely $a^{s}_{\rm sl}=(1.9\pm 0.3)\times 10^{-5}$ and
$a^{d}_{\rm sl}=(-4.1\pm 0.6)\times 10^{-4}$ [4]. More recently D0 published
measurements of $a^{d}_{\rm sl}=(0.68\pm 0.45\pm 0.14)\%$ [8], and $a^{s}_{\rm
sl}=(-1.12\pm 0.74\pm 0.17)\%$ [9], consistent both with the anomalous
asymmetry $A_{\rm sl}^{b}$ and the SM predictions for $a^{s}_{\rm sl}$ and
$a^{d}_{\rm sl}$. If the measured value of $A_{\rm sl}^{b}$ is confirmed, this
would demonstrate the presence of physics beyond the SM in the quark sector.
The $e^{+}e^{-}$ $B$-factory average asymmetry in $B^{0}$ decays is
$a^{d}_{\rm sl}=(0.02\pm 0.31)$% [10], in good agreement with the SM. A
measurement of $a_{\rm sl}^{s}$ with comparable accuracy is important to
establish whether physics beyond the SM influences flavour oscillations in the
$B^{0}_{s}$ system.
When measuring a semileptonic asymmetry at a $pp$ collider, such as the LHC,
particle-antiparticle production asymmetries, denoted as $a_{\rm P}$, as well
as detector related asymmetries, may bias the measured value of $a_{\rm
sl}^{s}$. We define $a_{\rm P}$ in terms of the numbers of produced
$b$-hadrons, $N(B)$, and anti $b$-hadrons, $N(\kern
1.79993pt\overline{\kern-1.79993ptB}{})$, as
$a_{\rm P}\equiv\frac{N(B)-N(\overline{B})}{N(B)+N(\overline{B})}~{},$ (3)
where $a_{\rm P}$ may in general be different for different species of
$b$-hadron.
In this paper we report the measurement of the asymmetry between
$D_{s}^{+}X\mu^{-}\overline{\nu}$ and $D_{s}^{-}X\mu^{+}\nu$ decays, with $X$
representing possible associated hadrons. We use the
$D_{s}^{\pm}\rightarrow\phi\pi^{\pm}$ decay. For a time-integrated measurement
we have, to first order in $a_{\rm sl}^{\rm s}$
$A_{\rm
meas}\equiv\frac{\Gamma[{D_{s}^{-}\mu^{+}}]-\Gamma[{D_{s}^{+}\mu^{-}}]}{\Gamma[{D_{s}^{-}\mu^{+}}]+\Gamma[{D_{s}^{+}\mu^{-}}]}=\frac{a_{\rm
sl}^{s}}{2}+\left[a_{\rm P}-\frac{a_{\rm
sl}^{s}}{2}\right]\frac{\int_{t=0}^{\infty}{e^{-\Gamma_{s}t}\cos(\Delta
M_{s}\,t)\epsilon(t)dt}}{\int_{t=0}^{\infty}{e^{-\Gamma_{s}t}\cosh(\frac{\Delta\Gamma_{s}\,t}{2})\epsilon(t)dt}},$
(4)
where $\Delta M_{s}$ and $\Gamma_{s}$ are the mass difference and average
decay width of the $B^{0}_{s}-\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson system, respectively,
and $\epsilon(t)$ is the decay time acceptance function for $B^{0}_{s}$
mesons. Due to the large value of $\Delta M_{s}$, 17.768 $\pm$0.024 ps-1 [11],
the oscillations are rapid and the integral ratio in Eq. (4) is approximately
0.2%. Since the production asymmetry within the detector acceptance is
expected to be at most a few percent [12, 13, 14], this reduces the effect of
$a_{\rm p}$ to the level of a few $10^{-4}$ for $B^{0}_{s}$ decays. This is
well beneath our target uncertainty of the order of $10^{-3}$, and thus can be
neglected, therefore yielding $A_{\rm meas}$=0.5 $a_{\rm sl}^{\rm s}$.
The measurement could be affected by a detection charge-asymmetry, which may
be induced by the event selection, tracking, and muon selection criteria. The
measured asymmetry can be written as
$A_{\rm meas}=A_{\mu}^{\rm c}-A_{\rm track}-A_{\rm bkg},$ (5)
where $A_{\mu}^{\rm c}$ is given by
$A_{\mu}^{\rm
c}=\frac{N(D^{-}_{s}\mu^{+})-N(D^{+}_{s}\mu^{-})\times\frac{\epsilon(\mu^{+})}{\epsilon(\mu^{-})}}{N(D^{-}_{s}\mu^{+})+N(D^{+}_{s}\mu^{-})\times\frac{\epsilon(\mu^{+})}{\epsilon(\mu^{-})}}.$
(6)
$N(D^{-}_{s}\mu^{+})$ and $N(D^{+}_{s}\mu^{-})$ are the measured yields of
$D_{s}\mu$ pairs, $\epsilon(\mu^{+})$ and $\epsilon(\mu^{-})$ are efficiency
corrections accounting for trigger and muon identification effects, $A_{\rm
track}$ is the track-reconstruction asymmetry of charged particles, and
$A_{\rm bkg}$ accounts for asymmetries induced by backgrounds.
## 2 The LHCb detector and trigger
We use a data sample corresponding to an integrated luminosity of 1.0
$\mbox{\,fb}^{-1}$ collected in 7 TeV $pp$ collisions with the LHCb detector
[15]. This detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift-tubes placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ that varies from
0.4% at 5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$.222We work in units with $c$=1. Charged
hadrons are identified using two ring-imaging Cherenkov (RICH) detectors [16].
Photon, electron and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and pre-shower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers
[17]. The LHCb coordinate system is a right handed Cartesian system with the
positive $z$-axis aligned with the beam line and pointing away from the
interaction point and the positive $x$-axis following the ground of the
experimental area, and pointing towards the outside of the LHC ring.
The trigger system [18] consists of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage which
applies a full event reconstruction. For the $D_{s}\mu$ signal samples, the
hardware trigger (L0) requires the detection of a muon of either charge with
transverse momentum $\mbox{$p_{\rm T}$}>1.64$ GeV. In the subsequent software
trigger, a first selection algorithm confirms the L0 candidate muon as a fully
reconstructed track, while the second level algorithm includes two possible
selections. One is based on the topology of the candidate muon and one or two
additional tracks, requiring them to be detached from the primary interaction
vertex. The second category is specifically designed to detect inclusive
$\phi\rightarrow K^{+}K^{-}$ decays. We consider all candidates that satisfy
either selection algorithm. We also study two mutually exclusive samples, one
composed of candidates that satisfy the second trigger category, and the other
satisfying the topological selection of events including a muon, but not the
inclusive $\phi$ algorithm. Approximately 40% of the data were taken with the
magnetic field up, oriented along the positive $y$-axis in the LHCb coordinate
system, and the rest with the opposite down polarity. We exploit the fact that
certain detection asymmetries cancel if data from different magnet polarities
are combined.
## 3 Selection requirements
Additional selection criteria exploiting the kinematic properties of
semileptonic $b$-hadron decays [19, 20, 21] are used. In order to minimize
backgrounds associated with misidentified muons, additional selection criteria
on muons are that the momentum, $p$, be between 6 and 100 GeV, that the
pseudorapidity, $\eta$, be between 2 and 5, and that they are inconsistent
with being produced at any primary vertex. Tracks are considered as kaon
candidates if they are identified by the RICH system, have $\mbox{$p_{\rm
T}$}>0.3$ GeV and $p>2$ GeV. The impact parameter (IP), defined as the minimum
distance of approach of the track with respect to the primary vertex, is used
to select tracks coming from charm decays. We require that the $\chi^{2}$,
formed by using the hypothesis that each track’s IP is equal to 0, which
measures whether a track is consistent with coming from the PV, is greater
than 9. To be reconstructed as a $\phi$ meson candidate, a $K^{+}K^{-}$ pair
must have invariant mass within $\pm$20 MeV of the $\phi$ meson mass.
Candidate $\phi$ mesons are combined with charged pions to make $D_{s}$ meson
candidates. The sum of the $p_{\rm T}$ of $K^{+}$, $K^{-}$ and $\pi^{\pm}$
candidates must be larger than 2.1 GeV. The vertex fit $\chi^{2}$ divided by
the number of degrees of freedom (ndf) must be less than 6, and the flight
distance $\chi^{2}$, formed by using the hypothesis that the $D^{+}_{s}$
flight distance is equal to 0, must be greater than 100. The $B^{0}_{s}$
candidate, formed from the $D_{s}$ and the muon, must have vertex fit
$\chi^{2}$/ndf $<6$, be downstream of the primary vertex, have $2<\eta<5$ and
have invariant mass between 3.1 and 5.1 GeV. Finally, we include some angular
selection criteria that require that the $B_{s}$ candidate have a momentum
aligned with the measured fight direction. The cosine of the angle between the
$D_{s}\mu$ momentum direction and the vector from the primary vertex to the
$D_{s}\mu$ origin must be larger than 0.999. The cosine of the angle between
the $D_{s}$ momentum and the vector from the primary vertex to the $D_{s}$
decay vertex must be larger than 0.99.
## 4 Analysis method
Signal yields are determined by fitting the $K^{+}K^{-}\pi^{+}$ invariant mass
distributions shown in Fig. 1. We fit both the signal $D^{+}_{s}$ and $D^{+}$
peaks with double Gaussian functions with common means. The $D^{+}$ channel is
used only as a component of the fit to the mass spectrum. The average mass
resolution is about 7.1 $\mathrm{\,Me\kern-1.00006ptV}$. The background is
modelled with a second-order Chebychev polynomial. The signal yields from the
fits are listed in Table 1.
Figure 1: Invariant mass distributions for: (a) $K^{+}K^{-}\pi^{+}$ and (b) $K^{+}K^{-}\pi^{-}$ candidates for magnet up, (c) $K^{+}K^{-}\pi^{+}$ and (d) $K^{+}K^{-}\pi^{-}$ candidates for magnet down with $K^{+}K^{-}$ invariant mass within $\pm$20 $\mathrm{\,Me\kern-0.90005ptV}$ of the $\phi$ meson mass. The $D^{+}_{s}$ [yellow (grey) shaded area] and $D^{+}$ [red (dark) shaded area] signal shapes are described in the text. The $\chi^{2}$/ndf for these fits are 1.28, 1.25, 1.53, and 1.27 respectively, the corresponding p-values are 7%, 8%, 4%, 7%. Table 1: Yields for $D^{+}_{s}\mu^{-}$ and $D^{-}_{s}\mu^{+}$ events separately for magnet up and down data. These yields contain very small contributions from prompt $D_{s}$ and b-hadron backgrounds. | magnet up | magnet down
---|---|---
$D^{-}_{s}\mu^{+}$ | $38\,742\pm 218$ | $53\,768\pm 264$
$D^{+}_{s}\mu^{-}$ | $38\,055\pm 223$ | $54\,252\pm 259$
The detection asymmetry is largely induced by the dipole magnet, which bends
particles of different charge in different detector halves. The magnet
polarity is reversed periodically, thus allowing the measurement and
understanding of the size of this effect. We analyze data taken with different
magnet polarities separately, deriving charge asymmetry corrections for the
two data sets independently. Finally, we average the two values in order to
cancel charge any residual effects. We use two calibration samples containing
muons to measure the relative trigger efficiencies of
$D_{s}^{+}\mu^{-}/D_{s}^{-}\mu^{+}$ events, and the relative $\mu^{-}/\mu^{+}$
identification efficiencies. The first sample contains
$b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})X$ decays triggered independently of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson, and where the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is selected by requiring two
particles of opposite charge have an invariant mass consistent with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass. This sample is called the
kinematically-selected (KS) sample. The second sample is collected by
triggering on one muon from a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
decay that is detached from the primary vertex. It is called muon selected
(MS) as it relies on the presence of a well identified muon.
In order to measure the relative $\pi^{+}$ and $\pi^{-}$ detection
efficiencies, we use the ratio of partially reconstructed and fully
reconstructed $D^{*+}\rightarrow\pi^{+}D^{0}$, $D^{0}\rightarrow
K^{-}\pi^{+}\pi^{+}(\pi^{-})$ decays. The former sample is gathered without
explicitly reconstructing the $\pi^{-}$ particle, and then the efficiency of
finding this track in the event is measured. The same procedure is applied to
the charge conjugate mode, so the relative $\pi^{+}$ to $\pi^{-}$ efficiency
is measured. A detailed description is given in Ref. [22].
Finally, a sample of $D^{+}(\rightarrow K^{-}\pi^{+}\pi^{+})\mu^{-}$
candidates is obtained using similar triggers to the $D_{s}\mu$ sample. This
sample is used to assess charge asymmetries induced by the software trigger.
The efficiency ratio $\epsilon_{\mu^{+}}/\epsilon_{\mu^{-}}$ in Eq. (6)
accounts for losses due to the muon identification efficiency algorithm and
the trigger requirements. We measure $\epsilon_{\mu^{+}}/\epsilon_{\mu^{-}}$
using the KS and MS calibration samples. There are about 0.6 million KS
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates selected in total,
and about 1.2 million MS ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidates. As the calibration muon spectra are slightly softer than that of
the signal, we subdivide the signal and calibration samples into subsamples
defined by the kinematic properties of the candidate muon. We define five muon
momentum bins: $6-20~{}\mathrm{\,Ge\kern-1.00006ptV}$,
$20-30~{}\mathrm{\,Ge\kern-1.00006ptV}$,
$30-40~{}\mathrm{\,Ge\kern-1.00006ptV}$,
$40-50~{}\mathrm{\,Ge\kern-1.00006ptV}$, and
$50-100~{}\mathrm{\,Ge\kern-1.00006ptV}$. We further subdivide the signal and
calibration samples with two binning schemes. In the first, each $\mu$
momentum bin is split into 10 rectangular regions in $qp_{x}$ and $p_{y}$,
where $q$ represents the muon charge and $p_{x}$ and $p_{y}$ are the Cartesian
components of the muon momentum in the directions perpendicular to the beam
axis. The second grid uses 8 regions of muon $p_{\rm T}$ and azimuthal angle
$\phi$ to reduce the sensitivity to differences in $\phi$ acceptance between
signal and calibration samples. In this case the first and third bins in
$\phi$ are flipped for negative charges, to symmetrize the acceptance in a
consistent manner with the $qp_{x}$ and $p_{y}$ binning. Signal and
calibration yields are determined separately in each of the intervals both for
magnet up and down data. Figure 2 shows the $\mu^{+}\mu^{-}$ invariant mass
distribution for the KS ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events
in magnet up data.
Figure 2: Invariant $\mu^{+}\mu^{-}$ mass distributions of the kinematically-
selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates in magnet
up data, where the red (open) circles represent entries where the muon
candidate, kinematically selected, is rejected and the black (filled) circles
those where it is accepted by the muon identification algorithm. The dashed
lines represent the combinatorial background.
The relative efficiencies for triggering and identifying muons in five
different momentum bins are shown in Fig. 3 for magnet up and magnet down data
using the KS calibration sample. They are consistent with being independent of
momentum. The small difference of approximately 1% between the two samples can
be attributed to the alignment of the muon stations, which affects
predominantly the hardware muon trigger.
Figure 3: Relative muon efficiency as a function of muon momentum determined
using the kinematically-selected ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ sample.
The $D_{s}^{+}\mu^{-}$ final state benefits from several cancellations of
potential instrumental asymmetries that can arise due to the different
interaction cross-sections in the detector material or to differences between
tracking reconstructions of negative and positive particles. The $\mu$ and
$\pi$ charged tracks have very similar reconstruction efficiencies. Using the
partially-reconstructed $D^{*+}$ calibration sample, we found that the
$\pi^{+}$ versus $\pi^{-}$ relative tracking efficiencies are independent of
momentum and transverse momentum [22]. This, along with the fact that
$\pi^{+}$ and $\pi^{-}$ interaction cross-sections on isoscalar targets are
equal, and that the detector is almost isoscalar, implies that the difference
between $\pi^{+}$ and $\pi^{-}$ tracking efficiencies depend only upon the
magnetic field orientation and the detector acceptance. Thus the charge
asymmetry ratios measured for pions are applicable to muons as well. In the
$\phi\pi^{+}\mu^{-}$ final states, the pion and muon have opposite signs, and
thus the charge asymmetry in the track reconstruction efficiency induced by
imperfect $\pi\mu$ cancellation, $A_{\rm track}^{\pi\mu}$, is small. Using the
efficiency ratios $\epsilon_{\pi^{+}}/\epsilon_{\pi^{-}}$ measured with the
$D^{*+}$ calibration sample, we obtain $A_{\rm track}^{\pi\mu}=(+0.01\pm
0.13)$%. A small residual sensitivity to the charge asymmetry in $K$ track
reconstruction is present due to a slight momentum mismatch between the two
kaons from $\phi$ decays arising from the interference with the S-wave
component. It is determined to be $A_{\rm track}^{KK}=(+0.012\pm 0.004)$%. The
efficiency ratios used in determining $A_{\rm track}^{KK}$ are based on
$\epsilon_{\pi^{+}}/\epsilon_{\pi^{-}}$ with a correction derived from the
comparison between the Cabibbo-favoured decays $D^{+}\rightarrow
K^{-}\pi^{+}\pi^{-}$ and $D^{+}_{s}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}$, accounting for additional charge asymmetry induced by $K$
interactions in the detector. Therefore, the total tracking asymmetry is
$A_{\rm track}=(+0.02\pm 0.13)$%.
## 5 Backgrounds
Backgrounds include prompt charm production, fake muons associated with real
$D_{s}^{+}$ particles produced in $b$-hadron decays, and $B\rightarrow DD_{s}$
decays where the $D$ hadron decays semileptonically. Here $B$ denotes any
meson or baryon containing a $b$ (or $\overline{b}$) quark, and similarly, $D$
denotes any hadron containing a $c$ (or $\overline{c}$) quark. The prompt
background is highly suppressed by the requirement of a well identified muon
forming a vertex with the $D^{+}_{s}$ candidate. The prompt yield is separated
from false $D_{s}$ backgrounds using a binned two-dimensional fit to the mass
and ln(IP/mm) of the $\phi\pi^{+}$ candidates. The method is described in
detail in Ref. [21]. Figure 4 shows the fit results for the magnet-down
$D_{s}^{+}\mu^{-}$ candidate sample. From the asymmetry in the prompt yield
normalized to the overall signal yield in the five momentum bins, we obtain an
asymmetry due to prompt background equal to (+0.14$\pm$0.07)% for magnet up
data, ($-0.05\pm 0.05)$% for magnet down data, with an average value of
(+0.04$\pm$0.04)%.
Figure 4: (a) Spectrum of the logarithm of the IP calculated with respect to
the primary vertex for $D^{+}_{s}$ candidates in combination with muons; the
insert shows a magnified view of the region where the prompt $D^{+}_{s}$
contribution peaks. The blue dashed line is the component coming from $B$
hadron decays, the black dashed line the false $D^{+}_{s}$ background, the red
line the prompt background, (b) the invariant mass distributions for
$D^{+}_{s}\rightarrow\phi\pi$ candidates. These distributions are for the
magnet down sample. (For interpretation of the reference to colour in this
figure legend, the reader is referred to the web version of this Letter.)
Samples of $D^{+}_{s}\pi^{-}X$ and $D^{+}_{s}K^{-}X$ events, where $X$
represents undetected particles from the same decay, are used to infer the
numbers of $D^{+}_{s}$-hadron combinations from $B$ decays that could be
mistaken for $D^{+}_{s}\mu^{-}$ events if the hadron is misidentified as a
muon. Kaons and pions are identified using the RICH. These numbers, combined
with knowledge of the probability that kaons or pions are mistaken for muons,
provide a measurement of the fake hadron background. These misidentification
probabilities are also calculated in the five momentum bins using
$D^{*+}\rightarrow\pi^{+}D^{0}$ decays, with $D^{0}$ decaying into the
$K^{-}\pi^{+}$ final state. The net effect on the asymmetry is below $10^{-4}$
and thus the $D^{+}_{s}$-hadron background can be ignored.
We also consider the background induced by $D_{s}^{+}\mu^{-}$ events deriving
from $b\rightarrow c\bar{c}s$ decays where the $D^{+}_{s}$ hadron originates
from the virtual $W^{+}$ boson and the muon originates from the charmed-hadron
semileptonic decay. These backgrounds are suppressed since the $D$ hadron
travels away from the $B$ vertex prior to its semileptonic decay. As these
decays are of opposite sign to the signal, they cause a background asymmetry
that is proportional to the production asymmetry of the background sources.
The $B^{0}$ production asymmetry has been measured in LHCb to be $(-0.1\pm
1.0)$% [13], and the $B^{+}$ production asymmetry to be $(+0.3\pm 0.9)$% by
comparing $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$
and $B^{-}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}$ decays
[23]. A small subset of this background is from $\mathchar
28931\relax_{b}^{0}$ decays, whose production asymmetry is not well known,
$a_{\rm P}=(-1.0\pm 4.0)$%, but is consistent with zero [24]. The $B^{0}$
final states include $D^{0}$ and $D^{+}$ hadrons, in proportions determined
according to the $D^{*+}/D^{+}$ ratio in the measured exclusive final states.
In addition, we consider backgrounds coming from $B^{0},B^{+}\rightarrow
D^{-}_{s}K\mu^{+}$ decays, that provide a background asymmetry with opposite
sign. We estimate this background asymmetry to be (+0.01$\pm$0.04)%. The
systematic uncertainty includes the limited knowledge of the inclusive
branching fraction of the $b$-hadrons, uncertainties in the $b$-hadron
production ratios, and in the charm semileptonic branching fractions, but is
dominated by the uncertainty in the production asymmetry. By combining these
estimates, we obtain $A_{\rm bkg}=(+0.05\pm 0.05)$%.
## 6 Results
We perform weighted averages of the corrected asymmetries $A_{\mu}^{c}$
observed in each $p_{\rm T}\phi$ and $p_{x}p_{y}$ subsample, using muon
identification corrections both in the KS and MS sample (see Fig. 5). In order
to cancel remaining detection asymmetry effects, the most appropriate way to
combine magnet up and magnet down data is with an arithmetic average [22]. We
then perform an arithmetic average of the four values of $A_{\mu}^{c}$
obtained with the two binning schemes chosen and with the two muon correction
methods, assuming the results to be fully statistically correlated, and obtain
$A_{\mu}^{c}=(+0.04\pm 0.25)$%. The results are shown in Table 2. Finally, we
correct for tracking efficiency asymmetries and background asymmetries, and
obtain
$A_{\rm meas}=(-0.03\pm 0.25\pm 0.18)\%,$
where the first uncertainty reflects statistical fluctuations in the signal
yield and the second reflects the systematic uncertainties. This gives
$a^{s}_{\rm sl}=(-0.06\pm 0.50\pm 0.36)\%.$
Figure 5: Asymmetries corrected for relative muon efficiencies,
$A_{\mu}^{\text{c}}$, examined in the five muon momentum intervals for (a)
magnet up data, (b) magnet down data and (c) average, using the KS muon
calibration method. Then (d) magnet up data, (e) magnet down data and (f)
average, using the MS muon calibration method in the two different binning
scheme. Table 2: Muon efficiency ratio corrected asymmetry $A_{\mu}^{c}$. The
errors account for the statistical uncertainties in the $B^{0}_{s}$ signal
yields.
$A_{\mu}^{c}~{}~{}[\%]$ | KS muon correction | MS muon correction | Average
---|---|---|---
Magnet | $p_{x}p_{y}$ | $p_{\rm T}\phi$ | $p_{x}p_{y}$ | $p_{\rm T}\phi$ |
Up | $+0.38\pm 0.38$ | $+0.30\pm 0.38$ | $+0.64\pm 0.37$ | $+0.63\pm 0.37$ | $+0.49\pm 0.38$
Down | $-0.17\pm 0.32$ | $-0.25\pm 0.32$ | $-0.60\pm 0.32$ | $-0.62\pm 0.32$ | $-0.41\pm 0.32$
Avg. | $+0.11\pm 0.25$ | $+0.02\pm 0.25$ | $+0.02\pm 0.24$ | $+0.01\pm 0.24$ | $+0.04\pm 0.25$
We consider several sources of systematic uncertainties on $A_{\rm meas}$ that
are summarized in Table 3. By examining the variations on the average
$A_{\mu}^{c}$ obtained with different procedures, we assign a 0.07%
uncertainty, reflecting three almost equal components: the fitting procedure,
the kinematic binning and a residual systematic uncertainty related to the
muon efficiency ratio calculation. We study the effect of the fitting
procedure by comparing results obtained with different models for signal and
background shapes. In addition, we consider the effects of the statistical
uncertainties of the efficiency ratios, assigning 0.08%, which is obtained by
propagating the uncertainties in the average $A_{\mu}^{\rm c}$. The
uncertainties affecting the background estimates are discussed in Sec. 5.
Possible changes in detector acceptance during magnet up and magnet down data
taking periods are estimated to contribute 0.01%. The software trigger
systematic uncertainty is mainly due to the topological trigger algorithm and
is estimated to be 0.05%. These uncertainties are considered uncorrelated and
added in quadrature to obtain the total systematic uncertainty.
Table 3: Sources of systematic uncertainty on $A_{\rm meas}$. Source | $\sigma(A_{\text{meas}})$[%]
---|---
Signal modelling and muon correction | 0.07
Statistical uncertainty on the efficiency ratios | 0.08
Background asymmetry | 0.05
Asymmetry in track reconstruction | 0.13
Field-up and field-down run conditions | 0.01
Software trigger bias (topological trigger) | 0.05
Total | 0.18
## 7 Conclusions
We measure the asymmetry $a^{s}_{\rm sl}$, which is twice the measured
asymmetry between $D_{s}^{-}\mu^{+}$ and $D_{s}^{+}\mu^{-}$ yields, to be
$a^{s}_{\rm sl}=(-0.06\pm 0.50\pm 0.36)\%.$
Figure 6 shows this measurement, the D0 measured asymmetries in dimuon decays
in 1.96 TeV $p\overline{p}$ collisions of $A_{\rm sl}^{b}=(-0.787\pm 0.172\pm
0.093)$% [5], $a^{d}_{\rm sl}=(0.68\pm 0.45\pm 0.14)\%$ [8], and $a^{s}_{\rm
sl}=(-1.12\pm 0.74\pm 0.17)\%$ [9], and the most recent average from
$B$-factories [10], namely $a_{\rm sl}^{d}=(0.02\pm 0.31)$%. Our result for
$a^{s}_{\rm sl}$ is currently the most precise measurement made and is
consistent with the SM.
Figure 6: Measurements of semileptonic decay asymmetries. The bands correspond
to the central values $\pm$1 standard deviation uncertainties, defined as the
sum in quadrature of the statistical and systematic errors. The solid dot
indicates the SM prediction.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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arxiv-papers
| 2013-08-05T17:31:28 |
2024-09-04T02:49:49.020207
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, C. Gotti, M. Grabalosa G\\'andara, R. Graciani Diaz,\n L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, M. Hess, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R.\n Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M.\n Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O.\n Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C.\n Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi,\n P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, T. Palczewski, M.\n Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson,\n G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P.\n Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, K. Petridis, A. Petrolini,\n A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S.\n Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A.\n Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A.\n Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, E.\n Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, J.\n Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, S. Swientek, V. Syropoulos, M. Szczekowski,\n P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti,\n A. Vallier, M. Van Dijk, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez\n Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B.\n Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong,\n A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, C. Wallace, R.\n Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley, J. Wishahi,\n W. Wislicki, M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Marina Artuso",
"url": "https://arxiv.org/abs/1308.1048"
}
|
1308.1055
|
# The Two-Loop Infrared Structure of Amplitudes with Mixed Gauge Groups
William B. Kilgore Physics Department, Brookhaven National Laboratory, Upton,
New York 11973, USA.
[[email protected]]
###### Abstract
The infrared structure of (multi-loop) scattering amplitudes is determined
entirely by the identities of the external particles participating in the
scattering. The two-loop infrared structure of pure QCD amplitudes has been
known for some time. By computing the two-loop amplitudes for
$\overline{f}\,f\longrightarrow X$ and $\overline{f}\,f\longrightarrow
V_{1}\,V_{2}$ scattering in an $SU(N)\times SU(M)\times U(1)$ gauge theory, I
determine the anomalous dimensions which govern the infrared structure for any
massless two-loop amplitude.
## I Introduction
The infrared structure of gauge theory amplitudes is governed by a set of
anomalous dimensions. The anomalous dimensions at a particular loop-level can
be computed directly or extracted from a small number of relatively simple
amplitude calculations. Once determined, these anomalous dimensions allow one
to predict, for any amplitude, no matter how complex, the complete infrared
structure to the given loop level Catani (1998); Sterman and Tejeda-Yeomans
(2003). In QCD, the anomalous dimensions are known completely, in both the
massless and massive cases for one and two loop amplitudes, and their
properties beyond the two-loop level are being actively studied Aybat et al.
(2006a, b); Mitov et al. (2009); Becher and Neubert (2009a); Gardi and Magnea
(2009a); Becher and Neubert (2009b, c); Gardi and Magnea (2009b); Dixon et al.
(2010); Mitov et al. (2010). Because of the many diagrams involved and the
complexity of the resulting amplitudes, foreknowledge of the infrared
structure is extremely valuable. This knowledge was an important guide for the
ground-breaking calculations of two-loop parton scattering amplitudes Bern et
al. (2001); Anastasiou et al. (2001a, b, c); Glover et al. (2001); Garland et
al. (2001); Anastasiou et al. (2002); Glover and Tejeda-Yeomans (2003).
Precision measurements in particle physics often involve the interaction of
more than one gauge group. In particular, at hadron colliders, nominally
electroweak processes always involve some interaction with QCD. Precision
calculations of such processes, therefore, require the computation of higher-
order corrections in mixed gauge groups Kilgore and Sturm (2012).
In the current letter, I consider a theory with the following structure: There
are three gauge interactions, obeying an $SU(N)\times SU(M)\times U(1)$
symmetry. Fermions occur in four different representations: $F_{l}$, which
carry $U(1)$ charge $Q_{l}$ and are singlets under $SU(N)$ and $SU(M)$;
$F_{n}$, which are in the fundamental representation of $SU(N)$, carry $U(1)$
charge $Q_{n}$ and are singlets under $SU(M)$; $F_{m}$, which are in the
fundamental representation of $SU(M)$, carry $U(1)$ charge $Q_{m}$ and are
singlets under $SU(N)$; and $F_{b}$, which are in the fundamental
representation of both $SU(N)$ and $SU(M)$ and carry $U(1)$ charge $Q_{b}$.
Note that this is precisely the structure of the (unbroken) Standard Model,
where the $SU(N)$ theory corresponds to QCD, the $SU(M)$ theory to the weak
$SU(2)_{L}$ and the $U(1)$ to the hypercharge interaction. Under this
identification, the $F_{l}$ multiplets correspond to the right-handed leptons,
the $F_{m}$ multiplets to the left-handed leptons, the $F_{n}$ multiplets to
the right-handed quarks and the $F_{b}$ multiplets to the left-handed quarks.
I will compute the two-loop amplitudes for
$\overline{f}_{x}\,f_{x}\longrightarrow X$ (where $X$ is a massive vector
boson, neutral under the $SU(N)\times SU(M)\times U(1)$ gauge symmetry) and
$\overline{f}_{x}\,f_{x}\longrightarrow V_{1}\,V_{2}$ for various combinations
of fermions and gauge bosons. These calculations will give me redundant
extractions of the anomalous dimensions for each particle type in the mixed
gauge structure. As a cross-check, I can compare my results for the anomalous
dimensions in a pure structure to the known results in the literature. All
calculations are performed in the conventional dimensional regularization
scheme Collins (1984).
## II The infrared structure of QCD amplitudes
The infrared structure of pure QCD interactions is well known. For a general
$n$-parton scattering process, I label the set of external partons by ${\bf
f}=\\{f_{i}\\}_{i=1\dots n}$. In the formulation of Refs. Sterman and Tejeda-
Yeomans (2003); Aybat et al. (2006a, b), a renormalized amplitude may be
factorized into three functions: the jet function ${\cal J}_{\bf f}$, which
describes the collinear dynamics of the external partons that participate in
the collision; the soft function ${\bf S_{f}}$, which describes soft exchanges
between the external partons; and the hard-scattering function $\left|H_{\bf
f}\right\rangle$, which describes the short-distance scattering process,
$\left|{\cal M}_{\bf
f}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),{\varepsilon}\right)\right\rangle={\cal
J_{\bf f}}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)\ {\bf
S_{f}}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),{\varepsilon}\right)\
\left|H_{\bf
f}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2})\right)\right\rangle\,.$
(1)
The notation indicates that $\left|H_{\bf f}\right\rangle$ is a vector and
${\bf S_{f}}$ is a matrix in color space Catani and Seymour (1996, 1997);
Catani (1998). As with any factorization, there is considerable freedom to
move terms about from one function to the others. It is convenient Aybat et
al. (2006a, b) to define the jet and soft functions, ${\cal J}_{\bf f}$ and
${\bf S_{f}}$, so that they contain all of the infrared poles but only contain
infrared poles, while all infrared finite terms, including those at higher-
order in ${\varepsilon}$, are absorbed into $\left|H_{\bf f}\right\rangle$.
### II.1 The jet function in QCD
The jet function ${\cal J}_{\bf f}$ is found to be the product of individual
jet functions ${\cal J}_{f_{i}}$ for each of the external partons,
${\cal J}_{\bf
f}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)=\prod_{i\in{\bf{f}}}\ {\cal
J}_{i}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)\,.$ (2)
Each individual jet function is naturally defined in terms of the anomalous
dimensions of the Sudakov form factor Sterman and Tejeda-Yeomans (2003),
$\begin{split}\ln{\cal
J}_{i}\left(\alpha_{s}(\mu^{2}),{\varepsilon}\right)&=-{\left(\frac{\alpha_{s}}{\pi}\right)}\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(1)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(1)}({\varepsilon})\right]\\\
&\quad+{\left(\frac{\alpha_{s}}{\pi}\right)}^{2}\left\\{\frac{{\beta_{0}}}{8}\frac{1}{{\varepsilon}^{2}}\left[\frac{3}{4\,{\varepsilon}}\gamma_{K\,i}^{(1)}+{\cal
G}_{i}^{(1)}({\varepsilon})\right]-\frac{1}{8}\left[\frac{\gamma_{K\,i}^{(2)}}{4\,{\varepsilon}^{2}}+\frac{{\cal
G}_{i}^{(2)}({\varepsilon})}{{\varepsilon}}\right]\right\\}+\dots\,,\end{split}$
(3)
where
$\begin{split}\gamma_{K\,i}^{(1)}&=2\,C_{i},\quad\gamma_{K\,i}^{(2)}=C_{i}\,K=C_{i}\left[C_{A}\left(\frac{67}{18}-\zeta_{2}\right)-\frac{10}{9}T_{f}\,N_{f}\right],\quad
C_{q}\equiv C_{F},\quad C_{g}\equiv C_{A}\,,\\\ {\cal
G}_{q}^{(1)}&=\frac{3}{2}C_{F}+\frac{{\varepsilon}}{2}C_{F}\left(8-\zeta_{2}\right),\qquad{\cal
G}_{g}^{(1)}=2\,{\beta_{0}}-\frac{{\varepsilon}}{2}C_{A}\,\zeta_{2}\,,\\\
{\cal
G}_{q}^{(2)}&=C_{F}^{2}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\,\zeta_{3}\right)+C_{F}\,C_{A}\left(\frac{2545}{432}+\frac{11}{12}\zeta_{2}-\frac{13}{4}\zeta_{3}\right)-C_{F}\,T_{f}\,N_{f}\left(\frac{209}{108}+\frac{1}{3}\zeta_{2}\right)\,,\\\
{\cal
G}_{g}^{(2)}&=4\,{\beta_{1}}+C_{A}^{2}\left(\frac{10}{27}-\frac{11}{12}\zeta_{2}-\frac{1}{4}\zeta_{3}\right)+C_{A}\,T_{f}\,N_{f}\left(\frac{13}{27}+\frac{1}{3}\zeta_{2}\right)+\frac{1}{2}C_{F}\,T_{f}\,N_{f}\,,\\\
{\beta_{0}}&=\frac{11}{12}C_{A}-\frac{1}{3}T_{f}\,N_{f}\,,\qquad{\beta_{1}}=\frac{17}{24}C_{A}^{2}-\frac{5}{12}C_{A}\,T_{f}\,N_{f}-\frac{1}{4}C_{F}\,T_{f}\,N_{f}\end{split}$
(4)
Although ${\cal G}_{i}$ and $\gamma_{K\,i}$ are defined through the Sudakov
form factor, they can be extracted from fixed-order calculations Gonsalves
(1983); Kramer and Lampe (1987); Matsuura and van Neerven (1988); Matsuura et
al. (1989); Harlander (2000); Moch et al. (2005a, b). $\gamma_{K\,i}$ is the
cusp anomalous dimension and represents a pure pole term. The ${\cal G}_{i}$
anomalous dimensions contain terms at higher order in ${\varepsilon}$, but I
only keep terms in the expansion that contribute poles to $\ln\left({\cal
J}_{i}\right)$. $\beta_{0}$ and $\beta_{1}$ are the first two coefficients of
the QCD $\beta$-function, $C_{F}=(N_{c}^{2}-1)/(2\,N_{c})$ denotes the Casimir
operator of the fundamental representation of SU($N_{c}$), while $C_{A}=N_{c}$
denotes the Casimir of the adjoint representation. $N_{f}$ is the number of
quark flavors and $T_{f}=1/2$ is the normalization of the QCD charge of the
fundamental representation. $\zeta_{n}=\sum_{k=1}^{\infty}1/k^{n}$ represents
the Riemann zeta-function of integer argument $n$.
### II.2 The soft function in QCD
The soft function is determined entirely by the soft anomalous dimension
matrix ${\bm{\Gamma}}_{S_{f}}$,
$\begin{split}{\bf
S_{f}}\left(p_{i},\tfrac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),{\varepsilon}\right)&=1+\frac{1}{2\,{\varepsilon}}{\left(\frac{\alpha_{s}}{\pi}\right)}{\bm{\Gamma}}_{S_{f}}^{(1)}+\frac{1}{8\,{\varepsilon}^{2}}{\left(\frac{\alpha_{s}}{\pi}\right)}^{2}{\bm{\Gamma}}_{S_{f}}^{(1)}\times{\bm{\Gamma}}_{S_{f}}^{(1)}\\\
&\qquad-\frac{{\beta_{0}}}{4\,{\varepsilon}^{2}}{\left(\frac{\alpha_{s}}{\pi}\right)}^{2}{\bm{\Gamma}}_{S_{f}}^{(1)}+\frac{1}{4\,{\varepsilon}}{\left(\frac{\alpha_{s}}{\pi}\right)}^{2}{\bm{\Gamma}}_{S_{f}}^{(2)}+\dots\,.\end{split}$
(5)
In the color-space notation of Refs. Catani and Seymour (1996, 1997); Catani
(1998), the soft anomalous dimension is given by Aybat et al. (2006a, b)
${\bm{\Gamma}}_{S_{f}}^{(1)}=\frac{1}{2}\,\sum_{i\in{\bf f}}\ \sum_{j\neq
i}{\bf T}_{i}\cdot{\bf
T}_{j}\,\ln\left(\frac{\mu^{2}}{-s_{ij}}\right),\qquad{\bm{\Gamma}}_{S_{f}}^{(2)}=\frac{K}{2}{\bm{\Gamma}}_{S_{f}}^{(1)}\,,$
(6)
where $K=C_{A}\left(67/18-\zeta_{2}\right)-10\,T_{f}\,N_{f}/9$ is the same
constant that relates the one- and two-loop cusp anomalous dimensions. The
${\bf T}_{i}$ are the color generators in the representation of parton $i$
(multiplied by $(-1)$ for incoming quarks and gluons and outgoing anti-
quarks).
## III The infrared structure of mixed gauge groups
When one includes additional gauge symmetries, the dominant effect on the
infrared structure is a replication of the QCD structure, with appropriate
changes accounting for the size of the gauge group and the Abelian character
of the $U(1)$. There are, however, new terms that correspond to intrinsically
mixed gauge interactions. It is these mixed terms I am interested in computing
in this letter. In reference Kilgore and Sturm (2012), some of the two-loop
anomalous dimensions for QCD $\times$QED amplitudes were determined, while the
forms of others, particularly those involving external gauge bosons, were
merely conjectured. The current calculation explicitly determines all of the
two-loop mixed anomalous dimensions.
In a theory with the $SU(N)\times SU(M)\times U(1)$ symmetry described above,
the jet function for an external parton of species $i$ is
$\begin{split}\ln{\cal
J}_{i}\left(\alpha_{N},\alpha_{M},\alpha_{U},{\varepsilon}\right)&=-{\left(\frac{\alpha_{N}}{\pi}\right)}\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(100)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(100)}({\varepsilon})\right]\\\
&\quad+{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}\left\\{\frac{{\beta^{N}_{200}}}{8}\frac{1}{{\varepsilon}^{2}}\left[\frac{3}{4\,{\varepsilon}}\gamma_{K\,i}^{(100)}+{\cal
G}_{i}^{(100)}({\varepsilon})\right]-\frac{1}{8}\left[\frac{\gamma_{K\,i}^{(200)}}{4\,{\varepsilon}^{2}}+\frac{{\cal
G}_{i}^{(200)}({\varepsilon})}{{\varepsilon}}\right]\right\\}\\\
&\quad-{\left(\frac{\alpha_{M}}{\pi}\right)}\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(010)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(010)}({\varepsilon})\right]\\\
&\quad+{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}\left\\{\frac{{\beta^{M}_{020}}}{8}\frac{1}{{\varepsilon}^{2}}\left[\frac{3}{4\,{\varepsilon}}\gamma_{K\,i}^{(010)}+{\cal
G}_{i}^{(010)}({\varepsilon})\right]-\frac{1}{8}\left[\frac{\gamma_{K\,i}^{(020)}}{4\,{\varepsilon}^{2}}+\frac{{\cal
G}_{i}^{(020)}({\varepsilon})}{{\varepsilon}}\right]\right\\}\\\
&\quad-{\left(\frac{\alpha_{U}}{\pi}\right)}\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(001)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(001)}({\varepsilon})\right]\\\
&\quad+{\left(\frac{\alpha_{U}}{\pi}\right)}^{2}\left\\{\frac{{\beta^{U}_{002}}}{8}\frac{1}{{\varepsilon}^{2}}\left[\frac{3}{4\,{\varepsilon}}\gamma_{K\,i}^{(001)}+{\cal
G}_{i}^{(001)}({\varepsilon})\right]-\frac{1}{8}\left[\frac{\gamma_{K\,i}^{(002)}}{4\,{\varepsilon}^{2}}+\frac{{\cal
G}_{i}^{(002)}({\varepsilon})}{{\varepsilon}}\right]\right\\}\\\
&\quad-{\left(\frac{\alpha_{N}}{\pi}\right)}\,{\left(\frac{\alpha_{M}}{\pi}\right)}\,\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(110)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(110)}({\varepsilon})\right]\\\
&\quad-{\left(\frac{\alpha_{N}}{\pi}\right)}\,{\left(\frac{\alpha_{U}}{\pi}\right)}\,\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(101)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(101)}({\varepsilon})\right]\\\
&\quad-{\left(\frac{\alpha_{M}}{\pi}\right)}\,{\left(\frac{\alpha_{U}}{\pi}\right)}\,\left[\frac{1}{8\,{\varepsilon}^{2}}\gamma_{K\,i}^{(011)}+\frac{1}{4\,{\varepsilon}}{\cal
G}_{i}^{(011)}({\varepsilon})\right]+\dots\,.\end{split}$ (7)
To deal with the multiplicity of gauge couplings, I have introduced some new
notations. $\alpha_{N}$, $\alpha_{M}$, $\alpha_{U}$, are the renormalized
gauge couplings of the $SU(N)$, $SU(M)$ and $U(1)$ symmetries respectively.
Their $\beta$-function coefficients are indexed by the powers of the gauge
couplings (in $N,M,U$ order) that multiply that coefficient. For example,
$\begin{split}\beta^{N}(\alpha_{N},\alpha_{M},\alpha_{U})&=\mu^{2}\frac{d}{d\mu^{2}}{\left(\frac{\alpha_{N}}{\pi}\right)}\\\
&=-{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}{\beta^{N}_{200}}-{\left(\frac{\alpha_{N}}{\pi}\right)}^{3}{\beta^{N}_{300}}-{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}{\left(\frac{\alpha_{M}}{\pi}\right)}{\beta^{N}_{210}}-{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}{\left(\frac{\alpha_{U}}{\pi}\right)}{\beta^{N}_{201}}+\dots\,,\\\
\end{split}$ (8)
where
$\begin{split}{\beta^{N}_{200}}&=\frac{11}{12}C_{A_{N}}-\frac{1}{6}\left(N_{f_{n}}+C_{A_{M}}\,N_{f_{b}}\right)\,,\qquad{\beta^{N}_{300}}=\frac{17}{24}C_{A_{N}}^{2}-\left(\frac{5}{24}C_{A_{N}}+\frac{1}{8}C_{F_{N}}\right)\left(\,N_{f_{n}}+C_{A_{M}}\,N_{f_{b}}\right)\,,\\\
{\beta^{N}_{210}}&=-\frac{1}{16}C_{A_{M}}\,N_{f_{b}}C_{F_{M}}\,,\qquad{\beta^{N}_{201}}=-\frac{1}{16}\left(\sum_{i=1}^{N_{f_{n}}}Q_{f_{n}^{i}}^{2}+C_{A_{M}}\sum_{i=1}^{N_{f_{b}}}Q_{f_{b}^{i}}^{2}\right)\,,\end{split}$
(9)
Similarly, the cusp ($\gamma_{K}$) and ${\cal G}$ anomalous dimensions are
indexed by the powers of the gauge couplings that multiply their leading
appearance in the jet functions. The explicit values of all of the anomalous
dimensions that appear through two loops are given in Appendix A.
The soft anomalous dimension of a mixed gauge structure, like the log of the
jet function, consists of the sum of the soft anomalous dimensions for each of
the separate gauge interactions, plus possible terms that are due exclusively
to the mixed interaction. The structure of such a mixed soft anomalous
dimension would have to involve (at least) pairs of generators from each of
the mixing gauge groups. The least complicated of such terms would be of the
form
$\begin{split}{\bm{\Gamma}}_{S_{f}}^{(110)}&=\frac{\digamma^{(110)}}{2}\,\sum_{i\in{\bf
f}}\ \sum_{j\neq i}\left({\bf T_{N}}_{i}\cdot{\bf T_{N}}_{j}\right)\left({\bf
T_{M}}_{i}\cdot{\bf
T_{M}}_{j}\right)\,\ln\left(\frac{\mu^{2}}{-s_{ij}}\right)\\\
{\bm{\Gamma}}_{S_{f}}^{(101)}&=\frac{\digamma^{(101)}}{2}\,\sum_{i\in{\bf f}}\
\sum_{j\neq i}\left({\bf T_{N}}_{i}\cdot{\bf
T_{N}}_{j}\right)Q_{i}\,Q_{j}\,\ln\left(\frac{\mu^{2}}{-s_{ij}}\right)\end{split}$
(10)
The resulting soft function is
$\begin{split}{\bf
S_{f}}=1&+{\left(\frac{\alpha_{N}}{\pi}\right)}\frac{1}{2\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(100)}+{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}\left(\frac{1}{8\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(100)}\times{\bm{\Gamma}}_{S_{f}}^{(100)}-\frac{{\beta^{N}_{200}}}{4\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(100)}+\frac{1}{4\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(200)}\right)\\\
&+{\left(\frac{\alpha_{M}}{\pi}\right)}\frac{1}{2\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(010)}+{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}\left(\frac{1}{8\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(010)}\times{\bm{\Gamma}}_{S_{f}}^{(010)}-\frac{{\beta^{M}_{020}}}{4\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(010)}+\frac{1}{4\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(020)}\right)\\\
&+{\left(\frac{\alpha_{U}}{\pi}\right)}\frac{1}{2\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(001)}+{\left(\frac{\alpha_{U}}{\pi}\right)}^{2}\left(\frac{1}{8\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(001)}\times{\bm{\Gamma}}_{S_{f}}^{(001)}-\frac{{\beta^{U}_{002}}}{4\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(001)}+\frac{1}{4\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(002)}\right)\\\
&+{\left(\frac{\alpha_{N}}{\pi}\right)}{\left(\frac{\alpha_{M}}{\pi}\right)}\left(\frac{1}{4\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(100)}\times{\bm{\Gamma}}_{S_{f}}^{(010)}+\frac{1}{4\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(110)}\right)\\\
&+{\left(\frac{\alpha_{N}}{\pi}\right)}{\left(\frac{\alpha_{U}}{\pi}\right)}\left(\frac{1}{4\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(100)}\times{\bm{\Gamma}}_{S_{f}}^{(001)}+\frac{1}{4\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(101)}\right)\\\
&+{\left(\frac{\alpha_{M}}{\pi}\right)}{\left(\frac{\alpha_{U}}{\pi}\right)}\left(\frac{1}{4\,{\varepsilon}^{2}}{\bm{\Gamma}}_{S_{f}}^{(010)}\times{\bm{\Gamma}}_{S_{f}}^{(001)}+\frac{1}{4\,{\varepsilon}}{\bm{\Gamma}}_{S_{f}}^{(011)}\right)\end{split}$
(11)
Any new terms that might arise from mixing are parameterized by
${\bm{\Gamma}}_{S_{f}}^{(110)}$, ${\bm{\Gamma}}_{S_{f}}^{(101)}$ and
${\bm{\Gamma}}_{S_{f}}^{(011)}$.
## IV Extracting the anomalous dimensions
I will extract the anomalous dimensions be performing a few, relatively
simple, explicit calculations. The anomalous dimensions associated with the
fermions can be extracted from a Sudakov-type calculation,
$\overline{f}_{x}\,f_{x}\longrightarrow X$, where $X$ is a massive vector
boson that is uncharged under the $SU(N)\times SU(M)\times U(1)$ symmetry. In
this case the infrared structure of the amplitude is uniquely associated with
the $f_{x}$ fermions. Alternatively, one could extract the fermion anomalous
dimensions from a set of calculations of the form
$\overline{f}_{l}\,f_{l}\longrightarrow\overline{f}_{x}\,f_{x}$. For instance,
because $f_{l}$ carries only the $U(1)$ charge, the mixed infrared structure
can again be uniquely associated with the $f_{x}$ fermions. This is the method
used in Ref. Kilgore and Sturm (2012), where the $SU(3)\times U(1)$ anomalous
dimensions were determined from the mixed corrections to
$\overline{q}q\longrightarrow l^{+}l^{-}$.
I could extract the boson anomalous dimensions from another Sudakov-type
calculation, that of “Higgs” production, $V_{i}\,V_{i}\longrightarrow H$. The
problem with this calculation is that the scalar must either carry quantum
numbers of the vector boson, in which case it contributes to the infrared
structure of the amplitude, or it must couple to the vectors through an
effective interaction, for which one would need to determine the
renormalization properties and Wilson coefficients. I will instead extract the
gauge boson anomalous dimensions from calculations of the more complicated
amplitudes, $\overline{f}_{x}\,f_{x}\longrightarrow V_{1}\,V_{2}$. The
extraction of the boson anomalous dimensions from these amplitudes is made
simpler by the fact that I have already determined the fermion anomalous
dimensions from Sudakov-type amplitudes.
### IV.1 Extracting the fermion anomalous dimensions
The fermion anomalous dimensions are extracted from calculations of the
Sudakov-type amplitudes $\overline{f}_{x}\,f_{x}\longrightarrow X$.
Figure 1: Sample diagrams of $\overline{f}_{b}\,f_{b}\longrightarrow X$
The Feynman diagrams (see Fig. 1) are essentially the same as for two-loop QCD
corrections to Drell-Yan production. I generate the Feynman diagrams using
QGRAF Nogueira (1993) and implement the Feynman rules and perform algebraic
manipulations with FORM Vermaseren (2000). The resulting loop integrals are
reduced to master integrals using the integration-by-parts (IBP) method
Chetyrkin and Tkachov (1981) in combination with Laporta’s algorithm Laporta
and Remiddi (1996); Laporta (2000) as implemented in the program REDUZE2 von
Manteuffel and Studerus (2012).
Figure 2: Master Integrals for two-loop Sudakov-type amplitudes.
There are only four master integrals (see Fig. 2) that contribute to these
processes and all can be evaluated in closed form by standard Feynman
parameter integrals. The results of the reduction to master integrals and the
values of the master integrals are inserted into the FORM program, and the
amplitude is evaluated as a Laurent series in the dimensional regularization
parameter ${\varepsilon}$. After renormalization, the poles in ${\varepsilon}$
are entirely infrared in origin. Most of the infrared terms can be readily
associated with pure $SU(N)$, $SU(M)$ or $U(1)$ interactions, or with the
overlap of two one-loop terms. Once these terms are accounted for, however,
one obtains the two-loop mixed contribution to the fermion anomalous
dimensions. I find that there are no mixed cusp anomalous dimensions for the
fermions, nor is there a mixed soft anomalous dimension involving only
fermions. There are, however, mixed ${\cal G}$ anomalous dimensions. The
results are collected in Appendix A.
### IV.2 Extracting the boson anomalous dimensions
The boson anomalous dimensions are extracted from two-loop, two-to-two fermion
to di-boson scattering amplitudes. Sample diagrams are shown in Fig. 3.
Figure 3: Sample diagrams of $\overline{f}_{b}\,f_{b}\longrightarrow
A_{N}\,A_{M}$
In addition to the four master integrals that contribute to two-loop Sudakov-
type diagrams, there are six more that contribute to massless two-to-two
scattering (see Fig. 4).
Figure 4: Master integrals for two-loop massless two-to-two scattering. The
double lines indicate a squared propagator.
In this case the infrared structure of the amplitudes involves the overlap of
the infrared structure of the fermions and the two gauge bosons. The soft
anomalous dimensions can be identified by their dependence on the logs of
kinematic invariants. The gauge boson contributions to the jet functions must
be determined by taking different combinations of the external gauge bosons
and accounting for the contributions of the already-determined quark anomalous
dimensions. As with the quarks, I find that there are no mixed cusp or soft
anomalous dimensions at two loops, but that there are non-vanishing mixed
${\cal G}$ anomalous dimensions.
## V Conclusion
I have computed the anomalous dimensions that govern the two-loop infrared
structure of mixed gauge interactions. I have presented results for a general
$SU(N)\times SU(M)\times U(1)$ gauge structure with fermions that lie in the
fundamental representations of both non-Abelian gauge groups ($F_{b}$), the
fundamental representation of one and the singlet representation of the other
($F_{n}$ and $F_{m}$), or are singlets under both non-Abelian gauge groups
($F_{l}$). All fermions are assumed to carry $U(1)$ charges. I note that this
is the gauge structure and fermion content of the unbroken Standard Model.
However, I have treated the fermions as vector-like, and therefore do not have
the chiral structure of the Standard Model. Since the chiral anomaly and
anomaly cancellation are ultraviolet issues, they should not affect the
infrared structure at all. If one were to make the fermion multiplets chiral,
so that $F_{b}$ and $F_{m}$ represent the left-handed quarks and leptons,
respectively, while $F_{n}$ and $F_{l}$ represent the right-handed quarks and
leptons, one would only need to weight factors of $N_{f_{x}}$ by a factor of
$1/2$ to account for the chiral projector in the fermion trace. Since I have
expressed the anomalous dimensions so that explicit factors of $N_{f_{x}}$
only appear in the coefficients of the $\beta$-functions, it is only there
that one would need to make this change. The rest of the formulæ in Appendix A
remain unchanged.
The connection of the current results to applications in QCD$\times$ QED is
more direct. Here, I can identify the $SU(N)$ symmetry as QCD, and the $U(1)$
as QED and drop the $SU(M)$ interaction. In this case, I need only $F_{n}$ and
$F_{l}$ vector-like representations of fermions. One can readily check that
the mixed ${\cal G}$ anomalous dimensions determined here agree with those
determined for the quarks in Reference Kilgore and Sturm (2012).
The results determined here are not surprising and were largely anticipated in
Reference Kilgore and Sturm (2012) by examining the structure of the QCD
anomalous dimensions. The argument was that there can be no non-Abelian
structure in the mixed terms because the generators of the different gauge
groups commute with one another and two-loop amplitudes are not sufficiently
complicated to allow both mixed interactions and non-Abelian structures of a
single gauge group in the same term. Therefore, all factors of $C_{A}$ that
appear in the two-loop QCD anomalous dimensions should be set to zero.
Furthermore, it was postulated that all factors of $N_{f}$ that appear should
be associated with coefficients of the $\beta$-functions. However,
contributions to two-loop anomalous dimensions that might arise from
corrections to one-loop terms would only involve leading coefficients of the
$\beta$-functions. Because of the Ward identity, mixing first appears in the
$\beta$-functions of gauge couplings at second order. Therefore, corrections
that are proportional to leading coefficients of the $\beta$-functions should
also be set to zero.
From this, one expects that there will be no mixed cusp or soft anomalous
dimensions at two-loops. The factor $K$ which governs the two-loop corrections
to both of these terms can be written as a linear combination of $C_{A}$ and
the leading coefficient of the $\beta$-function. Thus, by this reasoning, the
only mixed anomalous dimensions that one expects at two-loops are ${\cal G}$
terms. If I assume that the mixed ${\cal G}$ anomalous dimensions will have
essentially the same form as those of QCD, the only terms that remain are
proportional to $C_{F}^{2}$ or to $\beta_{1}$. The minimal possible change
that is consistent with the mixed terms is to change each factor of $C_{F}$ to
one of $\\{C_{F_{N}},C_{F_{M}},Q_{f}^{2}\\}$ and to change $\beta_{1}$ to the
appropriate one of
$\\{{\beta^{N}_{110}},{\beta^{N}_{101}},{\beta^{M}_{110}},{\beta^{M}_{011}},{\beta^{U}_{101}},{\beta^{U}_{011}}\\}$.
It turns out that these simple transformations give exactly the correct
result.
#### Acknowledgments:
This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.
## Appendix A Infrared Anomalous Dimensions
### A.1 $\beta$-Functions
The $\beta$-function of the $SU(N)$ coupling is
$\begin{split}\beta^{N}(\alpha_{N},\alpha_{M},\alpha_{U})&=\mu^{2}\frac{d}{d\mu^{2}}{\left(\frac{\alpha_{N}}{\pi}\right)}\\\
&=-{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}{\beta^{N}_{200}}-{\left(\frac{\alpha_{N}}{\pi}\right)}^{3}{\beta^{N}_{300}}-{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}{\left(\frac{\alpha_{M}}{\pi}\right)}{\beta^{N}_{210}}-{\left(\frac{\alpha_{N}}{\pi}\right)}^{2}{\left(\frac{\alpha_{U}}{\pi}\right)}{\beta^{N}_{201}}+\dots\,,\\\
\end{split}$ (12)
where
$\begin{split}{\beta^{N}_{200}}&=\frac{11}{12}C_{A_{N}}-\frac{1}{3}\,T_{f}\left(N_{f_{n}}+C_{A_{M}}\,N_{f_{b}}\right)\,,\qquad{\beta^{N}_{300}}=\frac{17}{24}C_{A_{N}}^{2}-\left(\frac{5}{12}C_{A_{N}}+\frac{1}{4}C_{F_{N}}\right)T_{f}\left(\,N_{f_{n}}+C_{A_{M}}\,N_{f_{b}}\right)\,,\\\
{\beta^{N}_{210}}&=-\frac{1}{8}C_{F_{M}}\,C_{A_{M}}\,T_{f}\,N_{f_{b}}\,,\qquad{\beta^{N}_{201}}=-\frac{1}{16}\left(\sum_{i=1}^{N_{f_{n}}}Q_{f_{n}^{i}}^{2}+C_{A_{M}}\sum_{i=1}^{N_{f_{b}}}Q_{f_{b}^{i}}^{2}\right)\,,\end{split}$
(13)
For the $SU(M)$ coupling,
$\begin{split}\beta^{M}(\alpha_{N},\alpha_{M},\alpha_{U})&=\mu^{2}\frac{d}{d\mu^{2}}{\left(\frac{\alpha_{M}}{\pi}\right)}\\\
&=-{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}{\beta^{M}_{020}}-{\left(\frac{\alpha_{M}}{\pi}\right)}^{3}{\beta^{M}_{030}}-{\left(\frac{\alpha_{N}}{\pi}\right)}{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}{\beta^{M}_{120}}-{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}{\left(\frac{\alpha_{U}}{\pi}\right)}{\beta^{M}_{021}}+\dots\,,\\\
\end{split}$ (14)
where
$\begin{split}{\beta^{M}_{020}}&=\frac{11}{12}C_{A_{M}}-\frac{1}{3}\,T_{f}\left(N_{f_{m}}+C_{A_{N}}\,N_{f_{b}}\right)\,,\qquad{\beta^{M}_{030}}=\frac{17}{24}C_{A_{M}}^{2}-\left(\frac{5}{12}C_{A_{M}}+\frac{1}{4}C_{F_{M}}\right)T_{f}\left(\,N_{f_{m}}+C_{A_{N}}\,N_{f_{b}}\right)\,,\\\
{\beta^{M}_{120}}&=-\frac{1}{8}C_{F_{N}}\,C_{A_{N}}\,T_{f}\,N_{f_{b}}\,,\qquad{\beta^{M}_{021}}=-\frac{1}{16}\left(\sum_{i=1}^{N_{f_{m}}}Q_{f_{m}^{i}}^{2}+C_{A_{N}}\sum_{i=1}^{N_{f_{b}}}Q_{f_{b}^{i}}^{2}\right)\,,\end{split}$
(15)
while for the $U(1)$,
$\begin{split}\beta^{U}(\alpha_{N},\alpha_{M},\alpha_{U})&=\mu^{2}\frac{d}{d\mu^{2}}{\left(\frac{\alpha_{M}}{\pi}\right)}\\\
&=-{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}{\beta^{U}_{020}}-{\left(\frac{\alpha_{M}}{\pi}\right)}^{3}{\beta^{U}_{030}}-{\left(\frac{\alpha_{N}}{\pi}\right)}{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}{\beta^{U}_{120}}-{\left(\frac{\alpha_{M}}{\pi}\right)}^{2}{\left(\frac{\alpha_{U}}{\pi}\right)}{\beta^{U}_{021}}+\dots\,,\\\
\end{split}$ (16)
where
$\begin{split}{\beta^{U}_{020}}&=-\frac{1}{3}\left(\sum_{i=1}^{N_{f_{l}}}Q_{f_{l}^{i}}^{2}+C_{A_{M}}\sum_{i=1}^{N_{f_{m}}}Q_{f_{m}^{i}}^{2}+C_{A_{N}}\sum_{i=1}^{N_{f_{n}}}Q_{f_{n}^{i}}^{2}+C_{A_{N}}C_{A_{M}}\sum_{i=1}^{N_{f_{b}}}Q_{f_{b}^{i}}^{2}\right)\,,\\\
{\beta^{U}_{030}}&=-\frac{1}{4}\left(\sum_{i=1}^{N_{f_{l}}}Q_{f_{l}^{i}}^{4}+C_{A_{M}}\sum_{i=1}^{N_{f_{m}}}Q_{f_{m}^{i}}^{4}+C_{A_{N}}\sum_{i=1}^{N_{f_{n}}}Q_{f_{n}^{i}}^{4}+C_{A_{N}}C_{A_{M}}\sum_{i=1}^{N_{f_{b}}}Q_{f_{b}^{i}}^{4}\right)\,,\\\
{\beta^{U}_{102}}&=-\frac{1}{8}C_{A_{N}}\left(\sum_{i=1}^{N_{f_{n}}}\,Q^{2}_{f_{n}^{i}}+C_{A_{M}}\,\sum_{i=1}^{N_{f_{b}}}\,Q^{2}_{f_{b}^{i}}\right)\,,\qquad{\beta^{U}_{012}}=-\frac{1}{8}C_{A_{M}}\left(\sum_{i=1}^{N_{f_{m}}}\,Q^{2}_{f_{m}^{i}}+C_{A_{N}}\,\sum_{i=1}^{N_{f_{b}}}\,Q^{2}_{f_{b}^{i}}\right)\,.\end{split}$
(17)
### A.2 The Cusp Anomalous Dimensions
$\begin{split}\gamma_{K\,f_{n}}^{(100)}&=\gamma_{K\,f_{b}}^{(100)}=2\,C_{F_{N}}\,\qquad\gamma_{K\,A_{N}}^{(100)}=2\,C_{A_{N}}\,\qquad\gamma_{K\,f_{m}}^{(100)}=\gamma_{K\,f_{l}}^{(100)}=\gamma_{K\,A_{M}}^{(100)}=\gamma_{K\,A_{U}}^{(100)}=0\\\
\gamma_{K\,f_{m}}^{(010)}&=\gamma_{K\,f_{b}}^{(010)}=2\,C_{F_{M}}\,\qquad\gamma_{K\,A_{M}}^{(010)}=2\,C_{A_{M}}\,\qquad\gamma_{K\,f_{n}}^{(010)}=\gamma_{K\,f_{l}}^{(010)}=\gamma_{K\,A_{N}}^{(010)}=\gamma_{K\,A_{U}}^{(010)}=0\\\
\gamma_{K\,f_{y}^{i}}^{(001)}&=2\,Q^{2}_{f_{y}^{i}}\
(y\in\\{l,m,n,b\\})\qquad\gamma_{K\,A_{N}}^{(001)}=\gamma_{K\,A_{M}}^{(001)}=\gamma_{K\,A_{U}}^{(001)}=0\\\
\gamma_{K\,x}^{(200)}&=\frac{K^{(200)}}{2}\gamma_{K\,x}^{(100)}\,,\qquad
K^{(200)}=C_{A_{N}}\left(\frac{2}{3}-\zeta_{2}\right)+\frac{10}{3}{\beta^{N}_{200}}\\\
\gamma_{K\,x}^{(020)}&=\frac{K^{(020)}}{2}\gamma_{K\,x}^{(010)}\,,\qquad
K^{(020)}=C_{A_{M}}\left(\frac{2}{3}-\zeta_{2}\right)+\frac{10}{3}{\beta^{M}_{020}}\\\
\gamma_{K\,x}^{(002)}&=\frac{K^{(002)}}{2}\gamma_{K\,x}^{(001)}\,,\qquad
K^{(002)}=\frac{10}{3}{\beta^{U}_{002}}\\\
\gamma_{K\,x}^{(110)}&=\gamma_{K\,x}^{(101)}=\gamma_{K\,x}^{(011)}=0\
(x\in\\{f_{l},f_{n},f_{m},f_{b},A_{N},A_{M},A_{U}\\}\,.\end{split}$ (18)
### A.3 The ${\cal G}$ Anomalous Dimensions
$\begin{split}{\cal G}_{f_{n}}^{(100)}&={\cal
G}_{f_{b}}^{(100)}=\frac{3}{2}\,C_{F_{N}}+\frac{{\varepsilon}}{2}C_{F_{N}}\left(8-\zeta_{2}\right)\,\qquad{\cal
G}_{A_{N}}^{(100)}=2\,{\beta^{N}_{200}}-\frac{{\varepsilon}}{2}C_{A_{N}}\,\zeta_{2}\,\qquad{\cal
G}_{f_{m,l}}^{(100)}={\cal G}_{A_{M,U}}^{(100)}=0\\\ {\cal
G}_{f_{m}}^{(010)}&={\cal
G}_{f_{b}}^{(010)}=\frac{3}{2}\,C_{F_{M}}+\frac{{\varepsilon}}{2}C_{F_{M}}\left(8-\zeta_{2}\right)\,\qquad{\cal
G}_{A_{M}}^{(010)}=2\,{\beta^{M}_{020}}-\frac{{\varepsilon}}{2}C_{A_{M}}\,\zeta_{2}\,\qquad{\cal
G}_{f_{n,l}}^{(010)}={\cal G}_{A_{N,U}}^{(010)}=0\\\ {\cal
G}_{f_{x}^{i}}^{(001)}&=\frac{3}{2}\,Q^{2}_{f_{x}^{i}}+\frac{{\varepsilon}}{2}Q^{2}_{f_{x}^{i}}\left(8-\zeta_{2}\right)\
(x\in\\{l,m,n,b\\})\,\qquad{\cal
G}_{A_{U}}^{(001)}=2\,{\beta^{U}_{002}}\,\qquad{\cal G}_{A_{M,N}}^{(001)}=0\\\
{\cal G}_{f_{n}}^{(200)}&={\cal
G}_{f_{b}}^{(200)}=C_{F_{N}}^{2}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\zeta_{3}\right)+C_{F_{N}}\,{\beta^{N}_{200}}\left(\frac{209}{36}+\zeta_{2}\right)+C_{F_{N}}\,C_{A_{N}}\left(\frac{41}{72}-\frac{13}{4}\zeta_{3}\right)\\\
{\cal
G}_{A_{N}}^{(200)}&=2\,{\beta^{N}_{300}}+C_{A_{N}}\,{\beta^{N}_{200}}\left(\frac{19}{18}-\zeta_{2}\right)+C_{A_{N}}^{2}\left(\frac{177}{216}-\frac{1}{4}\zeta_{3}\right)\,\qquad{\cal
G}_{f_{m,l}}^{(200)}={\cal G}_{A_{M,U}}^{(200)}=0\\\ {\cal
G}_{f_{m}}^{(020)}&={\cal
G}_{f_{b}}^{(020)}=C_{F_{M}}^{2}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\zeta_{3}\right)+C_{F_{M}}\,{\beta^{M}_{020}}\left(\frac{209}{36}+\zeta_{2}\right)+C_{F_{M}}\,C_{A_{M}}\left(\frac{41}{72}-\frac{13}{4}\zeta_{3}\right)\\\
{\cal
G}_{A_{M}}^{(020)}&=2\,{\beta^{M}_{030}}+C_{A_{M}}\,{\beta^{M}_{020}}\left(\frac{19}{18}-\zeta_{2}\right)+C_{A_{M}}^{2}\left(\frac{177}{216}-\frac{1}{4}\zeta_{3}\right)\,\qquad{\cal
G}_{f_{n,l}}^{(020)}={\cal G}_{A_{N,U}}^{(020)}=0\\\ {\cal
G}_{f_{x}^{i}}^{(002)}&=Q^{4}_{f_{x}^{i}}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\zeta_{3}\right)+Q^{2}_{f_{x}^{i}}\,{\beta^{U}_{002}}\left(\frac{209}{36}+\zeta_{2}\right)\,\qquad{\cal
G}_{A_{U}}^{(002)}=2\,{\beta^{U}_{003}}\,\qquad{\cal G}_{A_{N,M}}^{(002)}=0\\\
{\cal
G}_{f_{b}}^{(110)}&=C_{F_{N}}\,C_{F_{M}}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\zeta_{3}\right)\,\qquad{\cal
G}_{A_{N}}^{(110)}=2\,{\beta^{N}_{210}}\,\qquad{\cal
G}_{A_{M}}^{(110)}=2\,{\beta^{M}_{120}}\,\qquad{\cal
G}_{f_{n,m,l}}^{(110)}={\cal G}_{A_{U}}^{(110)}=0\\\ {\cal
G}_{f_{\\{b,n\\}}^{i}}^{(101)}&=C_{F_{N}}\,Q^{2}_{f_{\\{b,n\\}}^{i}}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\zeta_{3}\right)\
\quad{\cal G}_{A_{N}}^{(101)}=2\,{\beta^{N}_{201}}\,\qquad{\cal
G}_{A_{U}}^{(101)}=2\,{\beta^{U}_{102}}\,\qquad{\cal
G}_{f_{m,l}}^{(101)}={\cal G}_{A_{M}}^{(101)}=0\\\ {\cal
G}_{f_{\\{b,m\\}}^{i}}^{(011)}&=C_{F_{M}}\,Q^{2}_{f_{\\{b,m\\}}^{i}}\left(\frac{3}{16}-\frac{3}{2}\zeta_{2}+3\zeta_{3}\right)\,\quad{\cal
G}_{A_{M}}^{(011)}=2\,{\beta^{M}_{021}}\,\qquad{\cal
G}_{A_{U}}^{(011)}=2\,{\beta^{U}_{012}}\,\qquad{\cal
G}_{f_{n,l}}^{(011)}={\cal G}_{A_{N}}^{(011)}=0\,.\end{split}$ (19)
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|
arxiv-papers
| 2013-08-05T18:19:42 |
2024-09-04T02:49:49.030362
|
{
"license": "Public Domain",
"authors": "William B. Kilgore",
"submitter": "William Kilgore",
"url": "https://arxiv.org/abs/1308.1055"
}
|
1308.1092
|
∎
11institutetext: S. Alsid and M. Serna 22institutetext: 2354 Fairchild Drive,
Department of Physics
United States Air Force Academy, CO 80840
Tel.: +1-719-333-3510
Fax: +1-719-333-3182
22email: [email protected] 22email: [email protected]
# Unifying Geometrical Representations of Gauge Theory
Scott Alsid Mario Serna
(Received: date / Accepted: date)
###### Abstract
We unify three approaches within the vast body of gauge-theory research that
have independently developed distinct representations of a geometrical
surface-like structure underlying the vector-potential. The three approaches
that we unify are: those who use the compactified dimensions of Kaluza-Klein
theory, those who use Grassmannian models (also called gauge theory embedding
or $CP^{N-1}$ models) to represent gauge fields, and those who use a hidden
spatial metric to replace the gauge fields. In this paper we identify a
correspondence between the geometrical representations of the three schools.
Each school was mostly independently developed, does not compete with other
schools, and attempts to isolate the gauge-invariant geometrical surface-like
structures that are responsible for the resulting physics. By providing a
mapping between geometrical representations, we hope physicists can now
isolate representation-dependent physics from gauge-invariant physical results
and share results between each school. We provide visual examples of the
geometrical relationships between each school for $U(1)$ electric and magnetic
fields. We highlight a first new result: in all three representations a static
electric field (electric field from a fixed ring of charge or a sphere of
charge) has a hidden gauge-invariant time dependent surface that is underlying
the vector potential.
###### Keywords:
Kaluza Klein Gauge field theory: Composite Field theoretical model: $CP^{N-1}$
Gauge Geometry Embedding Grassmannian Models Hidden-spatial geometry
###### pacs:
04.20.Cv 11.15.-q 04.20.-q 12.38.Aw
## 1 Introduction
In this study, we unify three small but largely independently developed
schools within the vast body of gauge-theory research that have developed
distinct representations of a geometrical surface-like structure underlying
the vector-potential. By school we mean a grouping of conceptual approaches
which share a common methodology. The approaches are not in competition with
each other. They are simply our grouping of mathematical tools that make use
of a surface-like representation from which one can derive or induce a gauge
field. Each school has been employed by Fields Medalist and Nobel Prize
winners to extract or to separate gauge-invariant key physics from gauge-
dependent artifacts. Each school has been largely independently invented; each
school has had distinct strands of papers with very little reference to papers
of other schools. We highlight the easily overlooked commonalities of the
different strands within each school, and then we tie the geometric
representations of each school onto a common representation. This paper’s new
results are: the direct geometrical relationship between each school, the
explicit examples that we work out, and third we will show that in all three
representations, a static electric field has a hidden time dependence that is
not captured by our normal notation.
Although the ‘spell’ of gauge theory has captured most modern physicists, most
of the research on gauge theory does not fall into one of the three schools
that we describe. Fig. 1 shows a map of gauge theory and where this paper
contributes. Our contribution, as depicted in Fig. 1, is represented by the
dotted red lines.
Historically electric and magnetic fields were thought to be the fundamental
objects in the model. This is the top layer in the figure. Vector potentials
were introduced as a mathematical trick, but were not ascribed as physical
objects in the model. It was not until the the Aharonov-Bohm effect was
predicted and observed that the vector potential was elevated from a
representational convenience to something with predictive power. As depicted
in the Fig. 1 vector potentials are a layer deeper as we dig for the
foundations of gauge theory. Today we recognize that electromagnetic fields
are a curvature $2$-form that originates from the vector potential, which is a
connection $1$-form.
But what is the geometrical surface that gives rise to this $1$-form? This is
the deeper foundation of gauge theory that is represented on the third row
down on Fig. 1. There have been at least three schools providing possible
geometrical surface-like foundations to the vector potential. This paper
reviews these three schools and explores the relationships between them
depicted by the red dotted lines. In this paper we will observe that in all
three schools there is a gauge-invariant hidden time dependence to the
surface-like geometrical structures of electric fields that is not captured in
the connection $1$-form representation nor in the curvature $2$-form
representation.
Before going into details on the three representations, one might ask why
different representations might be expected to give new physics? Richard
Feynman once remarked: “every theoretical physicist who is any good knows six
or seven different theoretical representations for exactly the same physics.
He knows that they are all equivalent $\ldots$ but he keeps them in his head
hoping that they will give him different ideas for guessing (new physical
laws)” (feynman1994character, , pg 168). Therefore, we do not expect new
physics at this stage. We do hope that finding the commonalities between
deeper representations of gauge theory will provide insights that may help us
“guess” new physical laws. The time-dependence of electric fields in the
surface-like layer is one such insight.
Figure 1: Map of geometric foundations of gauge theory. This paper
contributes the connections depicted in the dotted lines.
The first attempt to find a surfrace underlying the vector potential was
started by Theodor Kaluza and Oskar Klein Kaluza:1921ar ; Klein:1926ar . They
use a $4+n$ dimensional space-time where extra dimensions are curled up and
result in a gauge theory (see Schwarz:1992zt ; Salam:1981xd for reviews).
This well-known school has over 1600 papers. The second school uses a
Grassmannian manifold to represent gauge fields using a type of gauge-theory
embedding Narasimhan1961 ; Narasimhan63 ; 79Atiyah ; Corrigan:1978ce ; Dubois-
Violette:1979it ; Felsager:1979fq ; Cahill:1993mp ; Cahill:1993uq ;
Cahill:1996yw ; Valtancoli:2001gx ; Bars:1978xy ; Bars:1979qd ; Stoll:1994cn ;
Stoll:1994vx ; PhysRevLett.52.2111 ; Serna:2002ux ; Serna:2005ar ;
Gliozzi:1978xe ; Eichenherr:1978qa ; Gava:1979sp ; 1980NuPhB.174..397D ;
Balakrishna:1993ja ; Palumbo:1993vu ; PhysRevD.66.025022 Marsh:2007qp ;
2006JPhA…39.9187G ; 2010JMP….51j3509H . The third school introduces
alternative variables for gauge theory that uncover a hidden spatial metric
which reproduces the gauge fields Goldstone:1978he ; Freedman:1993mu ;
Freedman:1994rg ; Lunev:1994ty ; Haagensen:1994sy ; Haagensen:1995py ;
Schiappa:1997yh ; Niemi:2010mw ; Zee:1988mc . Each of the schools start with a
different geometrical representation which then faithfully maps onto the
traditional gauge fields $A_{\mu}$. This paper directly unifies the
geometrical representations of the three schools without appealing to their
common gauge-field image-space. Our unified geometrical representation allows
physicists to better identify gauge-invariant foundations underlying the
physical results. As an example, we will discuss a hidden time dependence that
we reveal is present even in static electric fields. Our unification will also
help translate results, such as instanton solutions, between each independent
school.
Mathematicians describe both gauge theory and Riemannian manifolds with the
language of fiber bundles. Fiber bundles are not a geometrical representation,
but rather a rigorous lexicon used to describe a wide array of geometrical
structure. This language enables descriptions of gauge theories on
topologically non-trivial spaces. However, the power gained by abstraction to
the fiber bundle language often leaves out insight that may be gained from
explicit examples. Here we concern ourself only with local descriptions of
gauge fields on topologically trivial flat space-time; therefore, we will
avoid extensive use of the bundle language in favor of explicit examples.
This paper is organized as follows: In section 2 we review the development and
research activity of each school. Sections 3 and 4 contain our new research
results: they present the connections between the Kaluza-Klein and
Grassmannian school, and between the Hidden-Spatial-Geometry and the
Grassmannian school respectively. Finally in section 5 we provide examples
with familiar electric and magnetic fields. In our conclusion, we discuss the
hidden time dependence that is revealed to be in static electric fields.
## 2 Literature Survey of the Three Schools
(a) (b)
Figure 2: Shown is a geometrical representation of the magnetic field (left)
and the electric field (right). The angular change in the phase of a wave
function after parallel transport around a closed loop in space-time yields
electric and magnetic fields multiplied by the area of the loop enclosed. This
parallels Riemannian geometry where the Riemann tensor gives the rotation
matrix that results from parallel transport around a loop.
All the geometrized representations that we discuss emphasize non-integrable
phase factors to define the internal curvature Wu:1975es . The Wilson loop
$\Delta\theta=\oint{\vec{A}\cdot d\vec{r}}$ gives the phase-angle shift111We
have chosen to work in units where $\hbar=c=1$ and where we absorb the
electron’s charge $e$ into the definition of $A_{\mu}$. resulting from
parallel transport of the wave function around an infinitesimal loop. The non-
integrable phase is similar to how curvature is found in Riemannian geometry.
For example, the magnetic field is equal to the phase-angle change in the wave
function after parallel transporting the wave function around a closed loop on
a spatial slice of space-time. In the limit of an infinitesimal loop, the
magnetic field is given in terms of the phase-shift $\Delta\theta$ as
$B_{z}=\frac{\Delta\theta}{\Delta x\Delta
y}=\frac{\displaystyle\oint{\vec{A}\cdot d\vec{r}}}{\Delta x\Delta
y}=\frac{\displaystyle\int\int({\vec{\nabla}\times\vec{A}})\cdot(d\vec{x}\times
d\vec{y})}{|d\vec{x}\times d\vec{y}|}$ (1)
where we have used the Wilson loop and classic vector identities.
Figs. 2a and 2b show this non-integrable phase angle using the tools of the
Grassmannian school described in section 2.2. In this figure, the complex
plane on which the wave function lives is represented by the plane spanned by
the two red basis vectors. The complex plane is inserted into a trivial
internal space at each space-time point and is represented by a disk. A wave
function is shown as a vector (black or yellow) on the disk inserted at each
space-time point. We parallel transport the wave function along two paths (A
and B) represented by the black and yellow vectors. Comparing path A with path
B gives the non-integrable phase shift $\Delta\theta$. Because of this phase
shift, the wave front of a plane wave is pulled and the plane wave changes
directions.
Fig. 2a shows the case of the magnetic field where the loop is all spatial.
Likewise, Fig. 2b shows the electric field is equal to parallel transporting
the wave function or matter field around a closed loop on a part spatial and
part temporal slice of space-time.
As we delve into a review of the three schools, there will be a proliferation
of notations for each of the schools and the past papers. To help clarify
this, we provide a table in appendix A to help define the different variables
as we use it to express the basis vectors in each school and the various index
types.
### 2.1 The Kaluza-Klein School
Kaluza Klein theories unify classical electromagnetism with Einstein’s general
relativity Kaluza:1921ar Klein:1926ar . They posit extra spatial dimensions
that are compactified within ordinary space-time along a very small radius
$R$. All tensor quantities are independent of this fifth coordinate (the
cylinder condition).
In traditional Kaluza-Klein theory the line element of the five-dimensional
space is222Throughout this paper we use the convention that lower-case Latin
letters near the beginning of the alphabet $a,b,...$ will be gauge-theory
color indices, Greek letters $\mu,\nu,...$ will be space-time coordinates,
upper-case Latin letters $A,B,...$ will be used for Kaluza-Klein metric
indices, and lower-case Latin letters towards the middle of the alphabet
$i,j,...$ will be used for the variables corresponding to subspaces of space-
time and the embedding dimensions, where context will keep them distinct. The
Kaluza-Klein index values 0 through 3 are the usual space-time coordinates
$t,x,y,z$ and the index value 5 is the fifth dimension coordinate $x^{5}$,
which is used to parameterize the tiny compact dimension. The appendix
provides a summary.
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}+(R\,A_{\mu}\,dx^{\mu}+dx^{5})^{2},$ (2)
where we omit the dilaton field for simplicity of presentation. Here
$g_{\mu\nu}$ is the familiar four-dimensional metric from general relativity,
$A_{\mu}$ is the four-vector potential, $x^{5}$ is the fifth dimension’s
coordinate, and $R$ is the radius of the curled up fifth dimension.
In Kaluza-Klein theory charge is explained as motion of a neutral particle
along the fifth dimension, where the two directions it can go in $x^{5}$
explain the two different types of charge. Electric fields are four-
dimensional manifestations of the inertial-dragging effect in the fifth
dimension Gron85 Gron92 Gron:2005aw . Furthermore coordinate transformations
of the fifth dimension are shown to be $U(1)$ gauge transformations.
One pitfall of the classical theory is that there are no measurable new
predictions. Another pitfall occurs with quantum mechanics. The wave function
around the fifth dimension gives particles a mass-spectrum tower of
$m^{2}=(n/R)^{2}$, where $n$ is an arbitrary integer. For an $R$ near the
Planck scale, particles would be either massless or have Planck-scale masses,
which implies that the model must be modified to be used in new physical
theories. Modified Kaluza-Klein theories play a large role in string theory.
For a further review of Kaluza-Klein theory see references Salam:1981xd ;
Schwarz:1992zt and the references therein.
### 2.2 The Grassmannian School
Grassmannian representations of gauge fields started in 1961 when Narasimhan
and Ramanan showed that every $U(n)$ gauge theory could be represented by a
section of a Grassmannian $Gr(n,N)$ fiber bundle Narasimhan1961 ; Narasimhan63
. A Grassmannian manifold $Gr(n,N)$ is the set of orientations an $n$-plane
can take in a larger $N$-dimensional space with a fixed origin. Another way to
view $Gr(n,N)$ fiber bundle is as a $n$-plane embedded into a higher-
dimensional $N$-Euclidean space that is inserted into each point in space and
time. The Grassmannian school is essentially a gauge theory version of the
1956 Nash embedding theorem which proved that every Riemannian manifold could
be embedded in a higher-dimensional Euclidean space Nash56 .
In the language of bundles, Narasimhan and Ramanan proved for any $U(n)$ gauge
field and $d$ space-time dimensions, the gauge field can be constructed by
inserting a $\mathbb{C}^{n}$ vector bundle into a trivial $\mathbb{C}^{N}$
vector bundle if $N\geq(d+1)(2d+1)n^{3}$. Narasimhan’s condition guarantees us
an embedding for this $N$, but we can sometimes represent the embedding for
specific field configurations for smaller $N$ as we will do in section 5. For
an $O(n)$ gauge field $\mathbb{R}^{n}$ vector bundles are embedded in a
trivial $\mathbb{R}^{N}$ vector-bundle.
In the Grassmannian school, wave functions are sections of the $n$-dimensional
vector bundle. That is, they are a vector on the $\mathbb{C}^{n}$ or
$\mathbb{R}^{n}$ vector space. By definition the vector bundles have a fixed
origin. All the embedding-school approaches have a set of $n$ orthonormal
gauge basis vectors $\vec{e}_{a}$ that span the gauge fiber internal to each
space-time point. The dual basis vectors $\vec{e}^{\;a}$ satisfy
$\vec{e}^{\;a}\cdot\vec{e}_{b}=\delta^{a}_{b}$. There are $n$ vectors
$\vec{e}_{a}$ in a real or complex Euclidean $N$-dimensional embedding space.
The matter field (wave function) exists as a vector on the gauge fiber spanned
by the gauge-fiber basis vectors:
$\vec{\phi}=\phi^{a}\vec{e}_{a}.$ (3)
The projection operator is the outer product
$P^{j}_{k}={e}^{j}_{a}{e}^{\;a}_{k}$. The gauge field is then
$(A_{\mu})_{\;b}^{a}=i\vec{e}^{\;a}\cdot\partial_{\mu}\vec{e}_{b}.$ (4)
Notice that if $n=N$ then $A_{\mu}$ is a pure gauge with a vanishing
$F_{\mu\nu}$. In all the cases that we study here, $N>n$. Fig. 3 shows the
$Gr(2,3)$ Grassmannian model visually. The bubbles show the $N=3$ trivial
vector space inside each space-time point. The red vectors are the gauge-fiber
basis vectors $\vec{e}_{a}$ which span the displayed disk. The gauge fields
depend on all the ways one can orient the $R^{2}$ space within the trivial
$R^{3}$ space. The wave function or matter field $\vec{\phi}$ is the black
vector that lives on the disks.
Figure 3: A graphical representation of the Grassmannian school. A set of two
basis vectors span the internal vector space attached to every point in space-
time. How they vary determines the electromagnetic field.
A gauge transformation is a rotation of the basis vectors $\vec{e}_{a}$
accompanied by the inverse rotation on the matter vector coefficients
$\phi^{a}$ that preserves their inner product and leaves the wave function
$\vec{\phi}=\phi^{a}\vec{e}_{a}$ and the projection operator
$P^{j}_{k}=e^{j}_{a}{e}^{\;a}_{k}$ invariant. It is very central to our
argument to understand that gauge transformations leave two objects invariant:
(1) the plane spanned by the basis vectors $\vec{e}_{a}$, and (2) the vector
formed by the wave function $\vec{\phi}^{a}$. Global transformations on the
embedding space do not affect Eq.(4).
A long list of notable physicists have employed the Grassmannian-model as a
part of their gauge theory research. In the following review, we map the
notation used in these previous approaches onto the notation introduced above.
Atiyah in 1979 79Atiyah defined the linear maps
$u_{x}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}$, whose image was in the
trivial space $\mathbb{R}^{N}$. Atiyah’s $u$’s play the role of the gauge-
fiber basis vectors $\vec{e}_{a}$. The projection operator is written as
$P=uu^{*}$, with $u^{*}u=1$, and the gauge potential is
$A_{\mu}=u^{*}\partial_{\mu}u$, where $u^{*}$ is the dual to $u$. Atiyah,
Drineld, Hitchin, and Manin (ADHM) used the rectangular matrices of the
Grassmannian school as one of the tools in their construction of self-dual
instanton solutions in Euclidean Yang-Mills Theory Atiyah1978185 . Corrigan
and followers Corrigan:1978ce ; Alekseevsky:2002pi used the embedding
representation in finding Green’s functions for self-dual gauge fields.
Dubois-Violette Dubois-Violette:1979it created a formulation of gauge theory
using only globally defined complex $N\times n$ matrices $V$ (analogous to
$e^{j}_{\,a}$) such that $V^{\dagger}V=I$ and $VV^{\dagger}=P$, and
$A_{\mu}=V^{\dagger}(x)\partial_{\mu}V(x).$
An independent research line refers to the Grassmannian school as $CP^{N-1}$
models Eichenherr:1978qa ; Gava:1979sp ; 1980NuPhB.174..397D ;
Balakrishna:1993ja ; Palumbo:1993vu ; PhysRevD.66.025022 ; Marsh:2007qp ;
2006JPhA…39.9187G ; 2010JMP….51j3509H . In the $CP^{N-1}$ models a setup is
created with $z^{\dagger}\cdot z=1$ where $z$, which is sometimes called a
zweibein, is a complex $N$-vector. The gauge field
$A_{\mu}=z^{\dagger}\partial_{\mu}\cdot z$ is discovered in the equations of
motion. Here the complex vector $z$ plays the role of a gauge basis vector
$e^{j}_{a}$ with complex dimensions $1\times N$.
Felsager, Leinaas, and Gliozzi Gliozzi:1978xe ; Felsager:1979fq had a similar
approach. In a manner very similar to Fig. 3 and section 5, they geometrically
represented magnetic fields by use of plane bundles in $\mathbb{R}^{3}$, where
the distribution of the planes in each point was characterized by a curvature
related to the magnetic field strength. For two vectors
$\vec{e}_{1},\vec{e}_{2}$ orthonormal to each other and to the normal vector
of the plane, the vector potential is
$A_{j}=\lambda\vec{e}_{1}\cdot\nabla_{j}\vec{e}_{2},$ where the $\vec{e}$’s
play the same role as $\vec{e}_{a}$ introduced in the beginning of this
section, and $\lambda$ is a constant for dimensionality. Since the nineties,
Cahill Cahill:1993mp ; Cahill:1996yw ; Cahill:1993uq ; PhysRevD.88.125014 has
used gauge basis vectors $\vec{e}_{a}$ in lattice simulations and in his most
recent textbook (cahill2013physical, , Sections 11.51 and 11.52).
In finding projectors for the fuzzy sphere, Valtancoli Valtancoli:2001gx used
the connection $A_{n}^{\nabla}=\langle\psi_{n},d\psi_{n}\rangle$ for
$n$-monopoles. Here $|\psi\rangle$ plays the role of $\vec{e}_{a}$.
In another variant of the Grassmannian school, Bars Bars:1978xy ; Bars:1979qd
used a separate embedding for each of the spatial dimensions of the gauge
field (corner variables), as opposed to using a single embedding for all the
gauge-fiber basis vectors. He used $n\times n$ unitary matrices
$B_{13}^{ij},B_{23}^{ij}$ to rewrite the canonical variables $A_{i}^{a}$ and
$E_{i}^{a}$. For example, $A_{1}^{a}$ was written as
$T^{a}A_{1}^{a}=iB_{13}^{\dagger}\partial_{1}B_{13},$ where $T^{a}$ is a
generator of $SU(n)$. Stoll Stoll:1994vx ; Stoll:1994cn introduced angle
variables in the Hamiltonian formulation of QCD to investigate the low-energy
properties in terms of gauge invariant degrees of freedom. The angle variables
are similar to corner variables and are the exponents of $SU(n)$ matrices, and
the gauge fields are defined as
$A_{j}(x)=\frac{i}{g}V_{j}(x)\partial_{j}V_{j}^{\dagger}(x)\;{\rm{(no\;summation)}}$.
Zee and Wilczek, building on work by Simon, also independently developed a
Yang-Mills structure associated with Barry’s phase and degenerate spaces (see
Simon:1983mh ; PhysRevLett.52.2111 ; Zee:1988mc ; Zee:2003mt ). A given wave
function is expanded in terms of eigenfunctions spanning a degenerate subspace
$\Psi(t)=c_{a}\psi_{a}(t)$. One finds in the adiabatic limit that
$\frac{dc_{b}}{dt}=-A_{ba}c_{a}$, where
$A_{ba}(t)=i\langle\psi_{b}(t)|\frac{\partial\psi_{a}}{\partial t}\rangle$.
For a Hamiltonian $H(t)$ that depends on parameters
$\lambda^{1},...,\lambda^{d}$, when one traces out a path in the parameter
space the time derivative of $c_{b}$ becomes
$\frac{dc_{b}}{dt}=-(A_{\mu})_{ba}c_{a}\frac{d\lambda^{\mu}}{dt}$, where
$(A_{\mu})_{ba}=i\langle\psi_{b}|\partial_{\mu}\psi_{a}\rangle.$ In Zee and
Wilczek’s approach, $|\psi_{a}\rangle$ plays the role of $\vec{e}_{a}$.
In the early 2000’s one of us (MS) and Cahill used the Narasimhan and Ramanan
theorem to visualize the geometry of simple electromagnetic fields with an
$SO(2)$ gauge group. To gain some visual intuition they found $SO(2)$ gauge
basis vectors $\vec{e}_{a}$ embedded in an $\mathbb{R}^{3}$ trivial fiber for
certain vector potentials Serna:2002ux . For a matter vector on the gauge
fiber, as represented visually in their work, a clockwise rotation in the
momentum direction corresponded to a positive charge, while a counterclockwise
rotation corresponded to a negative charge. In addition to this they observed
an indication for a geometry-based explanation of charge quantization. This is
similar to the representation of charge in Kaluza-Klein. Although all free
fundamental particles have $\pm e$ charge, quarks have fractional charge. The
fractional charge would follow from a GUT gauge theory, such as that of
$U(1)\times SU(2)\times SU(3)\subset SU(5)\subset SU(10)$. These GUTs always
enable one to absorb $e\,A=A^{\prime}$. We can only absorb $e$ into $A_{\mu}$
if every field couples with the same coefficient as in most GUTs.
### 2.3 The Hidden-Spatial-Geometry School
The next school maps a hidden spatial geometry onto the gauge fields. The
gauge potential transforms inhomogeneously and makes unclear the physical
nature of the theory. In 1978, Goldstone and Jackiw Goldstone:1978he made the
electric field in an $SU(2)$ gauge theory diagonal, which made easier
separating the gauge-invariant parts of gauge angles. They wed these ideas to
a 4-space ‘spinning top’ analogy.
In 1994 Lunev Lunev:1994ty formulated a tetrad-based mapping from $A^{a}_{j}$
to a tetrad variable. In 1995 Freedman, Haagensen, Johnson, and Latorre also
introduced tetrad variables $u^{a}_{j}$ as a replacement to $A^{a}_{j}$
Freedman:1993mu ; Freedman:1994rg ; Haagensen:1994sy ; Haagensen:1995py ,
where the index $a$ denotes the color index and $j$ denotes the spatial index.
In our work we follow the notation of Haagensen and Johnson. The $u^{a}_{j}$
variables serve as a mapping from the basis vectors that span an internal
color space at a space-time point to the coordinate tangent vectors of a
hidden spatial metric at that space-time point. For this approach to work, the
color-space dimension of the tetrad must be equal to the space-time dimension
of a chosen slice.
Haagensen and Johnson used an $SU(2)$ gauge group, with structure constants
$f^{abc}=\varepsilon^{abc}$ for the color index and $GL(3,\mathbb{R})$ for the
spatial component. They worked in the temporal gauge $A^{a}_{0}=0$ so they
could map the three vector potentials $A^{j}_{a}$ to the three dimensions of
the space slice. The constraint imposed on $u_{j}^{a}$ was that the color
index had to transform as a covariant vector under $SU(2)$ and the spatial
index had to transform as $GL(3,\mathbb{R})$. The condition that $u^{a}_{j}$
transform as a vector leads to the gluon ‘spin’ operator constraint
$\varepsilon^{ijk}(\partial_{j}u^{a}_{k}+\varepsilon^{abc}A^{b}_{j}u^{c}_{k})=0,$
(5)
which is similar to the spinning top analogy given in Jackiw and Goldstone.
The end result is that, for a given set of tetrads $u^{a}_{j}$, a unique
vector potential $A^{a}_{j}$ can be found; however, the other direction is not
unique. For a given $A^{a}_{j}$ several $u^{a}_{j}$ exist. Given a set of
tetrad fields $u^{a}_{j}$, the $SO(3)$ vector potential is given by
$A^{a}_{j}=\frac{(\varepsilon^{nmk}\partial_{m}u^{b}_{k})(u^{a}_{n}u^{b}_{j}-\frac{1}{2}u^{b}_{n}u^{a}_{j})}{{\rm{det}}\;u}.$
(6)
In using the constraint of Eq. (5), a hidden spatial metric was implicitly
introduced. The anti-symmetric tensor in Eq. (5) implies
$\partial_{j}u^{a}_{k}+\varepsilon^{abc}A^{b}_{j}u^{c}_{k}=\Gamma^{s}_{jk}u^{a}_{s},$
(7)
where $\Gamma^{s}_{jk}$ is a quantity symmetric in the indices $j,k$. Notice
that Eq.(7) is the standard covariant derivative of a tetrad. It therefore
implicitly defines the relationship between spin-connection $A_{\mu}$, the
Levi-Civita-connection $\Gamma^{s}_{ij}$, and the tetrad $u^{a}_{k}$.
Standard manipulation shows that $\Gamma^{s}_{ij}$ is indeed the Christoffel
symbols of the Levi-Civita-connection for a Riemannian manifold:
$\Gamma^{i}_{jk}=\frac{1}{2}g^{im}(\partial_{j}g_{mk}-\partial_{k}g_{jm}-\partial_{m}g_{jk}),$
(8)
where $g_{ij}=u^{a}_{i}u^{a}_{j}$. Thus, imposing Eq. (5) implicitly
introduced a covariant derivative of a tetrad in Eq.(7), and therefore a
Riemannian geometry with a tetrad $u$ and a metric.
The matter fields are vectors in the tangent space of the manifold,
$\vec{\phi}=\phi^{i}\vec{t}_{i}.$ (9)
Towards the end of the nineties Schiappa adapted these local gauge-invariant
variables for supersymmetric gauge theory Schiappa:1997yh . An independent
variation of this school was pursued by Slizovskiy and Niemi Niemi:2010mw .
In summary, the variables $u^{a}_{j}$ map basis vectors that span the internal
color space to coordinate tangent vectors of a hidden spatial metric.
### 2.4 A Hidden Time Dependence to Electric Fields
At first it seems odd to suggest that a static electric field has a time
dependence. If we have a single non-accelerating charge, Coulomb’s law
produces a static electric field. As nothing is moving, one would not expect
any time dependence. When we introduce gauge fields, we can either choose to
describe a static electric field as the negative gradient of a static voltage
or as the time derivative of $\vec{A}$. The two descriptions are related by a
gauge transformation, but only one has an explicit time dependence. Is this
time dependence real or an artifact of poor choice of gauge?
The language of gauge theory has long suggested that the time dependence of an
the electric field is fundamental but often hidden. The electric field is
given by the $0$-$i$ component of the field strength tensor which
geometrically measures curvature of an internal space after parallel
transporting a wave function through a part spatial and part temporal space-
time loop. For this curvature to be non-zero, it seems that something must be
changing with time. The Lagrangian is typically written as the kinetic energy
minus the potential energy. In gauge theories the Lagrangian density
${\mathcal{L}}=\frac{1}{2}(E^{2}-B^{2})$ has the electric field play the role
of kinetic energy. This again suggests there may be a time-dependence to the
electric field.
The explicit time dependence for electric fields in the Grassmannian school
can be seen in the work Dubois-Violette and Georgelin Dubois-Violette:1979it .
They expressed the field strength $F_{\mu\nu}$ in terms of the projection
operators $P(x)^{j}_{k}=e^{j}_{a}(x)e^{\;a}_{k}(x)$ formed from the outer-
product of the Grassmannian school’s basis vectors. Their expression
$e_{a}^{\,k}(F_{\mu\nu})^{a}_{\,b}e^{b}_{\,j}=(P(x)[\partial_{\mu}P(x),\partial_{\nu}P(x)])^{k}_{\,j}$
(10)
shows that for $F_{0i}$ to be non-zero, then at a minimum $\partial_{0}P(x)$
must be non-zero. This means if there is a non-zero electric field, then the
vector-space spanned by the gauge fiber as seen in the Grassmannian school
will be time-varying.
Is this time dependence an artifact of the Grassmannian representation? What
does it look like? The explicit time-dependence is not unambiguously present
in the traditional field-strength description $F_{0i}$, nor in the vector
potential $A_{\mu}$, nor in the Kaluza-Klein representation, nor in the
Hidden-spatial metric representation. By providing the mapping between these
different geometrical representation schools in the following sections, we
hope to show that this time dependence should be taken more seriously. In the
subsequent examples, we’ll be able to visualize a few special cases of this
time-dependence in all three schools discussed in this paper.
## 3 Mapping the Grassmannian School onto the Kaluza-Klein School
We now wish to map the Grassmannian school of section 2.2 to the Kaluza-Klein
school of section 2.1. We begin with the Grassmannian school representation of
an $SO(2)$ gauge theory. We then construct an explicit isometric immersion
into an $(4+N)$-dimensional Lorentzian space. Finally, we calculate the
induced $5$-dimensional metric. This induced metric will be the Kaluza-Klein
$5$-dimensional metric with the the gauge field $A_{\mu}$ in the $\mu$ $5$
off-diagonal element of the metric
$g_{\mu 5}=\vec{t}_{\mu}\cdot\vec{t}_{5}\propto A_{\mu}$ (11)
as is required in Kaluza-Klein theory. The domain of the map is the
Grassmannian schools representation given by $\vec{e}_{a}(x)$ where the vector
potential is given by Eq.(4). Narasimhan and Ramanan Narasimhan1961 ;
Narasimhan63 and the additional references in section 2.2 showed that this
rectangular matrix can be found for any vector potential
$(A_{\mu})^{a}_{\,b}$. The final target or image of the map will be the
$5$-dimensional Kaluza-Klein metric. The generalization to non-abelian gauge
fields is straight forward.
The first step of the map is that we insert the traditional space-time
manifold and gauge fiber into an $SO(1,3+N)$ embedding with a Lorentzian
signature $\eta={\rm{diag}}(-1,1,1,\ldots,1)$. The explicit embedding being
considered is
$\vec{X}=\left(t,x,y,z,R\vec{e}_{\,1}\cos\frac{x_{5}}{R}+R\vec{e}_{\,2}\sin\frac{x_{5}}{R}\right),$
(12)
where $\vec{e}_{a}$ is the rectangular $N\times n$-dimensional matrix from
Grassmannian school explained in section 2.2. Because we are mapping an
$SO(2)$ gauge theory to a Kaluza-Klein metric, the index $a$ will only run
from $1$ to $2$. The variable $x^{5}$ is the fifth Kaluza-Klein space-time
coordinate which runs from $0$ to $2\,\pi\,R$ in our notation. As is true for
the Grassmannian school, the matrix $\vec{e}_{a}(x)$ depends only on the first
four space-time coordinates $x^{\mu}$. This inserts a ring in the embedding
space. The tangent vectors used in Eq. (11) are given by
$\vec{t}_{A}=\partial_{A}\vec{X}$. For the first 4 space-time coordinates the
tangent vectors $\vec{t}_{\mu}$ are given by
$t_{\mu}^{k}=\partial_{\mu}X^{k}=\delta^{k}_{\mu}+\Theta(k-5)\,R\,\left(\cos(\frac{x^{5}}{R})\partial_{\mu}e_{1}^{k-4}+\sin(\frac{x^{5}}{R})\partial_{\mu}e_{2}^{k-4}\right),$
(13)
where the discrete Heaviside function $\Theta(x-a)$ is defined to be $1$ if
$x\geq a$ and $0$ when $x<a$ and $k$ indexes the $4+N$ coordinates of the
embedding space. The tangent vector $\vec{t}_{5}$ is given by
$\displaystyle t^{k}_{5}$ $\displaystyle=$
$\displaystyle\Theta(k-5)\,\partial_{5}\left(R\,e^{k-4}_{1}\cos(\frac{x^{5}}{R})+R\,e^{k-4}_{2}\sin(\frac{x^{5}}{R})\right)$
(14) $\displaystyle=$
$\displaystyle\Theta(k-5)\,\left(-e^{k-4}_{1}\sin(\frac{x^{5}}{R})+e^{k-4}_{2}\cos(\frac{x^{5}}{R})\right).$
The resulting five-dimensional space-time metric $\tilde{g}_{AB}$ for this
embedding to first order in $R$ is
$g_{\mu\nu}=t^{k}_{\mu}t^{l}_{\nu}\eta_{kl}=\eta_{\mu\nu}+O(R^{2})$, $g_{\mu
5}=t^{k}_{\mu}t^{l}_{5}\eta_{kl}=RA_{\mu}$, and
$g_{55}=t^{k}_{5}t^{l}_{5}\eta_{kl}=1$ where we have used Eq. (4) from the
Grassmannian school applied to $SO(2)$ to relate
$A_{\mu}=\vec{e}^{\;2}\cdot\partial_{\mu}\vec{e}_{1}=-\vec{e}^{\;1}\cdot\partial_{\mu}\vec{e}_{2}$
and $\vec{e}_{a}=\vec{e}^{\;a}$. The dilaton field $\Phi(x)$ follows if we
allow the size of the curled up dimension to vary: $R\rightarrow R\Phi(x)$.
This $5$-dimensional metric is the target of this explicit map between these
previously defined geometrical representations of gauge theory.
A few general comments. The geometry of the embedding, which reproduced the
Kaluza-Klein metric, has a compactified ring at every point in space-time on
the same plane spanned by the Grassmannian-school’s basis vectors. Visual
examples will be shown in section 5.
Although we have shown a general map to first order in $R$ between the Kaluza-
Klein theory and the Grassmannian school, the Kaluza Klein school has a
different representation of the wave function. In the Grassmannian school
there is one wave function at a space-time point and it is a vector on the
tangent space spanned by the basis vectors $\vec{e}_{a}$. In the Kaluza-Klein
picture we see that the wave function is a function of each point in space-
time including $x_{5}$. It can vary circularly as we vary position of the
fifth coordinate. It is this feature that is responsible for the Kaluza-Klein
tower of masses $m^{2}=(n/R)^{2}$, where $n$ is again an integer. The other
schools lack such a mass tower.
Let us discuss the coordinate dependence and independence of the relationship
between the Kaluza-Klein and Grassmannian models. In the Kaluza-Klein school,
the value of the $x^{5}$ coordinate is the same as the $\theta$ that
delineates the angle on the ring inserted in the Grassmannian school. In the
immersion Eq.(12), coordinate changes such as gauge transformations leave the
surface formed by the immersion unchanged. This does not mean that the surface
in Eq.(12) is unique: there are many surfaces that lead to the same gauge
field $A_{\mu}$.333 Some specific many-to-one mappings will be provided in
section 5 in Eqs.(39) and (40). The many to one relationship does not mean
that there is a coordinate dependence to the mapping. Notice that all
coordinate transformations leave the surface formed by immersion unchanged.
For example, see Figs. 3 and 5 of Ref. Serna:2002ux . Fig. 3 of Ref.
Serna:2002ux shows two different Grassmannian representations for the
magnetic field of a solenoid. Both representations give the exact same gauge
field, but they are not related by a gauge transformation. Fig. 5 of Ref.
Serna:2002ux shows the same magnetic field in two different gauges. You can
see the surface-like structure is unchanged by the change of gauge.
## 4 Mapping the Hidden-Spatial-Geometry School onto the Grassmannian School
Next we show how the hidden-spatial-geometry school of section 2.3 is mapped
onto the Grassmannian school of section 2.2. The domain of the mapping is the
tetrads $u^{a}_{j}$ of section 2.3. The target space will be the Grassmannian
school representation.
The first step of the mapping is to use the Nash embedding theorem Nash56 to
define an immersion of the hidden spatial metric of section 2.2 into a larger-
dimensional Euclidean space. Nash guarantees that such an immersion exists for
any metric given the dimension $N$ of the Euclidean space is sufficiently
large. Given this guaranteed immersion, the coordinate tangent vectors
$\vec{t}_{j}$ of dimension $N\times n$ will reproduce the hidden-spatial-
geometry metric
$g_{jk}=u^{a}_{j}u^{a}_{k}=\vec{t}_{j}\cdot\vec{t}_{k}.$ (15)
Next, we identify the $N$-dimensional embedding-space dimensions that Nash
guarantees exist with the trivial $N$-dimensional vector bundle used by
Narasimhan and Ramanan in the Grassmannian representation. The tetrads
$u^{a}_{j}$ from the hidden-spatial-metric school will map the coordinate
tangent vectors $\vec{t}_{j}$ to the orthonormal frame $\vec{e}_{a}$:
$\vec{e}_{a}=u^{\;i}_{a}\vec{t}_{i},$ (16)
or its dual,
$\vec{e}^{\;a}=u^{a}_{i}\vec{t}^{\;i}.$ (17)
The tetrads $u^{i}_{a}$ may be obtained by using the Gram-Schmidt
orthogonalization process on the coordinate tangent vectors of the hidden
spatial metric.
The target space of the mapping is this orthonormal frame $\vec{e}^{a}_{j}$
which we will show is exactly the defining $N\times n$ rectangular matrix of
the Grassmannian school representation.
Repeating the definitions from the literature presented in section 2.3, we
note that the color-space dimension of the tetrad must be equal to the
dimension of the space-time slice under consideration. For $SO(3)$ we need a
three-dimensional slice of space-time to identify with the three color
dimensions of the $SO(3)$ gauge fiber. For the $SO(2)$ representation used in
section 5, we need a two-dimensional slice of space-time to identify with the
two real dimensions of $SO(2)$.
We have proposed that Eq.(16) maps the hidden-spatial-metric school to the
Grassmannian school. To verify this claim, we will use Eq.(16) in the
Grassmannian definition of the gauge field $A_{\mu}$ from Eq.(4). We will
check that it reproduces the defining relation in section 2.3. We express Eq.
(4) not in terms of the gauge basis vectors but the coordinate tangent vectors
$\vec{t}_{i}$ associated with the hidden spatial metric of the Grassmannian
school via Eq. (16), then Eq. (4) becomes
$(u^{a}_{i}\vec{t}^{\;i})\cdot\partial_{j}(u^{k}_{b}\vec{t}_{k})=-iA_{j\;\;b}^{\;\;a},$
(18)
$u^{a}_{i}\delta^{i}_{k}\partial_{j}u^{k}_{b}+iA_{j\;\;b}^{\;\;a}+u^{a}_{i}u^{k}_{b}\Gamma^{i}_{jk}=0.$
(19)
Multiplying by $-u^{b}_{l}$ and using the identity
$u^{k}_{b}\partial_{j}u^{a}_{k}+u^{a}_{k}\partial_{j}u^{k}_{b}=0$ yields
$\partial_{j}u^{a}_{k}-iA_{j\;\;b}^{\;\;a}u^{b}_{l}\delta^{l}_{k}-u^{a}_{i}\Gamma^{i}_{jk}=0.$
(20)
Now we specialize to $SU(2)$, where the form of the generators are444The
distinction between lower and upper indices are dropped in the epsilon term
for convenience (see Weinberg weinberg1996quantum , chapter 15 appendix A).
$(T^{c})^{a}_{\;\;b}=-i\varepsilon^{abc}.$ (21)
Thus Eq. (18) leads to
$\partial_{j}u^{a}_{k}+\varepsilon^{abc}A^{b}_{j}u^{c}_{k}=u^{a}_{i}\Gamma^{i}_{jk},$
(22)
which is Eq. (7) of the hidden-spatial-metric school, but this time derived by
the Grassmannian school’s methods. A similar relationship was independently
observed by Refs. Schuster:2003kt ; 2006JPhA…39.9187G but without noting the
generality of the relationships to the long research records of the two
schools.
As for the wave function in the Grassmannian school, it is a vector on the
gauge fiber in the internal space. In the hidden-spatial-metric school, it is
a vector on the tangent space. As the gauge fiber is the same vector space in
the two schools, then the wave functions are the same vector in these two
schools. This is in contrast to the Kaluza-Klein model, where the wave
function is a scalar function of each point in the five-dimensional space-
time.
Now let us discuss the coordinate dependence and independence of the
relationship between the Grassmannian school and the hidden-spatial-metric
school. When formulated at MIT, there was no embedding space associated with
the hidden-spatial-metric school. Johnson, Haagensen, Schiappa, _et.al._
highlighted that the metric formed by contracting over the color indicies in
the tetrad $g_{ij}=u^{a}_{i}\,u^{b}_{j}\,\delta_{ab}$ was invariant under
gauge transformations which only act on the internal color indices. They also
discussed the many-to-one relationship between metrics and gauge-fields.
Nash’s Nash56 embedding theorem guarantees that we can represent the metric
that represents the hidden-spatial-metric school as an isometric immersion
into a trivial embedding space. We observe that in the Grassmannian school,
the tangent plane to the coordinates of a space-time point of the hidden-
spatial metric, as viewed by the embedding, provide the element of the
Grassmannian that corresponds to that space-time point. Coordinate changes on
the space-time slice do not change the surface. Gauge-transformations do not
change the metric nor the element of the Grassmannian that represents that
point. There are no special coordinates that enable the relationship between
Eq.(7), which was derived from symmetry principles without an embedding space,
and Eq.(22), which was derived from a surface immersed in the embedding space
guaranteed by Nash. The mapping is general and does not depend on special
coordinates.
## 5 Examples in Electromagnetism
We now apply the geometric representations from the different schools to an
abelian $U(1)$ gauge theory, namely ordinary electromagnetism. We work with
$SO(2)$ (which is isomorphic to $U(1)$) so that everything is real. In
$SO(2)$, there is only one generator; the gauge potential is
$A_{j\;\;b}^{\;\;a}=T^{a}_{\;\;b}A_{j},$ (23)
where $T^{a}_{\;\;b}=-i\varepsilon^{ab}$.
Each electromagnetic field configuration has at least one (sometimes many)
geometric representations in each school. In order to demonstrate the hidden
spatial geometry, we need to select a space-time slice of equal dimension to
the dimension of the gauge fiber. For $SO(2)$ we will need to select two-
dimensional slices. We will analyze two-dimensional slices of space-time
denoted by $x^{\mu}(\sigma,\tau)$ and show each school’s representation in
this slice. We use the pullback to map the four-dimensional field-strength
tensor $F_{\mu\nu}$ and vector potential $A_{\mu}$ onto the 2-D slice of
space-time using
$F_{ij}=\frac{\partial x^{\mu}}{\partial x^{i}}\frac{\partial
x^{\nu}}{\partial x^{j}}F_{\mu\nu},\ \ \ \ A_{j}=\frac{\partial
x^{\mu}}{\partial x^{j}}A_{\mu}.$ (24)
The $SO(2)$ analog of Eq. (22) is
$\partial_{j}u^{a}_{k}+\varepsilon^{ab}A_{j}u^{b}_{k}-u^{a}_{i}\Gamma^{i}_{jk}=0.$
(25)
Solving this for $A_{j}$ gives
$A_{j}=\frac{-1}{2}\epsilon_{ac}\,g^{kl}u^{c}_{l}(\partial_{j}u^{a}_{k}-\Gamma^{i}_{jk}u^{a}_{i}).$
(26)
The Grassmannian and Kaluza-Klein school’s equations are unaltered in
specializing to $SO(2)$ examples.
We now proceed to illustrate the above connections for three elementary
electromagnetic field configurations: a $y$-polarized plane wave, an
electrically charged ring, and a spherical charge. These examples were chosen
for their familiarity and because their hidden spatial metrics correspond to a
sphere, a paraboloid, and a funnel-shaped object respectively.
### 5.1 The $Y$-Polarized Plane Wave
Consider the four-potential for a $y$-polarized plane wave traveling in the
$x$-direction
$A_{y}=A_{0}\cos(k(x-t)),$ (27)
where $A_{0}=\frac{E_{0}}{k}=\frac{B_{0}}{k}$. We take a $yt$ slice of space-
time. This is parameterized by
$t(\sigma,\tau)=\tau,x(\sigma,\tau)=x_{0},y(\sigma,\tau)=\sigma,$ and
$z(\sigma,\tau)=z_{0}$, where $x_{0}$ and $z_{0}$ are fixed coordinates.
Using the pullback the $SO(2)$ vector potential for the plane wave is
$\;\;A_{\sigma}=A_{0}\cos(k(x_{0}-\tau)).$ (28)
The question now is: What two-dimensional shape from the hidden-spatial-metric
school has this specific vector potential? Using trial and error, we
considered shapes until we found the ones whose tangent vectors led to Eq.
(28). The plane wave follows from a sphere parametrized as 555 We have reused
the variable name $X$ to parametrize each immersion. This is not not same
immersion as Eq.(12) nor the same as in the other examples.
$\vec{X}=\left(\begin{array}[]{c}\varrho\sin(k(x_{0}-\tau))\cos(A_{0}\sigma)\\\
\varrho\sin(k(x_{0}-\tau))\sin(A_{0}\sigma)\\\ \varrho\cos(k(x_{0}-\tau))\\\
\end{array}\right),$ (29)
where $\sigma=y$ and $\tau=t$, and $\varrho$ is a positive real value on which
$A_{i}$ and $F_{ij}$ do not depend.
Fig. 4a shows the domain of the variables $\sigma$ and $\tau$, which
parameterize the $yt$ slice of space-time. This shape in Fig. 4a helps map
space-time points to the corresponding locations in the other figures. There
are two lines, one in the direction of increasing $\sigma$ on the outer ring
and one in the direction of increasing $\tau$ on the inner ring. Fig. 4b shows
the diagram as it appears parameterized on the surface of the hidden spatial
geometry where we let $\varrho=1$ m, $k=1\;{\rm{m}}^{-1},$ and
$A_{0}=1\;{\rm{m}}^{-1}$. This corresponds to an average intensity beam of
about $5\times 10^{-17}\;{\rm{Watt/m^{2}}}$. Here we see that increasing
$\sigma$ corresponds to a line of longitude on the sphere, whereas increasing
$\tau$ corresponds to a line of latitude.
(a)
(b)
(c)
(d)
Figure 4: The geometry of a $y$-polarized plane wave for a $yt$ slice, with
$k=A_{0}=1\;{\rm{m^{-1}}},\varrho=1\;{\rm{m}}$. (a): Reference pattern which
will be shown parameterized on the geometries of the three schools. (b): A
hidden-spatial-geometry representation of the $y$-polarized plane wave, $yt$
slice. (c): A Grassmannian representation of the $y$-polarized plane wave,
$yt$ slice. (d): A hidden-spatial-geometry representation with the disks of
the Grassmannian representation mapped to their corresponding location of the
shape.
Let us verify that the embedding from Eq. (29) produces Eq. (28). The
coordinate tangent vectors, $\vec{t}_{\sigma}$ and $\vec{t}_{\tau}$, are found
by differentiating Eq. (29) by the respective coordinates
$\vec{t}_{\tau}=\partial_{\tau}\vec{X}$ and
$\vec{t}_{\sigma}=\partial_{\sigma}\vec{X}$. The hidden-spatial-metric
$g_{ij}$ is found by taking the dot products between each of the tangent
vectors $g_{ij}=\vec{t}_{i}\cdot\vec{t}_{j}$. The resulting line element is
$ds^{2}=k^{2}\varrho^{2}d\sigma^{2}+A_{0}^{2}\varrho^{2}\sin^{2}(k(x_{0}-\tau))d\tau^{2}.$
(30)
This is the metric of the surface shown in Fig. 4b. Notice that each space-
time points $(\sigma,\tau)$ correspond to points on the shape in Fig. 4b. The
shape as parameterized is not the curvature of space-time, but a surface whose
curvature represents the electric and magnetic fields of a plane wave. If we
were to perform a change of coordinates on
$(\sigma,\tau)\rightarrow(\sigma^{\prime},\tau^{\prime})$ neither the shape
nor the electric and magnetic fields would change. This is because a point on
surface of Fig. 4b maps to a point on space-time. Coordinate re-
parameterizations leave this mapping unchanged. If instead one were to map the
points on the surface to different space-time points, then the resulting
electric and magnetic fields would be potentially very different.
Next we find the Grassmannian school representation. The lack of off-diagonal
terms in the line-element of Eq.(30) means that the tangent vectors are
orthogonal (such is generally not the case for coordinate tangent vectors, the
sphere is kind enough to permit this simplicity). The gauge basis vectors
$\vec{e}_{a}$ on the gauge fibers are orthogonal and normalized. While
$\vec{t}_{\sigma}$ and $\vec{t}_{\tau}$ are orthogonal, they are not
normalized. The tetrads $u^{j}_{a}$, which map $\vec{t}_{j}$ to $\vec{e}_{a}$,
are $u^{\sigma}_{1}=\frac{1}{|\vec{t}_{\sigma}|}=\frac{1}{k\varrho}$,
$u^{\sigma}_{2}=0$, $u^{\tau}_{1}=0$, and
$u^{\tau}_{2}=\frac{1}{|\vec{t}_{\tau}|}=\frac{1}{A_{0}\varrho\sin(k\tau)}$.
Normalizing the tangent vectors gives the Grassmannian’s basis vectors
$\vec{e}_{1}=\left(\begin{array}[]{c}-\sin(A_{0}\sigma)\\\
\cos(A_{0}\sigma)\\\ 0\\\ \end{array}\right),\ \ \ \
\vec{e}_{2}=\left(\begin{array}[]{c}-\cos(k(x_{0}-\tau))\cos(A_{0}\sigma)\\\
-\cos(k(x_{0}-\tau))\sin(A_{0}\sigma)\\\ \sin(k(x_{0}-\tau))\\\
\end{array}\right).$ (31)
Now we map the Grassmannian representation back to the gauge field
representation. The only nonzero value of $A_{j}$ is
$(A_{\sigma})^{a}_{\
b}=i\vec{e}^{\;a}\cdot\partial_{\sigma}\vec{e}_{b}=-iA_{0}\cos(k(x_{0}-\tau))\varepsilon^{ab}$
(32)
which shows that this parameterization of the sphere leads to the geometry of
the $yt$ slice of the $y$-linearly-polarized plane wave. Likewise if we
calculate $A_{j}$ from the hidden-spatial-geometry school via the tetrads
$u^{a}_{j}$ and Eq.(26), we also find the plane wave.
Eq.(31) is visually displayed in Fig. 4c. We can see that each space-time
point $(\sigma,\tau)$ has the same tangent plane as the corresponding space-
time point in Fig. 4b. As we move toward smaller $\sigma$ on Fig. 4c, we see
the tangent planes approaching a common plane which maps to the north pole of
Fig. 4b. These vectors $\vec{e}_{1}$ and $\vec{e}_{2}$ are visualized in Fig.
4c as the red basis vectors which span the disks. The figure is to be
interpreted as in Fig. 3, but without the bubbles. The reference pattern from
Fig. 4a is again shown to help visualize the directions of $\sigma$ and $\tau$
in both spaces. A rotation of the red basis vectors on the disks corresponds
to a gauge transformation.
Next Fig. 4d shows the disks from the Grassmannian-school representation
rearranged into the shape associated with the hidden-spatial-metric school.
The reference pattern is removed, but one can see the tangent plane associated
with each space-time point mapped across all three Figs. 4 b, c, and d.
From the Kaluza-Klein picture, we have at each point in space-time a curled up
fifth dimension. This is represented by a ring in the Grassmannian’s space on
the same tangent plane. The ring’s embedding is parameterized by $x^{5}$ as
$\vec{r}(x^{5})=R\,\vec{e}_{1}\cos(\frac{x^{5}}{R})+R\,\vec{e}_{2}\sin(\frac{x^{5}}{R}),$
(33)
where $\vec{r}$ is a three-dimensional vector in a trivial Grassmannian’s
embedding space over space-time. Fig. 5 shows the representation of the
Kaluza-Klein school, where the rings are shown at several space-time points
for our given slice. We have changed the scale of $R$ to better see the ring.
Thus, Figs. 4b, 4c, and 5 are the three school’s geometrical representations
of the plane wave; all share a common set of tangent planes as represented in
the embedding.
Figure 5: A Kaluza-Klein representation of the $y$-polarized plane wave from a
$yt$ space-time slice.
A shape by-itself does not represent an electric or magnetic field
configuration. The points on the shape must be mapped to space-time points. If
we change the space-time point to which a point on the shape maps, then it
manifests as a different gauge field. For example if we map a shape to an $xy$
slice of space, it will manifest as a $z$-directed magnetic field. If we map a
shape to an $xt$ slice of space-time it will manifest as an $x$-directed
electric field.
As an example consider the $A_{\mu}$ of Eq. (27). Now we want to understand
the magnetic field. We consider the $xy$ slice of space-time:
$t(\sigma,\tau)=t_{0},x(\sigma,\tau)=\sigma,y(\sigma,\tau)=\tau,$ and
$z(\sigma,\tau)=z_{0}$ where $t_{0}$ and $z_{0}$ are fixed values. The
pullback of the vector potential on the $xy$ slice is
$\;\;A_{\sigma}=\frac{B_{0}}{k}\cos(k(\sigma-t_{0})).$ (34)
The shape corresponding to this slice is also a sphere ${}^{\ref{FootNoteX}}$:
$\vec{X}=\left(\begin{array}[]{c}\varrho\sin(k(\sigma-
t_{0}))\cos(A_{0}\tau)\\\ \varrho\sin(k(\sigma-t_{0}))\sin(A_{0}\tau)\\\
\varrho\cos(k(\sigma-t_{0}))\\\ \end{array}\right).$ (35)
To the best of our understanding, the similarity to Eq. (29) is not general.
### 5.2 The Electrically Charged Ring
(a)
(b)
(c)
Figure 6: The geometry of an electrically charged ring, along the $z$ axis
only, for $Q=2\pi$ and $b=\frac{1}{2}$ m. In SI units this is a charge of 1.24
$\mu$C. (a): A Grassmannian representation showing the $\mathbb{R}^{2}$
subspaces along points of space-time. (b): A Kaluza-Klein representation
showing the gauge subspaces at each space-time point for an electrically
charged ring. (c): A hidden-spatial-geometry representation of the
electrically charged ring.
Next consider the electric field due to a ring of charge $-Q$ and radius $b$
centered at the origin on the $xy$ plane:
$E_{z}=\frac{-Qz}{4\pi(b^{2}+z^{2})^{\frac{3}{2}}}.$ (36)
The associated vector potential pulled-back onto a $zt$ slice ($z=\sigma$,
$t=\tau$, $x=0$, $y=0$) is given by
$A_{\tau}=\frac{Q}{4\pi}\frac{1}{\sqrt{b^{2}+\sigma^{2}}}.$ (37)
We found the hidden-spatial-geometry shape to be a paraboloid parametrized as
${}^{\ref{FootNoteX}}$:
$\vec{X}=\left(\begin{array}[]{c}\frac{\sigma}{2b}\cos(\frac{Q}{4\pi
b}\tau)\\\ \frac{\sigma}{2b}\sin(\frac{Q}{4\pi b}\tau)\\\
(\frac{\sigma}{2b})^{2}\\\ \end{array}\right).$ (38)
After calculating the tangent vectors $\vec{t}_{j}$ and the tetrads
$u^{a}_{j}$ that create the basis vectors $\vec{e}_{a}$, we calculate the
associated vector potential $A_{\tau}$ to verify that it gives the same
$z$-directed electric field of the negatively charged ring.
Fig. 6a shows the charged ring from the Grassmannian school with the choice of
$b=\frac{1}{2}$ m and $Q=2\pi$. The associated electric-field pictured
corresponds to a 1.24 $\mu$C ring when converted to SI units. Fig. 6b shows
the gauge subspace from the Kaluza-Klein school, and Fig. 6c shows the hidden-
spatial-metric picture.
Now let us carefully study these figures. The electric field in Eq.(36) is
constant in time. The corresponding gauge field (voltage) shown in Eq.(37) is
also independent of time. If we had chosen a different gauge, _e.g._ the
temporal gauge with $A_{0}=0$, then there would be a linear time dependence.
Is this time-independence a gauge artifact then, or is it part of the actual
physics about our charged ring? By using Fig. 6 which shows a _gauge-
invariant_ representation of the electric field, we can uncover the underling
gauge-invariant time dependence common to all three representations.
In Fig. 6a one sees the tangent planes repeating a pattern as the coordinate
$c\tau$ advances. Notice that this time dependency cannot be eliminated by a
gauge transformation or clever coordinate choice. This corresponds to the
periodicity of the cosine function with respect to $\tau$ in Eq.(38). The
embedding space is a fixed reference which enables one to see the gauge-
invariant phenomena.
From the Kaluza-Klein picture in Fig. 6b, we also see a repetition in the disk
arrangements in the $\tau$ direction. This again follows from to the
periodicity of the cosine function with respect to $\tau$ in Eq.(38). Notice
that at the metric-level, the terms which represent the gauge field
$g_{5t}\propto\,R\,A_{0}$ do not depend on the time coordinate $\tau$. When we
view the Kaluza-Klein surface from an embedding, we can see that the off-
diagonal terms follow from the time dependency of the orientation of the ring
parametrized by $x^{5}$ as viewed from the embedding.
In the hidden-spatial-geometry school shown in Fig. 6c, we see that time
dependence corresponds to an oscillation around a paraboloid shape. The
direction of increasing $\sigma$ is along the vertical dimension of the
paraboloid. As $\sigma$ increases the shape becomes flatter which corresponds
to distances farther from the ring (with a weaker electric field along the
axis). The $\tau$ direction is along the circular dimension of the paraboloid.
Our position on the paraboloid changes with time showing the hidden-time
dependence from another vantage.
Note that the charged ring has an explicit time dependence in all three gauge-
invariant geometric representations, as shown by $t=\tau$ in Eq. (38). The
time dependence disappears when we project to the scalar potential and the
electric field. Although the charged ring gives a static electric field, the
geometrical representations makes clear there is a hidden gauge-invariant time
dependence.
### 5.3 The Spherically Charged Shell
(a)
(b)
(c)
Figure 7: The geometry of a spherical charge for $q=4\pi$ and
$\omega=1\,{\rm{nm}}^{-1}$. (a): A Grassmannian school representation of the
charged sphere. (b): A hidden-spatial-geometry representation of the charged
sphere. (c): A Kaluza Klein representation which shows rings at each space-
time point for a spherical charge.
For a bounded scalar potential $A_{t}=\Phi(\vec{x})$ of a static electric
field, a funnel-shaped surface can be found for a given two-dimensional slice
of space-time. In this case,
$A_{\mu}=\left(\begin{array}[]{cccc}\Phi,&0,&0,&0\\\ \end{array}\right)$, and
$A_{\tau}=\frac{\partial t}{\partial\tau}A_{t}$ is a function of $\sigma$
only. The first derivative of $A_{0}$ must be strictly negative, and
$0<A_{\tau}\leq\omega$. If we use the shape ${}^{\ref{FootNoteX}}$
$\vec{X}=\left(\begin{array}[]{c}\frac{A_{\tau}(\sigma)}{\omega}\sin(\omega\,\tau)\\\
\frac{A_{\tau}(\sigma)}{\omega}\cos(\omega\,\tau)\\\
\sqrt{1-(\frac{A_{\tau}(\sigma)}{\omega})^{2}}+\ln(\frac{\frac{A_{\tau}(\sigma)}{\omega}}{1+\sqrt{1-(\frac{A_{\tau}(\sigma)}{\omega})^{2}}})\\\
\end{array}\right)$ (39)
with a $t(\tau)=\tau,x^{i}=x^{i}(\sigma)$ slice of space. The variable
$\omega$ represents a continuous class of geometries which give rise to a
single $A_{\tau}$. Notice that $\omega$ must be nonzero and larger than the
maximum value of $A_{\mu}$ in the given domain.
Consider a spherical shell of charge $q$ with radius smaller than
$q/4\pi\omega$, and let $x(\sigma)=\sigma,y=0,z=0$. The only nonzero component
of the field tensor, when looking strictly along the $x$-axis, is
$F_{tx}=E_{x}=\frac{q}{4\pi x^{2}}$. The pullback gives
$F_{\sigma\tau}=\frac{q}{4\pi\sigma^{2}}$ and $A_{\tau}=\frac{q}{4\pi\sigma}.$
Using Eq. (39), we find that the surface that is associated with the charged
sphere, looking along the $x$-axis, is ${}^{\ref{FootNoteX}}$
$\vec{X}=\left(\begin{array}[]{c}\frac{q}{4\omega\pi\sigma}\sin(\omega\,\tau)\\\
\frac{q}{4\omega\pi\sigma}\cos(\omega\,\tau)\\\
\sqrt{1-(\frac{q}{4\omega\pi\sigma})^{2}}+\ln(\frac{\frac{q}{4\omega\pi\sigma}}{1+\sqrt{1-(\frac{q}{4\omega\pi\sigma})^{2}}})\\\
\end{array}\right).$ (40)
Letting $q=4\pi$ and $\omega=1\,{\rm{nm}}^{-1}$, Fig. 7a shows the spherical
charge from the Grassmannian school, Fig. 7 b is the hidden-spatial-metric
picture, and Fig. 7c shows it from the Kaluza-Klein picture. In SI units this
corresponds to the field of a $2.2\times 10^{-17}$ C charge where $\sigma>1$
nm. From the reference Fig. 7a, we see that increasing $\sigma$ is down toward
the tip of the funnel and increasing $\tau$ is on the circular dimension. This
makes geometrical sense, as $\sigma$ increases, we move towards the narrow
throat of the funnel, and the shape gets more cylinder-like. Given that we
have potential $A_{\tau}=\frac{q}{4\pi\sigma},$ as we move farther from the
sphere, the weaker the field becomes, leading to a less curved surface.
Finally, note that the value of $\omega\ [>{\rm{max}}(A_{\tau})]$ is arbitrary
in this case. The time dependence of $\omega$, which is clearly present in all
three gauge-invariant geometrical representations, vanishes in the scalar
potential and electric field. Again, the geometric relationships make it clear
that there is a hidden gauge-invariant time dependence in the electric field
of the charged sphere.
Let us study Fig. 7 more carefully. We are again showing a time-independent
electric field, but we see time dependence when we dig down to the surfaces
that underlie the 1-form connection. The figures show a cutout in time. If we
had continued the figures towards larger $\tau$, you would again see a
periodic pattern for all three schools. In the Grassmannian representation
shown in Fig. 7a, let us look at $\sigma=0.004$ as we vary $\tau$. If we
continued $\tau$ towards larger values, we would see the disks complete a
complete cycle and repeat. A change of gauge will change the choice of red
basis vectors that span the disks, but the disks (the actual element of the
Grassmannian) remains unchanged. Fig. 7b shows the hidden-spatial-geometry
where we see the cutout of a funnel shape explicitly. The $\sin\omega\tau$ and
$\cos\omega\tau$ in Eq.(40) show that if we continued to plot points of larger
$\tau$, we would fill out the funnel and begin to repeat. Fig. 7c show the
rings that live on the same tangent plane as the Grassmannian school. All
three figures show a surface-like geometrical representation that gives rise
to the 1-form connection. All three figures use the embedding space so that
the geometrical objects (shape and disks) are independent of the gauge choice.
In all three representations, we can see the time dependence that is absent
(or ambiguous) in the 1-form connection. This does not mean that the figures
here are unique. The freedom to choose $\omega$ is an example of the many-to-
one map that is associated with this phenomena.
## 6 Discussion and Conclusion
The field of gauge theory geometry is vast. Fig. 1 shows the curvature 2-form
electric and magnetic fields as the layer with which most physicists are
familiar. By digging down to find the connection 1-form that gives rise to the
curvature, physicists discovered the Aharonov-Bohm effect. We wish to dig one
layer deeper. We have grouped into three schools the past efforts to find a
surface-like layer that would give rise to the connection 1-form. Our paper
shows the dotted-red line connections between these past efforts.
We have shown how these three representations of gauge theory that isolate
gauge-invariant surface-like structures are related geometrically without
appealing to their common gauge-field image space. For the Kaluza-Klein school
every point in space-time has a bundled-up fifth dimension. With an immersion
given by Eq.(12) that inserts a ring on the Grassmannian school’s tangent-
planes, we can recover the Kaluza-Klein metric and visualize this 5th
dimension. The Grassmannian school uses vector bundles to describe gauge
fields. We can visualize the subspace represented by these disks at each point
in space-time using an embedding space inside at each space-time point.
Finally, by combining the embedding with a shape unearthed in the hidden-
spatial-metric school, we can associate a spatial geometry with a gauge field
configuration.
The similarities are deep. All three schools share a common gauge-invariant
tangent plane. The wave function and projection operator are invariants of the
gauge transformations. This is because gauge transformations correspond to a
rotation of the basis vectors that leave the tangent-plane unchanged. This
tangent-plane is the same plane in both the Grassmannian and hidden-spatial-
metric schools. Gauge transformations in Kaluza-Klein theory change the $0$
point of the $x^{5}$ coordinate on the ring defined in Eq.(12). The tetrads in
the hidden-spatial metric school change the coordinate basis vectors of the
hidden spatial geometry to orthonormal basis vectors on the gauge fiber. The
Kaluza-Klein ring lies along the gauge fiber. There are also similarities
related to charge: in the Grassmannian and hidden-spatial-metric schools a
positive or negative charge corresponds to a clockwise or counterclockwise
rotation of a matter field vector on the gauge fiber, and in the Kaluza-Klein
school a positive or negative charge corresponds to opposite movement along
the ring (which lies along the gauge fiber). The opposite charge corresponds
to reverse rotation in all three schools.
Each of the three schools considered in this paper uses a gauge-invariant
surface-like geometrical structure to induce the gauge field. Normally we must
work very hard to isolate what is gauge-invariant. For example, Killing-vector
methods identify symmetries that can help reveal gauge-invariant physical
results like conservation laws. We in contrast are digging down to reach the
gauge-invariant surfaces which induce the gauge-dependent gauge field. No
Killing-vector-like approach is needed. In each of the three schools, one
might have thought the resulting metric or Grassmannian representation was a
special mathematical trick, and didn’t say something very profound. However we
have shown how the surface-like structures underlying each school are related
to each other. This suggests that perhaps results regarding this surface-like
foundation have some meaning. As of now, they are just representations so no
new physics should be present. However, our new understanding of the mappings
between the representations may help us “better guess” new physical laws
feynman1994character .
In section 2.4 we showed how previously published work indicated a time
dependence for electric fields when represented in the Grassmannian school.
The mappings we provide between the three schools show that this gauge-
invariant time dependence exists for all three schools. This agreement
suggests the hidden time dependence should be taken more seriously and may be
a new feature at the foundation of physics.
In section 5, we showed examples of what this time dependence ‘looks’ like in
static electric fields. In the charged ring and the charged sphere, we show
that even static electric fields have a hidden, gauge-invariant time
dependence in the surface-like structures underling the gauge-field. In both
cases, it is a time-harmonic repeating wobble in the underlying surface. Is
this time dependence physical or an artifact of the geometric representation?
The surface-like structure is common to all three schools. The surface-like
structure does not change with a gauge transformation. The time-dependence in
the surface-like structures are therefore not an artifact of the
parametrization or the coordinate system choice.
If we take the generalized time-dependent gauge-invariant features here
seriously, there are several questions to pursue in future research. What is
the underlying surface that is moving in the case of gauge theories. What is
the time-dependent source of the electric field’s time dependence? The wave
function has a time dependent phase. It would causally make sense if the wave
function’s time dependence was the source for the electric field’s time
dependence, but these two time dependencies are currently uncorrelated. We
suspect that resolving this tension will force a deeper form of Guass’s Law.
Are there any new observable consequences to this deeper Gauss’s Law? A deeper
form of Gauss’s law may also help address the mysteries of the mass gap in
Yang-Mills theory.
Another area of development lies within the study of instanton and other less-
well-known semi-classical solutions. Many of the papers previously published
in each school were geared towards identifying and studying instanton
solutions. By using the mappings we identify, checking and categorizing
instanton solutions may become easier, and it would be another tool for
finding other semi-classical solutions that would need to be included in the
path integral quantization.666The authors would like to thank Ricardo Schiappa
for highlighting these research directions.
###### Acknowledgements.
The authors would like to thank Laura Serna, Kevin Cahill, Richard Cook, Matt
Robinson, Christian Wohlwend, Ricardo Schiappa, and Yang-Hui He for helpful
comments after reviewing the manuscript. We would also like to thank the
reviewers for helpful contributions increasing the quality of the final paper.
The views expressed in this paper are those of the authors and do not reflect
the official policy or position of the United States Air Force, Department of
Defense, or the US Government. DISTRIBUTION A: Approved for public release.
Distribution unlimited.
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## Appendix A Appendix: Variable definitions reference
$\vec{e}_{a}$ or $e_{a}^{j}$ | The basis vector for the Grassmannian school. The $a$ coordinate is an internal ‘color’ index. If there is a Latin index like $j$, it refers to the embedding space. Forms a rectangular matrix.
---|---
$\vec{t}_{\mu}$ or $t_{\mu}^{j}$ | Coordinate tangent vector for the coordinate $x^{\mu}$. Used in defining a metric. If there is a Latin index like $j$, it refers to the embedding-space dimension. Forms a rectangular matrix.
$u^{a}_{j}$ | The tetrad of the hidden-spatial-geometry school. Notice the $a$ index specifies the ‘frame’ in color space and $j$ is a ‘frame’ in a slice of space-time. This maps the color index $a$ to the space-time coordinate tangent vector $j$ of a spatial metric which represents the gauge field and corresponding electric and magnetic fields. Must be a square matrix.
$\vec{X}$ or $X^{j}$ | Is the generic vector used to denote an explicit isometric embedding which will be used to induce a metric. The Latin index $j$ refers to the embedding space.
$\phi^{a}$ | Coefficients of the basis element $\vec{e}_{a}$ which specify a vector in color-space. $\phi^{a}$ changes with a gauge transformation but the vector $\vec{\phi}=\phi^{a}\vec{e}_{a}=\phi^{\prime\,b}\vec{e}^{\prime}_{b}$ is gauge invariant.
$a,b,c$ | Lower-case Latin letters near the beginning of the alphabet will be gauge-theory color indices
$\mu,\nu,...$ | Greek letters will be space-time coordinates
$A,B,...$ | Upper-case Latin letters will be used for Kaluza-Klein metric indices. Kaluza-Klein index values 0 through 3 are the usual space-time coordinates $t,x,y,z$ and the index value 5 is the fifth dimension coordinate $x^{5}$, which is used to parameterize the tiny compact dimension.
$i,j,...$ | Variables corresponding to subspaces of space-time and the embedding dimensions, where context will keep them distinct.
|
arxiv-papers
| 2013-08-05T20:00:01 |
2024-09-04T02:49:49.039592
|
{
"license": "Public Domain",
"authors": "Scott T Alsid and Mario A Serna",
"submitter": "Mario A. Serna Jr",
"url": "https://arxiv.org/abs/1308.1092"
}
|
1308.1262
|
# Pattern recognition issues on anisotropic smoothed particle hydrodynamics
Eraldo Pereira Marinho Univ Estadual Paulista (UNESP/IGCE), Department of
Computing, Applied Mathematics and Statistics [email protected]
###### Abstract.
This is a preliminary theoretical discussion on the computational requirements
of the state of the art smoothed particle hydrodynamics (SPH) from the optics
of pattern recognition and artificial intelligence. It is pointed out in the
present paper that, when including anisotropy detection to improve resolution
on shock layer, SPH is a very peculiar case of unsupervised machine learning.
On the other hand, the free particle nature of SPH opens an opportunity for
artificial intelligence to study particles as agents acting in a collaborative
framework in which the timed outcomes of a fluid simulation forms a large
knowledge base, which might be very attractive in computational astrophysics
phenomenological problems like self-propagating star formation.
###### Key words and phrases:
agents - anisotropy - density estimator - SPH - k-NN
###### 1991 Mathematics Subject Classification:
Artificial Intelligence on Computational Fluids Dynamics
## 1\. Introduction
Smoothed particle hydrodynamics (SPH) has been a successful computer
simulation paradigm originated in computational astrophysics since 1977 [2,
5]. Nowadays SPH is used in other areas and has gained significant improvement
in accuracy and stability, not only on simulating compressible shock as also
on performing high resolution incompressible fluid, solids etc, e.g. [4].
One essential issue in SPH is anisotropy, which arises naturally on performing
adaptive interpolation, e.g. [7], where the dimensionality reduction to detect
critical surfaces, as shock layers might be included. Anisotropy is an
important subject in pattern recognition for feature extraction methods [1].
There are encouraging works on the application of artificial intelligence to
reproduce real systems, mainly in the context of complex systems as in
economics and many other social and life sciences, e.g. [3]. The collective
phenomena of star formation, unveiled in the intermittent pattern in the
spiral galaxies arms, have been proposed by means of the stochastic self-
propagating star formation model, in the sense that star formation is
contagious, e.g. [8].
The present work is a brief discussion on some aspects of SPH circumstanced by
the concepts of pattern recognition and artificial intelligence. Several
details are omitted to fulfill the limited space with no significative lost of
focus on reviewing the theory in the context of intelligent computing.
## 2\. SPH database and space representation
The SPH database comprises $N$ instances, usually thousand hundreds or even
millions particles, indexed by a descriptor table $\mathcal{P}_{N}$. Each
particle is addressed by a unique label, or descriptor $i$, and as much as
possible the particle object is referred as just $i$ or $i$-particle. Any
particle attribute, say $A$, is addressed by sub-indexing the same with the
particle label, $A_{i}$.
Since a specific SPH problem is headed by the mathematical, physical and
computational models, the adopted methods might be included in the database in
the form of classes and module libraries, described in a commonly used data
model language as for instance the XML, which may also improve the information
interchange between different SPH simulation bases.
The usual 3D space description in SPH uses hierarchical spatial tessellation,
as for instance by means of octrees, e.g. [6], which are the 3D version of
quadtrees. Other tree-based spatial tessellation schemes are also adopted. For
instance [7] proposes an approach to easily adapt earlier versions of octree-
based SPH codes to covariance-based octree tessellation to improve anisotropic
kernel computations, e.g. [11].
## 3\. Starring pattern recognition and AI in SPH
The $k$-nearest neighbor is a mathematical relation
$\mathcal{N}_{k}\subseteq\mathcal{P}_{N}\times\mathcal{P}_{N}$, which
associates $i\in\mathcal{P}_{N}$ with a subset
$\mathcal{N}_{k}(i)=\\{i_{1},\ldots,i_{k}\\}\subseteq\mathcal{P}_{N}$, so that
$j\in\mathcal{N}_{k}(i)$ if, and only if, the adopted distance
$d(\vec{x}_{i},\vec{x}_{j})$ from $i$ to $j$ obeys the inequality
$d(\vec{x}_{i},\vec{x}_{j})\leq\max\\{d(\vec{x}_{i},\vec{x}_{l})|\,l\in\mathcal{N}_{k}(i)\\}$.
The KNN algorithm is the method by which the relation $\mathcal{N}_{k}$ is
populated by ordered pairs $(i,j)$ in
$\subseteq\mathcal{P}_{N}\times\mathcal{P}_{N}$, given the particle-descriptor
table $\mathcal{P}_{N}$.
The $\mathcal{N}_{k}$ relation is asymmetric and reflexive. The later comes
from the fact that each point is the nearest neighbor of itself – this is
called improper neighbor. The former comes from the fact that if $a$ is the
proper nearest neighbor (not a reflection) of $b$, not necessarily $b$ is the
nearest neighbor of $a$. For example, $a$ could be closer to a third point $c$
than $b$, which is too faraway from any other point but $a$.
The KNN asymmetry reflects imperfections on writing simpler forms of the SPH
conservation equations, which require particle commutation symmetry. To
workaround the asymmetry issue is necessary to introduce the symmetric closure
of the KNN relation, which is known as effective neighbors
(3.1)
$\mathcal{E}_{k}=\mathcal{N}_{k}\cup\\{(i,j)\in\mathcal{P}_{N}\times\mathcal{P}_{N}\;|\;(j,i)\in\mathcal{N}_{k}\\}.$
Of course, the KNN algorithm requires a predesign metric in the 3D space. If
the metric is invariant under rotation, the KNN relation is isotropic. On the
other hand, the metric relation is said anisotropic. For instance the
Mahalanobis metric, as adopted in the KNN algorithm proposed by [7], is
anisotropic and is used to reveal biased structures like the arms in spiral
galaxy images.
The Mahalanobis distance $\xi_{ij}$ is defined in terms of the covariance
tensor $\mathbf{\Sigma}$:
(3.2)
$\xi_{ij}=(\vec{x}_{i}-\vec{x}_{j})\mathbf{\Sigma}^{-1}(\vec{x}_{i}-\vec{x}_{j})^{\mathrm{T}},$
where $(\vec{x}_{i}-\vec{x}_{j})^{\mathrm{T}}$ is the transpose of the matrix
representation of the relative position vector $(\vec{x}_{i}-\vec{x}_{j})$.
Of course, equation (3.2) is not the only way of defining anisotropic distance
in SPH. For instance, the positive-definite stress tensor $\mathbf{T}$ might
be eventually used to define the non-normalized anisotropic distance
$\xi_{ij}$:
(3.3)
$\xi_{ij}=(\vec{x}_{i}-\vec{x}_{j})\mathbf{T}^{-1}(\vec{x}_{i}-\vec{x}_{j})^{\mathrm{T}}.$
According to equations (3.2) or (3.3), the outermost boundary for the
$k$-nearest neighbors of the $i$-particle is an ellipsoid centered in the
query position $\vec{x}_{i}$, whose principal axes are set by the respective
tensor eigenvectors [7].
A cognitive interpretation for the well-known SPH interpolation formula can be
illustrated as follows: given an $i$-labeled particle, say $i$-particle, one
may suppose this particle has to make an estimation, $\tilde{A}_{i}$, of a
local fluid quantity, $A_{i}$, after hearing votes, e.g. [1], from its
effective neighbors what impression they get regarding the same quantity.
A democratic decision is made if the $i$-particle weights the individual
suggestions from its informants, giving more importance to the closest ones.
The importance, or weight, comes from a compact-support smoothing kernel,
which drops to zero outside the influence zone defined by the effective
neighbors and grows up as gets closer to $i$, reaching its maximum for $i$
itself.
Each $i$-particle, $i=1,\ldots,N$, has its own effective neighbors,
$\mathcal{E}_{k}(i)$. The $i$-particle asks each $j$-particle in
$\mathcal{E}_{k}(i)$ for suggestions, which answers accordingly to the
predefined protocol, $A_{j}{m_{j}}/{\rho_{j}}$, whose reliability is expressed
by a weight, or smoothing kernel $W_{ij}$.
The $i$-particle gets a conclusive perception $\tilde{A}_{i}$ from its
locality by adding together all of the weighted votes,
$W_{ij}A_{j}{m_{j}}/{\rho_{j}}$, received from its $k$-nearest neighbors:
(3.4) $\tilde{A}_{i}=\sum_{\forall
j\in\,\mathcal{E}_{k}(i)}W_{ij}A_{j}\frac{m_{j}}{\rho_{j}},$
where $W_{ij}=W(\vec{x}_{i}-\vec{x}_{j})$ is the smoothing kernel, whose
analytical profile might be an issue regarding accuracy and stability on SPH
simulations, but this particular subject will not be discussed here.
Similar election procedure applies on estimating the interpolated gradient,
$\vec{\nabla}_{i}{A}_{i}$, yielding
(3.5) $\vec{\nabla}_{i}{A}_{i}=\sum_{\forall
j\in\,\mathcal{E}_{k}(i)}\vec{\nabla}_{i}{W}_{ij}A_{j}\frac{m_{j}}{\rho_{j}},$
where $\vec{\nabla}_{i}{W}_{ij}=\vec{\nabla}_{i}{W}(\vec{x}_{i}-\vec{x}_{j})$
is the smoothing-kernel gradient.
If the kernel is symmetric, one finds from the effective neighbors symmetry
that $i\in\mathcal{E}_{k}(j)\Leftrightarrow j\in\mathcal{E}_{k}(i)$, and also
finds $W_{ij}=W_{ji}\neq 0$, and
$\vec{\nabla}_{i}{W}_{ij}=-\vec{\nabla}_{j}{W}_{ji}$ if and only if
$(i,j)\in\mathcal{E}_{k}$.
Densities are required to perform SPH interpolations, as in equations (3.4)
and (3.5), and they are estimated from equation (3.4) itself by means of a
self-consistent replacement $A_{j}\rightarrow\rho_{j}$, yielding
(3.6) $\tilde{\rho}_{i}=\sum_{\forall
j\in\,\mathcal{E}_{k}(i)}W_{ij}{m_{j}}=\rho_{i}.$
The SPH fluid equations of motion are derived from the actual fluid equations,
and they must be solved by means of some integration scheme regarding accuracy
and stability. The timed outcomes from the integration scheme express discrete
states of the particle description. Depending on the time-integration method,
each particle knows a brief history of its previous states.
The way as the SPH equations are presented usually requires rearrangement to
attend to subsidiary information concerning physics, chemistry etc. For
instance, in most astrophysical problems, the SPH momentum conservation
equation can be written as
(3.7) $\frac{\mathrm{d}\vec{v}_{i}}{\mathrm{d}t}=-\sum_{\forall
j\in\,{\mathcal{E}_{k}(i)}}\vec{\nabla}_{i}W_{ij}\Pi_{ij}+\vec{F}_{i},$
where the $\Pi_{ij}$-factor carries the pressure coefficients, which might
even include anisotropic pressure as the elastic stress tensor and the Maxwell
stress tensor, e.g. [6]. The $\vec{F}_{i}$-vector term is a non-hydrodynamic
acceleration as for instance the gravity field $\vec{F}_{i}=-\vec{g}_{i}$ on
$i$-particle.
Time-integration scheme plays the role of particle actuators modifying their
local environment, in response to the information received from their
effective neighbors. Every particle contributes to a global knowledge, which
might attend to a subsidiary simulation, as for example the qualitative
results of self-propagating star formation, e.g. [8], the SPH-data history
constitutes a knowledge-base [10] or even a more pretentious context as in
live tissue simulations [9].
Each SPH particle recognizes its surroundings by means of its effective
neighbors using pattern detection techniques to identify the neighborhood
morphology and consequent critical surfaces. However, particles obey a set of
transition rules, according to the physics model, to decide what action they
have to do against their local environment.
From the theory of intelligent agents, SPH particles might be classified as
simple reflex agents [10], acting as environment modifiers in function of what
they percept in their surrounds through their effective neighbors. The
particles act under the local physical conditions in response to the input
they receive from their effective neighbors, ignoring the long term history of
all their actions and percepts. Regarding the adopted time integration scheme
embedded as actuators, only the knowledge of a recent past is required.
## 4\. Conclusion
More than a numerical simulation technique, SPH is a very complex system that
can be studied not only under applied mathematics techniques but also under
the light of intelligent computing, where particles are individuals
cooperatively working in behalf of a collective objective of mimicking the
fluid behavior.
The SPH spirit resides in computationally reproducing the continuous fluid
flow using free particles. A fluid particle moves like a marker, accordingly
to the lagrangian equations of motion. Each particle is a data structure
storing the specific fluid properties as density, pressure, position, velocity
etc. Any particle knows its surroundings through its $k$-nearest neighbors
(KNN), which play the role of sensors, or informants. The information
mechanism is known as KNN-based kernel interpolation, which might be
interpreted as a weighted voting, from the machine learning viewpoint.
## References
* [1] R. O. Duda and P. E. Hart, _Pattern Classification and Scene Analysis_ , New York: Wiley & Sons, 1973.
* [2] R. A. Gingold and J. J. Monaghan, _Smoothed particle hydrodynamics - Theory and application to non-spherical stars_ , Monthly Noticies of the Royal Astronomical Society 181 (1977), 375–389.
* [3] Volker Grimm, Eloy Revilla, Uta Berger, Florian Jeltsch, Wolf M. Mooij, Steven F. Railsback, Hans-Hermann Thulke, Jacob Weiner, Thorsten Wiegand, and Donald L. DeAngelis, _Pattern-oriented modeling of agent-based complex systems: Lessons from ecology_ , Science Vol. 310 (2005), no. 5750, 987–991.
* [4] X. Guo, Y. Wei, Z. Jin, and D. Guo, _Simulation research on diamond cutting of mold steel using SPH method_ , Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 8202, November 2011.
* [5] L. B. Lucy, _A numerical approach to the testing of the fission hypothesis_ , Astronomical Journal 82 (1977), 1013–1024.
* [6] E. P. Marinho, C. M. Andreazza, and J. R. D. Lépine, _SPH simulations of clumps formation by dissipative collisions of molecular clouds_ , Astronomy and Astrophysics 379 (2001), no. 3, 1123–1137.
* [7] Eraldo P. Marinho and Carmen M. Andreazza, _Mecánica Computacional. Computational Geometry (A)_ , vol. XXIX, Mecánica Computacional, no. 60, ch. Anisotropic K-nearest Neighbor Search Using Covariance Quadtree, pp. 6045–6064, Asociación Argentina de Mecánica Computacional, Buenos Aires, Argentina, November 2010, Open Journal Systems.
* [8] T. Mineikis and V. Vansevičius, _Disk Galaxy Models Driven by Stochastic Self-Propagating Star Formation_ , Baltic Astronomy 19 (2010), 111–120.
* [9] Matthias Müller, Simon Schirm, and Matthias Teschner, _Interactive blood simulation for virtual surgery based on smoothed particle hydrodynamics_ , Technol. Health Care 12 (2004), no. 1, 25–31.
* [10] Stuart Russell and Peter Norvig, _Artificial intelligence: A modern approach, 3rd edition_ , 3rd ed., Prentice Hall Copyright ©2010, Dec 1 2009, ISBN-10: 0-13-604259-7, ISBN-13: 978-0-13-604259-4, Format: Cloth.
* [11] Thanh N. Tran, Ron Wehrens, and Lutgarde M.C. Buydens, _Knn-kernel density-based clustering for high-dimensional multivariate data_ , Computational Statisticas & Data Analysis 51 (2006), 513–525.
|
arxiv-papers
| 2013-08-06T13:04:28 |
2024-09-04T02:49:49.055297
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eraldo Pereira Marinho",
"submitter": "Eraldo Marinho",
"url": "https://arxiv.org/abs/1308.1262"
}
|
1308.1277
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-147 LHCb-PAPER-2013-034 October 8, 2013
Branching fraction and CP asymmetry of the decays $B^{+}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}\pi^{+}$ and $B^{+}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}K^{+}$
The LHCb collaboration†††Authors are listed on the following pages.
An analysis of $B^{+}\rightarrow K_{\rm\scriptscriptstyle S}^{0}\pi^{+}$ and
$B^{+}\rightarrow K_{\rm\scriptscriptstyle S}^{0}K^{+}$ decays is performed
with the LHCb experiment. The $pp$ collision data used correspond to
integrated luminosities of $1\mbox{\,fb}^{-1}$ and $2\mbox{\,fb}^{-1}$
collected at centre-of-mass energies of
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and
$\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}$, respectively. The ratio of
branching fractions and the direct CP asymmetries are measured to be
$\mathcal{B}(B^{+}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}K^{+})/\mathcal{B}(B^{+}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}\pi^{+})=0.064\pm 0.009\textrm{ (stat.)}\pm 0.004\textrm{ (syst.)}$,
$\mathcal{A}^{\it CP}(B^{+}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}\pi^{+})=-0.022\pm 0.025\textrm{ (stat.)}\pm 0.010\textrm{ (syst.)}$ and
$\mathcal{A}^{\it CP}(B^{+}\rightarrow K_{\rm\scriptscriptstyle
S}^{0}K^{+})=-0.21\pm 0.14\textrm{ (stat.)}\pm 0.01\textrm{ (syst.)}$. The
data sample taken at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ is used to
search for $B_{c}^{+}\rightarrow K_{\rm\scriptscriptstyle S}^{0}K^{+}$ decays
and results in the upper limit $(f_{c}\cdot\mathcal{B}(B_{c}^{+}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}K^{+}))/(f_{u}\cdot\mathcal{B}(B^{+}\rightarrow
K_{\rm\scriptscriptstyle S}^{0}\pi^{+}))<5.8\times 10^{-2}\textrm{ at 90\%
confidence level}$, where $f_{c}$ and $f_{u}$ denote the hadronisation
fractions of a $\bar{b}$ quark into a $B_{c}^{+}$ or a $B^{+}$ meson,
respectively.
Submitted to Phys. Lett. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S. Farry51, D.
Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S.
Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R.
Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph.
Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30,
A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Hess60, A.
Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-
March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, J. Maratas5, U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11,
G. Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D.
Martinez Santos41, D. Martins Tostes2, A. Martynov31, A. Massafferri1, R.
Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J.
McCarthy44, A. McNab53, R. McNulty12, B. McSkelly51, B. Meadows56,54, F.
Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, A. Mordà6, M.J.
Morello22,s, R. Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28,
B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1,
M. Needham49, S. Neubert37, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C.
Nguyen-Mau38,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A.
Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S.
Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M.
Otalora Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T.
Palczewski27, M. Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C.
Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N.
Patrick48, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez
Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L.
Pescatore44, E. Pesen61, K. Petridis52, A. Petrolini19,i, A. Phan58, E.
Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M.
Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A.
Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54, J.
Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38,
G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47,
A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35, G.
Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d,
V. Salustino Guimaraes2, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, K. Senderowska26, I.
Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42,
P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L. Shekhtman33, O.
Shevchenko42, V. Shevchenko30, A. Shires9, R. Silva Coutinho47, M. Sirendi46,
N. Skidmore45, T. Skwarnicki58, N.A. Smith51, E. Smith54,48, J. Smith46, M.
Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro18, D. Souza45, B. Souza De
Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O.
Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58, B. Storaci39, M.
Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S. Swientek9, V.
Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S. T’Jampens4,
M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D. Tonelli37, S. Topp-
Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M.
Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37,
A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P. Vazquez
Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S.
Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, S.A. Wotton46, S.
Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, R. Young49,
X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
Studies of charmless two-body $B$ meson decays allow tests of the Cabibbo-
Kobayashi-Maskawa picture of $C\\!P$ violation [1, 2] in the Standard Model
(SM). They include contributions from loop amplitudes, and are therefore
particularly sensitive to processes beyond the SM [3, 4, 5, 6, 7]. However,
due to the presence of poorly known hadronic parameters, predictions of
$C\\!P$ violating asymmetries and branching fractions are imprecise. This
limitation may be overcome by combining measurements from several charmless
two-body $B$ meson decays and using flavour symmetries [3]. More precise
measurements of the branching fractions and $C\\!P$ violating asymmetries will
improve the determination of the size of SU(3) breaking effects and the
magnitudes of colour-suppressed and annihilation amplitudes [8, 9].
In $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ and
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays,111The
inclusion of charge conjugated decay modes is implied throughout this Letter
unless otherwise stated. gluonic loop, colour-suppressed electroweak loop and
annihilation amplitudes contribute. Measurements of their branching fractions
and $C\\!P$ asymmetries allow to check for the presence of sizeable
contributions from the latter two [6]. Further flavour symmetry checks can
also be performed by studying these decays [10]. First measurements have been
performed by the BaBar and Belle experiments [11, 12]. The world averages are
${\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\right)=-0.015\pm 0.019$, ${\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)=0.04\pm 0.14$ and ${\cal
B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)/{\cal
B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\right)=0.050\pm 0.008$, where
$\displaystyle{\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma\left(B^{-}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{-}\right)-\Gamma\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\right)}{{\Gamma\left(B^{-}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{-}\right)+\Gamma\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)}}$ (1)
and ${\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}\right)$ is defined in an analogous way.
Since the annihilation amplitudes are expected to be small in the SM and are
often accompanied by other topologies, they are difficult to determine
unambiguously. These can however be measured cleanly in
$B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays, where
other amplitudes do not contribute. Standard Model predictions for the
branching fractions of pure annihilation $B_{c}^{+}$ decays range from
$10^{-8}$ to $10^{-6}$ depending on the theoretical approach employed [13].
In this Letter, a measurement of the ratio of branching fractions of
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ and
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays with the
LHCb detector is reported along with a determination of their $C\\!P$
asymmetries. The data sample corresponds to integrated luminosities of 1 and
2$\mbox{\,fb}^{-1}$, recorded during 2011 and 2012 at centre-of-mass energies
of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$, respectively. A search for the pure
annihilation decay $B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}$ based on the data collected at 7$\mathrm{\,Te\kern-1.00006ptV}$ is
also presented. The $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$
and $B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ signal
regions, along with the raw $C\\!P$ asymmetries, were not examined until the
event selection and the fit procedure were finalised.
## 2 Detector, data sample and event selection
The LHCb detector [14] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream.
The magnetic field polarity is regularly flipped to reduce the effect of
detection asymmetries. The $pp$ collision data recorded with each of the two
magnetic field polarities correspond to approximately half of the data sample.
The combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter
resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors [15]. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers.
Simulated samples are used to determine efficiencies and the probability
density functions (PDFs) used in the fits. The $pp$ collisions are generated
using Pythia 6.4 [16] with a specific LHCb configuration [17]. Decays of
hadronic particles are described by EvtGen [18], in which final state
radiation is generated using Photos [19]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [20, *Agostinelli:2002hh] as described in Ref. [22].
The trigger [23] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which performs a
full event reconstruction. The candidates used in this analysis are triggered
at the hardware stage either directly by one of the particles from the $B$
candidate decay depositing a transverse energy of at least
$3.6\mathrm{\,Ge\kern-1.00006ptV}$ in the calorimeters, or by other activity
in the event (usually associated with the decay products of the other
$b$-hadron decay produced in the $pp\rightarrow b\overline{}bX$ interaction).
Inclusion of the latter category increases the acceptance of signal decays by
approximately a factor two. The software trigger requires a two- or three-
particle secondary vertex with a high scalar sum of the $p_{\rm T}$ of the
particles and significant displacement from the primary $pp$ interaction
vertices (PVs). A multivariate algorithm [24] is used for the identification
of secondary vertices consistent with the decay of a $b$ hadron.
Candidate $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays are formed
by combining a $K^{0}_{\rm\scriptscriptstyle S}\\!\rightarrow\pi^{+}\pi^{-}$
candidate with a charged track that is identified as a pion or kaon,
respectively. Only tracks in a fiducial volume with small detection
asymmetries [25] are accepted in the analysis. Pions used to reconstruct the
$K^{0}_{\rm\scriptscriptstyle S}$ decays are required to have momentum
$\mbox{$p$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $\chi^{2}_{\rm IP}>9$, and
track segments in the VELO and in the downstream tracking chambers. The
$\chi^{2}_{\rm IP}$ is defined as the difference in $\chi^{2}$ of a given PV
reconstructed with and without the considered particle. The
$K^{0}_{\rm\scriptscriptstyle S}$ candidates have
$\mbox{$p$}>8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $\mbox{$p_{\rm
T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, a good quality vertex fit, a
mass within $\pm 15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known
value [26], and are well-separated from all PVs in the event. It is also
required that their momentum vectors do not point back to any of the PVs in
the event.
Pion and kaon candidate identification is based on the information provided by
the RICH detectors [15], combined in the difference in the logarithms of the
likelihoods for the kaon and pion hypotheses ($\mathrm{DLL}_{K\pi}$). A track
is identified as a pion (kaon) if $\mathrm{DLL}_{K\pi}\leq 3$
($\mathrm{DLL}_{K\pi}>3$), and
$\mbox{$p$}<110{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, a momentum beyond which
there is little separation between pions and kaons. The efficiencies of these
requirements are 95% and 82% for signal pions and kaons, respectively. The
misidentification probabilities of pions to kaons and kaons to pions are 5%
and 18%. These figures are determined using a large sample of
$D^{*+}\\!\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ decays
reweighted by the kinematics of the simulated signal decays. Tracks that are
consistent with particles leaving hits in the muon detectors are rejected.
Pions and kaons are also required to have $\mbox{$p_{\rm
T}$}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}>2$.
The $B$ candidates are required to have the scalar $p_{\rm T}$ sum of the
$K^{0}_{\rm\scriptscriptstyle S}$ and the $\pi^{+}$ (or $K^{+}$) candidates
that exceeds $4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, to have $\chi^{2}_{\rm
IP}<10$ and $\mbox{$p$}>25{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and to form a
good-quality vertex well separated from all the PVs in the event and displaced
from the associated PV by at least $1\rm\,mm$. The daughter
($K^{0}_{\rm\scriptscriptstyle S}$ or $\pi^{+}$/$K^{+}$) with the larger
$p_{\rm T}$ is required to have an impact parameter above $50\,\upmu\rm m$.
The angle $\theta_{\textrm{dir}}$ between the $B$ candidate’s line of flight
and its momentum is required to be less than 32$\rm\,mrad$. Background for
$K^{0}_{\rm\scriptscriptstyle S}$ candidates is further reduced by requiring
the $K^{0}_{\rm\scriptscriptstyle S}$ decay vertex to be significantly
displaced from the reconstructed $B$ decay vertex along the beam direction
($z$-axis), with $S_{z}\equiv(z_{K^{0}_{\rm\scriptscriptstyle
S}}-z_{B})/\sqrt{\sigma^{2}_{z,K^{0}_{\rm\scriptscriptstyle
S}}+\sigma^{2}_{z,B}}>2$, where $\sigma^{2}_{z,K^{0}_{\rm\scriptscriptstyle
S}}$ and $\sigma^{2}_{z,B}$ are the uncertainties on the $z$ positions of the
$K^{0}_{\rm\scriptscriptstyle S}$ and $B$ decay vertices
$z_{K^{0}_{\rm\scriptscriptstyle S}}$ and $z_{B}$, respectively.
Boosted decision trees (BDT) [27] are trained using the AdaBoost algorithm[28]
to further separate signal from background. The discriminating variables used
are the following: $S_{z}$; the $\chi^{2}_{\rm IP}$ of the
$K^{0}_{\rm\scriptscriptstyle S}$ and $\pi^{+}$/$K^{+}$ candidates; $p_{\rm
T}$, $\cos(\theta_{\textrm{dir}})$, $\chi^{2}_{\rm VS}$ of the $B$ candidates
defined as the difference in $\chi^{2}$ of fits in which the $B^{+}$ decay
vertex is constrained to coincide with the PV or not; and the imbalance of
$p_{\rm T}$, $A_{\mbox{$p_{\rm T}$}}\equiv(\mbox{$p_{\rm
T}$}(B)-\sum{\mbox{$p_{\rm T}$}})/(\mbox{$p_{\rm T}$}(B)+\sum{\mbox{$p_{\rm
T}$}})$ where the scalar $p_{\rm T}$ sum is for all the tracks not used to
form the $B$ candidate and which lie in a cone around the $B$ momentum vector.
This cone is defined by a circle of radius 1 unit in the pseudorapidity-
azimuthal angle plane, where the azimuthal angle is measured in radians.
Combinatorial background tends to be less isolated with smaller $p_{\rm T}$
imbalance than typical $b$-hadron decays. The background training samples are
taken from the upper $B$ invariant mass sideband region in data
($5450<m_{B}<5800{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$), while those of
the signal are taken from simulated $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays. Two discriminants are
constructed to avoid biasing the background level in the upper $B$ mass
sideband while making maximal use of the available data for training the BDT.
The $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and $K^{0}_{\rm\scriptscriptstyle
S}K^{+}$ samples are merged to prepare the two BDTs. They are trained using
two independent equal-sized subsamples, each corresponding to half of the
whole data sample. Both BDT outputs are found to be in agreement with each
other in all aspects and each of them is applied to the other sample. For each
event not used to train the BDTs, one of the two BDT outputs is arbitrarily
applied. In this way, both BDT discriminants are applied to equal-sized data
samples and the number of events used to train the BDTs is maximised without
bias of the sideband region and the simulated samples used for the efficiency
determination. The choice of the requirement on the BDT output (${\cal Q}$) is
performed independently for the $K^{0}_{\rm\scriptscriptstyle S}\pi^{\pm}$ and
$K^{0}_{\rm\scriptscriptstyle S}K^{\pm}$ samples by evaluating the signal
significance $N_{\rm S}/\sqrt{N_{\rm S}+N_{\rm B}}$, where $N_{\rm S}$
($N_{\rm B}$) denotes the expected number of signal (background) candidates.
The predicted effective pollution from mis-identified $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays in the $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ signal mass region is taken into account
in the calculation of $N_{\rm B}$. The expected signal significance is
maximised by applying ${\cal Q}>0.4$ (0.8) for $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ ($B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$) decays.
## 3 Asymmetries and signal yields
The $C\\!P$-summed $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$
and $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ yields are
measured together with the raw charge asymmetries by means of a simultaneous
unbinned extended maximum likelihood fit to the $B^{\pm}$ candidate mass
distributions of the four possible final states ($B^{\pm}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{\pm}$ and $B^{\pm}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{\pm}$). Five components contribute to each
of the mass distributions. The signal is described by the sum of a Gaussian
distribution and a Crystal Ball function (CB) [29] with identical peak
positions determined in the fit. The CB component models the radiative tail.
The other parameters, which are determined from fits of simulated samples, are
common for both decay modes. The width of the CB function is, according to the
simulation, fixed to be 0.43 times that of the Gaussian distribution, which is
left free in the fit.
Due to imperfect particle identification, $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ ($B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$) decays can be misidentified as
$K^{0}_{\rm\scriptscriptstyle S}K^{+}$ ($K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}$) candidates. The corresponding PDFs are empirically modelled with
the sum of two CB functions. For the $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ ($B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$) decay, the misidentification shape has
a significant high (low) mass tail. The parameters of the two CB functions are
determined from the simulation, and then fixed in fits to data.
Partially reconstructed decays, coming mainly from $B^{0}$ and $B^{+}$
(labelled $B$ in this section), and $B^{0}_{s}$ meson decays to open charm and
to a lesser extent from three-body charmless $B$ and $B^{0}_{s}$ decays, are
modelled with two PDFs. These PDFs are identical in the four possible final
states. They are modelled by a step function with a threshold mass equal to
$m_{\textrm{$B$}}-m_{\textrm{$\pi$}}$
($m_{\textrm{$B^{0}_{s}$}}-m_{\textrm{$\pi$}}$) [26] for $B$ ($B^{0}_{s}$)
decays, convolved with a Gaussian distribution of width
$20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to account for detector
resolution effects. Backgrounds from $\mathchar 28931\relax^{0}_{b}$ decays
are found to be negligible. The combinatorial background is assumed to have a
flat distribution in all categories.
The signal and background yields are varied in the fit, apart from those of
the cross-feed contributions, which are constrained using known ratios of
selection efficiencies from the simulation and particle identification and
misidentification probabilities. The ratio of $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ ($B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$) events reconstructed and selected as
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ ($K^{0}_{\rm\scriptscriptstyle
S}K^{+}$) with respect to $K^{0}_{\rm\scriptscriptstyle S}K^{+}$
($K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$) are $0.245\pm 0.018$ ($0.0418\pm
0.0067$), where the uncertainties are dominated by the finite size of the
simulated samples. These numbers appear in Gaussian terms inserted in the fit
likelihood function. The charge asymmetries of the backgrounds vary
independently in the fit, apart from those of the cross-feed contributions,
which are identical to those of the properly reconstructed signal decay.
Figure 1 shows the four invariant mass distributions along with the
projections of the fit. The measured width of the Gaussian distribution used
in the signal PDF is found to be approximately 20% larger than in the
simulation, and is included as a systematic uncertainty. The $C\\!P$-summed
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ signal yields are
found to be $N(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})=1804\pm 47$ and $N(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+})=90\pm 13$, with raw $C\\!P$ asymmetries
$\mathcal{A}_{\textrm{raw}}(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})=-0.032\pm 0.025$ and
$\mathcal{A}_{\textrm{raw}}(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+})=-0.23\pm 0.14$. All background asymmetries are found to be consistent
with zero within two standard deviations. By dividing the sample in terms of
data taking periods and magnet polarity, no discrepancies of more than two
statistical standard deviations are found in the raw $C\\!P$ asymmetries.
Figure 1: Invariant mass distributions of selected (a) $B^{-}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{-}$, (b) $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$, (c) $B^{-}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{-}$ and (d) $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ candidates. Data are points with error
bars, the $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$
($B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$) components are
shown as red falling hatched (green rising hatched) curves, combinatorial
background is grey dash-dotted, partially reconstructed $B^{0}_{s}$
($B^{0}$/$B^{+}$) backgrounds are dotted magenta (dashed orange).
## 4 Corrections and systematic uncertainties
The ratio of branching fractions is determined as
$\displaystyle\frac{{\cal B}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)}{{\cal B}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)}$ $\displaystyle=$
$\displaystyle\frac{N(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+})}{N(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+})}\cdot
r_{\textrm{sel}}\cdot r_{\textrm{PID}}\textrm{,}$ (2)
where the ratio of selection efficiencies is factorised into two terms
representing the particle identification,
$\displaystyle r_{\textrm{PID}}$ $\displaystyle\equiv$
$\displaystyle\frac{\varepsilon_{\textrm{PID}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})}{\varepsilon_{\textrm{PID}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+})},$ (3)
and the rest of the selection,
$\displaystyle r_{\textrm{sel}}$ $\displaystyle\equiv$
$\displaystyle\frac{\varepsilon_{\textrm{sel}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})}{\varepsilon_{\textrm{sel}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+})}.$ (4)
The raw $C\\!P$ asymmetries of the $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays are corrected for detection and
production asymmetries $\mathcal{A}_{\textrm{det+prod}}$, as well as for a
small contribution due to $C\\!P$ violation in the neutral kaon system
($\mathcal{A}_{\textrm{$K^{0}_{\rm\scriptscriptstyle S}$}}$). The latter is
assumed to be the same for both $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays. At first order, the
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ $C\\!P$ asymmetry
can be written as
$\displaystyle{\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)$ $\displaystyle\approx$
$\displaystyle\mathcal{A}_{\textrm{raw}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})-\mathcal{A}_{\textrm{det+prod}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})+\mathcal{A}_{\textrm{$K^{0}_{\rm\scriptscriptstyle S}$}}$
and similarly for $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$,
up to a sign flip in front of
$\mathcal{A}_{\textrm{$K^{0}_{\rm\scriptscriptstyle S}$}}$.
Selection efficiencies are determined from simulated samples generated at a
centre-of-mass energy of 8$\mathrm{\,Te\kern-1.00006ptV}$. The ratio of
selection efficiencies is found to be $r_{\textrm{sel}}=1.111\pm 0.019$, where
the uncertainty is from the limited sample sizes. To first order, effects from
imperfect simulation should cancel in the ratio of efficiencies. In order to
assign a systematic uncertainty for a potential deviation of the ratio of
efficiencies in 7$\mathrm{\,Te\kern-1.00006ptV}$ data with respect to
8$\mathrm{\,Te\kern-1.00006ptV}$, the $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ simulated events are reweighted by a
linear function of the $B$-meson momentum such that the average $B$ momentum
is 13% lower, corresponding to the ratio of beam energies. The 0.7% relative
difference between the nominal and reweighted efficiency ratio is assigned as
a systematic uncertainty. The distribution of the BDT output for simulated
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ events is found
to be consistent with the observed distribution of signal candidates in the
data using the _sPlot_ technique [30], where the discriminating variable is
taken to be the $B$ invariant mass. The total systematic uncertainty related
to the selection is 1.8%.
The determination of the trigger efficiencies is subject to variations in the
data-taking conditions and, in particular, to the ageing of the calorimeter
system. These effects are mitigated by regular changes in the gain of the
calorimeter system. A large sample of $D^{*+}\\!\rightarrow D^{0}(\rightarrow
K^{-}\pi^{+})\pi^{+}$ decays is used to measure the trigger efficiency in bins
of $p_{\rm T}$ for pions and kaons from signal decays. These trigger
efficiencies are averaged using the $p_{\rm T}$ distributions obtained from
simulation. The hardware stage trigger efficiencies obtained by this procedure
are in agreement with those obtained in the simulation within 1.1%, which is
assigned as systematic uncertainty on the ratio of branching fractions. The
same procedure is also applied to $B^{+}$ and $B^{-}$ decays separately, and
results in 0.5% systematic uncertainty on the determination of the $C\\!P$
asymmetries.
Particle identification efficiencies are determined using a large sample of
$D^{*+}\\!\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ decays. The
kaons and pions from this calibration sample are reweighted in 18 bins of
momentum and 4 bins of pseudorapidity, according to the distribution of signal
kaons and pions from simulated $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays. The ratio of efficiencies is
$r_{\textrm{PID}}=1.154\pm 0.025$, where the uncertainty is given by the
limited size of the simulated samples. The systematic uncertainty associated
with the binning scheme is determined by computing the deviation of the
average efficiency calculated using the nominal binning from that obtained
with a single bin in each kinematic variable. A variation of $0.7\%$ (1.3%) is
observed for pions (kaons). A systematic uncertainty of $0.5\%$ is assigned
due to variations of the efficiencies, determined by comparing results
obtained with the 2011 and 2012 calibration samples. All these contributions
are added in quadrature to obtain 2.7% relative systematic uncertainty on the
particle identification efficiencies. Charge asymmetries due to the PID
requirements are found to be negligible.
Uncertainties due to the modelling of the reconstructed invariant mass
distributions are assigned by generating and fitting pseudo-experiments.
Parameters of the signal and cross-feed distributions are varied according to
results of independent fits to the $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ simulated samples. The relative
uncertainty on the ratio of yields from mis-modelling of the signal (cross-
feed) is 2.4% (2.7%) mostly affecting the small $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ yield. The width of the Gaussian
resolution function used to model the partially reconstructed backgrounds is
increased by 20%, while the other fixed parameters of the partially
reconstructed and combinatorial backgrounds are left free in the fit, in turn,
to obtain a relative uncertainty of $3.3\%$. The total contribution of the fit
model to the systematic uncertainty is $4.9\%$. Their contribution to the
systematic uncertainties on the $C\\!P$ asymmetries is found to be negligible.
Detection and production asymmetries are measured using approximately one
million $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}$ decays collected in 2011 and 2012. Using a kinematic and
topological selection similar to that employed in this analysis, a high purity
sample is obtained. The raw $C\\!P$ asymmetry is measured to be
$\mathcal{A}(B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm})=(-1.4\pm 0.1)\%$ within
$20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $B^{+}$ meson mass. The
same result is obtained by fitting the reconstructed invariant mass with a
similar model to that used for the $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ and $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ fits. This asymmetry is consistent
between bins of momentum and pseudorapidity within 0.5%, which is assigned as
the corresponding uncertainty. The $C\\!P$ asymmetry in
$B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$
decays is ${\cal
A}^{C\\!P}\left(B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}\right)=(+0.5\pm 0.3)\%$, where the value is the weighted average
of the values from Refs. [26] and [31]. This leads to a correction of
$\mathcal{A}_{\textrm{det+prod}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+})=(-1.9\pm 0.6)\%$. The combined
production and detection asymmetry for $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays is expressed as
$\mathcal{A}_{\textrm{det+prod}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle
S}\pi^{+})=\mathcal{A}_{\textrm{det+prod}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+})+\mathcal{A}_{\textrm{$K\pi$}}$, where
the kaon-pion detection asymmetry is
$\mathcal{A}_{\textrm{$K\pi$}}\approx\mathcal{A}_{\textrm{$K$}}-\mathcal{A}_{\textrm{$\pi$}}=(1.0\pm
0.5)\%$ [32]. The assigned uncertainty takes into account a potential
dependence of the difference of asymmetries as a function of the kinematics of
the tracks. The total correction to ${\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)$ is
$\mathcal{A}_{\textrm{det+prod}}(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+})=(-0.9\pm 0.8)\%$.
Potential effects from $C\\!P$ violation in the neutral kaon system, either
directly via $C\\!P$ violation in the neutral kaon system [33] or via
regeneration of a $K^{0}_{\rm\scriptscriptstyle S}$ component through
interactions of a $K^{0}_{\rm\scriptscriptstyle L}$ state with material in the
detector [34], are also considered. The former is estimated [35] by fitting
the background subtracted [30] decay time distribution of the observed
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ decays and
contributes 0.1% to the observed asymmetry. The systematic uncertainty on this
small effect is chosen to have the same magnitude as the correction itself.
The latter has been studied [36] and is small for decays in the LHCb
acceptance and thus no correction is applied. The systematic uncertainty
assigned for this assumption is estimated by using the method outlined in Ref.
[34]. Since the $K^{0}_{\rm\scriptscriptstyle S}$ decays reconstructed in this
analysis are concentrated at low lifetimes, the two effects are of similar
sizes and have the same sign. Thus an additional systematic uncertainty equal
to the size of the correction applied for $C\\!P$ violation in the neutral
kaon system and 100% correlated with it, is assigned. It results in
$\mathcal{A}_{\textrm{$K^{0}_{\rm\scriptscriptstyle S}$}}=(0.1\pm 0.2)\%$. A
summary of the sources of systematic uncertainty and corrections to the
$C\\!P$ asymmetries are given in Table 1. Total systematic uncertainties are
calculated as the sum in quadrature of the individual contributions.
Table 1: Corrections (above double line) and systematic uncertainties (below double line). The relative uncertainties on the ratio of branching fractions are given in the first column. The absolute corrections and related uncertainties on the $C\\!P$ asymmetries are given in the next two columns. The last column gathers the relative systematic uncertainties contributing to $r_{B_{c}^{+}}$. All values are given as percentages. Source | $\mathcal{B}$ ratio | ${\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)$ | ${\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)$ | $B_{c}^{+}$
---|---|---|---|---
$\mathcal{A}_{\textrm{det+prod}}$ | - | $-0.9$ | $-1.9$ | -
$\mathcal{A}_{\textrm{$K^{0}_{\rm\scriptscriptstyle S}$}}$ | - | 0.1 | 0.1 | -
Selection | 1.8 | - | - | 6.1
Trigger | 1.1 | 0.5 | 0.5 | 1.1
Particle identification | 2.7 | - | - | 3.6
Fit model | 4.9 | - | - | 2.0
$\mathcal{A}_{\textrm{det+prod}}$ | - | 0.8 | 0.6 | -
$\mathcal{A}_{\textrm{$K^{0}_{\rm\scriptscriptstyle S}$}}$ | - | 0.2 | 0.2 | -
Total syst. uncertainty | 6.0 | 1.0 | 0.8 | 7.4
## 5 Search for $B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}$ decays
An exploratory search for $B_{c}^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays is performed with the data sample
collected in 2011, corresponding to an integrated luminosity of
1$\mbox{\,fb}^{-1}$. The same selection as for the $B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays is used, only adding a proton
veto $\mathrm{DLL}_{pK}<10$ to the $K^{+}$ daughter, which is more than $99\%$
efficient. This is implemented to reduce a significant background from baryons
in the invariant mass region considered for this search. The ratios of
selection and particle identification efficiencies are $r_{\rm sel}=0.306\pm
0.012$ and $r_{\rm PID}=0.819\pm 0.027$, where the uncertainties are from the
limited size of the simulated samples. The related systematic uncertainties
are estimated in a similar way as for the measurement of ${\cal
B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)/{\cal
B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)$.
The $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ yield is also
evaluated with the 2011 data only. The $B_{c}^{+}$ signal yield is determined
by fitting a single Gaussian distribution with the mean fixed to the
$B_{c}^{+}$ mass [26] and the width fixed to $1.2$ times the value obtained
from simulation to take into account the worse resolution in data. The
combinatorial background is assumed to be flat. The invariant mass
distribution and the superimposed fit are presented in Fig. 2 (left). Pseudo-
experiments are used to evaluate the biases in the fit procedure and the
systematic uncertainties are evaluated by assuming that the combinatorial
background has an exponential slope. A similar procedure is used to take into
account an uncertainty related to the assumed width of the signal
distribution. The $20\%$ correction applied to match the observed resolution
in data, is assumed to estimate this uncertainty.
Figure 2: (Left) Invariant mass distribution of selected
$B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ candidates.
Data are points with error bars and the curve represents the fitted function.
(Right) The number of events and the corresponding value of $r_{B_{c}^{+}}$.
The central value (dotted line) and the upper and lower 90% statistical
confidence region bands are obtained using the Feldman and Cousins approach
[37] (dashed lines). The solid lines includes systematic uncertainties. The
gray outline of the box shows the obtained upper limit of $r_{B_{c}^{+}}$ for
the observed number of 2.8 events.
The Feldman and Cousins approach [37] is used to build 90% confidence region
bands that relate the true value of $r_{B_{c}^{+}}=(f_{c}\cdot{\cal
B}\left(B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}\right))/(f_{u}\cdot{\cal B}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right))$ to the measured number of
signal events, and where $f_{c}$ and $f_{u}$ are the hadronisation fraction of
a $b$ into a $B_{c}^{+}$ and a $B^{+}$ meson, respectively. All of the
systematic uncertainties are included in the construction of the confidence
region bands by inflating the width of the Gaussian functions used to build
the ranking variable of the Feldman and Cousins procedure. The result is shown
in Fig. 2 (right) and gives the upper limit
$r_{B_{c}^{+}}\equiv\frac{f_{c}}{f_{u}}\cdot\frac{{\cal
B}\left(B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}\right)}{{\cal B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\right)}<5.8\times 10^{-2}\textrm{ at 90\% confidence level.}$
This is the first upper limit on a $B_{c}^{+}$ meson decay into two light
quarks.
## 6 Results and summary
The decays $B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}$ and
$B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ have been studied
using a data sample corresponding to an integrated luminosity of
3$\mbox{\,fb}^{-1}$, collected in 2011 and 2012 by the LHCb detector and the
ratio of branching fractions and $C\\!P$ asymmetries are found to be
$\displaystyle\frac{{\cal B}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)}{{\cal B}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)}$ $\displaystyle=$
$\displaystyle\phantom{-}0.064\pm 0.009\textrm{ (stat.)}\pm 0.004\textrm{
(syst.)},$ $\displaystyle{\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)$ $\displaystyle=$
$\displaystyle-0.022\pm 0.025\textrm{ (stat.)}\pm 0.010\textrm{ (syst.)},$
and
$\displaystyle{\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)$ $\displaystyle=$
$\displaystyle-0.21\pm 0.14\textrm{ (stat.)}\pm 0.01\textrm{ (syst.)}.$
These results are compatible with previous determinations [11, 12]. The
measurements of ${\cal A}^{C\\!P}\left(B^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)$ and ${\cal
B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}\right)/{\cal
B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\right)$ are
the best single determinations to date. A search for $B_{c}^{+}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}$ decays is also performed with a data
sample corresponding to an integrated luminosity of 1$\mbox{\,fb}^{-1}$. The
upper limit
$\displaystyle\frac{f_{c}}{f_{u}}\cdot\frac{{\cal
B}\left(B_{c}^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}K^{+}\right)}{{\cal B}\left(B^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\right)}<5.8\times 10^{-2}\textrm{ at 90\% confidence level}$
is obtained. Assuming $f_{c}\simeq 0.001$ [13], $f_{u}=0.33$ [26, 38, 39], and
${\cal B}\left(B^{+}\\!\rightarrow K^{0}\pi^{+}\right)=(23.97\pm 0.53\textrm{
(stat.)}\pm 0.71\textrm{ (syst.)})\cdot 10^{-6}$ [12], an upper limit ${\cal
B}\left(B_{c}^{+}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{0}K^{+}\right)<4.6\times
10^{-4}\textrm{ at 90\% confidence level}$ is obtained. This is about two to
four orders of magnitude higher than theoretical predictions, which range from
$10^{-8}$ to $10^{-6}$ [13]. With the large data samples already collected by
the LHCb experiment, other two-body $B_{c}^{+}$ decay modes to light quarks
such as $B_{c}^{+}\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}K^{+}$ and
$B_{c}^{+}\\!\rightarrow\phi K^{+}$ may be searched for.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-08-06T14:14:08 |
2024-09-04T02:49:49.065655
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, C. Gotti, M. Grabalosa G\\'andara, R. Graciani Diaz,\n L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, M. Hess, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R.\n Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M.\n Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O.\n Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C.\n Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi,\n P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, T. Palczewski, M.\n Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson,\n G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P.\n Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, K. Petridis, A. Petrolini,\n A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S.\n Playfer, M. Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A.\n Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A.\n Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, E.\n Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, J.\n Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, J.\n Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, S. Swientek, V. Syropoulos, M. Szczekowski,\n P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti,\n A. Vallier, M. Van Dijk, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez\n Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B.\n Viaud, D. Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong,\n A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, C. Wallace, R.\n Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D.\n Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G.\n Wilkinson, M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley, J. Wishahi,\n W. Wislicki, M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Aur\\'elien Martens",
"url": "https://arxiv.org/abs/1308.1277"
}
|
1308.1302
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-143 LHCb-PAPER-2013-036 11${}^{\textrm{th}}$ October 2013
Observation of $B^{0}_{s}$-$\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}$ mixing and
measurement of mixing frequencies
using semileptonic $B$ decays
The LHCb collaboration†††Authors are listed on the following pages.
The $B^{0}_{s}$ and $B^{0}$ mixing frequencies, $\Delta m_{s}$ and $\Delta
m_{d}$, are measured using a data sample corresponding to an integrated
luminosity of 1.0 fb-1 collected by the LHCb experiment in $pp$ collisions at
a centre of mass energy of $7$ TeV during 2011. Around 1.8$\times 10^{6}$
candidate events are selected of the type $B^{0}_{(s)}\to D^{-}_{(s)}\mu^{+}$
($+$ anything), where about half are from peaking and combinatorial
backgrounds. To determine the $B$ decay times, a correction is required for
the momentum carried by missing particles, which is performed using a
simulation-based statistical method. Associated production of muons or mesons
allows us to tag the initial-state flavour and so to resolve oscillations due
to mixing. We obtain
$\displaystyle\Delta m_{s}=(17.93\pm 0.22\,\textrm{(stat)}\pm
0.15\,\textrm{(syst)})\,\textrm{ps}^{-1},$ $\displaystyle\Delta
m_{d}=(0.503\pm 0.011\,\textrm{(stat)}\pm
0.013\,\textrm{(syst)})\,\textrm{ps}^{-1}.$
The hypothesis of no oscillations is rejected by the equivalent of 5.8
standard deviations for $B^{0}_{s}$ and 13.0 standard deviations for $B^{0}$.
This is the first observation of $B^{0}_{s}$ mixing to be made using only
semileptonic decays.
To be published in Eur. Phys. J. C
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
F. Andrianala37, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A.
Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S.
Bachmann11, J.J. Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J.
Barlow53, C. Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J.
Beddow50, F. Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30, E. Ben-Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A.
Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S.
Bifani44, T. Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38,
J. Blouw11, S. Blusk58, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15,
S. Borghi53, A. Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T.
Brambach9, J. van den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T.
Britton58, N.H. Brook45, H. Brown51, I. Burducea28, A. Bursche39, G.
Busetto21,q, J. Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez35,n, A. Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c,
G. Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L.
Carson52, K. Carvalho Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37,
Ch. Cauet9, R. Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N.
Chiapolini39, M. Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L.
Clarke49, M. Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J.
Cogan6, E. Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A.
Cook45, M. Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49,
D.C. Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S. Farry51, D.
Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S.
Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R.
Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph.
Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30,
A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Hess60, A.
Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O.
Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U.
Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T. Palczewski27, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B.
Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8,
G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N.
Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E.
Santovetti23,k, M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M.
Savrie16,e, D. Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M.
Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A.
Semennikov30, K. Senderowska26, I. Sepp52, N. Serra39, J. Serrano6, P.
Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T.
Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires9, R.
Silva Coutinho47, M. Sirendi46, T. Skwarnicki58, N.A. Smith51, E. Smith54,48,
J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D.
Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F.
Stagni37, S. Stahl11, O. Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V.
Vagnoni14, G. Valenti14, A. Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P.
Vazquez Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M.
Veltri17,g, G. Veneziano38, K. Vervink37, M. Vesterinen37, B. Viaud7, D.
Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45,
A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47,
R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D.
Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D.
Wiedner11, L. Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55,
F.F. Wilson48, J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, S.A.
Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3,
R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L.
Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
$B_{s}^{0}$ and $B^{0}$ mesons propagate as superpositions of particle and
antiparticle flavour states. For a flavour-specific decay process111In this
paper, charge conjugate modes are always implied. such as
$B^{0}~{}\to~{}D^{-}\mu^{+}\nu$, particle-antiparticle mixing lends a
sinusoidal component to the decay rates [1, 2]. To measure mixing, the flavour
state of the $B$ meson must be observed to change, which requires knowledge of
the state from at least two points in time. The experimentally accessible
times to determine the flavour are at production and decay. Neglecting $C\\!P$
violation in mixing, the decay rate $N$ at a proper decay time $t$ simplifies
to
$N_{\pm}(t)=N(0)\,\frac{e^{-\Gamma
t}}{2}\left[\cosh{(\Delta\Gamma_{\,}t/2)}\pm\cos{(\Delta m_{\,}t)}\right]\,,$
(1)
where $\Delta\Gamma$ and $\Delta m$ are the width and mass differences222The
mass difference is measured here as an angular frequency, in units of inverse
time. of the two mass eigenstates, and $\Gamma$ is the average decay width
[2]. The positive sign applies when the $B$ meson decays with the same flavour
as its production and the negative sign when the particle decays with opposite
flavour to its production, later referred to as “even” and “odd”. In this
study, a sample of semileptonic decays obtained with the LHCb detector is used
to measure the mixing frequencies $\Delta m_{s}$ and $\Delta m_{d}$ for the
$B^{0}_{s}$ and $B^{0}$ systems. These quantities have previously been
measured to high precision, usually in the combination of several channels,
relying heavily on hadronic decay modes (see for example Refs. [3, 4] and our
recent results, Refs. [5, 6, 7]). To date no observation of $B_{s}^{0}$ mixing
has been made using only semileptonic decay channels.
## 2 Experimental setup
The LHCb detector [8] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector consists of several dedicated
subsystems, organized successively further from the interaction region. A
silicon-strip vertex detector surrounds the $pp$ interaction region and
approaches to within 8 mm of the proton beams. The first of two ring-imaging
Cherenkov (RICH) detectors comes next, followed by the remainder of the
tracking system, which comprises, in order: a large-area silicon-strip
detector; a dipole magnet with a bending power of about $4{\rm\,Tm}$; and
three multilayer tracking stations, each with central silicon-strip detectors
and peripheral straw drift tubes. After this comes the second RICH detector,
the calorimeter and the muon stations.
The combined high-precision tracking system provides a momentum measurement
with relative uncertainty that varies from 0.4 % at 5 GeV$c^{-1}$ to 0.6 % at
100 GeV$c^{-1}$, and impact parameter333The impact parameter is the distance
of closest approach of a track to a primary interaction vertex. resolution of
20 $\,\upmu\rm m$ for tracks with high transverse momentum. By combining
information from the two RICH detectors [9] charged hadrons can be identified
across a wide range in momentum, around 2 to 150 GeV$c^{-1}$. The calorimeter
system consists of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter, allowing
identification of photon, electron and hadron candidates. Muons that pass
through the calorimeters are detected using a system of alternating layers of
iron and multiwire proportional chambers [10]. Triggering of events is
performed in two stages [11]: a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which performs
full event reconstruction.
## 3 Data selection and reconstruction
The LHCb dataset used in this analysis corresponds to an integrated luminosity
of 1.0 fb-1 collected in $pp$ collisions at a centre of mass energy of $7$ TeV
during the 2011 physics run at the LHC. Where simulation is required, Pythia
6.4 [12] is used, with a specific LHCb configuration [13]. Decays of hadronic
particles are described by EvtGen [14], in which final-state radiation is
generated using Photos [15]. The interaction of the generated particles with
the detector and the detector response are implemented using the Geant4
toolkit [16, *Agostinelli:2002hh] as described in Ref. [18]. Input to EvtGen
is taken from the best knowledge of branching fractions ($\cal B$) and form
factors at the time of the simulation [1]. The same reconstruction and
selection is applied on simulated and detector data.
A sample of events is selected in which a $D_{(s)}^{+}\to K^{+}K^{-}\pi^{+}$
candidate forms a vertex with a muon candidate. A cut-based selection is
applied to enhance the fraction of real $D^{+}_{(s)}$ mesons in this sample
that arise from $B^{0}_{(s)}$ semileptonic decays. Vertex and track
reconstruction qualities, momenta, invariant masses, flight distances and
particle identification (PID) variables are used. The selection was initially
optimized on simulated data to maximize the signal significance,
$S/\sqrt{(S+B)}$, where $S$ ($B$) denotes the number of selected signal
(background) candidates. The most important cuts for this analysis are those
on the PID and invariant masses. Combined information from the RICH detectors,
muon stations, calorimeters and tracking allows us to place stringent
requirements on a log-likelihood based PID parameter for each final-state
particle separately, ensuring at least 99 % purity in the muon sample, and
suppressing peaking backgrounds such as $D^{+}\to K^{-}\pi^{+}\pi^{+}$ decays,
where a pion has been misidentified as a kaon. To allow a simultaneous
measurement of $\Delta m_{s}$ and $\Delta m_{d}$, a broad mass window for the
$K^{+}K^{-}\pi^{+}$ system is used to cover both the $D^{+}$ and $D_{s}^{+}$
masses, $-0.2<M(K^{+}K^{-}\pi^{+})-M_{0}(D^{+}_{s})<0.1$ GeV$c^{-2}$, where
$M_{0}(D^{+}_{s})$ is the known mass of the $D^{+}_{s}$ meson [1]. Decays of
the type $D^{\ast}(2010)^{+}\to D^{0}\pi^{+}$ are additionally suppressed by
requiring that the invariant mass of the two kaons $M(K^{+}K^{-})\,<\,1.84$
GeV$c^{-2}$, and combinatorial background with slow collinear pions is
similarly removed with the mass requirement
$M(K^{+}K^{-}\pi^{+})-M(K^{+}K^{-})-M_{0}(\pi^{+})>15$ MeV$c^{-2}$.
Simulation studies indicate that the selected sample is dominated by
$B_{s}^{0}\to D_{s}^{-}\mu^{+}(\nu,\pi^{0},\gamma)$, $B^{0}\to
D^{-}\mu^{+}(\nu,\pi^{0},\gamma)$ and $B^{+}\to
D^{-}\mu^{+}(\nu,\pi^{+},\gamma)$ decays, where no specific intermediate
states are required other than those mentioned, and where at least one
neutrino will occur together with any number of the other particles in the
parentheses. These additional particles are ignored and so a clear $B$ mass
peak cannot be reconstructed. For simplicity, to quantify the measured mass,
$M(D\mu)$, within its possible range, we define a “normalized mass”, $n$,
relative to the known masses $(M_{0})$ of the $B$, $D$, and $\mu$:
$n=\frac{M(D\mu)-M_{0}(D)-M_{0}(\mu)}{M_{0}(B)-M_{0}(D)-M_{0}(\mu)}.$ (2)
We require $0.24<n<1.0$, where the lower cut mainly removes low-mass
combinatorial background candidates. The $K^{+}K^{-}\pi^{+}$ invariant mass
distribution and the normalized mass distribution ($n$) of the selected
candidates are shown in Fig. 1, in which the $D^{+}_{s}$ and $D^{+}$ peaks can
clearly be seen over the combinatorial background.
Determination of the initial-state flavour is performed using the standard
LHCb flavour-tagging algorithms, which are described in detail elsewhere [19,
6, 5]. These algorithms rely on the reconstruction of particles that were
produced in association with, and are flavour-correlated with, the signal
$B$-meson. The correlations arise either from fragmentation, which often
produces a kaon or pion of specific charge correlated with the signal, or from
“opposite-side” decays, where the decay products of the partner $b$ quark are
reconstructed (e.g. a muon). A neural network combines tagging decisions for
the best tagging power[6].
Figure 1: Mass distributions for all selected signal candidates. Left, the
$K^{+}K^{-}\pi^{+}$ invariant mass, where the known mass of the $D^{+}_{s}$
has been subtracted. Right, the $D\mu$ normalized mass as defined in Eq. 2.
Neutral candidates are those of the form $D^{\mp}\mu^{\pm}$, while double-
charged candidates are those of the form $D^{\pm}\mu^{\pm}$. The double-
charged candidates arise from several background sources, most of which are
also present in the neutral sample. In the left plot, the neutral sample
exhibits much larger $D$ mass peaks, indicative of the large $B$ signal
component.
A hypothesis is required for the nature of the reconstructed candidate, either
$B^{0}_{s}$ or $B^{0}$, in order to choose the tagging algorithms to be
applied and to select the appropriate mass with which to calculate $n$. A
split around the midpoint between the $D^{+}_{s}$ and $D^{+}$ peaks is used.
For the $B^{0}_{s}$ hypothesis all available tags are used. For the $B^{0}$
hypothesis only opposite-side tags are used, to reduce the difference between
$B^{+}$ and $B^{0}$ tagging performance and thus better constrain the $B^{+}$
background (see Secs. 5 and 6). The flavour-tagged dataset comprises 594,845
selected candidates.
Two techniques are employed to measure the mixing frequencies: (a)
multidimensional log-likelihood maximization, simultaneously fitting $\Delta
m_{s}$ and $\Delta m_{d}$; (b) model-independent Fourier analysis, used as a
cross-check, which determines $\Delta m_{s}$ with good precision, but $\Delta
m_{d}$ with a very poor precision. Both methods use a common determination of
the proper decay time and so share a portion of the corresponding systematic
effects.
## 4 Proper decay-time distributions
Figure 2: Input to obtain the $k$-factor correction from the fully-simulated
$B^{0}_{s}$ sample. For each event the ratio of reconstructed to generated
momentum, $p_{\textrm{rec}}/p_{\textrm{sim}}$ is plotted against the
normalized $D\mu$ mass ($n$ in Eq. 2). The curve shows a fourth-order
polynomial resulting from a fit to the mean of the distribution (in bins of
$n$).
To obtain the $B$-meson decay times, a correction is applied for the momentum
lost due to missing particles, using a $k$-factor method as employed in many
previous measurements (see, for example, Refs. [20] and [21]). The $k$-factor
[22] is a simulation-based statistical correction, where the average missing
momentum in a simulated sample is used to correct the reconstructed momentum
as a function of the reconstructed $D\mu$ mass (as shown in Fig. 2). In this
study we use a fourth-order polynomial to parameterize $k$ as a function of
the normalized $D\mu$ mass ($n$ from Eq. 2), which allows us to use the same
correction for $B^{0}_{s}$ and $B^{0}$. With this approach, both $\Delta
m_{s}$ and $\Delta m_{d}$ exhibit residual biases of around $1$ %; these
biases are known to good precision from the full simulation and are corrected
in the final results.
The experimental resolution of the proper decay time ($t$) reduces the
visibility of the oscillations, smearing Eq. 1 with a resolution function
$R(t,t^{\prime}-t)$, where $t$ is the true decay time and $t^{\prime}$ is the
measured value. The limited performance of the tagging also reduces the
visibility of the oscillations. Our selection requirements include variables
that are correlated with the decay time, leading to a time-dependent
efficiency function, $\varepsilon(t^{\prime})$. Thus Eq. 1 becomes
$\displaystyle N_{\pm}(t^{\prime})=N(0)\,\eta\,\frac{e^{-\Gamma
t}}{2}\left[\cosh{(\Delta\Gamma\,t/2)}\pm(1-2\omega)\cos{(\Delta
m\,t)}\right]\otimes R(t,t^{\prime}-t)\times\varepsilon(t^{\prime}),$ (3)
where $\eta$ is the tagging efficiency and $\omega$ is the mistag probability
(the fraction of tags that assign the wrong flavour). We parameterize the
time-dependent efficiency with an empirical “acceptance” function.
Specifically Gaussian functions are used as motivated by data and full
simulation studies [22],
$\varepsilon(t^{\prime})=1-f\,G(t^{\prime};\mu_{0},\sigma_{1})-(1-f)\,G(t^{\prime};\mu_{0},\sigma_{2})$,
where $G$ is the Gaussian function and the parameters are determined from fits
to the data (typical values are $\sigma_{1,2}<1$ ps and $\mu_{0}\approx 0.01$
ps).
Figure 3: Illustration of the decay time resolution obtained from a fully
simulated $B^{0}$ signal sample. The left plots demonstrate the Gaussian fits
(solid lines) using the full LHCb simulated data (filled), to determine the
decay time resolution. Each measured (reconstructed and corrected) time,
$t^{\prime}$, is compared to the corresponding simulated decay time, $t$. The
results are shown for several bins of $t^{\prime}$. The dependence on decay
time of the mean (bias, $\mu$) and width (standard deviation, $\sigma$) can be
fitted with a quadratic or cubic function of either $t$ or $t^{\prime}$. The
right hand plot shows a quadratic fit to the widths.
The $k$-factor is a relative correction for the average missing momentum at a
given value of $n$; as shown in Fig. 2, the range of missing momenta is broad
and varies from about 70 % at $n=0.2$ to zero at $n=1$. This large relative
uncertainty on the corrected momentum ($p^{\prime}$) dominates the decay time
resolution, i.e.
$\sigma(t^{\prime})/t^{\prime}\approx\sigma(p^{\prime})/p^{\prime}$. Hence
$\sigma(t^{\prime})$ is approximately proportional to $t^{\prime}$ (as seen in
Fig. 3) and the decay time resolution worsens as decay time increases. This
dependence is determined and parameterized from the full simulation. We may
choose between a parameterization in terms of either the generated (“true”)
decay time, using a numerical convolution, or in terms of the measured decay
time, using analytical methods; the latter is the default approach. The
resolution dependence is well-fitted with second or third order polynomials.
## 5 Multivariate fits to the data
Figure 4: Distribution of measured $K^{+}K^{-}\pi^{+}$ mass, where the known
mass of the $D^{+}_{s}$ has been subtracted. Black points show the data, and
the various lines overlay the result of the fit. The small step at $-50$
MeV$c^{-2}$ is the result of differences in tagging efficiency for the
$B^{0}_{s}$ and $B^{0}$ hypotheses.
A binned, multidimensional, log-likelihood fit to the data is made, using the
Root and embedded RooFit fitting frameworks [23, 24]. In order to improve the
resolution on the fitted value of $\Delta m_{s}$, the sample is divided into
two subsamples about normalized mass $n=0.56$ (with this value determined
using fast-simulation “pseudo-experiment” studies), and the two subsamples are
fitted simultaneously as described below. There are 101,000 bins over the
$K^{+}K^{-}\pi^{+}$ mass, the measured decay time ($t^{\prime}$), the
normalized mass ($n<0.56$ and $n>0.56$), and the tagging result (even and
odd). Seven categories of signal and background are assigned component
probability density functions (PDFs) whose fractions and shape parameters are
left free in the fits to the data. The backgrounds are categorized in terms of
their shapes in the mass and decay-time observables. Using the
$M(K^{+}K^{-}\pi^{+})$ distribution we separate out peaking $D_{(s)}^{+}$
components from combinatorial background components. Each of these categories
can be further divided into two based on their decay-time shape. We use the
term “prompt” to describe fake candidates containing particles exclusively
produced in the primary $pp$ interaction, and the term “detached” for
candidates that contain at least one daughter of a secondary decay and which
therefore tend to exhibit a significantly larger lifetime. Candidates for the
signal $B$-decays of interest must be both detached and peaking. The signal-
like decays are usually grouped together in the fit; however, we separate the
specific background contribution of $B^{+}$ within the $D^{+}$ peak and fit
that directly. These components are shown in together in Fig. 4 and separately
in different $M(K^{+}K^{-}\pi^{+})$ regions in Figs. 5 and 6.
Figure 5: Measured $B$ decay-time distribution, overlaid with projections of
the fit, for background-only regions. Top left: a region between the two
signal peaks, $-80$ to $-20$ MeV$c^{-2}$ (with respect to the known mass of
the $D_{s}^{+}$), showing only low decay times. Top right: a region to the
right of the signal peaks $20$ to $100$ MeV$c^{-2}$, showing only low decay
times. Bottom row: the same on an extended decay-time scale and logarithmic.
The legend is the same as in Fig. 4.
Each mass PDF is a Gaussian function or a Chebychev polynomial (Fig. 4), and
each background decay-time PDF is a simple exponential with an appropriate
acceptance function as previously described (Fig. 6). For the signal decay-
time shape we use the model described in Eq. 3, with one instance for each
peak. The majority of our sensitivity arises from the mixing asymmetry, whose
time-dependent fit in the signal regions is shown in Fig. 7. Any odd/even
asymmetry is assumed to be constant as a function of time for prompt
backgrounds and for backgrounds that are known not to mix
($B^{+},\Lambda_{b}$, etc.). Generic detached backgrounds are allowed to have
a time-dependent asymmetry varying as an arbitrary quadratic polynomial.
Figure 6: Measured $B$ decay-time distribution, overlaid with projections of
the fit, for signal regions. Top left: for odd-tags, high-$n$ and a region of
$\pm 20$ MeV$c^{-2}$ around the $D^{+}_{s}$ mass peak, showing only low decay
times, where $B^{0}_{s}$ oscillations can be clearly seen. Top right: for odd-
tags and all $n$ for a region of $\pm 20$ MeV$c^{-2}$ around the $D^{+}$ mass
peak, showing only low decay times. Bottom row: for both tags and all $n$ for
regions of $\pm 20$ MeV$c^{-2}$ around the $D^{+}_{s}$ (left) and $D^{+}$
(right) mass peaks. The legend is the same as in Fig. 4.
Figure 7: Tagged (mixing) asymmetry, $(N_{+}-N_{-})/(N_{+}+N_{-})$, as a
function of $B$ decay time. The left plot shows the asymmetry for events for a
region of $\pm 20$ MeV$c^{-2}$ around the $D^{+}_{s}$ mass peak, and the right
plot shows the corresponding asymmetry around the $D^{+}$ mass peak. The black
points show the data and the curves are projections of the fitted PDF. On the
left plot the fast oscillations of $B^{0}_{s}$ are gradually washed out by the
increasingly poor decay-time resolution.
The proportion of $B^{+}\to D^{-}\mu^{+}(\nu,\pi^{+},\gamma)$ with respect to
$B^{0}\to D^{-}\mu^{+}(\nu,\pi^{0},\gamma)$ is fixed to $11\,\%$ with a
${\pm}2\,\%$ uncertainty, using the ratio of known fragmentation functions and
branching fractions [1]. Based on the full LHCb simulation, this ratio is
corrected by $25\,\%$ to account for differences in the reconstruction and
tagging efficiencies, with the full value of this correction taken as a
systematic uncertainty. We fix $\Delta\Gamma_{s}$ using the result of a recent
LHCb analysis [25], and $\Delta\Gamma_{d}$ is fixed to zero.
Only the signal mass shapes and the parameters of interest, $\Delta m_{s}$ and
$\Delta m_{d}$, are shared between the two subsamples in $n$, which are fitted
simultaneously. The goodness of the fit is verified with a local density
method [26], which finds a $p$-value of $19.6\,\%$.
## 6 Fit results and systematic uncertainties
Table 1 gives the fitted values for some important quantities. In principle
the signal lifetimes are also measured, but these have very large systematic
uncertainties and so no results are quoted. The systematic uncertainties on
$\Delta m_{s}$ and $\Delta m_{d}$ are first discussed before the final results
are given.
Several sources of systematic uncertainty on the main measured quantities,
$\Delta m_{s}$ and $\Delta m_{d}$, are considered, as summarized in Table 2.
The majority of the systematic uncertainties are obtained from the data.
* •
The $k$-factor: the $k$-factor correction is a simulation-based method, and so
differences between the simulation and reality that modify the visible and
invisible momenta potentially invalidate the correction. Such differences
could for example be in $D^{**}$ branching fractions or form factors. Large-
scale pseudo-experiment studies are combined with full simulations to vary
these underlying distributions within their uncertainties and examine biases
produced on the fitted $\Delta m$ values. Small relative uncertainties are
found, $0.3\,\%$ for $\Delta m_{s}$ and $1.0$ % for $\Delta m_{d}$,
representing the ultimate limit of this technique without further knowledge of
the various sub-decays.
* •
Detector alignment: momentum scale, decay-length scale, and track position
uncertainties arise from known alignment uncertainties and result in
variations in reconstructed masses and lifetimes as functions of decay opening
angle. These uncertainties have been studied using detector survey data and
various control modes; they are well determined and small in comparison to the
statistical uncertainties [27].
* •
Values of $\Delta\Gamma$: The quantities $\Delta\Gamma_{d}$ and
$\Delta\Gamma_{s}$ are nominally constant in our fits. When they are varied,
within $\pm 5$ % for $\Delta\Gamma_{d}$ (chosen to well-cover the experimental
range given the lack of information on its sign [1]) and within the known
uncertainty on $\Delta\Gamma_{s}$ [25], our result is only marginally
affected.
* •
Model bias: a correction has been made for the 1 % residual frequency bias
seen in full simulation studies, as discussed in Sec. 4. This is taken
directly from simulation and half of the correction is assigned a systematic
uncertainty.
* •
Signal proper-time model: the fit is repeated with two different time-
resolution models. (a) When the resolution is parameterized as a function of
true rather than measured decay time, using full numerical convolution, a
(0.09, 0.002) ps-1 variation is seen in ($\Delta m_{s}$, $\Delta m_{d}$). (b)
When a time-independent (average) resolution is used, a 0.001 ps-1 variation
is seen in $\Delta m_{d}$ (this method is not applicable to the measurement of
$\Delta m_{s}$ due to many factors; crucially, within the time frame of any
single $B^{0}_{s}$ oscillation the decay time resolution worsens by an
appreciable fraction of the oscillation period, seen in Figs. 3 and 7). With
other modifications to the signal model (resolutions and acceptances) a larger
variation in $\Delta m_{d}$ of $0.007$ ps-1 is found.
Table 1: A selection of fitted parameter values, for which statistical uncertainties only are given. The $B^{0}_{s}$ signal fraction includes contributions from any detached $D^{+}_{s}$ production. When the omitted fractions (of combinatorial background components) are included, the total fraction sums to unity within each $n$ region separately. Quantity | Normalized mass region
---|---
| Low-$n$ | High-$n$
Fit fraction of: | |
\- $B^{0}_{s}$ signal | 0.3247$\pm$0.0029 | 0.3604$\pm$0.0023
\- $B^{0}$ signal | 0.0781$\pm$0.0017 | 0.0968$\pm$0.0022
\- prompt $D^{+}_{s}$ | 0.0410$\pm$0.0026 | 0.0444$\pm$0.0018
\- prompt $D^{+}$ | 0.0196$\pm$0.0018 | 0.0311$\pm$0.0024
Mistag probability $\omega$: | |
\- $B^{0}_{s}$ signal | 0.347$\pm$0.054 | 0.333$\pm$0.021
\- $B^{0}$ signal | 0.3567$\pm$0.0063 | 0.3319$\pm$0.0065
Total candidates | 368,965 | 225,880
* •
Other models and binning: the order of the Chebychev polynomial is varied,
Crystal Ball functions are used for the mass peak shapes, and the background
parameterizations and the binning schemes are varied. Out of these
modifications, the binning scheme has the largest effect. Resulting
uncertanties of $0.05$ ps-1 and $0.001$ ps-1 are assigned to $\Delta m_{s}$
and $\Delta m_{d}$, respectively.
* •
Assumptions on $B^{+}$ decays: The $\Delta m_{d}$ measurement is sensitive to
$\chi_{d}$, the integrated mixing probability, which in turn is sensitive to
the non-mixing $B^{+}$-background. We hold constant several $B^{+}$-background
parameters in the baseline fit, determined from the full simulation. Many
features of the $B^{+}$ background fit are varied to evaluate systematic
variations, including the fraction, the lifetime, and the corrections for
relative tagging performance. The largest uncertainty arises from tagging
performance corrections and for this a $0.008$ ps-1 uncertainty is assigned to
$\Delta m_{d}$. It is possible to leave one or more of these parameters free
during the fit, but the loss in statistical precision is prohibitive.
Table 2: Sources of systematic uncertainty on $\Delta m_{s}$ and $\Delta m_{d}$. “Simulation” implies a combination of full LHCb simulation and pseudo-experiment studies. Source of uncertainty | Method | Systematic uncertainty
---|---|---
| | $\Delta m_{s}$ [ps-1] | $\Delta m_{d}$ [ps-1]
$k$-factor | Simulation | 0.06 | 0.0052
Detector alignment | Calibration | 0.03 | 0.0008
Values of $\Delta\Gamma$ | Data refit | n/a | 0.0004
Model bias | Simulation | 0.09 | 0.0055
Signal proper-time model | Data refit | 0.09 | 0.007
Other models and binning | Data refit | 0.05 | 0.001
$B^{+}$ ($\cal B$, efficiency, tagging) | Data refit | n/a | 0.008
Total | Sum in quadrature | 0.15 | 0.013
For cross-checks the data are split by LHCb magnet polarity and LHCb trigger
strategies; no variations beyond the expected statistical fluctuations are
observed. We obtain
$\displaystyle\Delta m_{s}=(17.93\pm 0.22\,\textrm{(stat)}\pm
0.15\,\textrm{(syst)})\,\textrm{ps}^{-1},$ $\displaystyle\Delta
m_{d}=(0.503\pm 0.011\,\textrm{(stat)}\pm
0.013\,\textrm{(syst)})\,\textrm{ps}^{-1}.$
To obtain a measure for the significance of the observed oscillations, the
global likelihood minimum for the full fit is compared with the likelihood of
the hypotheses corresponding to the edges of our search window ($\Delta m=0$
or $\Delta m\geq 50$ ps-1). Both would result in almost flat asymmetry curves
(cf. Fig. 7) corresponding to no observed oscillations. We reject the null
hypothesis of no oscillations by the equivalent of $5.8$ standard deviations
for $B^{0}_{s}$ oscillations and $13.0$ standard deviations for $B^{0}$
oscillations.
## 7 Fourier analysis
Figure 8: Result of using Fourier transforms to search for the $\Delta
m_{s}$-peak. The image on the left is constructed from bins of the
$K^{+}K^{-}\pi^{+}$ mass which are 25 MeV$c^{-2}$ in width, analysed in steps
of 5 MeV$c^{-2}$ such that a smooth image is produced. The colour scale (blue-
green-yellow-red) is an arbitrary linear representation of the signal
intensity; dark blue is used for zero and below. The vertical dashed line is
drawn at $18.0$ ps-1. The apparent double-peak structure is an artifact of
this image. On the right a slice around the $D^{+}_{s}$ mass region shows only
the peak as used to measure the central value and rms width.
The full fit as described above was performed in the time domain, but
measurement of the mixing frequency can also be made directly in the frequency
domain as a cross-check, using well-established Fourier transform techniques
[28, 29, 30]. The cosine term in Eq. 3 has a different sign for the odd and
even samples, where the lifetime, acceptance, and other features are shared;
this simplifies the analysis in the frequency domain. Any Fourier components
not arising from mixing are suppressed by subtracting the odd Fourier spectrum
from the even spectrum and no parameterizations of the background shapes,
signal shapes, or decay-time resolution are required, allowing a model-
independent measurement of the mixing frequencies. We search for the $\Delta
m_{s}$ peak in the subtracted Fourier spectrum, shown in Fig. 8. Extensive
fast simulation pseudo-experiments have shown that the value of $\Delta m_{s}$
is obtained reliably and with a reasonable precision using this method;
however $\Delta m_{d}$ is heavily biased and has a large uncertainty, and so a
result is not quoted. Since residual components of the Fourier spectrum are of
much lower frequency than the $\Delta m_{s}$ component, and several complete
oscillation periods of $\Delta m_{s}$ are observable, the search for a
spectral peak is relatively free from complications. For $\Delta m_{d}$,
however, the relatively low frequency is similar to that of many other
features of the data, and only a single oscillation period is observed;
therefore the determination of $\Delta m_{d}$ is difficult with this simple
model-independent approach.
Taking the spectrum for events in a 25 MeV$c^{-2}$ bin around the $D^{+}_{s}$
mass, we find a clear and separated peak (Fig. 8, right). The rms width of the
peak is 0.4 ps-1, around a peak value of $17.95$ ps-1; the rms can be used as
a model-independent proxy for the statistical uncertainty. To further evaluate
the expected statistical fluctuation in the peak value, we perform a large set
of fast simulation pseudo-experiments taking the result of the multivariate
fit as a model for signal and background. The uncertainty found from the
simulation studies is 0.32 ps-1, slightly smaller than given by the rms. We
report $\Delta m_{s}=(17.95\pm 0.40\,\textrm{(rms)}\pm 0.11\,\textrm{(syst)})$
ps${}^{-1},$ in order to be model-independent. Systematic uncertainties arise
from the detector alignment and the $k$-factor correction method, common to
both measurement techniques, as quantified previously in Sec. 6.
## 8 Conclusion
The mixing frequencies for neutral $B$ mesons have been measured using
flavour-specific semileptonic decays. To correct for the momentum lost to
missing particles, a simulation-based kinematic correction, known as the
$k$-factor, was adopted. Two techniques were used to measure the mixing
frequencies: a multidimensional simultaneous fit to the $K^{+}K^{-}\pi^{+}$
mass distribution, the decay-time distribution, and tagging information; and a
simple Fourier analysis. The results of the two methods were consistent, with
the first method being more precise. We obtain
$\displaystyle\Delta m_{s}=(17.93\pm 0.22\,\textrm{(stat)}\pm
0.15\,\textrm{(syst)})\,\textrm{ps}^{-1},$ $\displaystyle\Delta
m_{d}=(0.503\pm 0.011\,\textrm{(stat)}\pm
0.013\,\textrm{(syst)})\,\textrm{ps}^{-1}.$
We reject the hypothesis of no oscillations by 5.8 standard deviations for
$B^{0}_{s}$ and 13.0 standard deviations for $B^{0}$. This is the first
observation of $B^{0}_{s}$-$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing to be made using only
semileptonic decays.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-08-06T15:17:20 |
2024-09-04T02:49:49.085744
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, F. Andrianala, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G.\n Busetto, J. Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale,\n A. Cardini, H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L.\n Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C.\n Craik, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, A.\n Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L.\n De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono,\n N. D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H.\n Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett,\n A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba,\n S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, T. Skwarnicki, N.A.\n Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro,\n D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek, V. Syropoulos,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G.\n Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, K.\n Vervink, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Rob Lambert",
"url": "https://arxiv.org/abs/1308.1302"
}
|
1308.1340
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-145 LHCb-PAPER-2013-043 August 6, 2013
Measurement of the $C\\!P$ asymmetry in ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
A measurement of the $C\\!P$ asymmetry in ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ decays is presented using $pp$ collision data,
corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, recorded
by the LHCb experiment during 2011 at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$. The measurement is performed in seven bins
of $\mu^{+}\mu^{-}$ invariant mass squared in the range
${0.05<q^{2}<22.00{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}}$, excluding
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ resonance
regions. Production and detection asymmetries are corrected for using the
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ decay as
a control mode. Averaged over all the bins, the $C\\!P$ asymmetry is found to
be ${{\cal A}_{C\\!P}=0.000\pm 0.033\mbox{ (stat.)}\pm 0.005\mbox{ (syst.)}\pm
0.007\mbox{ }({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+})}$, where the
third uncertainty is due to the $C\\!P$ asymmetry of the control mode. This is
consistent with the Standard Model prediction.
Published in Phys. Rev. Lett.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, E.
Cowie45, D.C. Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8,
P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O.
Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M.
Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F.
Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M.
Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra
Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53,
T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H.
Gordon37, C. Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado
Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S.
Gregson46, P. Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu.
Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C.
Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N.
Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, T. Head37, V.
Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, M. Hess60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44,
R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J.
Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O.
Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U.
Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T. Palczewski27, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B.
Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8,
G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N.
Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E.
Santovetti23,k, M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M.
Savrie16,e, D. Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M.
Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A.
Semennikov30, K. Senderowska26, I. Sepp52, N. Serra39, J. Serrano6, P.
Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T.
Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires9, R.
Silva Coutinho47, M. Sirendi46, N. Skidmore45, T. Skwarnicki58, N.A. Smith51,
E. Smith54,48, J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro38, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stevenson54, S.
Stoica28, S. Stone58, B. Storaci39, M. Straticiuc28, U. Straumann39, V.K.
Subbiah37, L. Sun56, S. Swientek9, V. Syropoulos41, M. Szczekowski27, P.
Szczypka38,37, T. Szumlak26, S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F.
Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M.
Tobin38, S. Tolk41, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E.
Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6,
P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, D. Urner53, A.
Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, M. Van
Dijk45, R. Vazquez Gomez18, P. Vazquez Regueiro36, C. Vázquez Sierra36, S.
Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B.
Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10,
D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C.
Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K.
Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J.
Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P.
Williams47,48, M. Williams55, F.F. Wilson48, J. Wimberley57, J. Wishahi9, W.
Wislicki27, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y.
Xie49,37, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
The rare decay ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ is a flavour-changing
neutral current process mediated by electroweak loop (penguin) and box
diagrams. The absence of tree-level diagrams for the decay results in a small
value of the Standard Model (SM) prediction for the branching fraction, which
is supported by a measurement of ${(4.36\pm 0.23)\times 10^{-7}}$ [1]. Physics
processes beyond the SM that may enter via the loop and box diagrams could
have large effects on observables of the decay. Examples include the decay
rate, the $\mu^{+}\mu^{-}$ forward-backward asymmetry [1, 2, 3], and the
$C\\!P$ asymmetry [2, 4], as functions of the $\mu^{+}\mu^{-}$ invariant mass
squared ($q^{2}$).
The $C\\!P$ asymmetry is defined as
${\cal A}_{C\\!P}=\frac{\Gamma(B^{-}\rightarrow
K^{-}\mu^{+}\mu^{-})-\Gamma({B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}})}{\Gamma(B^{-}\rightarrow
K^{-}\mu^{+}\mu^{-})+\Gamma({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}})},$ (1)
where $\Gamma$ is the decay rate of the mode. This asymmetry is predicted to
be of order $10^{-4}$ in the SM [5], but can be significantly enhanced in
models beyond the SM [6]. Current measurements including the dielectron mode,
${\cal A}_{C\\!P}({B\rightarrow K^{+}\ell^{+}\ell^{-}})$, from BaBar and Belle
give ${-0.03\pm 0.14}$ and ${0.04\pm 0.10}$, respectively [4, 2], and are
consistent with the SM. The $C\\!P$ asymmetry has already been measured at
LHCb in $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays [7], ${{\cal
A}_{C\\!P}=-0.072\pm 0.040}$. Assuming that contributions beyond the SM are
independent of the flavour of the spectator quark, ${\cal A}_{C\\!P}$ should
be similar for both ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ and
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays.
In this Letter, a measurement of ${\cal A}_{C\\!P}$ in ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ decays is presented using $pp$ collision data,
corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, recorded
at a centre-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$ at LHCb in
2011. The inclusion of charge conjugate modes is implied throughout unless
explicitly stated.
The LHCb detector [8] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors [9]. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [10].
Samples of simulated events are used to determine the efficiency of selecting
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ signal events and to study certain
backgrounds. In the simulation, $pp$ collisions are generated using Pythia 6.4
[11] with a specific LHCb configuration [12]. Decays of hadronic particles are
described by EvtGen [13], in which final-state radiation is generated using
Photos [14]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [15,
*Agostinelli:2002hh] as described in Ref. [17]. The simulated samples are
corrected to reproduce the data distributions of the $B^{+}$ meson $p_{\rm T}$
and vertex $\chi^{2}$, the track $\chi^{2}$ of the kaon, as well as the
detector IP resolution, particle identification and momentum resolution.
Candidate events are first required to pass a hardware trigger, which selects
muons with $\mbox{$p_{\rm T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ [18].
In the subsequent software trigger, at least one of the final-state particles
is required to have $\mbox{$p_{\rm
T}$}>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and IP $>100\,\upmu\rm m$ with
respect to all primary $pp$ interaction vertices (PVs) in the event. Finally,
the tracks of two or more of the final-state particles are required to form a
vertex that is displaced from the PVs.
An initial selection is applied to the ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ candidates to enhance signal decays and suppress
combinatorial background. Candidate $B^{+}$ mesons must satisfy requirements
on their direction and flight distance, to ensure consistency with originating
from the PV. The decay products must pass criteria regarding the
$\chi^{2}_{\rm IP}$, where $\chi^{2}_{\rm IP}$ is defined as the difference in
$\chi^{2}$ of a given PV reconstructed with and without the considered
particle. There is also a requirement on the vertex $\chi^{2}$ of the
$\mu^{+}\mu^{-}$ pair. All the tracks are required to have $p_{\rm T}$
$>250{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
Additional background rejection is achieved by using a boosted decision tree
(BDT) [19] that implements the AdaBoost algorithm [20]. The BDT uses the
$p_{\rm T}$ and $\chi^{2}_{\rm IP}$ of the muons and the $B^{+}$ meson
candidate, as well as the decay time, vertex $\chi^{2}$, and flight direction
of the $B^{+}$ candidate and the $\chi^{2}_{\rm IP}$ of the kaon. Data,
corresponding to an integrated luminosity of 0.1$\mbox{\,fb}^{-1}$, are used
to optimise this selection, leaving 0.9$\mbox{\,fb}^{-1}$ for the
determination of ${\cal A}_{C\\!P}$.
Following the multivariate selection, candidate events pass several
requirements to remove specific sources of background. Particle identification
(PID) criteria are applied to kaon candidates to reduce the number of pions
incorrectly identified as kaons. Candidates with $\mu^{+}\mu^{-}$ invariant
mass in the ranges
$2.95<m_{\mu\mu}<3.18{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$3.59<m_{\mu\mu}<3.77{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are removed to
reject backgrounds from tree level
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\rightarrow\mu^{+}\mu^{-})K^{+}}$ and
${B^{+}\rightarrow\psi{(2S)}(\rightarrow\mu^{+}\mu^{-})K^{+}}$ decays. Those
in the first range are selected as
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ decays,
which are used as a control sample. If
$m_{K\mu\mu}<5.22{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the vetoes are
extended downwards by 0.25 and 0.19${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
respectively, to remove the radiative tails of the resonant decays. If
$5.35<m_{K\mu\mu}<5.50{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ the vetoes are
extended upwards by 0.05${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to remove
misreconstructed resonant candidates that appear at large $m_{\mu\mu}$ and
$m_{K\mu\mu}$. Further vetoes are applied to remove
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ events
in which the kaon and a muon have been swapped, and contributions from decays
involving charm mesons such as ${B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(\rightarrow K^{+}\pi^{-})\pi^{+}}$
where both pions are misidentified as muons. After these selection
requirements have been applied, there are two sources of background that are
difficult to distinguish from the signal. These are ${B^{+}\rightarrow
K^{+}\pi^{+}\pi^{-}}$ and ${B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ decays,
which both contribute at the level of 1% of the signal yield. These peaking
backgrounds are accounted for during the analysis.
In order to perform a measurement of ${\cal A}_{C\\!P}$, the production and
detection asymmetries associated with the measurement must be considered. The
raw measured asymmetry is, to first order,
$\mathcal{A}_{\rm RAW}({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}})={\cal
A}_{C\\!P}({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}})+\mathcal{A}_{\rm
P}+\mathcal{A}_{\rm D},$ (2)
where the production and detection asymmetries are defined as
$\displaystyle\mathcal{A}_{\rm P}$ $\displaystyle\equiv$
$\displaystyle[R(B^{-})-R(B^{+})]/[R(B^{-})+R(B^{+})],$ (3)
$\displaystyle\mathcal{A}_{\rm D}$ $\displaystyle\equiv$
$\displaystyle[\epsilon(K^{-})-\epsilon(K^{+})]/[\epsilon(K^{-})+\epsilon(K^{+})],$
(4)
where $R$ and $\epsilon$ represent the $B$ meson production rate and kaon
detection efficiency, respectively. The detection asymmetry has two
components: one due to the different interactions of positive and negative
kaons with the detector material, and a left-right asymmetry due to particles
of different charges being deflected to opposite sides of the detector by the
magnet. The component of the detection asymmetry from muon reconstruction is
small and neglected. Since the LHCb experiment reverses the magnetic field,
about half of the data used in the analysis is taken with each polarity.
Therefore, an average of the measurements with the two polarities is used to
suppress significantly the second effect. To account for both the detection
and production asymmetries, the decay
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ is used,
which has the same final-state particles as ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ and very similar kinematic properties. The $C\\!P$
asymmetry in ${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}$ decays has been measured as $(1\pm 7)\times 10^{-3}$ [21, 22].
Neglecting the difference in the final-state kinematic properties of the kaon,
the production and detection asymmetries are the same for both modes, and the
value of the $C\\!P$ asymmetry can be obtained via
${\cal A}_{C\\!P}({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}})=\mathcal{A}_{\rm
RAW}({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}})-\mathcal{A}_{\rm
RAW}({B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}})+{\cal
A}_{C\\!P}({B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}).$ (5)
Differences in the kinematic properties are accounted for by a systematic
uncertainty.
In the data set, approximately 1330 ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$
and 218,000 ${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}$ signal decays are reconstructed. To measure any variation in
${\cal A}_{C\\!P}$ as a function of $q^{2}$, which improves the sensitivity of
the measurement to physics beyond the SM, the ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ dataset is divided into the seven $q^{2}$ bins used in
Ref. [1]. The measurement is also made in a bin of
$1<q^{2}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, which is of
particular theoretical interest. To determine the number of $B^{+}$ decays in
each bin, a simultaneous unbinned maximum likelihood fit is performed to the
invariant mass distributions of the ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$
and ${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$
candidates in the range
${5.10<m_{K\mu\mu}<5.60{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$. The signal
shape is parameterised by a Cruijff function [23], and the combinatorial
background is described by an exponential function. All parameters of the
signal and combinatorial background are allowed to vary freely in the fit.
Additionally, there is background from partially-reconstructed decays such as
${B^{0}\rightarrow K^{*0}(\rightarrow K^{+}\pi^{-})\mu^{+}\mu^{-}}$ or
${B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}(\rightarrow K^{+}\pi^{-})}$ where the pion is undetected. For the
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ distribution, these decays are fitted
by an ARGUS function [24] convolved with a Gaussian function to account for
detector resolution. For the
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ decays
the partially-reconstructed background is modelled by another Cruijff
function. The shapes of the peaking backgrounds, due to ${B^{+}\rightarrow
K^{+}\pi^{+}\pi^{-}}$ and ${B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ decays,
are taken from fits to simulated events.
In each $q^{2}$ bin, the
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ and
${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ data sets are divided according to
the charge of the $B^{+}$ meson and magnet polarity, providing eight distinct
subsets. These are fitted simultaneously with the parameters of the signal
Cruijff function common for all eight subsets. For each subset, the only
independent fitting parameters are the combined yield of the $B^{+}$ and
$B^{-}$ decays and the values of $\mathcal{A}_{\rm RAW}$ for the signal,
control and background modes for each magnet polarity. The fits to the
invariant mass distributions of the ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$
candidates in the full $q^{2}$ range are shown in Fig. 1.
The value of ${\cal A}_{C\\!P}$ for each magnet polarity is determined from
Eq. 5, and an average with equal weights is taken to obtain a single value for
the $q^{2}$ bin. To obtain the final value of ${\cal A}_{C\\!P}$ for the full
dataset, an average is taken of the values in each $q^{2}$ bin, weighted
according to the signal efficiency and the number of ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ decays in the bin,
${\cal A}_{C\\!P}=\frac{\sum^{7}_{i=1}(N_{i}{\cal
A}_{C\\!P}^{i})/\epsilon_{i}}{\sum^{7}_{i=1}N_{i}/\epsilon_{i}},$ (6)
where $N_{i}$, $\epsilon_{i}$, and ${\cal A}_{C\\!P}^{i}$ are the signal
yield, signal efficiency, and the fitted value of the $C\\!P$ asymmetry in the
$i\mathrm{th}$ $q^{2}$ bin.
Figure 1: Invariant mass distributions of ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ candidates for the full $q^{2}$ range. The results of
the unbinned maximum likelihood fits are shown with blue, solid lines. Also
shown are the signal component (red, short-dashed), the combinatorial
background (grey, long-dashed), and the partially-reconstructed background
(magenta, dot-dashed). The peaking backgrounds ${B^{+}\rightarrow
K^{+}\pi^{+}\pi^{-}}$ (green, double-dot-dashed) and
${B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ (teal, dotted) are also shown under
the signal peak. The four datasets are (a) $B^{+}$ and (b) $B^{-}$ for one
magnet polarity, and (c) $B^{+}$ and (d) $B^{-}$ for the other.
Several assumptions are made about the backgrounds. The partially-
reconstructed background is assumed to exhibit no $C\\!P$ asymmetry. For
${B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}}$, ${\cal A}_{C\\!P}$ is also assumed
to be zero [25]. For the ${B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}}$ decay,
${\cal A}_{C\\!P}$ in each $q^{2}$ bin is taken from a recent LHCb measurement
[26]. The effect of these assumptions on the result is investigated as a
systematic uncertainty.
Various sources of systematic uncertainty are considered. The analysis relies
on the assumption that the ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ and
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ decays
have the same final-state kinematic distributions, so that the relation in Eq.
5 is exact. To estimate the bias associated with this assumption, the
kinematic distributions of
${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ decays
are reweighted to match those of ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$, and
the value of $\mathcal{A}_{\rm RAW}$ is recalculated. The variables used are
the momentum, $p_{\rm T}$ and pseudorapidity of the $B^{+}$ and $K^{+}$
mesons, as well as the $B^{+}$ decay time and the position of the kaon in the
detector. The difference between the two values of $\mathcal{A}_{\rm RAW}$ for
each variable is taken as the systematic uncertainty. The total systematic
uncertainty associated to the different kinematic behaviour of the two decays
in each $q^{2}$ bin is calculated by adding each individual contribution in
quadrature.
Table 1: Systematic uncertainties on ${\cal A}_{C\\!P}$ from non-cancelling asymmetries arising from kinematic differences between ${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$ and ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ decays, and fit uncertainties arising from the choice of signal shape, mass fit range and combinatorial background shape, and from the treatment of the asymmetries in the ${B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ and partially-reconstructed (PR) backgrounds. The total is the sum in quadrature of each component. | Residual | Signal | Mass | Comb. | ${\cal A}_{C\\!P}$ in | ${\cal A}_{C\\!P}$ in |
---|---|---|---|---|---|---|---
$q^{2}$ bin (${\mathrm{\,Ge\kern-0.80005ptV^{2}\\!/}c^{4}}$) | asymmetries | shape | range | shape | ${B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ | PR | Total
$0.05<q^{2}<2.00$ | $0.005$ | $0.005$ | $0.002$ | $0.002$ | $0.004$ | $0.002$ | $0.008$
$2.00<q^{2}<4.30$ | $0.004$ | $0.001$ | $0.005$ | $0.009$ | $0.005$ | $0.001$ | $0.012$
$4.30<q^{2}<8.68$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.005$ | $0.002$ | $0.005$
$10.09<q^{2}<12.86$ | $0.003$ | $0.005$ | $0.023$ | $0.003$ | $0.003$ | $0.001$ | $0.024$
$14.18<q^{2}<16.00$ | $0.006$ | $0.001$ | $0.004$ | $0.003$ | $<0.001$ | $0.001$ | $0.008$
$16.00<q^{2}<18.00$ | $0.005$ | $0.007$ | $0.017$ | $<0.001$ | $<0.001$ | $0.001$ | $0.019$
$18.00<q^{2}<22.00$ | $0.008$ | $0.001$ | $0.014$ | $<0.001$ | $0.001$ | $0.001$ | $0.016$
Weighted average | $0.001$ | $<0.001$ | $0.003$ | $0.001$ | $0.003$ | $<0.001$ | $0.005$
$1.00<q^{2}<6.00$ | $0.002$ | $<0.001$ | $0.009$ | $0.002$ | $0.004$ | $0.002$ | $0.010$
The choice of fit model also introduces systematic uncertainties. The fit is
repeated using a different signal model, replacing the Cruijff function with
the sum of two Crystal Ball functions [27] that have the same mean and tail
parameters, but different Gaussian widths. The difference in the value of
${\cal A}_{C\\!P}$ using these two fits is assigned as the uncertainty. The
fit is also repeated using a reduced mass range of
${5.17<m_{K\mu\mu}<5.60{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$ to
investigate the effect of excluding the partially-reconstructed background.
The difference in results obtained by modelling the combinatorial background
using a second-order polynomial, rather than an exponential function, produces
a small systematic uncertainty.
Uncertainties also arise from the assumptions made about the asymmetries in
background events. Phenomena beyond the SM could cause the $C\\!P$ asymmetry
in ${B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}}$ decays to be large [25], and so
the analysis is performed again for values of ${{\cal
A}_{C\\!P}({B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}})=\pm 0.5}$, with the larger
of the two deviations in ${\cal A}_{C\\!P}({B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}})$ taken as the systematic uncertainty. As the partially-
reconstructed background can arise from $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decays, the value of ${\cal A}_{C\\!P}$ for this source
background is taken to be $-0.072$ [7], the value from the LHCb measurement,
neglecting any further $C\\!P$ violation in angular distributions. The
difference in the fit result compared to the zero ${\cal A}_{C\\!P}$
hypothesis is taken as the systematic uncertainty. Variations in ${\cal
A}_{C\\!P}({B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}})$ have a negligible effect
on the final result. A summary of the systematic uncertainties is shown in
Table 1. The value of ${\cal A}_{C\\!P}$ calculated by performing the fits on
the data set integrated over $q^{2}$ is consistent with that from the weighted
average of the $q^{2}$ bins.
The results for ${\cal A}_{C\\!P}$ in each $q^{2}$ bin and the weighted
average are displayed in Table 2, as well as in Fig. 2. The value of the raw
asymmetry in ${B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}}$ determined from the fit is ${-0.016\pm 0.002}$. The $C\\!P$
asymmetry in ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ decays is measured to be
${\cal A}_{C\\!P}=0.000\pm 0.033\mbox{ (stat.)}\pm 0.005\mbox{ (syst.)}\pm
0.007\mbox{ }({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}),$
where the third uncertainty is due to the uncertainty on the known value of
${\cal A}_{C\\!P}({B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}})$. This compares with the current world average of ${-0.05\pm
0.13}$ [21], and previous measurements including the dielectron final-state
[4, 2]. This result is consistent with the SM, as well as the
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decay mode, and improves the precision
of the current world average for the dimuon mode by a factor of four. With the
recent observation of resonant structure in the low-recoil region above the
$\psi{(2S)}$ resonance [28], care should be taken when interpreting the result
in this region. Interesting effects due to physics beyond the SM are possible
through interference with this resonant structure, and could be investigated
in a future update of the measurement of ${\cal A}_{C\\!P}$.
Table 2: Values of ${\cal A}_{C\\!P}$ and the signal yields in the seven $q^{2}$ bins, the weighted average, and their associated uncertainties. | | | Stat. | Syst.
---|---|---|---|---
$q^{2}$ bin (${\mathrm{\,Ge\kern-0.80005ptV^{2}\\!/}c^{4}}$) | ${B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}$ yield | ${\cal A}_{C\\!P}\left({B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}}\right)$ | uncertainty | uncertainty
$0.05<q^{2}<2.00$ | $164\pm 14$ | $-0.152$ | $0.085$ | $0.008$
$2.00<q^{2}<4.30$ | $167\pm 14$ | $-0.008$ | $0.094$ | $0.012$
$4.30<q^{2}<8.68$ | $339\pm 21$ | $0.070$ | $0.067$ | $0.005$
$10.09<q^{2}<12.86$ | $221\pm 17$ | $0.060$ | $0.081$ | $0.024$
$14.18<q^{2}<16.00$ | $145\pm 13$ | $-0.079$ | $0.091$ | $0.008$
$16.00<q^{2}<18.00$ | $145\pm 13$ | $0.100$ | $0.093$ | $0.019$
$18.00<q^{2}<22.00$ | $120\pm 13$ | $-0.070$ | $0.111$ | $0.016$
Weighted average | | $0.000$ | $0.033$ | $0.005$
$1.00<q^{2}<6.00$ | $362\pm 21$ | $-0.019$ | $0.061$ | $0.010$
Figure 2: Measured value of ${\cal A}_{C\\!P}$ in ${B^{+}\rightarrow
K^{+}\mu^{+}\mu^{-}}$ decays in bins of the $\mu^{+}\mu^{-}$ invariant mass
squared ($q^{2}$). The points are displayed at the mean value of $q^{2}$ in
each bin. The uncertainties on each ${\cal A}_{C\\!P}$ value are the
statistical and systematic uncertainties added in quadrature. The excluded
charmonium regions are represented by the vertical red lines, the dashed line
is the weighted average, and the grey band indicates the 1$\sigma$ uncertainty
on the weighted average.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-08-06T16:37:21 |
2024-09-04T02:49:49.104862
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, E. Cowie, D.C. Craik, S.\n Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P.\n Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin,\n U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez,\n P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Simon Wright",
"url": "https://arxiv.org/abs/1308.1340"
}
|
1308.1343
|
# Performance and Optimization Abstractions for Large Scale Heterogeneous
Systems in the Cactus/Chemora Framework
Erik Schnetter Perimeter Institute for Theoretical Physics, Waterloo,
Ontario, Canada
Department of Physics, University of Guelph, Guelph, Ontario, Canada
Center for Computation & Technology, Louisiana State University, Baton Rouge,
Louisiana, USA
Homepage: http://www.perimeterinstitute.ca/personal/eschnetter/
(July 15, 2013)
###### Abstract
We describe a set of lower-level abstractions to improve performance on modern
large scale heterogeneous systems. These provide portable access to system-
and hardware-dependent features, automatically apply dynamic optimizations at
run time, and target stencil-based codes used in finite differencing, finite
volume, or block-structured adaptive mesh refinement codes.
These abstractions include a novel data structure to manage refinement
information for block-structured adaptive mesh refinement, an iterator
mechanism to efficiently traverse multi-dimensional arrays in stencil-based
codes, and a portable API and implementation for explicit SIMD vectorization.
These abstractions can either be employed manually, or be targeted by
automated code generation, or be used via support libraries by compilers
during code generation. The implementations described below are available in
the Cactus framework, and are used e.g. in the Einstein Toolkit for
relativistic astrophysics simulations.
## I Introduction
Cactus [16, 12] is a software framework for high performance computing,
notably used e.g. in the Einstein Toolkit [20, 15] for relativistic
astrophysics. The Chemora project [11] aims at significantly simplifying the
steps necessary to move from a physics model to an efficient implementation on
modern hardware. Starting from a set of partial differential equations
expressed in a high level language, it automatically generates highly
optimized code suitable for parallel execution on heterogeneous systems. The
generated code is portable to many operating systems, and adopts widely used
parallel programming standards and programming models (MPI, OpenMP, SIMD
Vectorization, CUDA, OpenCL).
In this paper, we describe a set of lower-level abstractions available in the
Cactus framework, and onto which Chemora is building. These abstractions are
used by many Cactus components outside the Chemora project as well.
These abstractions are:
1. 1.
a novel data structure to manage refinement information for block-structured
adaptive mesh refinement (section II),
2. 2.
an iterator mechanism to efficiently traverse multi-dimensional arrays in
stencil-based codes, employing dynamic auto-tuning at run time (section III),
3. 3.
a portable API and implementation for explicit SIMD vectorization, including
operations necessary for stencil-based kernels (section IV).
These abstractions address issues we encountered when porting Cactus-based
applications to modern HPC systems such as Blue Waters (NCSA), Hopper (NERSC),
Kraken (NICS), Mira (ALCF), or Stampede (TACC). Of course, these abstractions
also improve performance on “regular” HPC systems, workstations, or laptops.
Below, we describe each of these abstractions in turn, and conclude with
general observations and remarks.
## II Efficient Bounding Box Algebra
When using adaptive mesh refinement (AMR), one needs to specify which regions
of a grid need to be refined. The shape of these regions can be highly
irregular. Some AMR algorithms (called cell-based AMR) allow this decision to
be made independently for every cell, others (called block-structured AMR)
require that refined points be clustered into non-overlapping, rectangular
regions for improved efficiency [10, 9]. These regions can then efficiently be
represented e.g. via Fortran-style arrays on which loop kernels operate. While
cell-based AMR algorithms require tree data structures to represent the
refinement hierarchy, block-structured AMR algorithms (such as available in
Carpet [22, 21, 13]) require data structures to represent _sets of bounding
boxes_ describing the regions that make up a particular refinement level.
A _bounding box_ (`bbox`) describes the location and shape of a rectangular
region, a _bounding box set_ (`bboxset`) describes a set of non-overlapping
bounding boxes. There is a direct connection between a bboxset and how data
for grid points are stored in memory. While a bboxset can in principle
describe any set of grid points (that may have arbitrary shape and may be
disconnected), one assumes that a bboxset comprises just a few rectangular
regions, which will then be handled more efficiently.
Since a bboxset is used to describe the grid points that make up a particular
refinement level, its points lie on a uniform grid; see figure 1(a) below for
an example. Each grid point can be described by its location, which can be
expressed as $x_{0}^{i}+n^{i}\cdot\Delta x^{i}$ where $x_{0}^{i}$ and $\Delta
x^{i}$ describe origin and spacing of the grid, and $n^{i}$ is a vector with
integer elements. (The abstract index $i$ denotes that these are vectors,
where $i\in[1\ldots D]$ in $D$ dimensions.)
Carpet not only uses bounding box sets to describe refined regions, it also
offers a full algebra for bounding box sets. This includes operations such as
set union, intersection, difference, complement, etc., and also includes
additional operations enabled by the grid structure such as `shift` (to move a
set by a certain offset) or `expand` (to grow a set in some directions), or to
change the grid spacing. It is also possible to convert a bboxset into a
normalized list of bboxes.
This full set algebra allows using bboxsets as a convenient base for
implementing many other operations, such as determining the AMR operators for
prolongation, boundary prolongation, restriction, or refluxing; distributing a
refinement level’s grid points onto MPI processes; determining the
communication schedule; or performing consistency checks in a simple-to-
express manner. The price one has to pay is that this requires an efficient
data structure for these operations, such as we describe below.
### II-A Background
There exist two simple approaches to describe sets: one can either enumerate
the elements of the set (e.g. in a list or a tree), or one can view the set as
a mapping from elements to a boolean (storing one boolean for each element
e.g. in an array or a map). The former is efficient if the sets contain few
elements and if the elements can be meaningfully ordered, the latter is
efficient if the number of possible set elements is small.
Unfortunately, neither is the case here: We intend to handle a refinement
level as a set of rectangular bboxes for efficiency; building a data structure
that disregards this structure and manages points individually will be much
less efficient. At the same time, the _possible_ number of points can be many
orders of magnitude larger than the _actual_ number of points in a region.
Thus neither enumerating the grid points making up a bboxset (e.g. via their
integer coordinates) makes sense, nor using a boolean array to describe which
points belong to a refinement. A more complex data structure is needed.
The literature describes a host of data structures for holding sets of points,
or to describe sets of regions. For example, GIS (Geographic Information
Systems) heavily rely on such data structures, and $R$-trees or $R^{*}$-trees
[23] find applications there. While it would be possible to design an
efficient bboxset data structure based on these, they do not quite fit our
problem description: They assume that the points making up the set are
unstructured (i.e. do not need to be located on a uniform grid), and make no
attempt to cluster these points into bboxes. On the other hand, $R^{*}$-trees
are able to handle regions with varying point density, which is not relevant
for a uniform grid.
Other block-structured AMR packages introduce data structure to handle sets of
points on uniform grids, but do not provide a full set algebra. Internally,
these bboxsets are often represented as a list of bounding boxes. For example,
AMROC [1] calls this structure `BBoxList`. (AMROC is a successor of DAGH,
which is in turn the intellectual predecessor of Carpet.) Efficient operations
may include creating a bboxset from a list of non-overlapping bboxes, while
adding an individual bbox to an existing bboxset may not be an efficient
operation. Specific operations required for an AMR algorithm are then
implemented efficiently, but other operations – such as calculating the
intersection between two bboxsets – are not. Most AMR operations acting on
bboxsets are then implemented in an ad-hoc fashion and may introduce arbitrary
restrictions, e.g. regarding the size of the individual bboxes, or the number
of ghost zones for inter-process communication.
Carpet’s previous bboxset data structure was based on a list of non-
overlapping bboxes. It did provide a full algebra of set operations, but with
reduced efficiency. For example, the list of bboxes was kept normalized,
requiring an $O(n^{2})$ normalization step after each set operation, where $n$
is the number of bboxes in the list. Many other set operations also had an
$O(n^{2})$ cost, as is common for set implementations based on lists. This
cost was acceptable for small numbers of bboxes (say, less than 1,000), but
began to dominate the grid setup time when using more than 1,000 MPI
processes, as the regions owned by MPI processes are described by bboxes.
An earlier attempt to improve the efficiency of bboxsets is described in [25].
Unfortunately, this work never left the demonstration stage.
We are not aware of other literature or source code describing a generic,
efficient data structure to handle sets of points lying on a uniform grid. To
our knowledge, this is a novel data structure for AMR applications.
### II-B Discrete Derivatives of Bounding Box Sets
Our data structure is based on storing the _discrete derivative_ of a bboxset.
An example is shown in figure 1. Algebraically, the discrete derivative in the
$i$-direction of a bboxset $R$ is given by
$\displaystyle\partial_{i}R$ $\displaystyle:=$ $\displaystyle
R\;\veebar\;\mathrm{shift}(R,-e^{i})$ (1)
where $\veebar$ is the symmetric set difference (exclusive or),
$\mathrm{shift}(R,v)$ shifts the bboxset $R$ by a certain offset $v$, and
$e^{i}$ is the unit vector in direction $i$ ($i$th component is 1, all other
components are 0).
(a) L-shaped region
(b) $x$-derivative of this region
(c) $xy$-derivative of this region
Figure 1: An L-shaped region and its derivatives. The $xy$-derivative consists
only of the “key points” of this shape, and can be stored very efficiently.
This discrete derivative is equivalent to a finite difference derivative,
applied to the boolean values describing the set interpreted as integer values
modulo 2, i.e. using the following arithmetic rules for addition: $0\oplus
0=0$, $0\oplus 1=1$, $1\oplus 1=0$.
Note that we choose to use a leftward finite difference; this is a convention
only and has no other relevance. We also note that these discrete derivatives
commute, so that the order in which they are applied does not matter. Given a
set derivative, the anti-derivative is uniquely defined, and the original set
can be readily recovered:
$\displaystyle R$ $\displaystyle:=$
$\displaystyle\partial_{i}R\;\veebar\;\mathrm{shift}(R,-e^{i})$ (2)
This implies that the anti-derivative should be calculated by scanning from
left to right.
The salient point about taking the derivative is that it reduces the number of
elements in a set, assuming that the set has a “regular” structure. For
example, in two dimensions (as shown above), an L-shaped region is described
by just six points, and in three dimensions, a cuboid (“3D rectangle”) is
described by just eight points. In fact, the number of points in the
derivative of a set increases with the number of bboxes required to describe
it – this is exactly the property we are looking for, as the efficiency of a
block-structured AMR algorithm already depends on the number of bboxes
required to represent the bboxset.
Since the number of elements in the derivative of a set is small, we store
these points (i.e. their locations) directly in a tree structure.
Instead of taking derivatives to identify boundaries, one could also use a
run-length encoding; the resulting algorithm would be very similar.
### II-C Implementation
We now describe an efficient algorithm for set operations based on storing
bboxsets as derivatives.
Most set operations cannot directly be applied to bboxsets stored as
derivatives. The notable exception for this is the symmetric difference, which
can directly be applied to derivatives:
$\displaystyle R\veebar S$ $\displaystyle=$ $\displaystyle\partial
R\veebar\partial S$ (3)
where we introduce the notation $\partial R$ to denote subsequent derivatives
in all direction, i.e. $\partial R:=\partial_{0}\partial_{1}R$ in two
dimensions, and $\partial R:=\partial_{0}\partial_{1}\partial_{2}R$ in three
dimensions. Property (3) follows directly from the definition of the
derivative above and the properties of the exclusive-or operator.
To efficiently reconstruct a bboxset from its derivative, we employ a
_sweeping algorithm_ [24]. Instead of directly taking the derivative of a
bboxset $R$ in all directions, we employ dimensional recursion. We represent a
$d$-dimensional bboxset by taking its derivative in direction $d$, and storing
the resulting set of $d-1$-dimensional bboxsets in a tree structure. We do
this recursively, until we arrive at $0$-dimensional bboxsets. These are
single points, corresponding to a single boolean value that we store directly,
ending the recursion.
Since this data structure represents a bboxset, it is irrelevant how the
bboxset is represented internally. In particular, from the $d$-dimensional
bboxset’s representation, it does not matter how the $d-1$-dimensional
bboxsets are internally represented, and from an algorithm design point of
view, the $d-1$-dimensional bboxsets are directly available for processing.
We now describe how to efficiently evaluate the result of a set operation
acting on two bboxsets $R$ and $S$, calculating $T:=R\odot S$ for an arbitrary
set operation $\odot$. The main idea is to sweep the domain in direction $d$,
keeping track of of the _current state_ of the $d-1$-dimensional subsets
$R_{d-1}$, $S_{d-1}$, and $T_{d-1}$ on the sweep line. (This “line” is a
$d-1$-dimensional hypersurface in general.) As the sweep line progresses, we
update $R_{d-1}$ and $S_{d-1}$ by calculating the anti-derivative from our
stored derivatives, calculate $T_{d-1}:=R_{d-1}\odot S_{d-1}$, and then
calculate and store the derivate of $T_{d-1}$ in a new bboxset structure.
The operation $T_{d-1}:=R_{d-1}\odot S_{d-1}$ needs to be re-evaluated
whenever $R_{d-1}$ or $S_{d-1}$ change, i.e. once for each element in the
stored derivatives of $R_{d}$ and $S_{d}$. Figure 2 lists the respective
algorithm.
$R_{d-1}:=\\{\\}$
$S_{d-1}:=\\{\\}$
$T_{d-1}:=\\{\\}$
$n:=0$
while find next $n$ for which $\partial_{d}R_{d}$ or $\partial_{d}S_{d}$ do
if $\partial_{d}R_{d}$ contains an element at $n$ then
$R_{d_{1}}:=R_{d-1}\veebar\partial_{d}R_{d}[n]$
end if
if $\partial_{d}S_{d}$ contains an element at $n$ then
$S_{d_{1}}:=S_{d-1}\veebar\partial_{d}S_{d}[n]$
end if
$T^{\prime}_{d-1}:=T_{d-1}$
$T_{d-1}:=R_{d-1}\odot S_{d-1}$
$\partial_{d}T_{d}[n]:=T_{d-1}\veebar T^{\prime}_{d-1}$
if $\partial_{d}T_{d}[n]$ not empty then
store $\partial_{d}T_{d}[n]$
end if
end while
Figure 2: Algorithm for traversing two bboxsets $R$ and $S$, calculating
$T:=R\odot S$ where $\odot$ is an arbitrary set operation. This algorithm
applies to a $d$-dimensional set, recursing to $d-1$ dimensions.
Given that accessing set elements stored in a tree has a cost of $O(\log n)$,
set operations implemented via the algorithm above have a cost that can be
bounded by $O([n_{d}\log n_{d}]^{d})$, where $d$ is the number of dimensions,
and $n_{d}$ is the maximum number of bboxes encountered by a scan line in
direction $d$. In non-pathological cases, $n_{d}\approx n^{1/d}$, leading to a
log-linear cost. Figure 3 demonstrates then scalability of Carpet for a weak
scaling benchmark, when this bboxset algorithm is used for all set operations.
Figure 3: Weak scaling benchmarks for a relativistic astrophysics application
with Carpet, using nine refinement levels (top) and a uniform grid (bottom).
Smaller times are better, ideal scaling is a horizontal line. Carpet’s AMR
implementation scales to 10k+ cores. With a uniform grid, Carpet scales to
250k+ cores.
### II-D Future Work
This derivative bboxset data structure and its associated algorithms are
serial, as the sweeping algorithm is sequential and does not lead to a natural
parallelization. (Of course, different sets can still be processed in
parallel.)
One parallelization approach would be to break each bboxset into several
independent pieces and to process these in parallel. This would then also
require stitching the results together after each set operation.
## III Dynamic Loop Optimizations
Most CPUs and accelerators (if present) of modern HPC systems are multi-core
systems with a deep memory hierarchy, where each core requires SIMD
vectorization to obtain the highest performance. In addition, in-order-
execution systems (i.e. accelerators, including Blue Gene/Q) require SMT
(symmetric multi-threading) to hide memory access and instruction latencies.
These architectures require significant programmer effort to achieve good
single-node performance, even when leaving distributed memory MPI programming
aside.111Single-_core_ performance is not really relevant here, since (a)
applications will use more than one core per node, and (b) the individual
cores interact at run time e.g. via cache access patterns.
Ignoring the issues of SIMD vectorization here (see section IV below), one
would hope that language standards and implementations such as OpenMP or
OpenCL allow programmers to ensure efficiency. However, this is not so, for
several reasons:
* •
neither OpenMP nor OpenCL allow distinguishing between SMT, where threads
share all caches, and coarse-grained multi-threading, where most cache levels
are not shared;
* •
it is very difficult, if not impossible to reliably predict performance of
compute kernels, so that dynamic (run-time) decisions regarding optimizations
are necessary;
* •
the most efficient multi-threading algorithms need to be aware of cache line
boundaries, which also needs to factor into how multi-dimensional arrays are
allocated; neither OpenMP nor OpenCL provide support for this.
Here we present _LoopControl_ , an iterator mechanism to efficiently loop over
multi-dimensional arrays for stencil-based loop kernels. LoopControl
parallelizes iterations across multiple threads, across multiple SMT threads,
performs loop tiling to improve cache efficiency, and honours SIMD vector
sizes to ensure an efficient SIMD vectorization is possible.
LoopControl monitors the performance of its loops, and dynamically adjusts its
parameters to improve performance. This not only immediately adapts to
different machines and to code modifications, but also to differing conditions
at run time such as changes to array sizes (e.g. due to AMR) or changes to the
physics behaviour in loop kernels. LoopControl uses a _random-restart hill-
climbing_ algorithm for this dynamic optimization.
The multi-threading is based on OpenMP threads, but employs a dynamic region
selection and load distribution mechanism to handle kernels with non-uniform
cost per iteration.
LoopControl employs _hwloc_ [6] to query the system hardware, and also queries
MPI and OpenMP about process/thread setups. hwloc also reports thread-to-core
and thread-to-cache bindings that are relevant for performance. All
information is gathered automatically, requiring no user setup to achieve good
performance.
LoopControl dynamically auto-tunes stencil codes at run time. This is
fundamentally different from traditional auto-tuning, which surveys the
parameter space for a set of optimizations ahead of time, and then re-uses
these survey results at run time. See e.g. [14] for a description of ahead-of-
time auto-tuning of stencil-based codes, or [8] for a description of ahead-of-
time auto-tuning search algorithms.
Ahead-of-time surveys have the disadvantage that they need to be repeated for
each machine on which the code runs, for each compiler version/optimization
setting, for each modification to the loop kernel, and also for different
array sizes. This makes it prohibitively expensive to use in a code that
undergoes rapid development, or where adaptive features such as AMR are used.
LoopControl does not have these limitations, and to our knowledge,
LoopControl’s dynamic auto-tuning algorithm is novel.
### III-A Loop Traversal
LoopControl assumes that each loop iteration is independent of the others, and
can thus be executed in parallel or in an arbitrary order.
Most architectures have several levels of caches. LoopControl implicitly
chooses one cache level for which it optimizes. The random-restart algorithm
(see below) will explore optimizing for other cache levels as well, and will
settle for that level that yields the largest performance gain. It would be
straightforward to implement support for multiple cache levels, but it is not
clear that this would significantly improve performance in practice.
LoopControl uses the following mechanisms, in this order, to split the index
space of a loop:
1. 1.
coarse-grained (non-SMT) multithreading (expecting no shared caches)
2. 2.
iterating over loop tiles (each expected to be small enough to fit into the
cache)
3. 3.
iterating within loop tiles
4. 4.
fine-grained (SMT) multithreading (expecting to share the finest cache level)
5. 5.
SIMD vectorization (see section IV below).
The index space is only known at run time. It is split multiple times to find
respective smaller index spaces for each of the mechanisms described above.
Each index space is an integer multiple of the next smaller index space, up to
boundary effects.
Certain index space sizes and offsets have to obey certain constraints:
* •
SIMD vectorization requires that its index space to be aligned with and have
the same size as the SIMD hardware vector size.
* •
The SMT multithreading index space should be a multiple of the vector size, so
that partial vector store operations are not required except at loop
boundaries (as some hardware does not offer thread-safe partial vector
stores).
* •
The number of SMT and non-SMT threads is determined by the operating system
upon program start, and are not modified (i.e. all threads are used).
* •
Loop tiles should be aligned with cache lines for efficiency.
Since LoopControl cannot influence how arrays are allocated, the programmer
needs to specify the array alignment, if any. The first array element is
expected be aligned with the vector size or the cache line size (which can
always be ensured when the array is allocated), and the array dimensions may
or may not be padded to multiples of the vector size or cache line size.
Higher alignment leads to more efficient execution on some hardware, since
edge effects such as partial vector stores or partial cache line writes can be
avoided. LoopControl offers support for all cases.
### III-B Random-Restart Hill-Climbing
Since each loop behaves differently (obviously), and since this performance
behaviour also depends on the loop bounds, LoopControl optimizes each _loop
setup_ independently. A loop setup includes the loop’s source code location,
index space, array alignment, and number of threads available.
Several execution parameters describe how a loop setup is executed, describing
how the index space is split according to the mechanisms described above.
Each newly encountered loop setup has its initial execution parameters chosen
heuristically. As timing measurements of the loop setup’s execution become
available, these parameter settings are optimized. It is well known that the
execution time of a loop kernel depends on optimization parameters in a highly
non-linear and irregular manner, with many threshold effects present.
Simplistic optimization algorithms will thus fail. For this optimization, we
use a random-restart hill-climbing algorithm as described in the following.
Our optimization algorithm has two competing goals: (1) for a given execution
parameter setting, quickly find the nearby local optimum, and (2) do not get
stuck in local optima; instead, explore the whole parameter space. To find a
local optimum, we use a _hill climbing_ algorithm: we explore the local
neighbourhood of a given parameter setting, and move to any setting that leads
to a shorter run time, discarding parameter settings that lead to longer
execution times. To explore the whole parameter space, we use a _random
restart_ method: once we arrived in a local optimum, we decide with a certain,
small probability to chose a random new parameter setting. After exploring the
neighbourhood of this new parameter setting, we either remain there (if it is
better), or return to the currently known best setting.
There is one major difference between an ahead-of-time exploration of the
parameter space, and a dynamic optimization at run time: The goal of an ahead-
of-time exploration is to find the best possible parameter setting, while the
goal of a run-time optimization is to reduce the overall run time. A bad
parameter setting can be significantly worse than a mediocre parameter
setting, and can easily have a running time that is an order of magnitude
higher. That means that exploring even one such bad parameter setting has a
cost that is only amortized if one executes hundreds of loops with good
parameter settings.
This makes it important to be cautious about exploring the parameter space,
and to very quickly abort any excursion that significantly worsens the run
time. It is much more important to find a mediocre improvement and to find it
quickly, than to find the optimum parameter choice and incurring a large
overhead. In particular, we find that genetic algorithms or simulated
annealing spend much too much time on bad parameter settings, and while they
may ultimately find good parameter settings, this comes at too great a cost to
be useful for a dynamic optimization to be applied at run-time.
For the relatively large kernels present in our astrophysics application, we
observe roughly a 10% improvement over a naive OpenMP parallelization via
`#pragma omp parallel for` for the first loop executions via our heuristic
parameter choices, and an additional approximately 10% improvement in the long
run via LoopControl’s dynamic optimizations.
### III-C Future Work
It may be worthwhile to save and restore execution parameter settings and
their respective timings. Although these may be invalidated by code
modifications or changes to the build setup, this would provide a way to
remember exceptionally good parameter settings that may otherwise be difficult
to re-discover.
In particular in conjunction with OpenCL, where it is simple to dynamically
re-compile a loop kernel, LoopControl’s optimizations could also include
compile-time parameter settings such as loop unrolling or prefetching.
In addition to these low-level loop execution optimizations, one can also
introduce optimizations at a higher level, such as e.g. loop fission or loop
fusion. These optimizations can have large impacts on performance if they make
code fit into instruction- or data-caches. Combining LoopControl’s optimizer
with a way to select between different (sets of) loop kernels would be
straightforward.
## IV SIMD Vectorization
Modern CPUs offer SIMD (short vector) instructions that operate on a small
number of floating point values simultaneously; the exact number (e.g. 2, 4,
or 8) is determined by the hardware architecture. To achieve good performance,
it is essential that SIMD instructions are used when compiling compute
kernels; not doing so will generally reduce the possible theoretical peak
performance by this factor. Of course, this is relevant only for compute-bound
kernels.
### IV-A Background
However, using SIMD instructions typically comes with a set of restrictions
that need to be satisfied; if not, SIMD instructions either cannot be used, or
lose a significant fraction of their performance. One of these restrictions is
that it is efficient to perform element-wise operations, but quite inefficient
to reduce across a vector. That is, while e.g. $a_{i}:=b_{i}+c_{i}$ is highly
efficient, the operation $s:=\sum_{i}a_{i}$ will be relatively expensive. This
means that one should aim to vectorize across calculations that are mutually
independent. As a rule of thumb, it is better to vectorize across different
loop iterations than to try and find independent operations within a single
iteration.
Another restriction concerns memory access patterns. Memory and cache
subsystems are these days highly vectorized themselves (with typical vector
sizes of e.g. 64 bytes), and efficient load/store operations for SIMD vectors
require that these vectors are _aligned_ in memory. Usually, a SIMD vector
with a size of $N$ bytes needs to be located at an address that is a multiple
of $N$. Unaligned memory accesses are either slower, or are not possible at
all and then need to be split into two aligned memory accesses and shift
operations.
Finally, if one vectorizes across loop iterations, the number of iterations
may not be a multiple of the vector size. Similarly, if one accesses an array
in a loop, then the first accessed array element may not be aligned with the
vector size. In both cases, one needs to perform operations involving only a
subset of the elements of an SIMD vector. This is known as _masking_ the
vector operations. The alternative – using scalar operations for these edge
cases – is very expensive if the vector size is large.
Unfortunately, the programming languages that are widely used in HPC today (C,
C++, Fortran) do not offer any constructs that would directly map to these
SIMD machine instructions, nor do they offer declarations that would ensure
the necessary alignment of data structure. It is left to the compiler to
identify kernels where SIMD instructions can be used to increase efficiency,
and to determine whether data structures have the necessary alignment. Often,
system-dependent source code annotations can be used to help the compiler,
such as e.g. `#pragma ivdep` or `#pragma simd` for loops, or
`__builtin_assume_aligned` for pointers.
Generally, compiler-based vectorization works fine for small loop kernels,
surrounded by simple loop constructs, contained in small functions. This
simplifies the task of analyzing the code, proving that vectorization does not
alter the meaning, and allowing estimating the performance of the generated
code to ensure that vectorization provides a benefit. However, we find that
the converse is also true: large compute kernels, kernels containing non-
trivial control flow (if statements), or using non-trivial math functions
(exp, log) will simply not be vectorized by a given compiler. “Convincing” a
certain compiler that a loop should be vectorized remains a highly system-
specific and vendor-specific (i.e. non-portable) task. In addition, if a loop
is vectorized, then the generated code may make pessimistic assumptions
regarding memory alignment that lead to sub-ideal performance, in particular
when stencil operations in multi-dimensional arrays are involved.
The root of the problem seems to be that the compiler’s optimizer does not
have access to sufficiently rich, high-level information about the employed
algorithms and their implementation to make good decisions regarding
vectorization. (The same often holds true for other optimizations as well,
such as e.g. loop fission/fusion, or cloning functions to modify their
interfaces.) We hope that the coming years will lead to widely accepted ways
to pass such information to the compiler, either via new languages or via
source code annotations. For example, the upcoming OpenMP 4.0 standard will
provide a `#pragma omp simd` to enforce vectorization, GCC is already
providing `__builtin_assume_aligned` for pointers, and Clang’s vectorizer has
as of version 3.3 arguably surpassed that of GCC 4.8, justifying our hope that
things are improving.
### IV-B Manual Vectorization
The hope for future compiler features expressed in the previous section does
not help performance today. Today, vectorizing a non-trivial code requires
using architecture-specific and sometimes compiler-specific _intrinsics_ that
provide C/C++ datatypes and function calls mapping directly to respective
vector types and vector instructions that are directly supported by the
hardware. This allows achieving very high performance, at the cost of
portability.
For example, the simple loop
for (int i=0; i<N; ++i) {
a[i] = b[i] * c[i] + d[i];
}
can be manually vectorized with Intel/AMD’s SSE2 intrinsics (for all 64-bit
Intel and AMD CPUs) as
#include <emmintrin.h>
for (int i=0; i<N; i+=2) {
__m128d ai, bi, ci, di;
bi = _mm_load_pd(&b[i]);
ci = _mm_load_pd(&c[i]);
di = _mm_load_pd(&d[i]);
ai = _mm_add_pd(_mm_mul_pd(bi, ci), ci);
_mm_store_pd(&a[i], ai);
}
or with IBM’s QPX intrinsics (for the Blue Gene/Q) as
#include <builtins.h>
for (int i=0; i<N; i+=4) {
vector4double ai, bi, ci, di;
bi = vec_lda(0, &b[i]);
ci = vec_lda(0, &c[i]);
di = vec_lda(0, &d[i]);
ai = vec_madd(bi, ci, di);
vec_sta(ai, 0, &a[i]);
}
These vectorized loops assume that the array size $N$ is a multiple of the
vector size, and that the arrays are aligned with the vector size. If this is
not the case, the respective vectorized code is more complex.
While the _syntax_ of the vectorized kernels looks quite different, the
_semantic_ transformations applied to the original kernel are quite similar.
Vector values are stored in variables that have a specific type (`__mm128d`,
`vector4double`), memory access operations have to be denoted explicitly
(`_mm_load_pd`, `vec_lda`), and arithmetic operations become function calls
(`_mm_add_pd`, `vec_madd`). Other architectures require code transformations
along the very same lines.
Note that QPX intrinsics support a fused multiply-add (_fma_) instruction that
calculates $a\cdot b+c$ in a single instruction (and presumably also in a
single cycle). Regular C or C++ code would express these via separate multiply
and add operations, and it would be the task of the compiler to synthesize
such fma operations when beneficial. When writing vectorized code manually,
the compiler will generally not synthesize vector fma instructions, and this
transformation has to be applied explicitly. Today, most CPU architectures
support fma instructions.
Vector architectures relevant for high-performance computing these days
include Intel’s and AMD’s SSE instructions, Intel and AMD’s AVX instructions
(both SSE and AVX exist in several variants), Intel’s Xeon Phi vector
instructions, IBM’s Altivec and VSX instructions for Power CPUs, and IBM’s QPX
instructions for the Blue Gene/Q. On low-power devices, ARM’s NEON
instructions are also important.
### IV-C An API for Explicit Vectorization
Based on architecture- and compiler-dependent intrinsics, we have designed and
implemented a portable, efficient API for explicit loop vectorization. This
API targets stencil-based loop kernels, as can e.g. be found in codes using
finite differences or finite volumes, possibly via block-structured adaptive
mesh refinement. Our implementation `LSUThorns/Vectors` uses C++ and supports
all major current HPC architectures [20, 15].
The API is intended to be applied to existing scalar codes in a relatively
straightforward manner. Data structures do not need to be reorganized,
although it may provide a performance benefit if they are, e.g. ensuring
alignment of data accessed by vector instructions, or choosing integer sizes
compatible with the available vector instructions.
The API consists of the following parts:
* •
data types holding vectors of real numbers (float/double), integers, and
booleans (e.g. for results of comparison operators, or for masks);
* •
the usual arithmetic operations (+ - * /, copysign, fma, isnan, signbit, sqrt,
etc.), including comparisons, boolean operations, and an if-then construct;
* •
“expensive” math functions, such as cos, exp, log, sin, etc. that are
typically not available as hardware instructions;
* •
memory load/store operations, supporting both aligned and unaligned access,
supporting masks, and possibly offering to bypass the cache to improve
efficiency;
* •
helper functions to iterate over a index ranges, generating masks, and
ensuring efficient array access, suitable in particular for stencil-based
kernels.
We describe these parts in more detail below.
The OpenCL C language already provides all but the last item. Once mature
OpenCL implementations become available for HPC platforms – that is, for the
CPUs on which the applications will be running, not only for accelerators that
may be available there – this API could be replaced by programming in OpenCL C
instead. We are actively involved in the _pocl_ (_Portable Computing
Language_) project [5] which develops a portable OpenCL implementation based
on the LLVM infrastructure [2].
#### IV-C1 Data Types and Arithmetic Operations
The first two items – data types and arithmetic operations – can be directly
mapped to the vector intrinsics available on the particular architecture. We
remark that vectorized integer operations are often not available, and that
vectorized boolean values are internally often represented and handled quite
differently from C or C++.
For each architecture, the available vector instruction set and vector sizes
are determined automatically at compile time, and the most efficient vector
size available is chosen. Both double and single precision vectors are
supported.
For several architectures, this API is implemented via macros instead of via
inline functions. Surprisingly, several widely used compilers for HPC systems
cannot handle inline functions efficiently. The most prominent consequence of
this is that operator overloading is not possible; instead, arithmetic
operations have to be expressed in a function call syntax such as
`vec_add(a,b)`. While straightforward, this unfortunately reduces readability
somewhat.
A trivial implementation, useful e.g. for debugging, maps this API to scalar
operations without any loss of efficiency.
Using our API, the example from above becomes
#include <vectors.h>
for (int i=0; i<N; i+=CCTK_REAL_VEC_SIZE) {
CCTK_REAL_VEC ai, bi, ci, ci;
bi = vec_load(&b[i]);
ci = vec_load(&c[i]);
di = vec_load(&d[i]);
ai = vec_madd(bi, ci, di);
vec_store(&a[i], ai);
}
This code is portable across many architectures. However, this example still
assumes that all arrays are aligned with the vector size, that the array size
is a multiple of the vector size, and does not include any cache
optimizations.
`if` statements require further attention when vectorizing, since different
elements of a vector may lead to different paths through the code. Similarly,
the logical operators `&&` and `||` cannot have shortcut semantics with vector
operands (see e.g. the OpenCL standard [3]). To translate `if` statements, we
provide a function `ifthen(cond, then, else)` with a definition very similar
to the `?:` operator, but without shortcut semantics. (This corresponds to the
OpenCL `select` function, except for the order of the arguments.)
To vectorize an `if` statement, it needs to be rewritten using this `ifthen`
function, taking into account that both the _then_ and the _else_ branches
will be evaluated for all vector elements. Often, declaring separate local
variables for the _then_ and the _else_ branches and moving all memory store
operations (if any) out of the `if` statement (and turning them into masked
store operations if necessary) make this transformation straightforward.
#### IV-C2 “Expensive” Math Functions
Some compilers (IBM, Intel) offer efficient implementations of the “expensive”
math functions that can be used (`mass_simd`, `mkl_vml`), while other
compilers (GCC, Clang) do not. To support system architectures other than
IBM’s and Intel’s, we have implemented an open-source library _Vecmathlib_ [7,
18] providing portable, efficient, vectorized math functions.222Under some
circumstances, this library is for scalar code faster than glibc on Intel/AMD
CPUs.
The OpenCL C language standard requires that these math functions be available
for vector types. For the pocl project’s OpenCL compiler, we thus use
Vecmathlib to implement these where no vendor library is available.
#### IV-C3 Memory Access
The API supports a variety of access modes for memory load/store operations
that are likely to occur in stencil-based codes. In particular, great care has
been taken to ensure that the most efficient code is generated depending on
either compile-time or run-time guarantees that the code can make regarding
alignment. Some code transformations, such as array padding of multi-
dimensional arrays, enable such guarantees and can thus improve performance.
Let us consider a slightly more complex example using stencil operations. The
code below calculates a derivative via a forward finite difference:
for (int i=0; i<N-1; ++i) {
a[i] = b[i+1] - b[i];
}
We assume that the arrays `a` and `b` are aligned with the vector size, and
that $N-1$ is a multiple of the vector size. This code can then be vectorized
to
#include <vectors.h>
for (int i=0; i<N-1; i+=CCTK_REAL_VEC_SIZE) {
CCTK_REAL_VEC ai, bi, bip;
bi = vec_load(&b[i]);
bip = vec_loadu_off(+1, &b[i+1]);
ai = vec_sub(bip, bi);
vec_store(&a[i], ai);
}
Here, the function `vec_loadu_off(offset, ptr)` loads a value from memory that
is located at an offset of $+1$ from an aligned value. This specification
expects that the offset is known at compile time, and allows the compiler to
generate the most efficient code for this case. A similar function
`vec_loadu(ptr)` allows loading unaligned values if the offset is unknown at
compile time. Equivalent functions exist for storing values.
#### IV-C4 Iterators
Finally, our API provides an “iterator” to simplify looping over index ranges.
Typically, only the innermost loop of a loop nest is vectorized, and it is
expected that this loop has unit stride. This iterator also sets a mask to
handle edge cases at the beginning and end of the index range. This is also
connected to shared memory parallelization such as via OpenMP, where one wants
to ensure that an OpenMP parallelization of the innermost loop does not
introduce unaligned loop bounds.
The scalar code below evaluates a centered finite difference:
for (int i=1; i<N-1; ++i) {
a[i] = 0.5 * (b[i+1] - b[i-1]);
}
We assume again that the arrays `a` and `b` are aligned with the vector size.
We also assume that the array is padded, so that we can access elements that
are “slightly” out of bounds without causing a segmentation fault. Both
conditions can easily be guaranteed by allocating the arrays correspondingly,
e.g. via `posix_memalign`. If the arrays’ alignment is not known at compile
time, then they need to be accessed via `vec_loadu` and `vec_storeu` functions
instead. We make no other assumptions, and the array can have an arbitrary
size. This leads to the following vectorized code:
#include <vectors.h>
VEC_ITERATE(i, 1, N-1) {
CCTK_REAL_VEC ai, bim, bip;
bim = vec_loadu_off(-1, &b[i-1]);
bip = vec_loadu_off(+1, &b[i+1]);
ai = vec_mul(vec_set1(0.5), vec_sub(bip, bim));
vec_store_nta_partial(&a[i], ai);
}
The macro `VEC_ITERATE(i, imin, imax)` expands to a loop that iterates the
variable `i` from `imin` to `imax` with a stride of the vector size. It also
ensures that `i` is always a multiple of the vector size, starting from a
lower value than `imin` if necessary. Additionally, it prepares an (implicitly
declared) mask in each iteration.
The suffix `_partial` in the vector store operation indicates that this mask
is taken into account when storing. The code is optimized for the case where
all vector elements are stored. The suffix `_nta` invokes a possible cache
optimization, if available. It indicates that the stored value will in the
near future not be accessed again (“non-temporal access”). This hint can be
used by the implementation to bypass the cache when storing the value.
Most CPU architectures do not support masking arbitrary vector operations,
while masking load/store operations may be supported. In the examples given
here, we only mask store operations, assuming that arrays are sufficiently
padded for load operations to always succeed. The unused vector elements are
still participating in calculations, but this does not introduce an overhead.
This iterator provides provides a generic mechanism to traverse arrays holding
scalar values via vectorized operations. It thus provides the basic framework
to enable vectorization for a loop, corresponding to a `#pragma simd`
statement. By implicitly providing masks that can be used when storing values,
aligned and padded arrays are handled efficiently.
Different from the previous items, this iterator is applicable even for OpenCL
C code, since no equivalent constructs exist in the language.
### IV-D Applications
The API described above allows explicitly vectorizing C++ code. While somewhat
tedious, it is in our experience straightforward to vectorize a large class of
scalar codes where vectorization is beneficial. There is special support for
efficient support of stencil-based codes on block-structured grids using
multi-dimensional arrays.
While manual vectorization is possible, this API also lends itself for
automated code generation. We use Kranc [17, 19] to create Cactus components
from partial differential equations written in Mathematica notation, and have
modified Kranc’s back-end to emit vectorized code. Mathematica’s pattern
matching capabilities are ideal to apply optimizations to the generated vector
expressions that the compiler is unwilling to perform.
## V Conclusion
This paper describes a set of abstractions to improve performance on modern
large scale heterogeneous systems targetting stencil-based codes. These
abstractions are available in the Cactus framework, and are used in “real-
world” applications, such as in relativistic astrophysics simulations via the
Einstein Toolkit.
Our implementations of these abstractions require access to low-level system
information. Especially hwloc [6] and PAPI [4] provide valuable information.
While hwloc is very portable and easy to use, we are less satisfied with the
state of PAPI installations; these are often not available (and neither are
alternatives), not even on freshly installed cutting-edge systems. We are
highly dissatisfied with this situation, which forces us to resort to crude
overall timing measurements to evaluate performance.
While our performance and optimization abstractions are portable, they are by
their very nature somewhat low-level, and using them directly e.g. from C++
code can be tedious, although straightforward. We anticipate that they will
see most use either via automated code generation (e.g. via Kranc [17, 19]),
or via including them into compiler support libraries (e.g. via pocl [5]).
## Acknowledgements
We thank Marek Blazewicz, Steve Brandt, Peter Diener, Ian Hinder, David
Koppelman, and Frank Löffler for valuable discussions and feedback. We also
thank the Cactus and the Einstein Toolkit developer community for volunteering
to test these implementations in their applications.
This work was supported by NSF award 0725070 _Blue Waters_ , NSF awards
0905046 and 0941653 _PetaCactus_ , NSF award 1212401 _Einstein Toolkit_ , and
an NSERC grant to E. Schnetter.
This work used computational systems at ALCF, NCSA, NERSC, NICS, Sharcnet,
TACC, as well as private systems at Caltech, LSU, and the Perimeter Institute.
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|
arxiv-papers
| 2013-08-06T16:40:47 |
2024-09-04T02:49:49.114804
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Erik Schnetter",
"submitter": "Erik Schnetter",
"url": "https://arxiv.org/abs/1308.1343"
}
|
1308.1428
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-144 LHCb-PAPER-2013-040 29 September 2013
First measurement of time-dependent
$C\\!P$ violation in $B^{0}_{s}\rightarrow K^{+}K^{-}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
Direct and mixing-induced $C\\!P$-violating asymmetries in
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decays are measured for the first time using
a data sample of $pp$ collisions, corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$, collected with the LHCb detector at a centre-of-mass
energy of 7$\mathrm{\,Te\kern-1.00006ptV}$. The results are $C_{KK}=0.14\pm
0.11\pm 0.03$ and $S_{KK}=0.30\pm 0.12\pm 0.04$, where the first uncertainties
are statistical and the second systematic. The corresponding quantities are
also determined for $B^{0}\rightarrow\pi^{+}\pi^{-}$ decays to be
$C_{\pi\pi}=-0.38\pm 0.15\pm 0.02$ and $S_{\pi\pi}=-0.71\pm 0.13\pm 0.02$, in
good agreement with existing measurements.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
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Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
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A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The study of $C\\!P$ violation in charmless charged two-body decays of neutral
$B$ mesons provides a test of the Cabibbo-Kobayashi-Maskawa (CKM) picture [1,
*Kobayashi:1973fv] of the Standard Model (SM), and is a sensitive probe to
contributions of processes beyond SM [3, 4, 5, 6, 7]. However, quantitative SM
predictions for $C\\!P$ violation in these decays are challenging because of
the presence of loop (penguin) amplitudes, in addition to tree amplitudes. As
a consequence, the interpretation of the observables requires knowledge of
hadronic factors that cannot be accurately calculated from quantum
chromodynamics at present. Although this represents a limitation, penguin
amplitudes may also receive contributions from non-SM physics. It is necessary
to combine several measurements from such two-body decays, exploiting
approximate flavour symmetries, in order to cancel or constrain the unknown
hadronic factors [3, 6].
With the advent of the BaBar and Belle experiments, the isospin analysis of
$B\rightarrow\pi\pi$ decays [8] has been one of the most important tools for
determining the phase of the CKM matrix. As discussed in Refs.[3, 6, 7], the
hadronic parameters entering the $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decays are related by the U-spin symmetry,
_i.e._ by the exchange of $d$ and $s$ quarks in the decay diagrams. Although
the U-spin symmetry is known to be broken to a larger extent than isospin, it
is expected that the experimental knowledge of $B^{0}_{s}\rightarrow
K^{+}K^{-}$ can improve the determination of the CKM phase, also in
conjunction with the $B\rightarrow\pi\pi$ isospin analysis [9].
Other precise measurements in this sector also provide valuable information
for constraining hadronic parameters and give insights into hadron dynamics.
LHCb has already performed measurements of time-integrated $C\\!P$ asymmetries
in $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays [10, 11], as well as measurements of branching fractions of charmless
charged two-body $b$-hadron decays [12].
In this paper, the first measurement of time-dependent $C\\!P$-violating
asymmetries in $B^{0}_{s}\rightarrow K^{+}K^{-}$ decays is presented. The
analysis is based on a data sample, corresponding to an integrated luminosity
of $1.0$$\mbox{\,fb}^{-1}$, of $pp$ collisions at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ collected with the LHCb detector. A new
measurement of the corresponding quantities for
$B^{0}\rightarrow\pi^{+}\pi^{-}$ decays, previously measured with good
precision by the BaBar [13] and Belle [14] experiments, is also presented. The
inclusion of charge-conjugate decay modes is implied throughout.
Assuming $C\\!PT$ invariance, the $C\\!P$ asymmetry as a function of time for
neutral $B$ mesons decaying to a $C\\!P$ eigenstate $f$ is given by
$\mathcal{A}(t)=\frac{\Gamma_{{\kern
1.25995pt\overline{\kern-1.25995ptB}{}}^{0}_{(s)}\rightarrow
f}(t)-\Gamma_{B^{0}_{(s)}\rightarrow f}(t)}{\Gamma_{{\kern
1.25995pt\overline{\kern-1.25995ptB}{}}^{0}_{(s)}\rightarrow
f}(t)+\Gamma_{B^{0}_{(s)}\rightarrow f}(t)}=\frac{-C_{f}\cos(\Delta
m_{d(s)}t)+S_{f}\sin(\Delta
m_{d(s)}t)}{\cosh\left(\frac{\Delta\Gamma_{d(s)}}{2}t\right)-A^{\Delta\Gamma}_{f}\sinh\left(\frac{\Delta\Gamma_{d(s)}}{2}t\right)},$
(1)
where $\Delta m_{d(s)}=m_{{d(s)},\,\mathrm{H}}-m_{{d(s)},\,\mathrm{L}}$ and
$\Delta\Gamma_{d(s)}=\Gamma_{{d(s)},\,\mathrm{L}}-\Gamma_{{d(s)},\,\mathrm{H}}$
are the mass and width differences of the $B^{0}_{(s)}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ system mass eigenstates. The
subscripts $\mathrm{H}$ and $\mathrm{L}$ denote the heaviest and lightest of
these eigenstates, respectively. The quantities $C_{f}$, $S_{f}$ and
$A^{\Delta\Gamma}_{f}$ are
$\begin{split}C_{f}=\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}},\,\,\,\,\,\,\,\,\,\,\,S_{f}=\frac{2{\rm
Im}\lambda_{f}}{1+|\lambda_{f}|^{2}},\,\,\,\,\,\,\,\,\,\,\,A^{\Delta\Gamma}_{f}=-\frac{2{\rm
Re}\lambda_{f}}{1+|\lambda_{f}|^{2}},\end{split}$ (2)
with $\lambda_{f}$ defined as
$\lambda_{f}=\frac{q}{p}\frac{\bar{A}_{f}}{A_{f}}.$ (3)
The two mass eigenstates of the effective Hamiltonian in the
$B^{0}_{(s)}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ system
are $p|B^{0}_{(s)}\rangle\pm q|\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rangle$, where $p$ and $q$
are complex parameters. The parameter $\lambda_{f}$ is thus related to
$B^{0}_{(s)}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ mixing
(via $q/p$) and to the decay amplitudes of the $B^{0}_{(s)}\rightarrow f$
decay ($A_{f}$) and of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow f$ decay
($\bar{A}_{f}$). Assuming, in addition, negligible $C\\!P$ violation in the
mixing ($|q/p|=1$), as expected in the SM and confirmed by current
experimental determinations [15, 16], the terms $C_{f}$ and $S_{f}$
parameterize direct and mixing-induced $C\\!P$ violation, respectively. In the
case of the $B^{0}_{s}\rightarrow K^{+}K^{-}$ decay, these terms can be
expressed as [3]
$C_{KK}=\frac{2\tilde{d}^{\prime}\sin\vartheta^{\prime}\sin\gamma}{1+2\tilde{d}^{\prime}\cos\vartheta^{\prime}\cos\gamma+\tilde{d}^{\prime
2}},$ (4)
$S_{KK}=\frac{\sin(2\beta_{s}-2\gamma)+2\tilde{d}^{\prime}\cos\vartheta^{\prime}\sin(2\beta_{s}-\gamma)+\tilde{d}^{\prime
2}\sin(2\beta_{s})}{1+2\tilde{d}^{\prime}\cos\vartheta^{\prime}\cos\gamma+\tilde{d}^{\prime
2}},$ (5)
where $\tilde{d}^{\prime}$ and $\vartheta^{\prime}$ are hadronic parameters
related to the magnitude and phase of the tree and penguin amplitudes,
respectively, $-2\beta_{s}$ is the $B^{0}_{s}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mixing phase, and $\gamma$ is
the angle of the unitarity triangle given by
$\arg\left[-\left(V_{ud}V_{ub}^{*}\right)/\left(V_{cd}V_{cb}^{*}\right)\right]$.
Additional information can be provided by the knowledge of
$A^{\Delta\Gamma}_{KK}$, determined from $B^{0}_{s}\rightarrow K^{+}K^{-}$
effective lifetime measurements [17, 18].
The paper is organized as follows. After a brief introduction on the detector,
trigger and simulation in Sec. 2, the event selection is described in Sec. 3.
The measurement of time-dependent $C\\!P$ asymmetries with neutral $B$ mesons
requires that the flavour of the decaying $B$ meson at the time of production
is identified. This is discussed in Sec. 4. Direct and mixing-induced $C\\!P$
asymmetry terms are determined by means of two maximum likelihood fits to the
invariant mass and decay time distributions: one fit for the
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decay and one for
$B^{0}\rightarrow\pi^{+}\pi^{-}$ decay. The fit model is described in Sec. 5.
In Sec. 6, the calibration of flavour tagging performances, realized by using
a fit to $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow
K^{-}\pi^{+}$ mass and decay time distributions, is discussed. The results of
the $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ and
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ fits are given in Sec. 7 and the
determination of systematic uncertainties discussed in Sec. 8. Finally,
conclusions are drawn in Sec. 9.
## 2 Detector, trigger and simulation
The LHCb detector [19] is a single-arm forward spectrometer covering the
pseudorapidity range between 2 and 5, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes placed downstream.
The combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
($d_{\mathrm{IP}}$) resolution of 20$\,\upmu\rm m$ for tracks with high
transverse momenta. The $d_{\mathrm{IP}}$ is defined as the minimum distance
between the reconstructed trajectory of a particle and a given $pp$ collision
vertex (PV). Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) detectors [20]. Photon, electron and hadron candidates are identified
by a calorimeter system consisting of scintillating-pad and preshower
detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons
are identified by a system composed of alternating layers of iron and
multiwire proportional chambers [21].
The trigger [22] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. Events selected by any hardware trigger decision
are included in the analysis. The software trigger requires a two-, three- or
four-track secondary vertex with a large sum of the transverse momenta of the
tracks and a significant displacement from the PVs. At least one track should
have a transverse momentum ($p_{\mathrm{T}}$) exceeding
$1.7$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any PV greater than 16. The $\chi^{2}_{\rm IP}$ is the difference
in $\chi^{2}$ of a given PV reconstructed with and without the considered
track.
A multivariate algorithm [23] is used for the identification of secondary
vertices consistent with the decay of a $b$ hadron. To improve the trigger
efficiency on hadronic two-body $B$ decays, a dedicated two-body software
trigger is also used. This trigger selection imposes requirements on the
following quantities: the quality of the reconstructed tracks (in terms of
$\chi^{2}$/ndf, where ndf is the number of degrees of freedom), their
$p_{\mathrm{T}}$ and $d_{\mathrm{IP}}$; the distance of closest approach of
the decay products of the $B$ meson candidate ($d_{\mathrm{CA}}$), its
transverse momentum ($p_{\mathrm{T}}^{B}$), impact parameter
($d_{\mathrm{IP}}^{B}$) and the decay time in its rest frame ($t_{\pi\pi}$,
calculated assuming decay into $\pi^{+}\pi^{-}$).
Simulated events are used to determine the signal selection efficiency as a
function of the decay time, and to study flavour tagging, decay time
resolution and background modelling. In the simulation, $pp$ collisions are
generated using Pythia 6.4 [24] with a specific LHCb configuration [25].
Decays of hadronic particles are described by EvtGen [26], in which final
state radiation is generated using Photos [27]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [28, *Agostinelli:2002hh] as described in Ref. [30].
## 3 Event selection
Events passing the trigger requirements are filtered to reduce the size of the
data sample by means of a loose preselection. Candidates that pass the
preselection are then classified into mutually exclusive samples of different
final states by means of the particle identification (PID) capabilities of the
RICH detectors. Finally, a boosted decision tree (BDT) algorithm [31] is used
to separate signal from background.
Three sources of background are considered: other two-body $b$-hadron decays
with a misidentified pion or kaon in the final state (cross-feed background),
pairs of randomly associated oppositely-charged tracks (combinatorial
background), and pairs of oppositely-charged tracks from partially
reconstructed three-body $B$ decays (three-body background). Since the three-
body background gives rise to candidates with invariant mass values well
separated from the signal mass peak, the event selection is mainly intended to
reject cross-feed and combinatorial backgrounds, which mostly affect the
invariant mass region around the nominal $B^{0}_{(s)}$ mass.
The preselection, in addition to tighter requirements on the kinematic
variables already used in the software trigger, applies requirements on the
largest $p_{\mathrm{T}}$ and on the largest $d_{\mathrm{IP}}$ of the $B$
candidate decay products, as summarized in Table 1.
Table 1: Kinematic requirements applied by the event preselection. Variable | Requirement
---|---
Track $\chi^{2}$/ndf | $<5$
Track $p_{\mathrm{T}}\,[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $>1.1$
Track $d_{\mathrm{IP}}\,[\\!\,\upmu\rm m]$ | $>120$
$\mathrm{max}\,p_{\mathrm{T}}\,[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $>2.5$
$\mathrm{max}\,d_{\mathrm{IP}}\,[\\!\,\upmu\rm m]$ | $>200$
$d_{\mathrm{CA}}\,[\\!\,\upmu\rm m]$ | $<80$
$p_{\mathrm{T}}^{B}\,\,[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}]$ | $>1.2$
$d_{\mathrm{IP}}^{B}\,[\\!\,\upmu\rm m]$ | $<100$
$t_{\pi\pi}\,\,[\textrm{ps}]$ | $>0.6$
$m_{\pi^{+}\pi^{-}}\,\,[\\!{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}]$ | $4.8$–$5.8$
The main source of cross-feed background in the
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$
invariant mass signal regions is the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ decay,
where one of the two final state particles is misidentified. The PID is able
to reduce this background to 15% (11%) of the $B^{0}_{s}\rightarrow
K^{+}K^{-}$ ($B^{0}\rightarrow\pi^{+}\pi^{-}$) signal. Invariant mass fits are
used to estimate the yields of signal and combinatorial components. Figure 1
shows the $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ invariant mass spectra after
applying preselection and PID requirements. The results of the fits, which use
a single Gaussian function to describe the signal components and neglect
residual backgrounds from cross-feed decays, are superimposed. The presence of
a small component due to partially reconstructed three-body decays in the
$K^{+}K^{-}$ spectrum is also neglected. Approximately $11\times 10^{3}$
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and $14\times 10^{3}$ $B^{0}_{s}\rightarrow
K^{+}K^{-}$ decays are reconstructed.
Figure 1: Fits to the (a) $\pi^{+}\pi^{-}$ and (b) $K^{+}K^{-}$ invariant mass
spectra, after applying preselection and PID requirements. The components
contributing to the fit model are shown.
A BDT discriminant based on the AdaBoost algorithm [32] is then used to reduce
the combinatorial background. The BDT uses the following properties of the
decay products: the minimum $p_{\mathrm{T}}$ of the pair, the minimum
$d_{\mathrm{IP}}$, the minimum $\chi^{2}_{\rm IP}$, the maximum
$p_{\mathrm{T}}$, the maximum $d_{\mathrm{IP}}$, the maximum $\chi^{2}_{\rm
IP}$, the $d_{\mathrm{CA}}$, and the $\chi^{2}$ of the common vertex fit. The
BDT also uses the following properties of the $B$ candidate: the
$p_{\mathrm{T}}^{B}$, the $d_{\mathrm{IP}}^{B}$, the $\chi^{2}_{\rm IP}$, the
flight distance, and the $\chi^{2}$ of the flight distance. The BDT is
trained, separately for the $B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decays, using simulated events to model
the signal and data in the mass sideband
($5.5<m<5.8$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) to model the
combinatorial background. An optimal threshold on the BDT response is then
chosen by maximizing $S/\sqrt{S+B}$, where $S$ and $B$ represent the numbers
of signal and combinatorial background events within $\pm
60$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ (corresponding to about $\pm
3\sigma$) around the $B^{0}$ or $B^{0}_{s}$ mass. The resulting mass
distributions are discussed in Sec. 7. A control sample of $B^{0}\rightarrow
K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decays is selected using
the BDT selection optimized for the $B^{0}\rightarrow\pi^{+}\pi^{-}$ decay,
but with different PID requirements applied.
## 4 Flavour tagging
The sensitivity to the time-dependent $C\\!P$ asymmetry is directly related to
the tagging power, defined as $\varepsilon_{\rm
eff}=\varepsilon(1-2\omega)^{2}$, where $\varepsilon$ is the probability that
a tagging decision for a given candidate can be made (tagging efficiency) and
$\omega$ is the probability that such a decision is wrong (mistag
probability). If the candidates are divided into different subsamples, each
one characterized by an average tagging efficiency $\varepsilon_{i}$ and an
average mistag probability $\omega_{i}$, the effective tagging power is given
by $\varepsilon_{\rm eff}=\sum_{i}\varepsilon_{i}(1-2\omega_{i})^{2}$, where
the index $i$ runs over the various subsamples.
So-called opposite-side (OS) taggers are used to determine the initial flavour
of the signal $B$ meson [33]. This is achieved by looking at the charge of the
lepton, either muon or electron, originating from semileptonic decays, and of
the kaon from the $b\rightarrow c\rightarrow s$ decay transition of the other
$b$ hadron in the event. An additional OS tagger, the vertex charge tagger, is
based on the inclusive reconstruction of the opposite $B$ decay vertex and on
the computation of a weighted average of the charges of all tracks associated
to that vertex. For each tagger, the mistag probability is estimated by means
of an artificial neural network. When more than one tagger is available per
candidate, these probabilities are combined into a single mistag probability
$\eta$ and a unique decision per candidate is taken.
The data sample is divided into tagging categories according to the value of
$\eta$, and a calibration is performed to obtain the corrected mistag
probability $\omega$ for each category by means of a mass and decay time fit
to the $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
spectra, as described in Sec. 6. The consistency of tagging performances for
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$, $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$,
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
decays is verified using simulation. The definition of tagging categories is
optimized to obtain the highest tagging power. This is achieved by the five
categories reported in Table 2. The gain in tagging power using more
categories is found to be marginal.
Table 2: Definition of the five tagging categories determined from the optimization algorithm, in terms of ranges of the mistag probability $\eta$. Category | Range for $\eta$
---|---
1 | $0.00-0.22$
2 | $0.22-0.30$
3 | $0.30-0.37$
4 | $0.37-0.42$
5 | $0.42-0.47$
## 5 Fit model
For each component that contributes to the selected samples, the distributions
of invariant mass, decay time and tagging decision are modelled. Three sources
of background are considered: combinatorial background, cross-feed and
backgrounds from partially reconstructed three-body decays. The following
cross-feed backgrounds play a non-negligible role:
* •
in the $K^{\pm}\pi^{\mp}$ spectrum, $B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ decays where one of the two final state
particles is misidentified, and $B^{0}\rightarrow K^{+}\pi^{-}$ decays where
pion and kaon identities are swapped;
* •
in the $\pi^{+}\pi^{-}$ spectrum, $B^{0}\\!\rightarrow K^{+}\pi^{-}$ decays
where the kaon is misidentified as a pion; in this spectrum there is also a
small component of $B^{0}_{s}\rightarrow\pi^{+}\pi^{-}$ which must be taken
into account [12];
* •
in the $K^{+}K^{-}$ spectrum, $B^{0}\\!\rightarrow K^{+}\pi^{-}$ decays where
the pion is misidentified as a kaon.
### 5.1 Mass model
The signal component for each two-body decay is modelled convolving a double
Gaussian function with a parameterization of final state QED radiation. The
probability density function (PDF) is given by
$g(m)=A\left[\Theta(\mu-m)\,\left(\mu-m\right)^{s}\right]\otimes
G_{2}(m;\,f_{1},\,\sigma_{1},\,\sigma_{2}),$ (6)
where $A$ is a normalization factor, $\Theta$ is the Heaviside function,
$G_{2}$ is the sum of two Gaussian functions with widths $\sigma_{1}$ and
$\sigma_{2}$ and zero mean, $f_{1}$ is the fraction of the first Gaussian
function, and $\mu$ is the $B$-meson mass. The negative parameter $s$ governs
the amount of final state QED radiation, and is fixed for each signal
component using the respective theoretical QED prediction, calculated
according to Ref. [34].
The combinatorial background is modelled by an exponential function for all
the final states. The component due to partially reconstructed three-body $B$
decays in the $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ spectra is modelled convolving
a Gaussian resolution function with an ARGUS function [35]. The
$K^{\pm}\pi^{\mp}$ spectrum is described convolving a Gaussian function with
the sum of two ARGUS functions, in order to accurately model not only $B^{0}$
and $B^{+}$, but also a smaller fraction of $B^{0}_{s}$ three-body decays
[11]. Cross-feed background PDFs are obtained from simulations. For each final
state hypothesis, a set of invariant mass distributions is determined from
pairs where one or both tracks are misidentified, and each of them is
parameterized by means of a kernel estimation technique [36]. The yields of
the cross-feed backgrounds are fixed by means of a time-integrated
simultaneous fit to the mass spectra of all two-body $B$ decays [11].
### 5.2 Decay time model
The time-dependent decay rate of a flavour-specific $B\rightarrow f$ decay and
of its $C\\!P$ conjugate ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\rightarrow\bar{f}$ is given by the PDF
$\begin{split}f\left(t,\,\psi,\,\xi\right)&=K\left(1-\psi
A_{C\\!P}\right)\left(1-\psi A_{f}\right)\times\\\
&\left\\{\\!\left[\left(1\\!-\\!A_{\mathrm{P}}\right)\\!\Omega_{\xi}^{B}\\!+\\!\left(1\\!+\\!A_{\mathrm{P}}\right)\\!\bar{\Omega}_{\xi}^{B}\right]\\!H_{+}\left(t\right)\\!+\\!\psi\\!\left[\left(1\\!-\\!A_{\mathrm{P}}\right)\\!\Omega_{\xi}^{B}\\!-\\!\left(1\\!+\\!A_{\mathrm{P}}\right)\\!\bar{\Omega}_{\xi}^{B}\right]\\!H_{-}\left(t\right)\\!\right\\},\end{split}$
(7)
where $K$ is a normalization factor, and the variables $\psi$ and $\xi$ are
the final state tag and the initial flavour tag, respectively. This PDF is
suitable for the cases of $B^{0}\rightarrow K^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decays. The variable $\psi$ assumes the
value $+1$ for the final state $f$ and $-1$ for the final state $\bar{f}$. The
variable $\xi$ assumes the discrete value $+k$ when the candidate is tagged as
$B$ in the $k$-th category, $-k$ when the candidate is tagged as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}$ in the $k$-th category, and zero for
untagged candidates. The direct $C\\!P$ asymmetry, $A_{C\\!P}$, the asymmetry
of final state reconstruction efficiencies (detection asymmetry), $A_{f}$, and
the $B$ meson production asymmetry, $A_{\mathrm{P}}$, are defined as
$\displaystyle A_{C\\!P}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{B}\left({\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\rightarrow\bar{f}\right)-\mathcal{B}\left(B\rightarrow
f\right)}{\mathcal{B}\left({\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\rightarrow\bar{f}\right)+\mathcal{B}\left(B\rightarrow
f\right)},$ (8) $\displaystyle A_{f}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{\mathrm{rec}}\left(\bar{f}\right)-\varepsilon_{\mathrm{rec}}\left(f\right)}{\varepsilon_{\mathrm{rec}}\left(\bar{f}\right)+\varepsilon_{\mathrm{rec}}\left(f\right)},$
(9) $\displaystyle A_{\mathrm{P}}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{R}\left({\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\right)-\mathcal{R}\left(B\right)}{\mathcal{R}\left({\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\right)+\mathcal{R}\left(B\right)},$
(10)
where $\mathcal{B}$ denotes the branching fraction,
$\varepsilon_{\mathrm{rec}}$ is the reconstruction efficiency of the final
state $f$ or $\bar{f}$, and $\mathcal{R}$ is the production rate of the given
$B$ or ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}$ meson. The parameters
$\Omega_{\xi}^{B}$ and $\bar{\Omega}_{\xi}^{B}$ are the probabilities that a
$B$ or a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}$ meson is tagged as
$\xi$, respectively, and are defined as
$\begin{split}\Omega_{k}^{B}=\varepsilon_{k}\left(1-\omega_{k}\right),\qquad\Omega_{-k}^{B}=\varepsilon_{k}\omega_{k},\qquad\Omega_{0}^{B}=1-\sum_{j=1}^{5}\varepsilon_{j},\\\
\bar{\Omega}_{k}^{B}=\bar{\varepsilon}_{k}\bar{\omega}_{k},\qquad\bar{\Omega}_{-k}^{B}=\bar{\varepsilon}_{k}\left(1-\bar{\omega}_{k}\right),\qquad\bar{\Omega}_{0}^{B}=1-\sum_{j=1}^{5}\bar{\varepsilon}_{j},\end{split}$
(11)
where $\varepsilon_{k}$ ($\bar{\varepsilon}_{k}$) is the tagging efficiency
and $\omega_{k}$ ($\bar{\omega}_{k}$) is the mistag probability for signal $B$
($\kern 1.79993pt\overline{\kern-1.79993ptB}{}$) mesons that belong to the
$k$-th tagging category. The functions $H_{+}\left(t\right)$ and
$H_{-}\left(t\right)$ are defined as
$\displaystyle H_{+}\left(t\right)$ $\displaystyle=$
$\displaystyle\left[e^{-\Gamma_{d(s)}t}\cosh{\left(\Delta\Gamma_{d(s)}t/2\right)}\right]\otimes
R\left(t\right)\varepsilon_{\mathrm{acc}}\left(t\right),$ (12) $\displaystyle
H_{-}\left(t\right)$ $\displaystyle=$
$\displaystyle\left[e^{-\Gamma_{d(s)}t}\cos{\left(\Delta
m_{d(s)}t\right)}\right]\otimes
R\left(t\right)\varepsilon_{\mathrm{acc}}\left(t\right),$ (13)
where $\Gamma_{d(s)}$ is the average decay width of the $B^{0}_{(s)}$ meson,
$R$ is the decay time resolution model, and $\varepsilon_{\mathrm{acc}}$ is
the decay time acceptance.
In the fit to the $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow
K^{-}\pi^{+}$ mass and decay time distributions, the decay width differences
of $B^{0}$ and $B^{0}_{s}$ mesons are fixed to zero and to the value measured
by LHCb, $\Delta\Gamma_{s}=0.106$${\rm\,ps^{-1}}$ [37], respectively, whereas
the mass differences are left free to vary. The fit is performed
simultaneously for candidates belonging to the five tagging categories and for
untagged candidates.
If the final states $f$ and $\bar{f}$ are the same, as in the cases of
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$ decays,
the time-dependent decay rates are described by
$f\left(t,\,\xi\right)=K\left\\{\left[\left(1\\!-\\!A_{\mathrm{P}}\right)\\!\Omega_{\xi}^{B}\\!+\\!\left(1\\!+\\!A_{\mathrm{P}}\right)\\!\bar{\Omega}_{\xi}^{B}\right]\\!I_{+}\left(t\right)\\!+\\!\left[\left(1\\!-\\!A_{\mathrm{P}}\right)\\!\Omega_{\xi}^{B}\\!-\\!\left(1\\!+\\!A_{\mathrm{P}}\right)\\!\bar{\Omega}_{\xi}^{B}\right]\\!I_{-}\left(t\right)\right\\},$
(14)
where the functions $I_{+}\left(t\right)$ and $I_{-}\left(t\right)$ are
$\displaystyle I_{+}\left(t\right)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\left\\{e^{-\Gamma_{d(s)}t}\\!\left[\cosh\\!{\left(\Delta\Gamma_{d(s)}t/2\right)}\\!-\\!A_{f}^{\Delta\Gamma}\sinh\\!{\left(\Delta\Gamma_{d(s)}t/2\right)}\right]\\!\right\\}\\!\otimes\\!R\left(t\right)\varepsilon_{\mathrm{acc}}\left(t\right),$
(15) $\displaystyle I_{-}\left(t\right)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\left\\{e^{-\Gamma_{d(s)}t}\\!\left[C_{f}\cos\\!{\left(\Delta
m_{d(s)}t\right)}\\!-\\!S_{f}\sin\\!{\left(\Delta
m_{d(s)}t\right)}\right]\\!\right\\}\\!\otimes\\!R\left(t\right)\varepsilon_{\mathrm{acc}}\left(t\right).$
(16)
In the $B^{0}_{s}\rightarrow K^{+}K^{-}$ fit, the average decay width and mass
difference of the $B^{0}_{s}$ meson are fixed to the values
$\Gamma_{s}=0.661$${\rm\,ps^{-1}}$ [37] and $\Delta
m_{s}=17.768$${\rm\,ps^{-1}}$ [38]. The width difference $\Delta\Gamma_{s}$ is
left free to vary, but is constrained to be positive as expected in the SM and
measured by LHCb [39], in order to resolve the invariance of the decay rates
under the exchange
$\left(\Delta\Gamma_{d(s)},\,A_{f}^{\Delta\Gamma}\right)\rightarrow\left(-\Delta\Gamma_{d(s)},\,-A_{f}^{\Delta\Gamma}\right)$.
Moreover, the definitions of $C_{f}$, $S_{f}$ and $A_{f}^{\Delta\Gamma}$ in
Eq. (2) allow $A_{f}^{\Delta\Gamma}$ to be expressed as
$A_{f}^{\Delta\Gamma}=\pm\sqrt{1-C_{f}^{2}-S_{f}^{2}}.$ (17)
The positive solution, which is consistent with measurements of the
$B^{0}_{s}\rightarrow K^{+}K^{-}$ effective lifetime [17, 18], is taken. In
the case of the $B^{0}\rightarrow\pi^{+}\pi^{-}$ decay, where the width
difference of the $B^{0}$ meson is negligible and fixed to zero, the ambiguity
is not relevant. The mass difference is fixed to the value $\Delta
m_{d}=0.516$${\rm\,ps^{-1}}$ [40]. Again, these two fits are performed
simultaneously for candidates belonging to the five tagging categories and for
untagged candidates.
The dependence of the reconstruction efficiency on the decay time (decay time
acceptance) is studied with simulated events. For each simulated decay, namely
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$, $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$,
$B^{0}\\!\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$,
reconstruction, trigger requirements and event selection are applied, as for
data. It is empirically found that $\varepsilon_{\mathrm{acc}}\left(t\right)$
is well parameterized by
$\varepsilon_{\mathrm{acc}}\left(t\right)=\frac{1}{2}\left[1-\frac{1}{2}\mathrm{erf}\left(\frac{p_{1}-t}{p_{2}\,t}\right)-\frac{1}{2}\mathrm{erf}\left(\frac{p_{3}-t}{p_{4}\,t}\right)\right],$
(18)
where $\mathrm{erf}$ is the error function, and $p_{i}$ are free parameters
determined from simulation.
The expressions for the decay time PDFs of the cross-feed background
components are determined from Eqs. (7) and (14), assuming that the decay time
calculated under the wrong mass hypothesis resembles the correct one with
sufficient accuracy. This assumption is verified with simulations.
The parameterization of the decay time distribution for combinatorial
background events is studied using the high-mass sideband from data, defined
as $5.5<m<5.8$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Concerning the
$K^{\pm}\pi^{\mp}$ spectrum, for events selected by the
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ BDT, it is empirically found that the PDF
can be written as
$f\left(t,\,\xi,\,\psi\right)=K\Omega_{\xi}^{\mathrm{comb}}\left(1-\psi
A_{CP}^{\mathrm{comb}}\right)\left[g\,e^{-\Gamma_{1}^{\mathrm{comb}}t}+\left(1-g\right)e^{-\Gamma_{2}^{\mathrm{comb}}t}\right]\varepsilon^{\mathrm{comb}}_{\mathrm{acc}}(t),$
(19)
where $A_{C\\!P}^{\mathrm{comb}}$ is the charge asymmetry of the combinatorial
background, $g$ is the fraction of the first exponential component, and
$\Gamma_{1}^{\mathrm{comb}}$ and $\Gamma_{2}^{\mathrm{comb}}$ are two free
parameters. The term $\Omega_{\xi}^{\mathrm{comb}}$ is the probability to tag
a background event as $\xi$. It is parameterized as
$\Omega_{k}^{\mathrm{comb}}=\varepsilon_{k}^{\mathrm{comb}},\qquad\Omega_{-k}^{\mathrm{comb}}=\bar{\varepsilon}_{k}^{\mathrm{comb}},\qquad\Omega_{0}^{\mathrm{comb}}=1-\sum_{j}^{5}{\left(\varepsilon_{j}^{\mathrm{comb}}+\bar{\varepsilon}_{j}^{\mathrm{comb}}\right)},$
(20)
where $\varepsilon_{k}^{\mathrm{comb}}$
($\bar{\varepsilon}_{k}^{\mathrm{comb}}$) is the probability to tag a
background event as a $B$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}$) in
the $k$-th category. The effective function
$\varepsilon^{\mathrm{comb}}_{\mathrm{acc}}\left(t\right)$ is the analogue of
the decay time acceptance for signal decays, and is given by the same
expression of Eq. (18), but characterized by independent values of the
parameters $p_{i}$. For the $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ spectra, the
same expression as in Eq. (19) is used, with the difference that the charge
asymmetry is set to zero and no dependence on $\psi$ is needed.
The last case to examine is that of the three-body partially reconstructed
backgrounds in the $K^{\pm}\pi^{\mp}$, $\pi^{+}\pi^{-}$, and $K^{+}K^{-}$
spectra. In the $K^{\pm}\pi^{\mp}$ mass spectrum there are two components,
each described by an ARGUS function [35]. Each of the two corresponding decay
time components is empirically parameterized as
$f\left(t,\,\xi,\,\psi\right)=K\Omega_{\xi}^{\mathrm{part}}\left(1-\psi
A_{C\\!P}^{\mathrm{part}}\right)e^{-\Gamma^{\mathrm{part}}t}\varepsilon_{\mathrm{acc}}^{\mathrm{part}}\left(t\right),$
(21)
where $A_{C\\!P}^{\mathrm{part}}$ is the charge asymmetry and
$\Gamma^{\mathrm{part}}$ is a free parameter. The term
$\Omega_{\xi}^{\mathrm{part}}$ is the probability to tag a background event as
$\xi$, and is parameterized as in Eq. (20), but with independent tagging
probabilities. For the $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ spectra, the same
expression as in Eq. (21) is used, with the difference that the charge
asymmetry is set to zero and no dependence on $\psi$ is needed.
The accuracy of the combinatorial and three-body decay time parameterizations
is checked by performing a simultaneous fit to the invariant mass and decay
time spectra of the high- and low-mass sidebands. The combinatorial background
contributes to both the high- and low-mass sidebands, whereas the three-body
background is only present in the low-mass side. In Fig. 2 the decay time
distributions are shown, restricted to the high and low $K^{\pm}\pi^{\mp}$,
$\pi^{+}\pi^{-}$, and $K^{+}K^{-}$ mass sidebands. The low-mass sidebands are
defined by the requirement
$5.00<m<5.15$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for $K^{\pm}\pi^{\mp}$
and $\pi^{+}\pi^{-}$, and by the requirement
$5.00<m<5.25$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for $K^{+}K^{-}$,
whereas in all cases the high-mass sideband is defined by the requirement
$5.5<m<5.8$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Figure 2: Decay time distributions corresponding to (a, b, c) high- and (d, e,
f) low-mass sidebands from the (a and d) $K^{\pm}\pi^{\mp}$, (b and e)
$\pi^{+}\pi^{-}$ and (c and f) $K^{+}K^{-}$ mass spectra, with the results of
fits superimposed. In the bottom plots, the combinatorial background component
(dashed) and the three-body background component (dotted) are shown.
### 5.3 Decay time resolution
Large samples of $J/\psi\rightarrow\mu^{+}\mu^{-}$,
$\psi(\mathrm{2S})\rightarrow\mu^{+}\mu^{-}$,
$\Upsilon(\mathrm{1S})\rightarrow\mu^{+}\mu^{-}$,
$\Upsilon(\mathrm{2S})\rightarrow\mu^{+}\mu^{-}$ and
$\Upsilon(\mathrm{3S})\rightarrow\mu^{+}\mu^{-}$ decays can be selected
without any requirement that biases the decay time. Maximum likelihood fits to
the invariant mass and decay time distributions allow an average resolution to
be derived for each of these decays. A comparison of the resolutions
determined from data and simulation yields correction factors ranging from
$1.0$ to $1.1$, depending on the charmonium or bottomonium decay considered.
On this basis, a correction factor $1.05\pm 0.05$ is estimated. The simulation
also indicates that, in the case of $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B_{s}^{0}\rightarrow K^{+}K^{-}$ decays, an additional dependence of the
resolution on the decay time must be considered. Taking this dependence into
account, we finally estimate a decay time resolution of $50\pm 10$$\rm\,fs$.
Furthermore, from the same fits to the charmonium and bottomium decay time
spectra, it is found that the measurement of the decay time is biased by less
than $2$$\rm\,fs$. As a baseline resolution model, $R(t)$, a single Gaussian
function with zero mean and $50$$\rm\,fs$ width is used. Systematic
uncertainties on the direct and mixing-induced $C\\!P$-violating asymmetries
in $B^{0}_{s}\rightarrow K^{+}K^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$
decays, related to the choice of the baseline resolution model, are discussed
in Sec. 8.
## 6 Calibration of flavour tagging
In order to measure time-dependent $C\\!P$ asymmetries in
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$
decays, simultaneous unbinned maximum likelihood fits to the invariant mass
and decay time distributions are performed. First, a fit to the
$K^{\pm}\pi^{\mp}$ mass and time spectra is performed to determine the
performance of the flavour tagging and the $B^{0}$ and $B^{0}_{s}$ production
asymmetries. The flavour tagging efficiencies, mistag probabilities and
production asymmetries are then propagated to the
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$
fits by multiplying the likelihood functions with Gaussian terms. The flavour
tagging variables are parameterized as
$\begin{split}\varepsilon_{k}=\varepsilon_{k}^{\mathrm{tot}}\left(1-A_{k}^{\varepsilon}\right),\qquad\bar{\varepsilon}_{k}=\varepsilon_{k}^{\mathrm{tot}}\left(1+A_{k}^{\varepsilon}\right),\\\
\omega_{k}=\omega_{k}^{\mathrm{tot}}\left(1-A_{k}^{\omega}\right),\qquad\bar{\omega}_{k}=\omega_{k}^{\mathrm{tot}}\left(1+A_{k}^{\omega}\right),\end{split}$
(22)
where $\varepsilon_{k}^{\mathrm{tot}}$ ($\omega_{k}^{\mathrm{tot}}$) is the
tagging efficiency (mistag fraction) averaged between $B^{0}_{(s)}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ in the $k$-th category, and
$A_{k}^{\varepsilon}$ ($A_{k}^{\omega}$) measures a possible asymmetry between
the tagging efficiencies (mistag fractions) of $B^{0}_{(s)}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ in the $k$-th category.
To determine the values of $A_{k}^{\varepsilon}$, $\omega_{k}^{\mathrm{tot}}$
and $A_{k}^{\omega}$, we fit the model described in Sec. 5 to the
$K^{\pm}\pi^{\mp}$ spectra. In the $K^{\pm}\pi^{\mp}$ fit, the amount of
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\\!\rightarrow K^{+}K^{-}$
cross-feed backgrounds below the $B^{0}\\!\rightarrow K^{+}\pi^{-}$ peak are
fixed to the values obtained by performing a time-integrated simultaneous fit
to all two-body invariant mass spectra, as in Ref. [11]. In Fig. 3 the
$K^{\pm}\pi^{\mp}$ invariant mass and decay time distributions are shown.
Figure 3: Distributions of $K^{\pm}\pi^{\mp}$ (a) mass and (b) decay time,
with the result of the fit overlaid. The main components contributing to the
fit model are also shown.
In Fig. 4 the raw mixing asymmetry is shown for each of the five tagging
categories, by considering only candidates with invariant mass in the region
dominated by $B^{0}\\!\rightarrow K^{+}\pi^{-}$ decays,
$5.20<m<5.32$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The asymmetry
projection from the full fit is superimposed.
Figure 4: Raw mixing asymmetries for candidates in the $B^{0}\\!\rightarrow
K^{+}\pi^{-}$ signal mass region, corresponding to the five tagging
categories, with the result of the fit overlaid.
The $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$
event yields determined from the fit are $N(B^{0}\rightarrow
K^{+}\pi^{-})=49\hskip 1.42262pt356\pm 335\,\mathrm{(stat)}$ and
$N(B^{0}_{s}\rightarrow K^{-}\pi^{+})=3917\pm 142\,\mathrm{(stat)}$,
respectively. The mass differences are determined to be $\Delta m_{d}=0.512\pm
0.014\,\mathrm{(stat)}$${\rm\,ps^{-1}}$ and $\Delta m_{s}=17.84\pm
0.11\,\mathrm{(stat)}$${\rm\,ps^{-1}}$. The $B^{0}$ and $B^{0}_{s}$ average
lifetimes determined from the fit are $\tau(B^{0})=1.523\pm
0.007\,\mathrm{(stat)}$${\rm\,ps}$ and $\tau(B^{0}_{s})=1.51\pm
0.03\,\mathrm{(stat)}$${\rm\,ps}$. The signal tagging efficiencies and mistag
probabilities are summarized in Table 3. With the present precision, there is
no evidence of non-zero asymmetries in the tagging efficiencies and mistag
probabilities between $B^{0}_{(s)}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ mesons. The average
effective tagging power is $\varepsilon_{\mathrm{eff}}=(2.45\pm 0.25)\%$.
Table 3: Signal tagging efficiencies, mistag probabilities and associated
asymmetries, corresponding to the five tagging categories, as determined from
the $K^{\pm}\pi^{\mp}$ mass and decay time fit. The uncertainties are
statististical only.
Efficiency (%) | Efficiency asymmetry (%) | Mistag probability (%) | Mistag asymmetry (%)
---|---|---|---
$\varepsilon_{1}^{\mathrm{tot}}=1.92\pm 0.06$ | $A_{1}^{\varepsilon}=-8\pm 5$ | $\omega_{1}^{\mathrm{tot}}=20.0\pm 2.8$ | $A_{1}^{\omega}=\phantom{-}0\pm 10$
$\varepsilon_{2}^{\mathrm{tot}}=4.07\pm 0.09$ | $A_{2}^{\varepsilon}=\phantom{-}0\pm 4$ | $\omega_{2}^{\mathrm{tot}}=28.3\pm 2.0$ | $A_{2}^{\omega}=\phantom{-}5\pm 5\phantom{0}$
$\varepsilon_{3}^{\mathrm{tot}}=7.43\pm 0.12$ | $A_{3}^{\varepsilon}=\phantom{-}2\pm 3$ | $\omega_{3}^{\mathrm{tot}}=34.3\pm 1.5$ | $A_{3}^{\omega}=-1\pm 3\phantom{0}$
$\varepsilon_{4}^{\mathrm{tot}}=7.90\pm 0.13$ | $A_{4}^{\varepsilon}=-2\pm 3$ | $\omega_{4}^{\mathrm{tot}}=41.9\pm 1.5$ | $A_{4}^{\omega}=-2\pm 2\phantom{0}$
$\varepsilon_{5}^{\mathrm{tot}}=7.86\pm 0.13$ | $A_{5}^{\varepsilon}=\phantom{-}0\pm 3$ | $\omega_{5}^{\mathrm{tot}}=45.8\pm 1.5$ | $A_{5}^{\omega}=-4\pm 2\phantom{0}$
From the fit, the production asymmetries for the $B^{0}$ and $B^{0}_{s}$
mesons are determined to be $A_{\mathrm{P}}\left(B^{0}\right)=(0.6\pm 0.9)\%$
and $A_{\mathrm{P}}\left(B^{0}_{s}\right)=(7\pm 5)\%$, where the uncertainties
are statistical only.
## 7 Results
The fit to the mass and decay time distributions of the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ candidates determines the $C\\!P$
asymmetry coefficients $C_{KK}$ and $S_{KK}$, whereas the
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ fit determines $C_{\pi\pi}$ and
$S_{\pi\pi}$. In both fits, the yield of $B^{0}\\!\rightarrow K^{+}\pi^{-}$
cross-feed decays is fixed to the value obtained from a time-integrated fit,
identical to that of Ref. [11]. Furthermore, the flavour tagging efficiency
asymmetries, mistag fractions and mistag asymmetries, and the $B^{0}$ and
$B^{0}_{s}$ production asymmetries are constrained to the values measured in
the fit described in the previous section, by multiplying the likelihood
function with Gaussian terms.
The $K^{+}K^{-}$ invariant mass and decay time distributions are shown in Fig.
5.
Figure 5: Distributions of $K^{+}K^{-}$ (a) mass and (b) decay time, with the
result of the fit overlaid. The main components contributing to the fit model
are also shown.
The raw time-dependent asymmetry is shown in Fig. 6 for candidates with
invariant mass in the region dominated by signal events,
$5.30<m<5.44$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and belonging to the
first two tagging categories.
Figure 6: Time-dependent raw asymmetry for candidates in the
$B^{0}_{s}\\!\rightarrow K^{+}K^{-}$ signal mass region with the result of the
fit overlaid. In order to enhance the visibility of the oscillation, only
candidates belonging to the first two tagging categories are used. The offset
$t_{0}=0.6$${\rm\,ps}$ corresponds to the preselection requirement on the
decay time.
The $B^{0}_{s}\rightarrow K^{+}K^{-}$ event yield is determined to be
$N(B^{0}_{s}\rightarrow K^{+}K^{-})=14\hskip 1.42262pt646\pm
159\,(\mathrm{stat})$, while the $B^{0}_{s}$ decay width difference from the
fit is $\Delta\Gamma_{s}=0.104\pm 0.016\,\mathrm{(stat)}$${\rm\,ps^{-1}}$. The
values of $C_{KK}$ and $S_{KK}$ are determined to be
$C_{KK}=0.14\pm 0.11\,\mathrm{(stat)},\qquad S_{KK}=0.30\pm
0.12\,\mathrm{(stat)},$
with correlation coefficient $\rho\left(C_{KK},\,S_{KK}\right)=0.02$. The
small value of the correlation coefficient is a consequence of the large
$B^{0}_{s}$ mixing frequency. An alternative fit, fixing the value of
$\Delta\Gamma_{s}$ to $0.106$${\rm\,ps^{-1}}$ [37] and leaving
$A^{\Delta\Gamma}_{KK}$ free to vary, is also performed as a cross-check.
Central values and statistical uncertainties of $C_{KK}$ and $S_{KK}$ are
almost unchanged, and $A^{\Delta\Gamma}_{KK}$ is determined to be $0.91\pm
0.08\,\mathrm{(stat)}$.
Although very small, a component accounting for the presence of the
$B_{s}^{0}\rightarrow\pi^{+}\pi^{-}$ decay [12] is introduced in the
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ fit. This component is described using the
signal model, but assuming no $C\\!P$ violation. The $\pi^{+}\pi^{-}$
invariant mass and decay time distributions are shown in Fig. 7.
Figure 7: Distributions of $\pi^{+}\pi^{-}$ (a) mass and (b) decay time, with
the result of the fit overlaid. The main components contributing to the fit
model are also shown.
The raw time-dependent asymmetry is shown in Fig. 8 for candidates with
invariant mass in the region dominated by signal events,
$5.20<m<5.36$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Figure 8: Time-dependent raw asymmetry for candidates in the
$B^{0}\\!\rightarrow\pi^{+}\pi^{-}$ signal mass region with the result of the
fit overlaid. Tagged candidates belonging to all tagging categories are used.
The $B^{0}\rightarrow\pi^{+}\pi^{-}$ event yield is determined to be
$N(B^{0}\rightarrow\pi^{+}\pi^{-})=9170\pm 144\,\mathrm{(stat)}$, while the
$B^{0}$ average lifetime from the fit is $\tau(B^{0})=1.55\pm
0.02\,\mathrm{(stat)}$${\rm\,ps}$. The values of $C_{\pi\pi}$ and $S_{\pi\pi}$
are determined to be
$C_{\pi\pi}=-0.38\pm 0.15\,\mathrm{(stat)},\qquad S_{\pi\pi}=-0.71\pm
0.13\,\mathrm{(stat)},$
with correlation coefficient $\rho\left(C_{\pi\pi},\,S_{\pi\pi}\right)=0.38$.
## 8 Systematic uncertainties
Several sources of systematic uncertainty that may affect the determination of
the direct and mixing-induced $C\\!P$-violating asymmetries in
$B^{0}_{s}\rightarrow K^{+}K^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$ decays
are considered. For the invariant mass model, the accuracy of PID efficiencies
and the description of mass shapes for all components (signals, combinatorial,
partially reconstructed three-body and cross-feed backgrounds) are
investigated. For the decay time model, systematic effects related to the
decay time resolution and acceptance are studied. The effects of the external
input variables used in the fits ($\Delta m_{s}$, $\Delta m_{d}$,
$\Delta\Gamma_{s}$ and $\Gamma_{s}$), and the parameterization of the
backgrounds are also considered. To estimate the contribution of each single
source the fit is repeated after having modified the baseline
parameterization. The shifts from the relevant baseline values are accounted
for as systematic uncertainties.
The PID efficiencies are used to compute the yields of cross-feed backgrounds
present in the $K^{\pm}\pi^{\mp}$, $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ mass
distributions. In order to estimate the impact of imperfect PID calibration,
unbinned maximum likelihood fits are performed after having altered the number
of cross-feed background events present in the relevant mass spectra,
according to the systematic uncertainties associated to the PID efficiencies.
An estimate of the uncertainty due to possible mismodelling of the final-state
radiation is determined by varying the amount of emitted radiation [34] in the
signal shape parameterization, according to studies performed on simulated
events, in which final state radiation is generated using Photos [27]. The
possibility of an incorrect description of the signal mass model is
investigated by replacing the double Gaussian function with the sum of three
Gaussian functions, where the third component has fixed fraction ($5\%$) and
width ($50$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$), and is aimed at
describing long tails, as observed in simulation. The systematic uncertainties
related to the parameterization of the invariant mass shape for the
combinatorial background are investigated by replacing the exponential shape
with a straight line function. For the case of the cross-feed backgrounds, two
distinct systematic uncertainties are estimated: one due to a relative bias in
the mass scale of the simulated distributions with respect to the signal
distributions in data, and another to account for the difference in mass
resolution between simulation and data.
Systematic uncertainties associated to the decay time resolution are
investigated by altering the resolution model in different ways. The width of
the single Gaussian model used in the baseline fit is changed by $\pm
10$$\rm\,fs$. Effects due to a possible bias in the decay time measurement are
accounted for by repeating the fit with a bias of $\pm 2$$\rm\,fs$. Finally,
the single Gaussian model is substituted by a triple Gaussian model, where the
fractions of the Gaussian functions are taken from simulation and the widths
are rescaled to match the average width of $50$$\rm\,fs$ used in the baseline
fit.
To estimate systematic uncertainties arising from the choice of
parameterization for backgrounds, fits with alternative parameterizations are
performed. To account for possible inaccuracies in the decay time acceptances
determined from simulation, the fits are repeated fixing $\Gamma_{d}$ to
$0.658$${\rm\,ps^{-1}}$ and $\Delta\Gamma_{s}$ to $0.106$${\rm\,ps^{-1}}$, and
leaving the acceptance parameters $p_{i}$ free to vary.
Systematic uncertainties related to the use of external inputs are estimated
by varying the input quantities by $\pm 1\sigma$ of the corresponding
measurements. In particular, this is done in the $B^{0}\rightarrow
K^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{-}\pi^{+}$ fit for
$\Delta\Gamma_{s}$ ($\pm 0.013$${\rm\,ps^{-1}}$), in the
$B^{0}\rightarrow\pi^{+}\pi^{-}$ fit for $\Delta m_{d}$ ($\pm
0.006$${\rm\,ps^{-1}}$), and in the $B^{0}_{s}\rightarrow K^{+}K^{-}$ fit for
$\Delta m_{s}$ ($\pm 0.024$${\rm\,ps^{-1}}$) and $\Gamma_{s}$ ($\pm
0.007$${\rm\,ps^{-1}}$).
Following the procedure outlined above, we also estimate the systematic
uncertainties affecting the flavour tagging efficiencies, mistag probabilities
and production asymmetries, and propagate these uncertainties to the
systematic uncertainties on the direct and mixing-induced $C\\!P$ asymmetry
coefficients in $B^{0}_{s}\rightarrow K^{+}K^{-}$ and
$B^{0}\rightarrow\pi^{+}\pi^{-}$ decays. The final systematic uncertainties on
these coefficients are summarized in Table 4. They turn out to be much smaller
than the corresponding statistical uncertainties reported in Sec. 7.
Table 4: Systematic uncertainties affecting the $B^{0}_{s}\rightarrow
K^{+}K^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$ direct and mixing-induced
$C\\!P$ asymmetry coefficients. The total systematic uncertainties are
obtained by summing the individual contributions in quadrature.
Systematic uncertainty | $C_{KK}$ | $S_{KK}$ | $C_{\pi\pi}$ | $S_{\pi\pi}$
---|---|---|---|---
Particle identification | $0.003$ | $0.003$ | $0.002$ | $0.004$
Flavour tagging | $0.008$ | $0.009$ | $0.010$ | $0.011$
Production asymmetry | $0.002$ | $0.002$ | $0.003$ | $0.002$
Signal mass: | final state radiation | $0.002$ | $0.001$ | $0.001$ | $0.002$
shape model | $0.003$ | $0.004$ | $0.001$ | $0.004$
Bkg. mass: | combinatorial | $<0.001\phantom{11}$ | $<0.001\phantom{11}$ | $<0.001\phantom{11}$ | $<0.001\phantom{11}$
cross-feed | $0.002$ | $0.003$ | $0.002$ | $0.004$
Sig. decay time: | acceptance | $0.010$ | $0.018$ | $0.002$ | $0.003$
resolution width | $0.020$ | $0.025$ | $<0.001\phantom{11}$ | $<0.001\phantom{11}$
resolution bias | $0.009$ | $0.007$ | $<0.001\phantom{11}$ | $<0.001\phantom{11}$
resolution model | $0.008$ | $0.015$ | $<0.001\phantom{11}$ | $<0.001\phantom{11}$
Bkg. decay time: | cross-feed | $<0.001\phantom{11}$ | $<0.001\phantom{11}$ | $0.005$ | $0.002$
combinatorial | $0.008$ | $0.006$ | $0.015$ | $0.011$
three-body | $0.001$ | $0.003$ | $0.003$ | $0.005$
Ext. inputs: | $\Delta m_{s}$ | $0.015$ | $0.018$ | - | -
$\Delta m_{d}$ | - | - | $0.013$ | $0.010$
$\Gamma_{s}$ | $0.004$ | $0.005$ | - | -
Total | $0.032$ | $0.042$ | $0.023$ | $0.021$
## 9 Conclusions
The measurement of time-dependent $C\\!P$ violation in $B^{0}_{s}\rightarrow
K^{+}K^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$ decays, based on a data
sample corresponding to an integrated luminosity of 1.0 fb-1, has been
presented. The results for the $B^{0}_{s}\rightarrow K^{+}K^{-}$ decay are
$\begin{split}C_{KK}=0.14\pm 0.11\,\mathrm{(stat)}\pm
0.03\,\mathrm{(syst)},\\\ S_{KK}=0.30\pm 0.12\,\mathrm{(stat)}\pm
0.04\,\mathrm{(syst)},\end{split}$
with a statistical correlation coefficient of $0.02$. The results for the
$B^{0}\rightarrow\pi^{+}\pi^{-}$ decay are
$\begin{split}C_{\pi\pi}=-0.38\pm 0.15\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)},\\\ S_{\pi\pi}=-0.71\pm 0.13\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)},\end{split}$
with a statistical correlation coefficient of $0.38$.
Dividing the central values of the measurements by the sum in quadrature of
statistical and systematic uncertainties, and taking correlations into
account, the significances for $(C_{KK},\,S_{KK})$ and
$(C_{\pi\pi},\,S_{\pi\pi})$ to differ from $(0,\,0)$ are determined to be
2.7$\sigma$ and 5.6$\sigma$, respectively. The parameters $C_{KK}$ and
$S_{KK}$ are measured for the first time. The measurements of $C_{\pi\pi}$ and
$S_{\pi\pi}$ are in good agreement with previous measurements by BaBar [13]
and Belle [14], and those of $C_{KK}$ and $S_{KK}$ are compatible with
theoretical SM predictions [41, 42, 43, 7]. These results, together with those
from BaBar and Belle, allow the determination of the unitarity triangle angle
$\gamma$ using decays affected by penguin processes [3, 9]. The comparison to
the value of $\gamma$ determined from tree-level decays will provide a test of
the SM and constrain possible non-SM contributions.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-08-06T21:47:39 |
2024-09-04T02:49:49.126943
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, E. Cowie, D.C. Craik, S.\n Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P.\n Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin,\n U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez,\n P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss},\n H. Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Vincenzo Maria Vagnoni",
"url": "https://arxiv.org/abs/1308.1428"
}
|
1308.1438
|
EPJ Web of Conferences INPC 2013
11institutetext: Brookhaven National Laboratory, Physics Department, Upton, NY
11973–5000, USA
# Profiling hot and dense nuclear medium with high transverse momentum hadrons
produced in d+Au and Au+Au collisions by the PHENIX experiment at RHIC
Takao Sakaguchi 11 [email protected] for the PHENIX collaboration
###### Abstract
PHENIX measurements of high transverse momentum ($p_{T}$) identified hadrons
in $d$+Au and Au+Au collisions are presented. The nuclear modification factors
($R_{d{\rm A}}$ and $R_{\rm AA}$) for $\pi^{0}$ and $\eta$ are found to be
very consistent in both collision systems, respectively. Using large amount of
$p+p$ and Au+Au datasets, the fractional momentum loss ($S_{\rm loss}$) and
the path-length dependent yield of $\pi^{0}$ in Au+Au collisions are obtained.
The hadron spectra in the most central $d$+Au and the most peripheral Au+Au
collisions are studied. The spectra shapes are found to be similar in both
systems, but the yield is suppressed in the most peripheral Au+Au collisions.
## 1 Introduction
The interaction of hard scattered partons with the medium created by heavy ion
collisions (i.e., quark-gluon plasma, QGP) has been of interest since the
beginning of the RHIC running Wang:1998bha . A large suppression of the yields
of high transverse momentum ($p_{T}$) hadrons which are the fragments of such
partons was observed, suggesting that the matter is sufficiently dense to
cause parton-energy loss prior to hadronization Adler:2003qi . Absence of the
hadron suppression in $d$+Au collisions supported the parton-energy loss
scenario Adler:2003ii . After accumulating a large amount of $p+p$, $d$+Au,
and Au+Au collision events, we substantially extended the degree of freedom in
high $p_{T}$ hadron measurements. In this paper, we show the recent studies of
the QGP using high $p_{T}$ hadrons by the PHENIX experiment.
## 2 $\pi^{0}$ and $\eta$ measurements in d+Au and Au+Au collisions
The PHENIX experiment Adcox:2003zm has been exploring the highest $p_{T}$
region with single $\pi^{0}$ and $\eta$ mesons. They are leading hadrons of
jets, and thus provide a good measure of momentum of hard scattered partons.
Here, we present the results obtained from $d$+Au collisions collected in the
RHIC Year-2008 run (80 nb-1) and Au+Au collisions in the Year-2007 run (0.81
nb-1). Figure 1 shows the nuclear modification factors ($R_{d{\rm
A}}\equiv(dN_{d{\rm A}}/dydp_{T})/(\langle T_{d{\rm A}}\rangle
d\sigma_{pp}/dydp_{T})$) for $\pi^{0}$, $\eta$ and fully-reconstructed jets in
$d$+Au collisions at $\sqrt{s_{NN}}$=200 GeV.
Figure 1: $R_{d{\rm A}}$ for $\pi^{0}$, $\eta$ and fully-reconstructed jets in
$d$+Au collisions.
They are very consistent each other, and also consistent with unity at low
$p_{T}$ in both most central and peripheral collisions. However, at high
$p_{T}$, the yields are suppressed in most central collisions and enhanced in
most peripheral collisions.
The consistency of $\pi^{0}$ and $\eta$ are also seen in $R_{\rm AA}$
($\equiv(dN_{\rm AA}/dydp_{T})/(\langle T_{\rm AA}\rangle
d\sigma_{pp}/dydp_{T})$) in 200 GeV Au+Au collisions as shown in Figure 2(a)
Adare:2010dc .
Figure 2: (a, left) $R_{\rm AA}$ for $\pi^{0}$ and $\eta$ in minimum bias
Au+Au collisions. (b, right) $R_{\rm AA}$ for $\pi^{0}$ from the RHIC
Year-2004 run and Year-2007 run.
Because $\eta$ has four times larger mass compared to that of $\pi^{0}$, one
can resolve two photons decaying from $\eta$ up to four times larger $p_{T}$
of $\pi^{0}$, resulting in a higher $p_{T}$ reach with smaller systematic
errors with $\eta$. Figure 2(b) demonstrates that the $\pi^{0}$ from the
Year-2007 run has smaller errors and is consistent with that from the
Year-2004 run Adare:2012wg .
The recent result of single electron measurement shows that the $R_{d{\rm A}}$
and $R_{\rm AA}$ for light hadrons and electrons from heavy flavor hadrons
have similar trend of enhancement and suppression, except for low $p_{T}$
region, where soft production is dominant Adare:2012qb . This fact suggests
that the interaction of light hadrons and heavy hadrons with medium has same
system dependence.
## 3 Fractional momentum loss of hadrons in Au+Au collisions
The large amount of events collected in $p+p$ and Au+Au collisions made us
possible to quantify the energy loss effect from a different aspect.
Experiments have been looking at the suppression of the yield to see the
effect. However, the suppression is primarily the consequence of the reduction
of momentum of hadrons which have exponential $p_{T}$ distributions. We have
statistically extracted the fractional momentum loss ($S_{\rm
loss}\equiv\delta p_{T}/p_{T}$) of the partons using the hadron $p_{T}$
spectra measured in $p+p$ and Au+Au collisions Adare:2012wg . Figure 3(a)
depicts the method to compute the $S_{\rm loss}$.
Figure 3: (a, left) Method of calculating average $S_{\rm loss}$. (b, middle)
$S_{\rm loss}$ for $\pi^{0}$ for 0-10 % centrality 39, 62, and 200 GeV Au+Au
collisions. (c, right) $S_{\rm loss}$ for $\pi^{0}$ in 200 GeV Au+Au
collisions and charged hadrons in 2.76 TeV Pb+Pb collisions.
Using this method, we computed the $S_{\rm loss}$ in Au+Au collisions at
$\sqrt{s_{NN}}=$39, 62, and 200 GeV as shown in Figure 3(b) Adare:2012uk . We
also computed the $S_{\rm loss}$ in 2.76 TeV Pb+Pb collisions using charged
hadron spectra measured by the ALICE experiment Aamodt:2010jd as shown in
Figure 3(c). $S_{\rm loss}$’s vary by a factor of six from 39 GeV Au+Au to
2.76 TeV Pb+Pb collisions.
## 4 Path-length and collision system dependence of parton energy loss
With larger statistics, we were able to measure the $R_{\rm AA}$ of $\pi^{0}$
for in- and out-of event planes. Figure 4 shows the ones for $\pi^{0}$s in
20-30 % central 200 GeV Au+Au collisions Adare:2012wg .
Figure 4: $R_{\rm AA}(\phi)$ of $\pi^{0}$ in 20â30 % centrality for in-plane
and out-of-plane. (a, left) Data are compared with a pQCD-inspired model, and
(b, right) an AdS/CFT-inspired model.
The difference of the yield provides path-length dependence of yield
modification. Depending on the energy loss models, the powers of the path-
length dependence change. The data favors an AdS/CFT-inspired (strongly
coupled) model rather than pQCD-inspired (weakly coupled) model, implying that
the energy loss is $L^{3}$ dependent rather than $L^{2}$ dependence, where $L$
denotes the path-length of partons in the medium.
We note that the $N_{\rm coll}$ and $N_{\rm part}$ values are quite consistent
in certain central $d$+Au and peripheral Au+Au collisions. The ratio of
$N_{\rm coll}$ in 0-20% $d$$+$Au to that in 60-92% Au$+$Au is 1.02 $\pm$ 0.22,
and the same ratio for $N_{\rm part}$ values is 1.04 $\pm$ 0.21. Motivated by
this fact, we took the ratio of the spectra in 60-92 % Au$+$Au to 0-20 %
$d$$+$Au collisions for identified particles as shown in Figure 5
Adare:2013esx .
Figure 5: Ratio of invariant yield of particles in peripheral Au+Au (60â92 %)
to central $d$+Au (0â20 %) collisions as a function of $p_{T}$.
The ratios tend to the same value of roughly 0.65 for each particle species at
and above 2.5-3 GeV/$c$. This universal scaling is strongly suggestive of a
common particle production mechanism between peripheral Au$+$Au and central
$d$$+$Au collisions. The trend of overall rise in low $p_{T}$ may come from
rapidity shift in asymmetric collisions in $d$+Au. There is also mass
dependence of the rise seen in lower $p_{T}$. Assuming that the cold nuclear
effect scales with $N_{\rm coll}$ or $N_{\rm part}$, the ratio 0.65 may be
attributable to the parton energy loss in peripheral Au$+$Au collisions.
## 5 Summary
PHENIX measurement of high $p_{T}$ identified hadrons in $d$+Au and Au+Au
collisions are presented. The $R_{\rm AA}$ for $\pi^{0}$ and $\eta$ are found
to be very consistent in both collision systems, respectively. The $S_{\rm
loss}$’s of high $p_{T}$ hadrons are computed from 39 GeV Au+Au over to 2.76
TeV Pb+Pb, and found that they vary by a factor of six. The path-length
dependent $\pi^{0}$ yield deduced that the energy loss of partons is $L^{3}$
dependent. It was found that the hadron production mechanism in central $d$+Au
and peripheral Au+Au is similar, but the ratio of the yields is $\sim$0.65
which may be attributable to the parton energy loss in peripheral Au$+$Au
collisions.
## References
* (1) X. -N. Wang, Phys. Rev. C 58, 2321 (1998).
* (2) S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 072301 (2003).
* (3) S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 072303 (2003).
* (4) K. Adcox et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 469 (2003).
* (5) A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 82, 011902 (2010).
* (6) A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 87, 034911 (2013).
* (7) A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 109, 242301 (2012).
* (8) A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 109, 152301 (2012).
* (9) K. Aamodt et al. [ALICE Collaboration], Phys. Lett. B 696, 30 (2011).
* (10) A. Adare et al. [PHENIX Collaboration], arXiv:1304.3410 [nucl-ex], in press.
|
arxiv-papers
| 2013-08-06T22:57:02 |
2024-09-04T02:49:49.139441
|
{
"license": "Public Domain",
"authors": "Takao Sakaguchi (for the PHENIX collaboration)",
"submitter": "Takao Sakaguchi",
"url": "https://arxiv.org/abs/1308.1438"
}
|
1308.1443
|
# Category of asynchronous systems and polygonal morphisms
A. A. Husainov, [email protected]
###### Abstract
A weak asynchronous system is a trace monoid with a partial action on a set. A
polygonal morphism between weak asynchronous systems commutes with the actions
and preserves the independence of events. We prove that the category of weak
asynchronous systems and polygonal morphisms has all limits and colimits.
2010 Mathematics Subject Classification 18A35, 18A40, 18B20, 68Q85
Keywords: trace monoid, partial monoid action, limits, colimits, asynchronous
transition system.
## Introduction
Mathematical models of parallel systems find numerous applications in parallel
programming. They are applied for the development and verification of
programs, for searching deadlocks and estimation of runtime. These models are
widely applied to the description of semantics and the development of
languages of parallel programming [17].
There are various models of parallel computing systems [18]. For example, for
the solution of the dining philosophers problem, it is convenient to use
higher dimensional automata [7], but for a readers/writers problem, it is
better to consider asynchronous systems [12]. For comparing the models, the
adjoint functors between categories of these models are constructed [9], [10],
[11], [19].
But, at comparision of asynchronous transition systems and higher dimensional
automata, we face the open problem, whether there are colimits in the category
of asynchronous systems. We propose avoid this obstacle by constructing a
cocomplete category of asynchronous systems, and it allows us to build adjoint
functors in the standard way.
The asynchronous system is a model of the computing system consisting of
events (instructions, machine commands) and states. The states are defined by
values of variables (or cells of memory). Some events can occur
simultaneously. The category of asynchronous systems for the first time has
been studied by M. Bednarczyk [1]. Class of morphisms was extended in [2].
We consider asynchronous system as set with partial trace monoid action. We
represent the action as total, adding to asynchronous system a state “at
infinity”. Morphisms between trace monoids acting on the pointed sets lead to
polygonal morphisms of weak asynchronous systems.
These morphisms have great value for studying homology groups of the
asynchronous systems, introduced in [12]. They also help in the studying
homology groups of the Mazurkiewicz trace languages and Petri nets [14]. The
review of the homology of asynchronous systems is contained in [13].
The paper consist of three sections. In the first, the category $FPCM$ of
trace monoids and basic homomorhisms is investigated. It is proved, that in
this category, there are limits (Theorem 1.6) and colimits (Theorem 1.8)
though even finite products do not coincide with Cartesian products. The
subcategory $FPCM^{\|}\subset FPCM$ with independence preserving morphisms is
studied. It is proved, that this subcategory is complete (Theorem 1.14) and
cocomplete (Theorem 1.15). In the second section, the conditions of existence
of limits and colimits in a category of diagrams over the fixed category are
studied. The third section is devoted to a category of weak asynchronous
systems and polygonal . Main results about completeness and cocompleteness of
a category of weak asynchronous systems and polygonal morphisms are proved
(Theorems 3.12 and 3.13).
## 1 Categories of trace monoids
Bases of the trace monoid theory have been laid in [4]. Applications in
computer science belong to A. Mazurkiewicz [16], V. Diekert, Y. Métivier [6].
We shall consider a trace monoid category and basic homomorphisms and its
subcategory consisting of independence preserving homomorphisms.
The diagram is functor defined on a small category. Our objective is research
of a question on existence of limits and colimits of diagrams in these
categories.
### 1.1 Trace monoids
A map $f:M\to M^{\prime}$ between monoids is homomorphism, if $f(1)=1$ and
$f(\mu_{1}\mu_{2})=f(\mu_{1}\mu_{2})$ for all $\mu_{1},\mu_{2}\in M$. Denote
by $Mon$ the category of all monoids and homomorphisms.
Let $E$ be an arbitrary set. An independence relation on $E$ is subset
$I\subseteq E\times E$ satisfying the following conditions:
* •
$(\forall a\in E)~{}(a,a)\notin I$,
* •
$(\forall a,b\in E)~{}(a,b)\in I\Rightarrow(b,a)\in I$.
Elements $a,b\in E$ are independent, if $(a,b)\in I$.
Let $E^{*}$ be the monoid of all words $a_{1}a_{2}\cdots a_{n}$ where
$a_{1},a_{2},\cdots,a_{n}\in E$ and $n\geqslant 0$, with operation of
concatenation
$(a_{1}\cdots a_{n})(b_{1}\cdots b_{m})=a_{1}\cdots a_{n}b_{1}\cdots b_{m}\,.$
The identity $1$ is the empty word.
Let $I$ be an independence relation on $E$. We define the equivalence relation
$\equiv_{I}$ on $E^{*}$ putting $w_{1}\equiv_{I}w_{2}$ if $w_{2}$ can be
receive from $w_{1}$ by a finite sequence of adjacent independent elements.
---
$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{e\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d}$
Figure 1: Independence graph
For example, for the set $E=\\{a,b,c,d,e\\}$ and for the relation $I$ given by
the adjacency graph drawn in Figure 1 the sequence of permutations
$eadcc\stackrel{{\scriptstyle(e,a)}}{{\to}}aedcc\stackrel{{\scriptstyle(e,d)}}{{\to}}adecc\stackrel{{\scriptstyle(e,c)}}{{\to}}adcec\stackrel{{\scriptstyle(d,c)}}{{\to}}acdec\stackrel{{\scriptstyle(e,c)}}{{\to}}acdce\stackrel{{\scriptstyle(d,c)}}{{\to}}accde$
shows that $adecc\equiv_{I}accde$.
For every $w\in E^{*}$, its equivalence class $[w]$ is called the trace.
###### Definition 1.1
Let $E$ be a set and let $I$ be an independence relation. A trace monoid
$M(E,I)$ is the set of equivalence classes $[w]$ of all $w\in E^{*}$ with the
operation $[w_{1}][w_{2}]=[w_{1}w_{2}]$ for $w_{1},w_{2}\in E^{*}$.
We emphasize that the set $E$ can be infinite.
In some cases, we omit the square brackets in the notations for elements of
$M(E,I)$. If $I=\emptyset$, then $M(E,I)$ is equal to the free monoid $E^{*}$.
If $I=((E\times E)\setminus\\{(a,a)|a\in E\\})$, then $M(E,I)$ is the free
commutative monoid. In this case, we denote it by $M(E)$.
### 1.2 The category of trace monoids and basic homomorphisms
Let us introduce basic homomorphisms and we shall show, that the category of
trace monoids and basic homomorphism is complete and cocomplete.
###### Definition 1.2
A homomorphism $f:M(E,I)\to M(E^{\prime},I^{\prime})$ is basic if
$f(E)\subseteq E^{\prime}\cup\\{1\\}$.
If $w=e_{1}\cdots e_{n}\in M(E,I)$ for some $e_{1}\in E$, …, $e_{n}\in E$,
then $n$ is called the length of the trace $w$. It is easy to see, that a
homomorphism will be basic, if and only if it does not increase length of
elements of $M(E,I)$. Let $FPCM$ be a category of trace monoids and basic
homomorphisms.
Consider the problem on existence of the products in $FPCM$. The Cartesian
product $M(E_{1},I_{1})\times M(E_{2},I_{2})$ will not be the product in
$FPCM$. Thus for building products and other constructions, we shall consider
partial maps as total, adding to them the element $*$.
Let $E_{*}=E\sqcup\\{*\\}$. We assign to each partial map
$f:E_{1}{\rightharpoonup}E_{2}$, a total map $f_{*}:{E_{1}}_{*}\to{E_{2}}_{*}$
defined as
$f_{*}(a)=\left\\{\begin{array}[]{cc}f(a),&\mbox{ if }f(a)\mbox{ defined},\\\
,&\mbox{ otherwise.}\end{array}\right.$
Any basic homomorphism $M(E_{1},I_{1})\to M(E_{2},I_{2})$ can be given by some
partial map $f:E_{1}{\rightharpoonup}E_{2}$. We consider it as the pointed
total map $f_{*}:{E_{1}}_{*}\to{E_{2}}_{*}$ which brings any pair $(a,b)\in
I_{1}\cup(E_{1}\times\\{*\\})\cup(\\{*\\}\times E_{1})$ to the pair
$(f_{*}(a),f_{*}(b))\in I_{2}\cup(E_{2}\times\\{*\\})\cup(\\{*\\}\times
E_{2})$. It is clear that $f_{*}$ bring the elements of
$\Delta_{{E_{1}}_{*}}=\\{(a,a)|a\in{E_{1}}_{*}\\}$ to
$(f_{*}(a),f_{*}(a))\in\Delta_{{E_{2}}_{*}}$.
Let $ComRel$ be the category of pairs $(E_{*},T)$ where each pair consists of
a pointed set $E_{*}$ and binary relation of commutativity $T\subseteq
E_{*}\times E_{*}$ satisfying the following conditions
1. (i)
$(\forall a\in E_{*})~{}(a,*)\in T~{}\&~{}(*,a)\in T$ (commutativity with
$1$),
2. (ii)
$(\forall a\in E_{*})~{}(a,a)\in T$ (reflexivity),
3. (iii)
$(\forall a,b\in E_{*})(a,b)\in T\Rightarrow(b,a)\in T$ (symmetry).
Morphisms $({E_{1}}_{*},T_{1})\stackrel{{\scriptstyle
f}}{{\to}}({E_{2}}_{*},T_{2})$ in the category $ComRel$ are poinded maps
$f:{E_{1}}_{*}\to{E_{2}}_{*}$ satisfying $(a_{1},b_{1})\in
T_{1}\Rightarrow(f(a_{1}),f(b_{1}))\in T_{2}$.
###### Proposition 1.3
The category $FPCM$ is isomorphic to $ComRel$.
Proof. Define the functor $FPCM\to ComRel$ on objects by
$M(E,I)\mapsto(E_{*},T)$ where $T=I\cup(E\times\\{*\\})\cup(\\{*\\}\times
E)\cup\Delta_{E_{*}}$. The functor transforms basic homomorphisms
$f:M(E_{1},I_{1})\to M(E_{2},I_{2})$ into the maps
$f_{*}:{E_{1}}_{*}\to{E_{2}}_{*}$ assigning to pairs $(a_{1},b_{1})\in T_{1}$
the pairs $(f_{*}(a_{1}),f_{*}(b_{1}))\in T_{2}$.
An inverse functor assigns to each object $(E_{*},T)$ of the category $ComRel$
the trace monoid $M(E,I)$, where
$I=T\setminus\left(\\{(a,a)|a\in E_{*}\\}\cup\\{(a,*)|a\in
E\\}\cup\\{(*,a)|a\in E\\}\right),$ (1)
and to any morhism $({E_{1}}_{*},T_{1})\stackrel{{\scriptstyle
f}}{{\to}}({E_{2}}_{*},T_{2})$ the homomorphism
$\widetilde{f}:M(E_{1},I_{1})\to M(E_{2},I_{2})$ given at basic elements as
$\widetilde{f}(e)=f(e)$ if $f(e)\in E_{2}$, and $\widetilde{f}(e)=1$, if
$f(e)=*$. $\Box$
Consider a family of trace monoids $\\{M(E_{j},I_{j})\\}_{j\in J}$. Transform
it to family of pointed sets with commutativity relations
$\\{({E_{j}}_{*},T_{j})\\}_{j\in J}$. The product of this family in the
category $ComRel$ equals the Cartesian product $(\prod\limits_{j\in
J}{E_{j}}_{*},\prod\limits_{j\in J}T_{j})$. The category $FPCM$ is isomorphic
to $ComRel$. Therefore, we obtain the following
###### Proposition 1.4
The category $FPCM$ has the products.
Any object $(E_{*},T)$ of $ComRel$ corresponds to a trace monoid $M(E,I)$ with
the set $E=E_{*}\setminus\\{*\\}$ and independence relation defined by formula
(1).
It follows that the product of $M(E_{j},I_{j})$, $j\in J$ has the set of
generators $E=(\prod\limits_{j\in J}E_{j*})\setminus\\{(*)\\}$ where
$(*)\in\prod\limits_{j\in J}E_{j*}$ denotes a family of elements each of which
equals $*\in{E_{j}}_{*}$. Let
$T_{j}=I_{j}\cup(\\{(a,a)|a\in{E_{j}}_{*}\\}\cup\\{(a,*)|a\in
E_{j}\\}\cup\\{(*,a)|a\in E_{j}\\}).$
The relation $I$ is received from $T=\prod\limits_{j\in J}T_{j}$ by the
formula (1).
###### Example 1.5
Let $J=\\{1,2\\}$, $E_{1}=\\{e_{1}\\},E_{2}=\\{e_{2}\\}$,
$I_{1}=I_{2}=\emptyset$. Then $M(E_{1},I_{1})\cong
M(E_{2},I_{2})\cong{\,\mathbb{N}}$ are isomorphic to the monoid generated by
one element. Compute $M(E,I)=M(E_{1},I_{1})\prod M(E_{2},I_{2})$. The set
$E_{*}$ equals ${E_{1}}_{*}\times{E_{2}}_{*}$. In following picture at the
left, it is shown the graph of the relation $T\subseteq E_{*}\times E_{*}$ and
on the right it is shown the graph of the relation $I$ obtained by the formula
(1).
$\textstyle{(*,*)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(e_{1},*)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(*,e_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(e_{1},e_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{(e_{1},*)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(*,e_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(e_{1},e_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
We see, that the product is isomorphic to a free commutative monoid generated
by three elements.
###### Theorem 1.6
For each diagram $D$ in $FPCM$, there is the limit.
Proof. Since $FPCM$ has all products, it is enough existence of equalizers.
Consider a pair
$\textstyle{M(E_{1},I_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{M(E_{2},I_{2})}$
of basic homomorphisms. Let $E=\\{e\in E_{1}~{}|~{}f(e)=g(e)\\}$. The
submonoid of $M(E_{1},I_{1})$ generated by $E$ is a trace monoid $M(E,I)$ with
the independence relation $I=I_{1}\cap(E\times E)$. Consider an arbitrary
basic homomorphism $h^{\prime}:M(E^{\prime},I^{\prime})\to M(E_{1},I_{1})$
such that $g(h^{\prime}(e^{\prime}))=f(h^{\prime}(e^{\prime}))$ for all
$e^{\prime}\in E^{\prime}$. Obtain $h^{\prime}(e^{\prime})\in E\cup\\{1\\}$.
It follows that $h^{\prime}$ maps $M(E^{\prime},I^{\prime})$ into $M(E,I)$ and
the following triangle is commutative:
---
$\textstyle{M(E,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{M(E_{1},I_{1})}$$\textstyle{M(E^{\prime},I^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$
Therefore, the inclusion $M(E,I)$ into $M(E_{1},I_{1})$ is equalizer of the
pair $(f,g)$. $\Box$
###### Proposition 1.7
Let $\operatorname{{\rm Ob}}{Mon}\to\operatorname{{\rm Ob}}FPCM$ be the map
carried each monoid $M$ to a trace monoid $M(M\setminus\\{1\\},I_{M})$ with
$I_{M}=\\{(\mu_{1},\mu_{2})\in(M\setminus\\{1\\})\times(M\setminus\\{1\\})|~{}\mu_{1}\not=\mu_{2}~{}\&~{}\mu_{1}\mu_{2}=\mu_{2}\mu_{1}\\}.$
This map can be extended to a functor $R:{Mon}\to FPCM$ right adjoint to the
inclusion $U:FPCM\to{Mon}$.
Proof. Define a homomorphism $\varepsilon_{M}:M(M\setminus\\{1\\},I_{M})\to M$
setting $\varepsilon(\mu)=\mu$ on the generators of
$M(M\setminus\\{1\\},I_{M})\to M$. It easy to see that for each homomorphism
$f:M(E,I)\to M$, there exists unique basic homomorphism $\overline{f}$ making
the following diagram commutative
---
$\textstyle{M(M\setminus\\{1\\},I_{M})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon_{M}}$$\textstyle{M}$$\textstyle{M(E,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$$\scriptstyle{f}$
It is defined by $\overline{f}(e)=f(e)$ on elements $e\in E$. This
homomorphism is couniversal arrow. By the universal property, the map
$M\mapsto(M(M\setminus\\{1\\},I_{M}),\varepsilon_{M})$ uniquely extends up to
the right adjoint functor. $\Box$
###### Theorem 1.8
The category $FPCM$ is cocomplete and the inclusion functor $FPCM$ into the
category ${Mon}$ preserves all colimits.
Proof. Let $D:J\to FPCM$ be a diagram with values $D(j)=M(E_{j},I_{j})$.
Consider $\underrightarrow{\lim}^{J}D$ in the category $Mon$ of all monoids.
The colimit is isomorphic to a quotient monoid $\coprod_{j\in
J}M(E_{j},I_{j})/\equiv$ obtained from the coproduct in $Mon$ by
identifications of elements $e_{j}\equiv D(j\to k)e_{j}$. It follows that the
colimit is generated by the disjoint union $\coprod\limits_{j\in J}E_{j}$ and
represented by the following equations:
1. (i)
for all $j\in J$, $(e,e^{\prime})\in I_{j}$ it is true $ee^{\prime}\equiv
e^{\prime}e$,
2. (ii)
if $e^{\prime}_{k}=D(j\to k)(e_{j})$ for some $e_{j}\in E_{j}$,
$e^{\prime}_{k}\in E_{k}$, then $e_{j}\equiv e^{\prime}_{k}$,
3. (iii)
$e_{j}\equiv 1$ if $M(j\to k)(e_{j})=1$.
This monoid is generated by a set $E$ received of a quotient set of
$\coprod\limits_{j\in J}E_{j}$ under the equivalence relation containing pairs
type (ii) by removing the classes containing elements $e_{j}\equiv 1$. The
equations (i) give the relation $I$. We obtain the trace monoid
$\underrightarrow{\lim}^{J}D=M(E,I)$. The morphisms of colimiting cone are
basic homomorhisms sending to every $e_{j}$ its equivalence class or $1$. For
any other cone $f_{j}:M(E_{j},I_{j})\to M(E^{\prime},I^{\prime})$ consisting
of basic homomorphisms, the morphism $\underrightarrow{\lim}^{J}D\to
M(E^{\prime},I^{\prime})$ assigns to each class $[e_{j}]$ the element
$f_{j}(e_{j})$.
---
$\textstyle{M(E,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{M(E_{j},I_{j})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{j}}$$\scriptstyle{f_{j}}$$\textstyle{M(E^{\prime},I^{\prime})}$
Therefore, $FPCM$ has all colimits.
It follows from 1.7 that the inclusion $FPCM\subset Mon$ preserves all
colimits as having right adjoint [15]. $\Box$
###### Example 1.9
Consider the free commutative monoid $M(\\{a,b\\})$ and the trace monoid
$M=M(\\{c,d,e\\},\\{(c,d),(d,c),(d,e),(e,d)\\})$. Let $f,g:M\\{a,b\\}\to M$ be
two homomorphisms defined as $f(a)=c$, $g(b)=d$, $g(a)=d$, $g(b)=c$.
$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{e}$$\textstyle{1}$
The coequalizer of $f,g$ is the trace monoid generated by $c,d,e$ with
equations $c=a=d=b$, $cd=dc$, $de=ed$. In the picture in the top line, it is
shown the independence relation for $M(\\{a,b\\})$, and in bottom for $M$.
Consequently, the coequalizer is equal to the free commutative monoid
generated by one element.
### 1.3 Independence preserving basic homomorphisms
We prove that the category of trace monoids and independence preserving
homomorphisms has all limits and colimits.
###### Definition 1.10
A basic homomorphism $f:M(E,I)\to M(E^{\prime},I^{\prime})$ is called
independence preserving if for all $a,b\in E$, the following implication is
carried out
$(a,b)\in I\Rightarrow(f(a)\not=f(b))~{}\vee~{}(f(a)=f(b)=1)~{}.$
It is easy to see, that this implication is equivalent to the condition
$(a,b)\in I\Rightarrow(f(a),f(b))\in
I^{\prime}~{}\vee~{}f(a)=1~{}\vee~{}f(b)=1~{}.$
It follows that the class of independence preserving homomorphisms is closed
under composition. Let $FPCM^{\|}\subset FPCM$ be the subcategory consisting
of all trace monoids and independence preserving basic homomorphisms.
Let us prove the existence of the products in the $FPCM^{||}$. For this
purpose we introduce the following partial independence relation.
###### Definition 1.11
Let $E$ be a set. A partial independence relation on $E$ is a subset
$R\subseteq E_{*}\times E_{*}$ satisfying the followng conditions:
1. (i)
$(\forall a\in E_{*})~{}(a,*)\in R~{}\&~{}(*,a)\in R$;
2. (ii)
$(\forall a\in E_{*})~{}(a,a)\in R\Rightarrow a=*$;
3. (iii)
$(\forall a,b\in E_{*})~{}(a,b)\in R\Leftrightarrow(b,a)\in R$.
Let $IndRel$ be the category of pairs $(E_{*},R)$ consisting of pointed sets
$E_{*}$ and partial independence relations $R\subseteq E_{*}\times E_{*}$. Its
morphisms $(E_{*},R)\stackrel{{\scriptstyle
f}}{{\to}}(E_{*}^{\prime},R^{\prime})$ defined as pointed maps $f:E_{*}\to
E_{*}^{\prime}$ satisfying the following conditions:
$(a,b)\in R\Rightarrow(f(a),f(b))\in R^{\prime}.$
###### Proposition 1.12
The category $FPCM^{||}$ is isomorphic to $IndRel$.
Proof. Define the functor $FPCM^{||}\to IndRel$ as sending $M(E,I)$ to
$(E_{*},R)$ where $R=I\cup(E_{*}\times\\{*\\})\cup(\\{*\\}\times E_{*})$. The
inverse functor $IndRel\to IndRel$ carries any object $(E_{*},R)$ to the
monoid $M(E,I)$ where $I=R\setminus(E_{*}\times\\{*\\})\cup(\\{*\\}\times
E_{*})$. This functor send morphisms of $IndRel$ to independence preserving
morphisms. $\Box$
###### Corollary 1.13
The category $FPCM^{||}$ has all products.
Moreover, it is true the following
###### Theorem 1.14
The category $FPCM^{\|}$ has all limits. The inclusion functor
$FPCM^{\|}\subset FPCM$ preserves equalizers.
Proof. Since $FPCM^{\|}$ has products, it it enough to prove the existence
equalizers. For any pair of basic homorphisms
$\textstyle{M(E_{1},I_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{M(E_{2},I_{2})}$
in the category $FPCM$ its equalizer is the inclusion $M(E,I)\subseteq
M(E_{1},I_{1})$, where $E=\\{e\in E_{1}|f(e)=g(e)\\}$ and $I=I_{1}\cap(E\times
E)$. Inclusion preserves independence. Consider a trace monoid
$M(E^{\prime},I^{\prime})$ with a independence preserving homomorphism
$h:M(E^{\prime},I^{\prime})\to M(E_{1},I_{1})$ satisfying $fh=gh$. Since
$h(E^{\prime})\subseteq E$, there is a basic homomorphism $k$ drawn by dashed
arrow in the diagram:
|
---|---
$\textstyle{M(E,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{M(E_{1},I_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{M(E_{2},I_{2})}$$\textstyle{M(E^{\prime},I^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{k}$
We have $k(e^{\prime})=h(e^{\prime})$ for all $e^{\prime}\in E^{\prime}$. The
homomorphism $h$ preserves independence. Hence, for all
$(a^{\prime},b^{\prime})\in I^{\prime}$, the condition $k(a^{\prime})=1\vee
k(b^{\prime})=1\vee(k(a^{\prime}),k(b^{\prime}))\in I_{1}$ holds. Thus, $k$
preserves independence. Equalizers is constructed in the category $FPCM$.
Therefore the inclusion $FPCM^{\|}\subset FPCM$ preserves equivalizers. $\Box$
We now turn to the colimit.
###### Theorem 1.15
The category $FPCM^{\|}$ is cocomplete.
Proof. The coproduct of trace monoids $\\{M(E_{i},I_{i})\\}_{i\in J}$ is a
monoid given by generators $\coprod\limits_{i\in J}E_{i}$ and relations
$ab=ba$ for all $(a,b)\in\coprod\limits_{i\in J}I_{i}$. It is easy to see that
it is coproduct in the category $FPCM^{||}$. Hence, it is sufficient to prove
the existence coequalizers. For this purpose, consider an arbitrary pair of
morphisms
$\textstyle{M(E_{1},I_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{M(E_{1},I_{1})}$
in the category $FPCM^{\|}$. Let $h:M(E_{2},I_{2})\to M(E,I)$ be the
coequalizer in the category $FPCM$. For each $h^{\prime}:M(E_{2},I_{2})\to
M(E^{\prime},I^{\prime})$, there exists a unique $k$ making commutative
triangle in the following diagram
---
$\textstyle{M(E_{1},I_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{M(E_{1},I_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{\prime}}$$\scriptstyle{h}$$\textstyle{M(E,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exists!k}$$\textstyle{M(E^{\prime},I^{\prime})}$
If $h^{\prime}$ preserves independence, then the following implication is
true:
$(\forall(a,b)\in I_{2})(h^{\prime}(a)=h^{\prime}(b)\Rightarrow
h^{\prime}(a)=1~{}\&~{}h^{\prime}(b)=1).$ (2)
Let $\equiv_{h}$ be the smallest congruence relation for which
$h(a)\equiv_{h}1$ and $h(b)\equiv_{h}1$ if $(a,b)\in I_{2}$ satisfies
$h(a)=h(b)$. Denote by $cls:M(E,I)\to M(E,I)/\equiv_{h}$ the homomorphism
assigning to any $e\in E_{*}$ its class $cls(e)$ of the congruence. If
$h^{\prime}$ preserves independence, then it follows from (2) and
$kh=h^{\prime}$ that
$(\forall(a,b)\in I_{2})k(h(a))=k(h(b))\Rightarrow
k(h(a))=1~{}\&~{}k(h(b))=1.$
We see that $k$ has constant values on each congruence class $cls(e)$ where
$e\in E_{*}$. Hence, we can define a map $k^{\prime}:M(E,I)/\equiv_{h}\to
M(E^{\prime},I^{\prime})$ by $k^{\prime}(cls(e))=k(e)$ for all $e\in E_{*}$.
The homomorphism $k^{\prime}$ is unique for which $k^{\prime}\circ cls\circ
h=h^{\prime}$. Therefore, $cls\circ h:M(E_{2},I_{2})\to M(E,I)/\equiv_{h}$ is
the coequalizer of $(f,g)$. $\Box$
In Example 1.9, we have $h(c)=h(d)=h(e)$. Since $(c,d)\in I_{2}$ and $(d,e)\in
I_{2}$, we have $cls\circ h(c)=1$, $cls\circ h(d)=1$, $cls\circ h(e)=1$.
Therefore, the coequalizer equals $\\{1\\}$.
## 2 Category of diagrams with various domains
This Section is auxiliary also does not contain new results. A diagram in a
category ${\mathcal{A}}$ is a functor ${\mathscr{C}}\to{\mathcal{A}}$ defined
on some small category ${\mathscr{C}}$. We shall consider categories of the
diagrams accepting values in some fixed category. Let us study the conditions
providing completeness or cocompleteness of this category.
### 2.1 Morphisms and objects in a digram category
Let ${\mathcal{A}}$ be a category and let $F:{\mathscr{C}}\to{\mathcal{A}}$ be
a diagram. Denote this diagram by $({\mathscr{C}},F)$ specifying its domain
${\mathscr{C}}$.
Let $({\mathscr{C}},F)$ and $({\mathscr{D}},G)$ be diagrams in
${\mathcal{A}}$. A morphism of the diagrams
$(\Phi,\xi):({\mathscr{C}},F)\to({\mathscr{D}},G)$ is given by a pair
$(\Phi,\xi)$ consisting of a functor $\Phi:{\mathscr{C}}\to{\mathscr{D}}$ and
natural transformation $\xi:F\to G\Phi$
|
---|---
$\textstyle{{\mathscr{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{F}$$\scriptstyle{\xi\nearrow}$$\scriptstyle{G\Phi}$$\textstyle{{\mathscr{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{\mathcal{A}}$
Define the identity morphism by the formula
$1_{({\mathscr{C}},F)}=(1_{{\mathscr{C}}},1_{F})$ where
$1_{{\mathscr{C}}}:{\mathscr{C}}\to{\mathscr{C}}$ is the identity functor and
$1_{F}:F\to F$ is the identity natural transformation. The composition of
morphisms
$({\mathscr{C}},F)\stackrel{{\scriptstyle(\Phi,\xi)}}{{\to}}({\mathscr{D}},G)\stackrel{{\scriptstyle(\Psi,\eta)}}{{\to}}({\mathscr{E}},H)$
is defined as a pair $(\Psi\Phi,(\eta*\Phi)\cdot\xi)$ where
$\eta*\Phi:G\Phi\to H\Psi\Phi$ is a natural transformation given by a family
of morphisms specified as a family of morphisms
$(\eta*\Phi)_{c}=\eta_{\Phi(c)}:G(\Phi(c))\to
H(\Psi(\Phi(c))),~{}c\in\operatorname{{\rm Ob}}{\mathscr{C}},$
and $(\eta*\Phi)\cdot\xi$ is the composition of natural transformations
$F\stackrel{{\scriptstyle\xi}}{{\to}}G\Phi\stackrel{{\scriptstyle\eta*\Phi}}{{\to}}H\Psi\Phi$.
The composition is associative.
Let $Cat$ be the category of small categories and functors. Denote by
$(Cat,{\mathcal{A}})$ the category of diagrams in ${\mathcal{A}}$ and
morphisms of diagrams.
For any subcategory ${\mathfrak{C}}\subseteq Cat$, we consider diagrams
$F:{\mathscr{C}}\to{\mathcal{A}}$ defined on categories
${\mathscr{C}}\in{\mathfrak{C}}$. Such diagrams with morphisms
$(\Phi,\xi):({\mathscr{C}},F)\to({\mathscr{D}},G)$ where
$\Phi\in\operatorname{{\rm Mor}}{\mathfrak{C}}$, will be make a subcategory of
$(Cat,{\mathcal{A}})$. Denote this subcategory by
$({\mathfrak{C}},{\mathcal{A}})$.
### 2.2 Limits in a category of diagram
Let $J$ be a small category. In some cases, the diagrams are conveniently
denoted, specifying their values on objects. For example, we will denote by
$\\{A_{i}\\}_{i\in J}$ the diagram $J\to{\mathcal{A}}$ with values $A_{i}$ on
objects $i\in J$ and $A_{\alpha}:A_{i}\to A_{j}$ on morphisms $\alpha:i\to j$
of $J$. We say that a category ${\mathcal{A}}$ has $J$-limits if every diagram
$\\{A_{i}\\}_{i\in J}$ in ${\mathcal{A}}$ has a limit. If ${\mathcal{A}}$ has
$J$-limits for all small categories $J$, then ${\mathcal{A}}$ is said to be a
complete category or a category with all limits.
We will consider subcategories ${\mathfrak{C}}\subseteq Cat$ with $J$-limits.
But the $J$-limits in ${\mathfrak{C}}$ need not be isomorphic to the
$J$-limits in $Cat$.
###### Proposition 2.1
Let ${\mathcal{A}}$ be a complete category and let $J$ be a small category. If
a subcategory ${\mathfrak{C}}\subseteq Cat$ has $J$-limits, then the category
$({\mathfrak{C}},{\mathcal{A}})$ has $J$-limits. In particular, if
${\mathfrak{C}}\subseteq Cat$ is a complete category, then the category
$({\mathfrak{C}},{\mathcal{A}})$ is complete.
Proof. Let $\\{({\mathscr{C}}_{i},F_{i})\\}_{i\in J}$ be a diagram in
$({\mathfrak{C}},{\mathcal{A}})$. One given by a diagram
$\\{{\mathscr{C}}_{i}\\}_{i\in J}$ with functors
${\mathscr{C}}_{\alpha}:{\mathscr{C}}_{i}\to{\mathscr{C}}_{j}$ and natural
transformations $\varphi_{\alpha}:F_{i}\to F_{j}{\mathscr{C}}_{\alpha}$. Let
$p_{i}:\underleftarrow{\lim}_{J}\\{{\mathscr{C}}_{i}\\}\to{\mathscr{C}}_{i}$
is the limit cone of the diagram $\\{{\mathscr{C}}_{i}\\}_{i\in J}$ in
${\mathfrak{C}}$. The compositions $F_{i}\circ p_{i}$ belong to the category
${\mathcal{A}}^{\underleftarrow{\lim}_{J}\\{{\mathscr{C}}_{i}\\}}$. The
natural transformations
$F_{i}p_{i}\stackrel{{\scriptstyle\varphi_{\alpha}*p_{i}}}{{\to}}F_{j}{\mathscr{C}}_{\alpha}p_{i}\stackrel{{\scriptstyle=}}{{\to}}F_{j}p_{j}$
give the functor
$J\to{\mathcal{A}}^{\underleftarrow{\lim}_{J}\\{{\mathscr{C}}_{i}\\}}$. Let
$\underleftarrow{\lim}_{J}\\{F_{i}p_{i}\\}\in{\mathcal{A}}^{\underleftarrow{\lim}_{J}\\{{\mathscr{C}}_{i}\\}}$
be its limit. Denote by $\pi_{i}:\underleftarrow{\lim}_{J}\\{F_{i}p_{i}\\}\to
F_{i}p_{i}$ the limit cone. It easy to see that morphisms
$(p_{i},\pi_{i}):(\underleftarrow{\lim}_{J}\\{{\mathscr{C}}_{i}\\},\underleftarrow{\lim}_{J}\\{F_{i}p_{i}\\})\to({\mathscr{C}}_{i},F_{i})$
of diagrams make the cone over the diagram in
$({\mathfrak{C}},{\mathcal{A}})$. Considering an another cone
$(r_{i},\xi_{i}):({\mathscr{C}},F)\to({\mathscr{C}}_{i},F_{i})$ it can be seen
that there exists the unique morphism $(r,\xi)$ making the commutative
triangle
---
$\textstyle{({\mathscr{C}},F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(r_{i},\xi_{i})}$$\scriptstyle{(r,\xi)}$$\textstyle{({\mathscr{C}}_{i},F_{i})}$$\textstyle{(\underleftarrow{\lim}_{J}\\{{\mathscr{C}}_{i}\\},\underleftarrow{\lim}_{J}\\{F_{i}p_{i}\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(p_{i},\pi_{i})}$
It follows that the limit is isomorphic to
$(\underleftarrow{\lim}_{J}\\{{\mathscr{C}}_{i}\\},\underleftarrow{\lim}_{J}\\{F_{i}p_{i}\\})$.
$\Box$
### 2.3 Colimits in a category of diagrams
Let ${\mathfrak{C}}\subseteq Cat$ be a subcategory. Consider an arbitrary
category ${\mathcal{A}}$. We shall prove that if the colimits exist in
${\mathfrak{C}}$, then those exist in $({\mathfrak{C}},{\mathcal{A}})$. For
any functor $\Phi:{\mathscr{C}}\to{\mathscr{D}}$, we denote by
$\operatorname{{\rm
Lan}}^{\Phi}:{\mathcal{A}}^{{\mathscr{C}}}\to{\mathcal{A}}^{{\mathscr{D}}}$
the left Kan extension functor [15]. Its properties are well described in
[15]. This functor is characterized as a left adjoint to the functor
$\Phi^{*}:\ mA^{{\mathscr{D}}}\to{\mathcal{A}}^{{\mathscr{C}}}$ assigning to
each diagram $F:{\mathscr{D}}\to{\mathcal{A}}$ the composition $F\circ\Phi$,
and to the natural transformation $\eta:F\to G$ the natural transformation
$\eta*\Phi$.
###### Proposition 2.2
Let ${\mathfrak{C}}\subseteq Cat$ be a category with all colimits. Then, for
any cocomplete category ${\mathcal{A}}$, the category
$({\mathfrak{C}},{\mathcal{A}})$ has all colimits.
Proof. Consider a diagram $\\{({\mathscr{C}}_{i},F_{i})\\}_{i\in J}$ in
$({\mathfrak{C}},{\mathcal{A}})$. As above, each morphism $\alpha:i\to j$ is
mapped to the natural transformation $\varphi_{\alpha}:F_{i}\to
F_{j}{\mathscr{C}}_{\alpha}$. Let
$\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}$ be the colimit of the
diagram in ${\mathfrak{C}}$. Denote by
$q_{i}:{\mathscr{C}}_{i}\to\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}$
morphisms of the colimit cone. Consider the Kan extensions $\operatorname{{\rm
Lan}}^{q_{i}}F_{i}$ and the units of adjunction
---
$\textstyle{{\mathscr{C}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{i}}$$\scriptstyle{F_{i}}$$\scriptstyle{\nearrow\eta_{i}}$$\textstyle{\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}}$$\textstyle{\mathcal{A}}$
We get the diagram in the category
${\mathcal{A}}^{\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}}$ consisting
of objects $\operatorname{{\rm Lan}}^{q_{i}}F_{i}$ and morphisms given at
$\alpha:i\to j$ by the compositions
$\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\stackrel{{\scriptstyle\operatorname{{\rm
Lan}}^{q_{i}}(\varphi_{\alpha})}}{{\to}}\operatorname{{\rm
Lan}}^{q_{i}}F_{j}{\mathscr{C}}_{\alpha}\stackrel{{\scriptstyle=}}{{\to}}\operatorname{{\rm
Lan}}^{q_{j}}\operatorname{{\rm
Lan}}^{{\mathscr{C}}_{\alpha}}F_{j}{\mathscr{C}}_{\alpha}\stackrel{{\scriptstyle\operatorname{{\rm
Lan}}^{q_{j}}(\varepsilon_{\alpha})}}{{\to}}\operatorname{{\rm
Lan}}^{q_{j}}F_{j}$
where $\varepsilon_{\alpha}:\operatorname{{\rm
Lan}}^{{\mathscr{C}}_{\alpha}}(F_{j}{\mathscr{C}}_{\alpha})\to F_{j}$ are
counits of adjunction. Let $\underrightarrow{\lim}^{J}\\{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\\}$ be the colimit of this diagram.
Prove that
$(\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\},\underrightarrow{\lim}^{J}\\{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\\})$ is a colimit of the diagram
$\\{({\mathscr{C}}_{i},F_{i})\\}_{i\in J}$ in
$({\mathfrak{C}},{\mathcal{A}})$. For this purpose, consider an arbitrary
(direct) cone $({\mathscr{C}}_{i},F_{i})\to({\mathscr{C}},F)$ over
$\\{({\mathscr{C}}_{i},F_{i})\\}_{i\in J}$ in the category
$({\mathfrak{C}},{\mathcal{A}})$. One is given by some functors
---
$\textstyle{{\mathscr{C}}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{i}}$$\scriptstyle{F_{i}}$$\scriptstyle{\nearrow\psi_{i}}$$\textstyle{{\mathscr{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\mathcal{A}}$
and natural transformations $\psi_{i}:F_{i}\to Fr_{i}$ for which the following
diagrams are commutative
---
$\textstyle{({\mathscr{C}}_{i},F_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathscr{C}}_{\alpha},\varphi_{\alpha})}$$\scriptstyle{(r_{i},\psi_{i})}$$\textstyle{({\mathscr{C}},F)}$$\textstyle{({\mathscr{C}}_{j},F_{j})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(r_{j},\psi_{j})}$
$\textstyle{F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{i}}$$\scriptstyle{\varphi_{\alpha}}$$\textstyle{Fr_{i}}$$\textstyle{F_{j}{\mathscr{C}}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{j}*{\mathscr{C}}_{\alpha}}$$\textstyle{Fr_{j}{\mathscr{C}}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Since $\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}$ is the colimit in
${\mathfrak{C}}$, the unique functor
$r:\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}\to{\mathscr{C}}$ is
corresponded to the functors of this cone
$r_{i}:{\mathscr{C}}_{i}\to{\mathscr{C}}$, such that $r_{i}=rq_{i}$ dor all
$i\in J$ where
$q_{i}:{\mathscr{C}}_{i}\to\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}$
is the colimit cone.
For any $i\in J$, the functor $\operatorname{{\rm Lan}}^{q_{i}}$ is left
adjoint to $q_{i}^{*}$. Hence, there exists a bijection between natural
transformations
$F_{i}\stackrel{{\scriptstyle\psi_{i}}}{{\to}}Fr_{i}=Frq_{i}\quad\mbox{ and
}\quad\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\stackrel{{\scriptstyle\overline{\psi_{i}}}}{{\to}}Fr\,.$
This bijection maps each commutative triangle in
${\mathcal{A}}^{{\mathscr{C}}_{i}}$ to the commutative triangle in
${\mathcal{A}}^{\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}_{i\in J}}$:
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
13.62534pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-7.87436pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.05951pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\psi_{i}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 43.37631pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-19.63664pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi_{\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.43999pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.37631pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Frq_{i}}$}}}}}}}{\hbox{\kern-13.62534pt\raise-39.2733pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{F_{j}{\mathscr{C}}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.25325pt\raise-45.75226pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38214pt\hbox{$\scriptstyle{\psi_{j}*{\mathscr{C}}_{\alpha}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
37.62534pt\raise-39.2733pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.62534pt\raise-39.2733pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Frq_{j}{\mathscr{C}}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces}}}}\ignorespaces\quad\mapsto\quad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
18.72676pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\\\crcr}}}\ignorespaces{\hbox{\kern-18.1788pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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Lan}}^{q_{i}}F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.3911pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
20.51558pt\raise 5.83888pt\hbox{{}\hbox{\kern 0.0pt\raise
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0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{\psi_{i}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.72676pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
42.72676pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Fr}$}}}}}}}{\hbox{\kern-18.72676pt\raise-40.42996pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{{\rm
Lan}}^{q_{j}}F_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
24.79097pt\raise-26.05386pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.83888pt\hbox{$\scriptstyle{\overline{\psi_{j}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 48.16985pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces}}}}\ignorespaces$
For the diagram
|
---|---
$\textstyle{F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{i}}$$\scriptstyle{\varphi_{\alpha}}$$\textstyle{Frq_{i}=Frq_{j}{\mathscr{C}}_{\alpha}}$$\textstyle{F_{j}{\mathscr{C}}_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{j}*{\mathscr{C}}_{\alpha}}$
we have the commutative diagram in ${\mathcal{A}}^{{\mathscr{C}}_{j}}$
|
---|---
$\textstyle{\operatorname{{\rm
Lan}}^{{\mathscr{C}}_{\alpha}}F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\varphi_{\alpha}}}$$\scriptstyle{\overline{\psi_{i}}}$$\textstyle{Frq_{j}}$$\textstyle{F_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{j}}$
Applying $\operatorname{{\rm Lan}}^{q_{j}}$, we obtain the commutative diagram
| |
---|---|---
$\textstyle{\operatorname{{\rm Lan}}^{q_{j}}\operatorname{{\rm
Lan}}^{{\mathscr{C}}_{\alpha}}F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{\rm
Lan}}^{q_{j}}{\overline{\varphi_{\alpha}}}}$$\scriptstyle{\operatorname{{\rm
Lan}}^{q_{j}}{\overline{\psi_{i}}}}$$\textstyle{\operatorname{{\rm
Lan}}^{q_{j}}Frq_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\varepsilon_{j}}_{Fr}}$$\textstyle{Fr}$$\textstyle{\operatorname{{\rm
Lan}}^{q_{j}}F_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{{\rm
Lan}}^{q_{j}}({\psi_{j}})}$
which leads us to the (direct) cone over the diagram $\\{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\\}_{i\in J}$
---
$\textstyle{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Fr}$$\textstyle{\operatorname{{\rm
Lan}}^{q_{j}}F_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
This cone gives the morphism $\underrightarrow{\lim}^{J}\\{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\\}\to Fr$ in
${\mathcal{A}}^{\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\}}$ which
define a unique morphism in $({\mathfrak{C}},{\mathcal{A}})$ making
commutative triangles
---
$\textstyle{(\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\},\underrightarrow{\lim}^{J}\\{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\mathscr{C}},F)}$$\textstyle{({\mathscr{C}}_{i},F_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Therefore, the diagram
$(\underrightarrow{\lim}^{J}\\{{\mathscr{C}}_{i}\\},\underrightarrow{\lim}^{J}\\{\operatorname{{\rm
Lan}}^{q_{i}}F_{i}\\})$ in ${\mathcal{A}}$ is the colimit of the diagram
$\\{({\mathscr{C}}_{i},F_{i})\\}_{i\in J}$ in the category
$({\mathfrak{C}},{\mathcal{A}})$. $\Box$
## 3 Category of pointed polygons on trace monoids
We apply auxiliary propositions from section 2 to categories
${\mathcal{A}}={\rm Set}_{*}$, ${\mathfrak{C}}=FPCM$ and
${\mathfrak{C}}=FPCM^{\|}$. Then we shall establish communications between a
category of asynchronous systems and categories of right $M(E,I)$-sets and we
investigate a category of asynchronous systems and polygonal morphisms.
### 3.1 Category of state spaces
A state space $(M(E,I),X)$ consists of a trace monoid $M(E,I)$ with an action
on a pointed set $X$ by some operation $\cdot:X\times M(E,I)\to X$, $x\mapsto
x\cdot w$ for $x\in X$, $w\in M(E,I)$. Since the monoid is a category with a
unique object, we can consider the state space as a functor $X:M(E,I)^{op}\to
Set_{*}$ sending the unique object to the pointed set $X$ and morphisms $w\in
M(E,I)$ to maps $X(w):X\to X$ given as $X(w)(x)=x\cdot w$. Here we denote by
$X$ the pointed set on which the monoid acts as well as functor defined by
this action.
###### Definition 3.1
A morphism of state spaces
$(M(E,I),X)\to(M(E^{\prime},I^{\prime}),X^{\prime})$
is a pair $(\eta,\sigma)$ where $\eta:M(E,I)\to M(E^{\prime},I^{\prime})$ is a
basic homomorphism and $\sigma:X\to X^{\prime}\circ\eta^{op}$ is a natural
transformation.
A morphism of state spaces is possible to represent by means of the diagram
|
---|---
$\textstyle{M(E,I)^{op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X}$$\scriptstyle{\sigma\nearrow}$$\scriptstyle{\eta^{op}}$$\textstyle{M(E^{\prime},I^{\prime})^{op}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X^{\prime}}$$\textstyle{Set_{*}}$
The category of state space is isomorphic to $(FPCM,Set_{*})$.
By Proposition 2.1, if a subcategory ${\mathfrak{C}}\subseteq Cat$ has
$J$-limits, then $({\mathfrak{C}},{\rm Set}_{*})$ has $J$-limits. For
${\mathfrak{C}}=FPCM$ and for discrete category $J$ with $\operatorname{{\rm
Ob}}(J)=\\{1,2\\}$, it follows from Proposition 2.1, the following
###### Proposition 3.2
Let $(M(E_{1},I_{1}),X_{1})$ and $(M(E_{2},I_{2}),X_{2})$ be state spaces.
Their product in $(FPCM,Set_{*})$ is a state space
$(M(E_{1},I_{1})\prod M(E_{2},I_{2}),X_{1}\circ\pi_{1}^{op}\times
X_{2}\circ\pi_{2}^{op})$
where $\pi_{i}:M(E_{1},I_{1})\prod M(E_{2},I_{2})\to M(E_{i},I_{i})$ are the
projections of the product in the category $FPCM$ for $i\in\\{1,2\\}$.
###### Definition 3.3
A morphism $(\eta,\sigma):(M(E,I),X)\to(M(E^{\prime},I^{\prime}),X^{\prime})$
of state spaces is independence preserving if $\eta:M(E,I)\to
M(E^{\prime},I^{\prime})$ is independence preserving.
Let $(FPCM^{\|},Set_{*})\subset(FPCM,Set_{*})$ be the subcategory of all state
spaces and independence preserving morphisms.
###### Proposition 3.4
The categories $(FPCM,Set_{*})$ and $(FPCM^{\|},Set_{*})$ are complete.
Proof. The category $FPCM$ is complete by Theorem 1.6 and $FPCM^{\|}$ is
complete by Theorem 1.14. Proposition 2.1 gives completeness of
$(FPCM,Set_{*})$ and $(FPCM^{\|},Set_{*})$.
###### Proposition 3.5
The categories $(FPCM,Set_{*})$ and $(FPCM^{\|},Set_{*})$ are cocomplete.
Proof. First statement follows from Theorem 1.8 and Proposition 2.2 applied to
${\mathfrak{C}}=FPCM$ and ${\mathcal{A}}=Set_{*}$. The second statement
follows from Theorem 1.15 and Proposition 2.2. $\Box$.
### 3.2 Category of weak asynchronous system and polygonal morphisms
###### Definition 3.6
The weak asynchronous system ${\mathcal{A}}=(S,s_{0},E,I,\operatorname{{\rm
Tran}})$ consist of a set $S$ which elements called states, an initial state
$s_{0}\in S_{*}$, a set $E$ of events, the irreflective symmetric relation
$I\subseteq E\times E$ of independence, satisfying the conditions
* •
If $(s,a,s^{\prime})\in\operatorname{{\rm Tran}}$ $\&$
$(s,a,s^{\prime\prime})\in\operatorname{{\rm Tran}}$, then
$s^{\prime}=s^{\prime\prime}$.
* •
If $(a,b)\in I~{}\&~{}(s,a,s^{\prime})\in\operatorname{{\rm
Tran}}~{}\&~{}(s^{\prime},b,s^{\prime\prime})\in\operatorname{{\rm Tran}}$,
then there exists $s_{1}\in S$ such that $(s,b,s_{1})\in\operatorname{{\rm
Tran}}$ $\&$ $(s_{1},a,s^{\prime\prime})\in\operatorname{{\rm Tran}}$.
If we add to Definition 3.6 the conditions $s_{0}\in S$ and $S\not=\emptyset$,
then we obtain asynchronous systems in the sense of M. Bednarczyk [1]. If more
than that, we require the condition $(\forall e\in E)(\exists e,e^{\prime}\in
S)~{}(s,e,s^{\prime})\in\operatorname{{\rm Tran}}$, then we get an
asynchronous transition system [19].
###### Lemma 3.7
Every weak asynchronous system $(S,s_{0},E,I,\operatorname{{\rm Tran}})$ gives
a state space $(M(E,I),S_{*})$ with a distinguished element $s_{0}\in S_{*}$
wherein the action is defined by
$(s,[e_{1}\cdots e_{n}])\mapsto(\ldots((s\cdot e_{1})\cdot e_{2})\ldots\cdot
e_{n}),$
for all $s\in S_{*}$ and $e_{1}$, …, $e_{n}\in E$. Here for $s\in S$, $e\in
E$, we let
$s\cdot e=\left\\{\begin{array}[]{cl}s^{\prime},&\mbox{ if
}(s,e,s^{\prime})\in\operatorname{{\rm Tran}};\\\ ,&\mbox{ if there is no
}s^{\prime}\mbox{ such that }(s,e,s^{\prime})\in\operatorname{{\rm
Tran}}.\end{array}\right.$
This correspondence is one-to-one. The inverse map takes any state space
$(M(E,I),S_{*})$ and $s_{0}\in S_{*}$ to an asyncronous system
$(S,s_{0},E,I,\operatorname{{\rm Tran}})$ where $\operatorname{{\rm
Tran}}=\\{(s,e,s\cdot e)~{}|~{}s\in S~{}\&~{}s\cdot e\in S\\}$.
In other words, the weak asynchronous system and hence the asynchronous
transition system can be viewed as the state space $(M(E,I),S_{*})$ with
distinguished $s_{0}\in S_{*}$.
###### Definition 3.8
A morphism of weak asynchronous systems
$(f,\sigma):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ consists of partial maps
$f:E{\rightharpoonup}E^{\prime}$ and $\sigma:S{\rightharpoonup}S^{\prime}$
satifying the following conditions
1. (i)
$\sigma(s_{0})=s^{\prime}_{0}$;
2. (ii)
for any triple $(s_{1},e,s_{2})\in\operatorname{{\rm Tran}}$, there is an
alternative
$\left\\{\begin{array}[]{cl}(\sigma(s_{1}),f(e),\sigma(s_{2}))\in\operatorname{{\rm
Tran}}^{\prime},&\mbox{ if }f(e)\mbox{ is defined},\\\
\sigma(s_{1})=\sigma(s_{2}),&\mbox{ if }f(e)\mbox{ is
undefined},\end{array}\right.$
3. (iii)
for each pair $(e_{1},e_{2})\in I$ such that $f(e_{1})$ and $f(e_{2})$ are
defined, the pair $(f(e_{1}),f(e_{2}))$ must belong to $I^{\prime}$.
If $s_{0}\not=*$, $s^{\prime}_{0}\not=*$ and $\sigma:S\to S^{\prime}$ is
defined on the whole $S$, then these conditions gives a morphism of
asynchronous systems in the sense of [1]. Following [1] denote by
${\mathcal{A}S}$ the category of asynchronous systems.
###### Definition 3.9
A morphism of weak asynchronous systems
$(f,\sigma):{\mathcal{A}}\to{\mathcal{A}}^{\prime}$ is polygonal if
$(f,\sigma)$ defines the independence preserving morphism of the corresponding
state spaces.
Denote by ${\mathcal{A}S^{\flat}}$ the category of asynchronous systems and
polygonal morphisms. We show that the category ${\mathcal{A}S}$ is not a
subcategory of ${\mathcal{A}S^{\flat}}$.
###### Proposition 3.10
A morphism $(\eta,\sigma):(S,s_{0},E,I,\operatorname{{\rm
Tran}})\to(S^{\prime},s^{\prime}_{0},E^{\prime},I^{\prime},\operatorname{{\rm
Tran}}^{\prime})$ in the category ${\mathcal{A}S}$ is polygonal if and only if
for any $s_{1}\in S$, $e\in E$, $s^{\prime}_{2}\in S^{\prime}$ the following
implication holds
$(\sigma(s_{1}),\eta(e),s^{\prime}_{2})\in\operatorname{{\rm
Tran}}^{\prime}~{}\Rightarrow~{}(\exists s_{2}\in
S)(s_{1},e,s_{2})\in\operatorname{{\rm Tran}}.$
Proof. If $(\eta,\sigma)$ is a polygonal morphism, then for any $s_{1}\in S$
and $e\in E$ such those $s_{1}\cdot e=*$, we have
$\sigma(s_{1})\cdot\eta(e)=\sigma(s_{1}\cdot e)=*$. It follows that a morphism
of asynchronous systems is polygonal if and only if for all $s_{1}\in S$ and
$e\in E$ the following implication holds $s_{1}\cdot
e=*\Rightarrow\sigma(s_{1})\cdot\eta(e)=*$. By the law of contraposition, we
obtain for all $s_{1}\in S$ and $e\in E$ that
$(\exists s^{\prime}_{2}\in
S^{\prime})(\sigma(s_{1}),\eta(e),s^{\prime}_{2})\in\operatorname{{\rm
Tran}}^{\prime}\Rightarrow(\exists s_{2}\in
S)(s_{1},e,s_{2})\in\operatorname{{\rm Tran}}.$ (3)
Taking out from the formula (3) the variable $s^{\prime}_{2}$ with the
quantifier, we get
$(\forall s^{\prime}_{2}\in
S^{\prime})\left((\sigma(s_{1}),\eta(e),s^{\prime}_{2})\in\operatorname{{\rm
Tran}}^{\prime}\Rightarrow(\exists s_{2}\in
S)(s_{1},e,s_{2})\in\operatorname{{\rm Tran}}\right).$
Adding to the formula the quantifiers $(\forall s_{1}\in S)(\forall e\in E)$,
we obtain the required assertion. $\Box$
Let ${\rm pt}_{*}=\\{p,*\\}$ be a state space with the monoid
$M(\emptyset,\emptyset)=\\{1\\}$. Associating with weak asynchronous system
the morphism of state spaces ${\rm pt}_{*}\to(M(E,I),S_{*})$ defined as
$p\mapsto s_{0}$, we obtain
###### Proposition 3.11
${\mathcal{A}S^{\flat}}$ is isomorphic to the comma category ${\rm
pt}_{*}/(FPCM^{\|},{\rm Set}_{*})$.
For any complete category ${\mathcal{A}}$ and object $A\in\operatorname{{\rm
Ob}}{\mathcal{A}}$, the comma-category $A/{\mathcal{A}}$ is complete. It
follows from 3.4 and 3.11 the following
###### Theorem 3.12
The category ${\mathcal{A}S^{\flat}}$ is complete.
The completeness of ${\mathcal{A}S}$ is shown in [1]. It follows from
Propositions 3.5 and 3.11 the following
###### Theorem 3.13
The category ${\mathcal{A}S^{\flat}}$ is cocomplete.
## 4 Conclusion
There are possible applications of the results related with building adjoint
functors between the category of ${\mathcal{A}S^{\flat}}$ and the category of
higher dimensional automata. Unlabeled semiregular higher dimensional
automation [10] is a contravariant functor from the category of cubes into the
category ${\rm Set}$. Let $\Upsilon_{sr}$ be a category of unlabeled
semiregular higher dimensional automata and natural transformations. By [8,
Proposition II.1.3] for each functor from the category of cubes to the
category $(FPCM^{||},{\rm Set})$, there exists a pair of adjoint functors
between the categories $\Upsilon_{sr}$ and $(FPCM^{||},{\rm Set})$. We can
take the functor assigning to $n$-dimensional cube the state space
$({\,\mathbb{N}}^{n},h_{{\,\mathbb{N}}^{n}})$ where
$h_{{\,\mathbb{N}}^{op}}:{\,\mathbb{N}}^{op}\to{\rm Set}$ is the contravariant
functor of morphisms. So, we get left adjoint to the composition
$(FPCM^{||},{\rm Set}_{*})\to(FPCM^{||},{\rm Set})\to\Upsilon_{sr}$. Taking
initial point, we obtain adjoint functors between ${\mathcal{A}S^{\flat}}$ and
the category of higher dimensional automata with the initial point.
Considering the comma categories, we can compare the labelled asynchronous
systems with labelled higher dimensional automata.
Event structures and Petri nets can be considered as asynchronous systems.
Therefore, applications of polygonal morphisms for the study of Petri nets and
event structures are possible.
## References
* [1] M. Bednarczyk, Categories of Asynchronous Systems, University of Sussex, Brighton, 1987. – 230p.
* [2] M. A. Bednarczyk, L. Bernardinello, B. Caillaud, W. Pawlowski, L. Pomello, “Modular System Development with Pullbacks”, Applications and Theory of Petri Nets 2003, Lecture Notes in Computer Science, 2679, Springer-Verlag, Berlin, 2003, 140–160
* [3] M. A. Bednarczyk, A. M. Borzyszkowski, R. Somla, “Finite Completeness of Categories of Petri Nets”, Fundamenta Informaticae, 43 (2000) 21 -48.
* [4] P. Cartier, D. Foata, Problèmes combinatories de commutation et réarrangements, Lecture Notes in Math., 85, Springer-Verlag, Berlin, 1969.
* [5] V. Diekert, Combinatorics on Traces, Lecture Notes in Computer Science, 454, Springer-Verlag, Berlin, 1990.
* [6] V. Diekert, Y. Métivier, Partial Commutation and Traces, Handbook of formal languages, 3, Springer-Verlag, New York, 1997, 457–533.
* [7] L. Fajstrup, E. Goubault, M. Raußen, “Detecting Deadlocks in Concurrent Systems”, Concur’98, Lecture Notes in Computer Science, 1466, Springer-Verlag, Berlin, 1998, 332–346
* [8] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory. Springer, Berlin (1967)
* [9] E. Goubault, “Labeled cubical sets and asynchronous transitions systems: an adjunction”, In Preliminary Proceedings CMCIM’02, 2002.
http://www.lix.polytechnique.fr/$\tilde{~{}}$goubault/papers/cmcim02.ps.gz
* [10] E. Goubault, The Geometry of Concurrency, Ph.D. Thesis, Ecole Normale Supérieure, 1995, 349 p.; http://www.dmi.ens.fr/$\widetilde{~}{}$goubault
* [11] E. Goubault and S. Mimram. “Formal relationships between geometrical and classical models for concurrency.” Electronic Notes in Theoretical Computer Science 283 (2012): 77-109.
* [12] A. A. Husainov, “On the homology of small categories and asynchronous transition systems”, Homology Homotopy Appl., 6:1 (2004), 439–471. http://www.rmi.acnet.ge/hha
* [13] A. A. Husainov, The cubical homology of trace monoids, Far Eastern Math. Journal 12:1 (2012) 108–122
http://mi.mathnet.ru/eng/dvmg/v12/i1/p108
* [14] A. A. Khusainov, “Homology groups of asynchronous systems, Petri nets, and trace languages.” Sibirskie Élektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] 9 (2012): 13-44.
* [15] S. Mac Lane, Categories for the Working Mathematician. Graduate texts in mathematics, vol. 5. Springer, New York (1998)
* [16] A. Mazurkiewicz, “Basic notions of trace theory”, Linear time, branching time and partial order in logics and models for concurrency, Lecture Notes in Computer Science, 354, Springer-Verlag, Berlin, 1989, 285–363
* [17] R. Milner, Communication and concurrency, International Series in Computer Science. Prentice Hall, New York, 1989.
* [18] M. Nielsen, “Models for concurrency”, Mathematical Foundations in Computer Science 1991, Lecture Notes in Computer Science, 520, Springer-Verlag, Berlin, 1991, 43–46
* [19] G. Winskel and M. Nielsen, Models for Concurrency, Handbook of Logic in Computer Science, Vol. IV, ed. Abramsky, Gabbay and Maibaum. Oxford University Press, 1995\. P.1–148.
|
arxiv-papers
| 2013-08-06T23:19:59 |
2024-09-04T02:49:49.145699
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmet A. Husainov",
"submitter": "Ahmet Husainov A.",
"url": "https://arxiv.org/abs/1308.1443"
}
|
1308.1494
|
# Tuning exciton and biexciton transition energies and fine structure
splitting through hydrostatic pressure in single InGaAs quantum dots
Xuefei Wu State Key Laboratory for Superlattices and Microstructures,
Institute of Semiconductors, Chinese Academy of Sciences, Beijing, 100083,
People’s Republic of China Hai Wei Key Laboratory of Quantum Information,
University of Science and Technology of China, Hefei, 230026, People’s
Republic of China Xiuming Dou State Key Laboratory for Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,
Beijing, 100083, People’s Republic of China Kun Ding State Key Laboratory
for Superlattices and Microstructures, Institute of Semiconductors, Chinese
Academy of Sciences, Beijing, 100083, People’s Republic of China Ying Yu
State Key Laboratory for Superlattices and Microstructures, Institute of
Semiconductors, Chinese Academy of Sciences, Beijing, 100083, People’s
Republic of China Haiqiao Ni State Key Laboratory for Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,
Beijing, 100083, People’s Republic of China Zhichuan Niu State Key
Laboratory for Superlattices and Microstructures, Institute of Semiconductors,
Chinese Academy of Sciences, Beijing, 100083, People’s Republic of China Yang
Ji State Key Laboratory for Superlattices and Microstructures, Institute of
Semiconductors, Chinese Academy of Sciences, Beijing, 100083, People’s
Republic of China Shushen Li State Key Laboratory for Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,
Beijing, 100083, People’s Republic of China Desheng Jiang State Key
Laboratory for Superlattices and Microstructures, Institute of Semiconductors,
Chinese Academy of Sciences, Beijing, 100083, People’s Republic of China
Guang-can Guo Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei, 230026, People’s Republic of China Lixin He
[email protected] Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei, 230026, People’s Republic of China Baoquan
Sun [email protected] State Key Laboratory for Superlattices and
Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,
Beijing, 100083, People’s Republic of China
###### Abstract
We demonstrate that the exciton and biexciton emission energies as well as
exciton fine structure splitting (FSS) in single (In,Ga)As/GaAs quantum dots
(QDs) can be efficiently tuned using hydrostatic pressure in situ in an
optical cryostat at up to 4.4 GPa. The maximum exciton emission energy shift
was up to 380 meV, and the FSS was up to 180 $\mu$eV. We successfully produced
a biexciton antibinding-binding transition in QDs, which is the key
experimental condition that generates color- and polarization-
indistinguishable photon pairs from the cascade of biexciton emissions and
that generates entangled photons via a time-reordering scheme. We perform
atomistic pseudopotential calculations on realistic (In,Ga)As/GaAs QDs to
understand the physical mechanism underlying the hydrostatic pressure-induced
effects.
###### pacs:
78.67.Hc, 07.35.+k, 78.55.Cr, 42.50.-p
Self-assembled semiconductor quantum dots (QDs) have considerable potential
for use as fundamental building blocks in future quantum information
applications. However, so far it is impossible to use QD growth techniques for
precisely controlling QD properties, which is essential for such applications.
Therefore, externally tuning the QD properties post-growth is extremely
important. One the most prominent examples is polarization-entangled photon
pair emission through a biexciton (XX) cascade process in QDs, which requires
that the different polarized photons are energetically indistinguishable.
However, the underlying asymmetry for self-assembled (In,Ga)As/GaAs QDs leads
to splitting in degenerate bright exciton (X) states (fine structure
splitting, FSS), which is typically tens of $\mu$eV Gammon et al. (1996a);
Bayer et al. (2002); Gammon et al. (1996b); Bester et al. (2003), and much
larger than the radiative linewidth ($\sim$ 1.0 $\mu$eV); therefore, photon
entanglement is destroyed Stevenson et al. (2006); Hafenbrak et al. (2007).
Tuning techniques, such as electric Bennett et al. (2010); Gerardot et al.
(2007); Vogel et al. (2007), magnetic Hudson et al. (2007), and strain fields
Trotta et al. (2012); Ding et al. (2010); Jöns et al. (2011); Seidl et al.
(2006); Dou et al. (2008); Gong et al. (2011); Wang et al. (2012) used to
erase the FSS have been explored. Uniaxial and biaxial stresses have been used
to tune the QD structural symmetry, exciton binding energies and FSS. However,
the strain that can be generated using such techniques is limited to
approximately tens of MPa, which corresponds to a spectral shift by only
several meVs for QD peak emissions Ding et al. (2010); Jöns et al. (2011).
Herein, we report high-pressure research (up to tens of GPa) for individual
QDs using the diamond anvil cell (DAC), which has been widely used to study
metal-semiconductor transitions, electronic structures, and optical
transitions in bulk crystals and microstructures Jayaraman (1983); Ma et al.
(2004); Itskevich et al. (1998); Li et al. (1994). An exciton emission line
shift in ensemble InAs/GaAs QDs is approximately 500 meV at 8 GPa Ma et al.
(2004), which is much larger than the QD peak shift induced by a piezoelectric
actuator PMN-PT Ding et al. (2010). However, tuning the QD structural
symmetry, exciton transition states and FSS for an individual QD using the DAC
has not been reported.
In this work, we demonstrate that the exciton (X) emission energy, FSS and
biexciton (XX) binding energy can be successively tuned for extremely large
ranges using hydrostatic pressure at up to 4.4 GPa. The emission energy, FSS
and XX binding energy almost increase linearly with increased pressure. The
maximum exciton emission energy shift and FSS change can extend to 380 meV and
180 $\mu$eV, respectively, which is considerably greater than through other
techniques. By tuning the applied pressure, color-indistinguishable photons
from the biexciton and exciton emission decay through a cascade, and across
generation color coincidence for biexciton and exciton transitions are
generated. Therefore, entangled photon pairs are generated via the proposed
“time reordering” scheme Avron et al. (2008). We also perform atomistic
pseudopotential calculations on realistic (In,Ga)As/GaAs QDs under hydrostatic
stress to discern the physical mechanisms underlying the effects induced by
the hydrostatic pressure.
The investigated (In,Ga)As/GaAs QD samples with low QD density were grown
using molecular beam epitaxy (MBE) on a semi-insulating GaAs substrate with
excitonic emission energies at 1.35-1.43 eV. Figure 1(a) shows the DAC
pressure device used for tuning QD photoluminescence (PL) in situ using the
optical cryostat. To fit the QD samples into the DAC chamber [indicated in
Fig. 1(b)], the samples were mechanically thinned to a total thickness of
approximately 20 $\mu$m and then cut into pieces approximately 100$\times$100
$\mu$m2. Condensed argon was used as the pressure-transmitting medium in the
DAC, which can be used to apply hydrostatic pressure up to 9 GPa Jayaraman
(1983); Shimizu et al. (2001). The initial pressure can be adjusted at room
temperature by driving screws and can be determined in situ using the ruby R1
fluorescence line shift. To successively tune the X and XX transition energies
and fine structure splitting (FSS) by pressure at a low temperature, a novel
and easily controlled version of the DAC shown in Fig. 1(a) was developed by
combining the well-known DAC with a piezo actuator. This device can
successively generate pressures up to several GPa for the QD samples studied
at a low temperature using PL measurements, and the maximum applied pressure
depends on the actuator stroke length. The QD sample in the DAC is cooled to
20 K through a continuous-flow liquid helium cryostat and excited by a He-Ne
laser at the wavelength 632.8 nm. The excitation laser was focused to a $\sim$
2 $\mu$m spot on the sample using a microscope objective (NA: 0.35). The PL
was collected using the same objective, spectrally filtered through a 0.5 m
monochromator, and detected using a silicon-charge coupled device (CCD). A
$\lambda$/2 wave plate and linear polarizer were used to distinguish
horizontal (H) and vertical (V) linear polarization for PL components. By
carefully following the changing X and XX PL energies using the polarization
angle, we measure FSS with an $\sim$ 10 $\mu$eV accuracy by fitting the
experimental data to a sinusoidal function Ghali et al. (2012). Figure 1(c)
displays the measured pressure values and excitonic emission energies at 20 K
as a function of actuator voltage (Piezo-ceramics: PSt 150/10$\times$10/40).
We clearly show that pressure can be successively tuned in situ using an
optical cryostat from 0.5 to 4.4 GPa through a piezo actuator, and the
corresponding blue shift for the excitonic PL peak energy is $\sim$ 310 meV.
Figure 2(a) depicts the exciton emission energies as a function of the
hydrostatic pressure from 0 to 4.4 GPa for QD1-QD5. At 0 GPa, the QD exciton
emission energies are 1.401, 1.349, 1.406, 1.432 and 1.394 eV, respectively.
The exciton emission energies for the five QDs studied herein increased
linearly with the applied pressure. The blue shift for the QD1 peak energy at
4.22 GPa is approximately 330 meV, which is much larger than previously
reported shifts ( $\sim$ 10 meV) from uniaxial or biaxial stresses using
conventional methods Ding et al. (2010); Dou et al. (2008); Jöns et al.
(2011); Seidl et al. (2006); Trotta et al. (2012). The experimental data were
linearly fit, which generated pressure coefficients for QD1-QD5 of 82, 87, 93,
81 and 85 meV/GPa, respectively; such values are consistent with the reported
pressure coefficients for ensemble quantum dot Ma et al. (2004).
Figure 2(b) shows the biexciton binding energies for QD1-QD5 as a function of
hydrostatic pressure at up to 4.4 GPa. The biexciton binding energy is defined
as $E_{B}(XX)$=$E_{X}$-$E_{XX}$, where $E_{X}$ and $E_{XX}$ are the X and XX
emission energies, respectively. When $E_{B}(XX)>0$, the biexciton is in the
“binding” state, wherein the two excitons are attracted. When $E_{B}(XX)<0$,
the biexciton is in the “antibinding” state, wherein the two excitons are
repulsive. For the QDs studied herein, $E_{B}(XX)$ increases as a function of
hydrostatic pressure up to 4.4 GPa. For QD1 and QD3, the biexcitons are in an
antibinding state at zero pressure and gradually progress to the binding state
at approximately 1 and 2 GPa, respectively (i.e., $E_{B}(XX)$=0), where the
exciton and biexciton are “color-indistinguishable”.
To demonstrate the biexciton antibinding-binding transitions under pressure in
greater detail, we plotted the polarization-resolved PL spectra for the QD1 X
and XX emission lines under different hydrostatic pressures, as shown in Fig.
3(a)-(e), wherein the red and black lines correspond to the horizontal (H) and
vertical (V) polarized photons, respectively. At zero pressure, both XX
emission energies, $E(H2)$ and $E(V2)$, are higher than the X emission energy,
$E(H1)$ and $E(V1)$ [see also the scheme in Fig. 3(f)]. In addition, $E(H2)$
is slightly larger than $E(V2)$ at FSS $\sim$ 50 $\mu$eV. Under pressure, the
blue shift for the X emission energy (82 meV/GPa) is more rapid than for XX
(81 meV/GPa). Therefore, with increasing pressure, the V-polarized XX and X
emission lines first degenerate at 1.62 GPa, as shown in Fig. 3(b), and then
the H-polarized emission lines degenerate at 2.07 GPa, as shown in Fig. 3(d).
In such instances, color-indistinguishable photon pairs are generated by an
XX-X cascade emission at 1.62 GPa for V-polarized photons or at 2.07 GPa for
H-polarized photons. Therefore, it is expected that the indistinguishable two-
photon streams will be produced by adjusting a time delay between the XX and X
emissions, wherein the time delay is approximately 0.4 ns Chang et al. (2009).
Remarkably, at 1.97 GPa, across generation color coincidence for XX and X
transition energies was generated (i.e., $E(H1)$=$E(V2)$ and $E(V1)$=$E(H2)$).
This is a key condition for entangled photon generation via the proposed time
reordering scheme Avron et al. (2008). When pressure was further increased,
the separation between the XX and X emission lines again increased, as shown
in Fig. 3(e) at 3.66 GPa. Ding and coworkers demonstrated that biaxial strain
can also tune the biexciton binding energies Ding et al. (2010). However,
because their experiment generated a relatively small strain, biexciton
antibinding-binding progression was not observed.
FSS tuning by uniaxial strain has been studied experimentally Seidl et al.
(2006); Trotta et al. (2012); Kuklewicz et al. (2012) and theoreticallySingh
and Bester (2010); Gong et al. (2011); Wang et al. (2012), which has shown
that the maximum tuned FSS value is approximately 20 $\mu$eV. It is
interesting to measure the FSS change under hydrostatic pressure. Figure 2(c)
depicts the FSS for QD4 and QD5 as a function of pressure at 20 K in the range
0.5 to 4.4 GPa. The figure clearly demonstrates that increasing pressure
produces an approximately linear increase in FSS with the slope 44 and 28
$\mu$eV/GPa for QD4 and QD5, respectively, which generates a total FSS shift
as large as $\sim$ 180 and 100 $\mu$eV for QD4 and QD5, respectively. Similar
results were observed from other investigated (In,Ga)As/GaAs QDs, which
indicates that such a large shift is typical for FSS under hydrostatic
pressure.
To understand the experimental results, we calculated the electronic and
optical properties for the In1-xGaxAs/GaAs QDs under hydrostatic pressure
using an atomistic empirical pseudopotential method (EPM) Williamson et al.
(2000). The optimized QD structures are obtained by the valence force filed
method Keating (1966). We then calculate the electron/hole single-particle
energies and wave functions using the linear combination of bulk bands (LCBB)
method Wang and Zunger (1999). The exciton and biexciton energies are
calculated via the configuration interaction (CI) method Franceschetti et al.
(1999). Herein, we present results for three QDs: (i) a lens-shaped InAs/GaAs
QDs with the height $h$=1.5 nm and base diameter $b$=12 nm; (ii) a lens-shaped
In0.8Ga0.2As/GaAs QDs with $h$=1.5 nm and $b$=12, 15 nm; and (iii)
In0.8Ga0.2As/GaAs QDs with $h$=2.5 and the elliptical major (minor) axis
$a$=10 nm ($b$=7.5 nm) along the [1$\mathrm{bar}{1}$0] ([110]) crystal
direction.
The calculated exciton emission energies under pressure are shown in Fig. 4(a)
and produce blue energy shifts at approximately 76 meV/GPa, which is
consistent with the experimental values. To understand the emission energy
blue shift, we analyzed the band offsets and confinement potentials for the
QDs under pressure, which strongly depend on the strain distribution in the
dots and matrix. When hydrostatic pressure is applied, the lattice constant
for the matrix material GaAs decreases, which effectively increases the
lattice mismatch between the dot material InAs and GaAs matrix. As a result,
both the (absolute values of) isotropic and biaxial strain inside the dots
increase. The averaged isotropic strain $I$=-0.072-0.011 $P$ and biaxial
strain $\epsilon_{zz}-\epsilon_{xx}$= 0.12+0.0014 $P$, where $P$ is the
applied hydrostatic pressure in GPa. Figure 4(b) depicts the strain-modified
band offsets for the conduction band (e), heavy hole (HH), light hole (LH) and
spin-orbit (SO) bands through the dot center under P =0, 2 and 4 GPa. Whereas
the band offset change is small for holes, the band offset changes
dramatically for the conduction band. Under pressure, the electron bands move
significantly toward the higher energy. The confinement potential also
increased dramatically with increasing pressure, which is the major reason for
the observed experimental results.
Because the electron-hole Coulomb energy change is relatively small [see Fig.
4(d)], the change in exciton emission energy can be estimated using the
electron-hole single-particle gap $E_{g}$, which can be written as follows:
$E_{g}(\tensor{\epsilon})=E_{g}(0)+a_{g}I+b_{v}(\epsilon_{zz}-\epsilon_{xx})\,,$
(1)
where $\tensor{\epsilon}$ is the strain tensor inside the InAs dots,
$a_{g}$=$-$6.08 eV is the hydrostatic deformation potential for the band gap,
and $b_{v}$=$-$1.8 eV is the biaxial deformation potential for the valence
band maximum. Therefore, we estimated that the exciton PL blue shift under
hydrostatic pressure is 82 meV/GPa for pure InAs/GaAs QDs, which is consistent
with the experimental values and EPM calculations. We note that the
hydrostatic pressure is much more efficient at tuning the exciton emission
energy than uniaxial stress ($\sim$10 $\mu$eV/MPa) Jöns et al. (2011);
Kuklewicz et al. (2012); Wang et al. (unpublished).
The calculated XX binding energies $E_{B}(XX)$ are presented in Fig. 4(c). We
found that the biexciton tends toward antibinding in small QDs under zero
pressure. When the pressure increases, the $E_{B}(XX)$ for the dots calculated
increased. The binding energy tends to be saturated at very high pressure. The
XX binding energy for the In0.8Ga0.2As/GaAs QDs with $b$=12 nm and $h$=1.5 nm
is consistent with the experimental QD1.
In the calculation, we found that it is important to include many
electron/hole energy levels for the correct XX binding energies using the CI
calculations, and the XX binding energy change as a function of pressure can
be observed only using the lowest energy conduction and valence bands (i.e.,
Hartree-Fock approximation). Such observations suggest that the XX binding
energies did not change due to the correlated energies; primarily, such
changes are due to changes in the direct Coulomb integrals between the lowest
electron and hole states, as follows:
$\Delta E_{B}(XX)\approx 2\Delta J_{eh}-\Delta J_{ee}-\Delta J_{hh}\,,$ (2)
where $J_{ee}$, $J_{hh}$ and $J_{eh}$ are the direct electron-electron, hole-
hole and electron-hole Coulomb integrals, respectively. As shown in Fig. 4(d),
whereas $J_{ee}$, and $J_{eh}$ increase rapidly with pressure, $J_{hh}$ is
approximately flat. The solid purple line describes the changing exciton
binding energy calculated using Eq. (2), which is consistent with the dashed
purple line from the EPM calculations. To understand how the Coulomb integrals
change under pressure, we compared the lowest electron and hole wave functions
in Fig. 4(f) at 0.0 GPa and 4.0 GPa for the 12$\times$1.5 nm In0.8Ga0.2As/GaAs
QDs. We found that, whereas the hole wave function shape primarily does not
change, the electron becomes much more localized due to the band offset
changes shown in Fig. 4(b), which explains the Coulomb integral changes under
pressure.
Finally, we examined FSS under hydrostatic pressure. The FSS calculated as a
function of pressure is shown in Fig. 4(e), which increases dramatically with
applied pressure and is consistent with the experimental data in Fig. 2(c). It
is surprising that FSS changes under hydrostatic pressure, which does not
change the QD symmetry. However, because the electron wave functions are more
localized under pressure, an electron-hole would have a larger effective
overlap under pressure, which increases the exchange energies (e.g., the dark-
bright splitting $\Delta_{bd}$, which is also shown in Fig. 4(e)). It has been
shown that FSS can be roughly estimated as $\sim 2\eta\Delta_{bd}$ Wang et al.
(unpublished), where $\eta$ is the HH-LH mixing parameter; therefore, as
$\Delta_{bd}$ increases, FSS increases, as clearly demonstrated in Fig. 4(e).
To summarize, we experimentally and theoretically investigated the effects of
hydrostatic pressure on the exciton and biexciton transition energies as well
as FSS in single InGaAs QDs. The excitonic emission energies and FSS can be
tuned in situ by applying hydrostatic pressure in an optical cryostat for
changes over a wide energy range. The observed exciton emission energy blue
shift and FSS change were as large as $\sim$ 380 meV and $\sim$ 180 $\mu$eV,
respectively, which is greater than the values from other strain-adjusting
techniques. Tuning the QD optical properties over such a larger spectral range
yields great advantages for future QD applications, such as for generating
color-indistinguishable photon pairs from the biexciton and exciton emission
decay cascades or generating entangled photon pairs via a time-reordering
scheme. Furthermore, it expected that photon antibunching for optical
communication band QD emission can be measured using the pressure-induced blue
shift into the spectral range detected by sufficiently powerful silicon
avalanche photodiodes.
BS and LH acknowledge support from the Chinese National Fundamental Research
Program (Grant Nos. 2013CB922304, 2013CB933304, 2011CB921200, 2009CB929301),
Chinese National Natural Science Funds (Grant No. 90921015), and National
Natural Science Funds for Distinguished Young Scholars.
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Figure 1: (Color online) (a) Schematic drawing of the successively applied
pressure device of the diamond anvil cell (DAC), where the DAC and piezo
actuator are assembled together via home-made copper cylinder. (b) The DAC
chamber, showing the positions of the QD sample and ruby in the DAC. (c)
Measured pressure in the DAC chamber (solid black circles) and the
corresponding QD excitonic PL peak energy (solid green circles) as a function
of actuator voltage. At zero voltage, an initial pressure of 0.5 GPa was
generated by four driven screws. Figure 2: (Color online) (a) Exciton
emission energies as a function of pressure for QD1-QD5. (b) Biexciton binding
energies as a function of pressure for QD1-QD5, showing biexciton antibinding-
binding transitions under pressure for QD1 and QD3. (c) FSS of QD4 and QD5 as
a function of pressure.
Figure 3: (Color online) (a)-(e) Polarization-resolved PL spectra for the QD1
X and XX emission lines under different pressures, where red lines (black
lines) correspond to the horizontal (H) and vertical (V) polarized photons,
respectively. At 1.62 (b) and 2.07 (d) GPa, color-indistinguishable photon
pairs are generated by an XX-X cascade emissions for V- and H- polarized
photons, respectively. At 1.97 (c) GPa, across generation color coincidence
for XX and X transition energies is achieved. (f) Level schemes showing the
XX-X cascade at 0 GPa. Figure 4: (Color online) The calculated results of:
(a) The exciton emission energies of (In,Ga)As/GaAs QDs as a function of
hydrostatic pressure. (b) The band offsets of the InAs/GaAs dot under the 0.0,
2.0 and 4.0 GPa hydrostatic pressure. The black, green, blue and red lines
indicate conduction (e), heavy-hole (HH), light-hole (LH) and spin-orbit (SO)
bands, respectively. (c) The biexciton binding energies as a function of
hydrostatic pressure in (In,Ga)As/GaAs QDs. (d) The changes of direct Coulomb
integrals of the lowest electron and hole states in the 12$\times$1.5 nm
In0.8Ga0.2As/GaAs QD as a function of the applied pressure. (e) The FSS in the
10$\times$7.5$\times$2.5 nm In0.8Ga0.2As/GaAs QD as a function of the applied
pressure. (f) The wave functions of the lowest electron (e0) and hole (h0)
states under 0.0 and 4.0 GPa hydrostatic pressure in the 12$\times$1.5 nm
In0.8Ga0.2As/GaAs QD.
|
arxiv-papers
| 2013-08-07T07:46:03 |
2024-09-04T02:49:49.155496
|
{
"license": "Public Domain",
"authors": "Xuefei Wu, Hai Wei, Xiuming Dou, Kun Ding, Ying Yu, Haiqiao Ni,\n Zhichuan Niu, Yang Ji, Shushen Li, Desheng Jiang, Guangcan Guo, Lixin He and\n Baoquan Sun",
"submitter": "Xuefei Wu",
"url": "https://arxiv.org/abs/1308.1494"
}
|
1308.1556
|
# On the Independent Set and Common Subgraph Problems in Random Graphs
Yinglei Song
School of Computer Science and Engineering, Jiangsu University of Science and
Technology
Zhenjiang, Jiangsu 212003, China
[email protected]
###### Abstract
In this paper, we develop efficient exact and approximate algorithms for
computing a maximum independent set in random graphs. In a random graph $G$,
each pair of vertices are joined by an edge with a probability $p$, where $p$
is a constant between $0$ and $1$. We show that, a maximum independent set in
a random graph that contains $n$ vertices can be computed in expected
computation time $2^{O(\log_{2}^{2}{n})}$. Using techniques based on
enumeration, we develop an algorithm that can find a largest common subgraph
in two random graphs in $n$ and $m$ vertices ($m\leq n$) in expected
computation time $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$. In
addition, we show that, with high probability, the parameterized independent
set problem is fixed parameter tractable in random graphs and the maximum
independent set in a random graph in $n$ vertices can be approximated within a
ratio of $\frac{2n}{2^{\sqrt{\log_{2}{n}}}}$ in expected polynomial time.
## 1 Introduction
In computer science, many optimization problems can be reduced to the
optimization of objectives that are formulated and described in a graph. The
development of efficient exact or approximate algorithms for graph
optimization problems thus constitute an important part of the research in
combinatorial optimization. However, a large number of graph optimization
problems have been shown to be NP-hard [18], which suggests that it is
unlikely to develop algorithms that can solve these problems in polynomial
time. A well known example is the Maximum Independent Set problem. Given a
graph $G=(V,E)$, a vertex set $I\subseteq V$ is an independent set if there is
no edge between any pair of two vertices in $I$. The goal of the Maximum
Independent Set problem is to find an independent set of the largest size in a
given graph $G$. The problem can be trivially solved in time $2^{O(n)}$ by
enumerating and checking all possible vertex subsets in the graph. Although
intensive research has been performed to improve the computation time needed
to find an optimal solution [2, 6, 14, 17, 36, 23, 27, 30, 31, 32, 33, 34, 37,
38], an algorithm that needs subexponential time is not yet available for this
problem. Recently, it is proposed that this problem is unlikely to be solved
in subexponential time [7, 8].
Due to the difficulty of developing efficient algorithms that can find optimal
solutions for these problems, a large number of algorithms have been developed
to generate approximate solutions that are close to optimal ones in polynomial
time [24]. Solutions provided by these algorithms are often guaranteed to be
within a ratio of the optimal solution and thus can be useful in practice. For
example, the Minimum Vertex Cover problem can be approximated by a simple
polynomial time algorithm within a ratio of $2.0$. However, it has been shown
that it is NP-hard to approximate this problem within a ratio of $1.362$ [9].
A well known inapproximability result regarding the Maximum Independent Set
problem is that it is NP-hard to approximate the maximum independent set in a
graph within a ratio of $n^{1-\epsilon}$, where $0<\epsilon<1$ is a constant
and $n$ is the number of vertices in the graph [21]. This result suggests that
an approximate solution with a guaranteed constant approximate ratio cannot be
obtained in polynomial time for the Maximum Independent Set problem unless
NP=P. So far, the best known approximation ratio that has been achieved for
this problem in general graphs is
$O(\frac{n\log_{2}^{2}{\log_{2}{n}}}{\log_{2}^{3}{n}})$ [15].
For those problems that cannot be even approximated within a good
approximation ratio in polynomial time, such as the Maximum Independent Set
problem, heuristics that can efficiently generate approximate solutions are
often employed in practice to solve them [3, 26, 20]. However, solutions
generated by heuristics are not guaranteed to be close to the optimal ones and
their applications are thus restricted to scenarios where the accuracy of
solutions is not a crucial issue.
Parameterized computation provides another potentially practical solution for
some problems that are computationally intractable. In particular, one or a
few parameters in some intractable problems can be identified and
parameterized computation studies whether efficient algorithms exist for these
problems while all parameters are small. A parameterized problem may contain a
few parameters $k_{1},k_{2},\cdots,k_{l}$ and the problem is fixed parameter
tractable if it can be solved in time $O(f(k_{1},k_{2},\cdots,k_{l})n^{c})$,
where $f$ is a function of $k_{1},k_{2},\cdots,k_{l}$, $n$ is the size of the
problem and $c$ is a constant independent of all parameters. For example, the
Vertex Cover problem is to determine whether a graph $G=(V,E)$ contains a
vertex cover of size at most $k$ or not. The problem is NP-complete. However,
a simple parameterized algorithm can solve the problem in time $O(2^{k}|V|)$
[11]. In practice, this algorithm can be used to efficiently solve the Vertex
Cover problem when the parameter $k$ is fixed and small. On the other hand,
some problems do not have known efficient parameterized solutions and are
therefore parameterized intractable. Similar to the conventional complexity
theory, a hierarchy of complexity classes has been constructed to describe the
parameterized complexity of these problems [11]. For example, the Independent
Set problem is to decide whether a graph contains an independent set of size
$k$ or not and has been shown to be W[1]-complete [12]. It cannot be solved
with an efficient parameterized algorithm unless all problems in W[1] are
fixed parameter tractable. A thorough investigation on these parameterized
complexity classes are provided in [10].
In this paper, we develop exact and approximate algorithms for the Maximum
Independent Set problem where the underlying graph is a random graph generated
based on the Erdős Rényi model [13]. Such a random graph is generated by
treating each pair of vertices independently and adding an edge to join them
with a probability of $p$ ($0<p<1$), where $p$ is a constant. Recent research
in molecular biology has shown that the protein side chain interaction network
conforms remarkably well to random graphs generated by the Erdős Rényi model
[5]. Therefore, efficient algorithms for some NP-hard problems in random
graphs, if exist, may significantly improve the computational efficiency for
some important optimization problems related to protein structure prediction.
In [19, 25], it has been shown that with high probability, the maximum
independent set in a random graph is of size $O(\log_{2}{n})$. However, this
result does not directly lead to an algorithm that can compute the maximum
independent set in a random graph in expected subexponential time. In [16], a
polynomial time algorithm that can compute a maximum independent set in a
sparse random graph with high probability is developed. However, the algorithm
is based on a large independent set that is embedded in the graph and thus
cannot be used for all graphs. We show that the maximum independent set in a
random graph can be computed in expected computation time
$2^{O(\log_{2}^{2}{n})}$, where $n$ is the number of vertices in the graph.
This result significantly improves the best known time complexity
$O(2^{\frac{n}{4}})$ for finding a maximum independent set in general graphs
[34].
Using techniques based on enumeration, we develop an algorithm that can
compute a largest common subgraph of two random graphs of $n$ and $m$ vertices
($n\geq m$) in expected computation time
$2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$. This result significantly
improves on the best known time complexity $2^{O(m\log_{2}{n})}$ for this
problem when $m=O(n)$. In addition, we show that, with high probability, the
parameterized independent set problem is fixed parameter tractable in random
graphs. For approximate algorithms, we develop an algorithm that can achieve
an approximation ratio of $\frac{2n}{2^{\sqrt{\log_{2}{n}}}}$ in expected
polynomial time, which is a significant improvement compared with the best
known approximate ratio that can be achieved in general graphs [1, 35].
## 2 Maximum Independent Set in Random Graphs
A random graph $G(V,p)$, where $0<p<1$, is a graph obtained by independently
adding edges between each pair of vertices in $V$ with a probability $p$.
Given a vertex $v\in V$, the degree of $v$ in $G$ is the number of vertices
that are connected to $v$ by an edge in G. We use $deg_{G}(v)$ to denote the
degree of vertex $v$ in graph $G$ and $N_{G}(v)$ to denote the set of vertices
that are connected to $v$ by an edge in $G$. A vertex subset $I\subseteq V$ is
an independent set in $G$ if there is no edge between any pair of vertices in
$I$. The goal of the Maximum Independent Set problem is to find an independent
set of the largest size in a given graph.
In [19, 25], it is shown that, with high probability, the size of a maximum
independent set in a random graph $G(V,p)$ is
$\frac{2\log_{2}{n}}{\log_{2}{\frac{1}{1-p}}}$, where $n$ is the number of
vertices in $G$. A straightforward algorithm by exhaustively enumerating all
vertex subsets of size $\frac{2\log_{2}{n}}{\log_{2}{\frac{1}{1-p}}}$ can thus
compute a maximum independent set in most random graphs in time
$n^{O(\log_{2}{n})}$. However, to compute a maximum independent set in all
random graphs, the algorithm must be able to cope with the cases where the
graph contains an independent set of size larger than $O(\log_{2}{n})$. The
algorithm needs time $2^{O(n)}$ to compute a maximum independent set in these
cases. The best known upper bound of the probability for a random graph to
have a maximum independent set larger than $O(\log_{2}{n})$ is
$\frac{1}{n^{O(1)}}$ [19, 25], the expected time complexity of this
enumeration based algorithm is thus $2^{O(n)}$.
We show that the maximum independent set in a random graph $G=(V,p)$ can be
computed in expected subexponential time.
###### Lemma 2.1
Given a random graph $G=(V,p)$ where $n=|V|$ and a sufficiently small constant
$\epsilon$ such that $\epsilon<p$, there exists a vertex $v\in V$ such that
$deg_{G}(v)\geq(p-\epsilon)n$ with probability at least $1-2^{-\mu n^{2}}$,
where $\mu$ is a positive constant that only depends on $\epsilon$ and $p$.
Proof. If such a vertex does not exist, the number of edges $n(E)$ in $G$ is
at most $\frac{(p-\epsilon)n^{2}}{2}$ since the degree of each vertex is at
most $(p-\epsilon)n$. However, from the construction of graph $G$, the
expected number of edges in $G$ can be obtained as follows
$E(n(E))=\frac{pn(n-1)}{2}$ (1)
From Chernoff bound, we can bound the probability for
$n(E)<\frac{(p-\epsilon)n^{2}}{2}$ by
$Pr(n(E)<\frac{(p-\epsilon)n^{2}}{2})<\exp{(-\frac{pn(n-1)\delta^{2}}{4})}$
(2)
where $\delta=\frac{n\epsilon-p}{p(n-1)}$. For sufficiently large $n$, we have
$\displaystyle\delta$ $\displaystyle>$ $\displaystyle\frac{\epsilon}{2p}$ (3)
$\displaystyle n-1$ $\displaystyle>$ $\displaystyle\frac{n}{2}.$ (4)
We can thus immediately obtain
$\displaystyle Pr(n(E)$ $\displaystyle<$
$\displaystyle\frac{(p-\epsilon)n^{2}}{2})$ (5) $\displaystyle<$
$\displaystyle\exp{(-\frac{\epsilon^{2}n^{2}}{32p})}$ (6) $\displaystyle=$
$\displaystyle 2^{-\frac{\epsilon^{2}n^{2}}{32p\ln{2}}}.$ (7)
We then let $\mu=\frac{\epsilon^{2}}{32p\ln{2}}$ and we conclude that with
probability at least $1-2^{-\mu n^{2}}$, there exists vertex $v\in V$ such
that $deg_{G}(v)\geq(p-\epsilon)n$.
The proof of Lemma 2.1 relies on the fact that $p$ is a constant independent
of $n$, the Lemma does not hold if the value of $p$ depends on $n$. A random
graph $G=(V,p)$ in $n$ vertices is good if it contains at least one vertex
whose degree is at least $(p-\epsilon)n$. Given a random graph, the algorithm
starts by finding a vertex $v$ such that $deg_{G}(v)$ is at least
$(p-\epsilon)n$. If such a vertex does not exist, the algorithm enumerates all
subsets of $V$ and returns an independent set of the largest size. If $v$
exists, the algorithm branches on two possible cases on whether $v$ is
contained in $I$ or not. In particular, if $v\in I$, $v$ and vertices in
$N(v)$ are deleted from $G$ and the resulting graph is $G_{1}$; if $v\notin
I$, $v$ is deleted from $G$ and the resulting graph is $G_{2}$. The algorithm
is then recursively applied on both $G_{1}$ and $G_{2}$ to compute a maximum
independent set in each of them. We use $I_{1}$ and $I_{2}$ to denote the
maximum independent sets in $G_{1}$ and $G_{2}$ found by the algorithm
respectively. $I_{2}$ is returned as a maximum independent set in $G$ if
$|I_{2}|\geq|I_{1}|+1$ and $I_{1}\cup\\{v\\}$ is returned otherwise. We show
that this algorithm terminates in expected time $2^{O(\log_{2}^{2}{n})}$.
###### Theorem 2.1
A maximum independent set in a random graph $G=(V,p)$ with $n$ vertices can be
computed in expected computation time $2^{O(\log_{2}^{2}{n})}$.
Proof. We show that the algorithm described above terminates in expected time
$2^{O(\log_{2}^{2}{n})}$. In particular, the algorithm is recursive and for
each step of recursion, we have the following recursion relation for the
computation time if the underlying graph is good and contains $m$ vertices
$T(m)\leq T((1-p+\epsilon)m)+T(m-1)+O(m^{2})$ (8)
where $T(m)$ is the computation time needed by the algorithm in a graph on $m$
vertices. The term $O(m^{2})$ is the computation time needed to find a vertex
whose degree is at least $(p-\epsilon)m$, since the time needed to compute the
degree of a vertex is $O(m)$ and the algorithm may need to check $m$ vertices
to find such a vertex. If the underlying graph is not good, the algorithm
exhaustively enumerates all subsets in the graph and finds an independent set
of the largest size. The computation time is $2^{O(m)}$.
We are now ready to establish the expected computation time for the algorithm.
In particular, we use $ET(m)$ to denote the expected computation time of the
algorithm on a graph that contains $m$ vertices. From Lemma 2.1, an underlying
graph $G^{\prime}$ in $m$ vertices is good with a probability of at least
$1-2^{-\mu m^{2}}$. We thus can immediately obtain the following recursion for
$ET(m)$.
$\displaystyle ET(m)$ $\displaystyle\leq$ $\displaystyle
ET((1-p+\epsilon)m)+ET(m-1)+O(m^{2})+2^{O(m)-\mu m^{2}}$ (9)
$\displaystyle\leq$ $\displaystyle ET((1-p+\epsilon)m)+ET(m-1)+O(m^{2})$ (10)
where the second inequality is due to the fact that $2^{O(m)-\mu m^{2}}$ is
bounded by a constant for all positive integers $m$.
We then show that $ET(m)\leq 2^{c\log_{2}^{2}{m}}$, where $c$ is a positive
constant. We show this by induction. First, for a sufficiently large positive
integer $m_{0}$ whose value will be specified later, we let $c_{0}=\max_{1\leq
t\leq m_{0}}{\\{\frac{\log_{2}{ET(t)}}{\log_{2}^{2}{t}}\\}}$ and choose
$c=\max{\\{c_{0},\frac{2}{\log_{2}{\frac{1}{1-p+\epsilon}}},1\\}}$. It is not
difficult to see that $ET(l)\leq 2^{c\log_{2}^{2}{l}}$ if $1\leq l\leq m_{0}$.
We then assume this holds for all positive integers less than $m$. From the
above recursion relation on $ET(m)$, we can obtain
$\displaystyle ET(m)$ $\displaystyle\leq$ $\displaystyle
2^{c\log_{2}^{2}{((1-p+\epsilon)m)}}+2^{c\log_{2}^{2}{(m-1)}}+Bm^{2}$ (11)
$\displaystyle\leq$ $\displaystyle
sm^{-l}2^{c\log_{2}^{2}{m}}+2^{c\log_{2}^{2}{m}}+(2^{c\log_{2}^{2}{(m-1)}}-2^{c\log_{2}^{2}{m}})+Bm^{2}$
(12) $\displaystyle\leq$ $\displaystyle
sm^{-l}2^{c\log_{2}^{2}{m}}+2^{c\log_{2}^{2}{m}}-\frac{\log_{2}{m}}{24m}2^{c\log_{2}^{2}{m}}+Bm^{2}$
(13) $\displaystyle\leq$ $\displaystyle 2^{c\log_{2}^{2}{m}}$ (14)
where $B$ is a positive constant independent of $c,p,\epsilon$ and $s$, $q$,
$l$ are some positive constants that depend on $c,p,\epsilon$ only. The first
inequality is obtained from the assumption for induction. The second one is
due to the fact that
$\log_{2}^{2}{((1-p+\epsilon)m)}=\log_{2}^{2}{(1-p+\epsilon)}+2\log_{2}{(1-p+\epsilon)}\log_{2}{m}+\log_{2}^{2}{m}$
and we can let $l=2c\log_{2}{\frac{1}{1-p+\epsilon}}$ ,
$s=2^{c\log_{2}^{2}{(1-p+\epsilon)}}$.
To establish the third inequality, we have
$\displaystyle\log_{2}^{2}{(m-1)}-\log_{2}^{2}{m}$ $\displaystyle=$
$\displaystyle(\log_{2}{m}+\log_{2}{(1-\frac{1}{m})})^{2}-\log_{2}^{2}{m}$
(15) $\displaystyle\leq$
$\displaystyle(\log_{2}{m}-\frac{1}{6m})^{2}-\log_{2}^{2}{m}$ (16)
$\displaystyle\leq$ $\displaystyle-\frac{\log_{2}{m}}{6m}$ (17)
$\displaystyle\leq$ $\displaystyle-\frac{\log_{2}{m}}{6cm}$ (18)
when $m\geq 16$, we can obtain
$\displaystyle 2^{c\log_{2}^{2}{(m-1)}}-2^{c\log_{2}^{2}{m}}$ $\displaystyle=$
$\displaystyle
2^{c\log_{2}^{2}{m}}(2^{c(\log_{2}^{2}{(m-1)}-\log_{2}^{2}{m})}-1)$ (19)
$\displaystyle\leq$ $\displaystyle
2^{c\log_{2}^{2}{m}}(2^{-\frac{\log_{2}{m}}{6m}}-1)$ (20) $\displaystyle\leq$
$\displaystyle-\frac{\log_{2}{m}}{24m}2^{c\log_{2}^{2}{m}}$ (21)
the third inequality thus follows.
From the fact that $c\geq\frac{2}{\log_{2}{\frac{1}{1-p+\epsilon}}}$, we have
$l\geq 4$. We let
$\displaystyle c^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2}{\log_{2}{\frac{1}{1-p+\epsilon}}}$ (22) $\displaystyle
s^{\prime}$ $\displaystyle=$ $\displaystyle
2^{c^{\prime}\log_{2}^{2}{((1-p+\epsilon)m)}}$ (23) $\displaystyle l^{\prime}$
$\displaystyle=$ $\displaystyle 2c^{\prime}\log_{2}{\frac{1}{1-p+\epsilon}}$
(24)
we now consider the function
$F(m)=(s^{\prime}{m}^{-l^{\prime}}-\frac{\log_{2}{m}}{24m})2^{c^{\prime}\log_{2}^{2}{m}}+B{m}^{2}$.
Since $s^{\prime}$, $l^{\prime}$, $c^{\prime}$, and $B$ are independent of $m$
and $l^{\prime}\geq 4$, there exists a positive integer $m_{1}(p,\epsilon)$
such that $F(m)\leq 0$ when $m\geq m_{1}(p,\epsilon)$. $m_{0}$ can be
determined as follows
$m_{0}=\max\\{m_{1}(p,\epsilon),\frac{1}{\sqrt{1-p+\epsilon}},16\\}$ (25)
It is not difficult to see that when $c\geq c^{\prime}$ and $m\geq m_{0}$, we
have $s^{\prime}m^{-l^{\prime}}-\frac{\log_{2}{m}}{24m}\leq 0$. In addition,
we can further verify that
$sm^{-l}=2^{c\log_{2}{(1-p+\epsilon)}\log_{2}{(m^{2}(1-p+\epsilon))}}$ (26)
since $c\geq c^{\prime}$, $m\geq\frac{1}{\sqrt{1-p+\epsilon}}$, and
$\log_{2}{(1-p+\epsilon)}\leq 0$, we can immediately obtain
$\displaystyle sm^{-l}$ $\displaystyle=$ $\displaystyle
2^{c\log_{2}{(1-p+\epsilon)}\log_{2}{(m^{2}(1-p+\epsilon))}}$ (27)
$\displaystyle\leq$ $\displaystyle
2^{c^{\prime}\log_{2}{(1-p+\epsilon)}\log_{2}{(m^{2}(1-p+\epsilon))}}$ (28)
$\displaystyle=$ $\displaystyle s^{\prime}m^{-l^{\prime}}$ (29)
the following thus holds
$\displaystyle(sm^{-l}-\frac{\log_{2}{m}}{24m})2^{c\log_{2}^{2}{m}}+B{m}^{2}$
$\displaystyle\leq$
$\displaystyle(s^{\prime}m^{-l^{\prime}}-\frac{\log_{2}{m}}{24m})2^{c\log_{2}^{2}{m}}+B{m}^{2}$
(30) $\displaystyle\leq$
$\displaystyle(s^{\prime}m^{-l^{\prime}}-\frac{\log_{2}{m}}{24m})2^{c^{\prime}\log_{2}^{2}{m}}+Bm^{2}$
(31) $\displaystyle=$ $\displaystyle F(m)$ (32) $\displaystyle\leq$
$\displaystyle 0$ (33)
the fourth inequality thus follows. From the principle of induction, the
theorem has been proved.
## 3 Parameterized Algorithm for Independent Set Problem
The parameterized independent set problem is to decide whether a given graph
$G=(V,E)$ contains an independent set of size $k$ or not. The problem is known
to be W[1]-hard [10, 11, 12] and cannot be solved in time $n^{o(k)}$ in
general graphs unless W[2]=FPT [7, 8]. We show that if the underlying graph
$G$ is a random graph, the problem can be solved in expected time
$2^{O(k^{2})}+O(n^{3})$, where $n$ is the number of vertices in the graph. We
need the following lemma to analyze the time complexity of the algorithm.
###### Lemma 3.1
Given a random graph $G=(V,p)$ where $n=|V|$ and a sufficiently small constant
$\epsilon$ such that $p+\epsilon<1$, there exists vertex $u\in V$ such that
$deg_{G}(u)\leq(p+\epsilon)n$ with a probability of at least $1-2^{-\mu
n^{2}}$, where $\mu$ is a positive constant that only depends on $\epsilon$
and $p$,
Proof. The proof is similar to the proof of Lemma 2.1. If such a vertex does
not exist, the degree of every vertex in $G$ is at least $(p+\epsilon)n$. The
graph thus contains at least $\frac{(p+\epsilon)n^{2}}{2}$ edges. The expected
number of edges in $G$ is $\frac{pn(n-1)}{2}$. We use $n(E)$ to denote the
number of the edges in $G$. From Chernoff bound, we can bound the probability
for $G$ to contain at least $\frac{(p+\epsilon)n^{2}}{2}$ edges.
$\displaystyle Pr(n(E)$ $\displaystyle\geq$
$\displaystyle\frac{(p+\epsilon)n^{2}}{2})$ (35) $\displaystyle<$
$\displaystyle\exp{(-\frac{\epsilon^{2}n^{2}}{64p})}$ (36) $\displaystyle=$
$\displaystyle 2^{-\frac{\epsilon^{2}n^{2}}{64p\ln{2}}}$ (37)
the lemma immediately follows by letting $\mu=\frac{\epsilon^{2}}{64p\ln{2}}$.
The proof of Lemma 3.1 relies on the fact that $p$ is a constant independent
of $n$, the Lemma does not hold if the value of $p$ depends on $n$.
###### Theorem 3.1
Given a random graph $G=(V,p)$, there exists an algorithm that can decide
whether $G$ contains an independent set of size $k$ in expected time
$2^{O(k^{2})}+O(n^{3})$.
Proof. We start the proof by comparing the values of $k$ and
$L(n)=\frac{1}{3}\log_{\frac{1}{1-p-\epsilon}}{n}$, if $k>L(n)$, we can
enumerate all possible vertex subsets of size $k$ in $G$ and check whether one
of them is an independent set of size $k$ or not. The enumeration and checking
needs at most $O(k^{2}n^{k})$ time. However, since $k>L(n)$, we can obtain
$n<(\frac{1}{1-p-\epsilon})^{3k}$, the computation time needed to determine
whether $G$ contains an independent set of size $k$ or not is thus at most
$O(k^{2}(\frac{1}{1-p-\epsilon})^{3k^{2}})=2^{O(k^{2})}$ in this case.
We then consider the case where $k\leq L(n)$. We use the following procedure
to generate an independent set $I$. We start with the vertex $u$ with the
minimum degree in $G$, we include $u$ in $I$ and remove $u$ and all its
neighbors in $G$ from $G$. We denote the resulting graph by $G_{1}$. The
procedure can be repeatedly executed until there are at most $n^{\frac{2}{3}}$
vertices left in the graph. We use $G_{0}=G,G_{1},G_{2},G_{3},\cdots,G_{l}$ to
denote the intermediate graphs generated during this iterative procedure. It
is not difficult to see that vertices in $I$ form an independent set in $G$.
We show that the above procedure can generate an independent set $I$ of size
at least $L(n)$ with high probability. We use $G_{1},G_{2},G_{3},\cdots,G_{l}$
to denote the resulting graph in each iterative step and $n(G_{i})$ to denote
the number of vertices in graph $G_{i}$. From Lemma 3.1, the following holds
with a probability of at least 1-$2^{-\mu n^{2}(G_{i})}$ for each $i$ between
$0$ and $l$.
$n(G_{i+1})\geq(1-p-\epsilon)n(G_{i})$ (38)
Since $n(G_{i})>n^{\frac{2}{3}}$, the probability for this inequality to hold
for all $i$’s between $0$ and $l$ is at least $1-n2^{-\mu n^{\frac{4}{3}}}$.
If this inequality holds for all $i$’s between $0$ and $l$. We can immediately
obtain
$\displaystyle l$ $\displaystyle\geq$
$\displaystyle\log_{\frac{1}{1-p-\epsilon}}{(\frac{n}{n^{\frac{2}{3}}})}$ (39)
$\displaystyle=$ $\displaystyle\frac{1}{3}\log_{\frac{1}{1-p-\epsilon}}{n}$
(40) $\displaystyle=$ $\displaystyle L(n)$ (41)
$I$ thus contains at least $L(n)$ vertices. With a probability of at least
$1-n2^{-\mu n^{\frac{4}{3}}}$, the above iterative procedure generates an
independent set of size $L(n)$. Since $k<L(n)$, the algorithm returns “yes” if
$I$ indeed contains $L(n)$ independent vertices, otherwise, the algorithm
simply enumerates all vertex subsets in $G$ and checks whether one of them is
an independent set of size at least $k$. Since the procedure for generating
$I$ needs $O(n^{3})$ time, the expected computation time needed for this is at
most
$O(n^{3})(1-n2^{-\mu n^{\frac{4}{3}}})+2^{O(n)}n2^{-\mu
n^{\frac{4}{3}}}=O(n^{3})$ (42)
where the equality is due to the fact that the second term is bounded by a
constant when $n$ is sufficiently large. The algorithm thus needs an expected
time $2^{O(k^{2})}+O(n^{3})$, the theorem has been proved.
## 4 The Largest Common Subgraph Problem
Given two graphs $G$, $H$, a common subgraph of $G$ and $H$ is a third graph
$K$ such that both $G$ and $H$ contain an induced subgraph that is isomorphic
to $K$. The largest common subgraph problem is to compute a common subgraph
that contains the largest number of vertices. The problem has important
applications in computational biology. For example, it is often desirable to
identify common subgraphs in the protein interaction networks of two
homologous organisms since proteins in these common subgraphs often together
play important roles for certain biological functions [28].
Unfortunately, the problem is NP hard when both of the underlying graphs are
general graphs [18]. The asymptotically best known algorithm for this problem
needs time $O^{*}((m+1)^{n})$ [1, 35] and little progress has been made to
improve the asymptotical time complexity of this problem. We show that, given
two random graphs $G$ and $H$ in $n$ and $m$ vertices, where $n\geq m$, the
largest common subgraph problem in $G$ and $H$ can be computed in expected
time $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$.
###### Lemma 4.1
The largest common subgraph problem can be solved in computation time
$O(m^{2}2^{m}n^{m})\leq 2^{hm\log_{2}{n}}$, where $h$ is some positive
constant that does not depend on $n$ or $m$.
Proof. We can solve the largest common subgraph problem with the following
simple algorithm. For each positive integer $l$ not greater than $m$, we
enumerate all vertex subsets that contain $l$ vertices in $G$. For each such
vertex subset $S_{1}$, we enumerate all vertex subsets of size $l$ in graph
$H$ and for each such vertex subset $S_{2}$, we enumerate all possible one to
one mappings between vertices in $S_{1}$ and those in $S_{2}$. We then check
whether there exists a one to one mapping that can establish the isomorphism
between the subgraph induced by $S_{1}$ in $G$ and the subgraph induced by
$S_{2}$ in $H$. The algorithm can find all common subgraphs and return one
that is of the largest size.
The number of vertex subsets of size $l$ in $G$ is ${n\choose l}$ and the
number of vertex subsets of size $l$ in $H$ is ${m\choose l}$. The number of
one to one mappings between $S_{1}$ and $S_{2}$ is $l!$ and the computation
time needed to check whether the two subgraphs induced by $S_{1}$ and $S_{2}$
are isomorphic under a particular mapping is at most $O(l^{2})$. The total
computation time needed to find and return the largest common subgraph is thus
at most
$\displaystyle\sum_{l=1}^{m}{C{n\choose l}{m\choose l}l!l^{2}}$
$\displaystyle\leq$ $\displaystyle\sum_{l=1}^{m}{Cn^{m}{m\choose l}m^{2}}$
(43) $\displaystyle\leq$ $\displaystyle C2^{m}n^{m}m^{2}$ (44)
$\displaystyle\leq$ $\displaystyle 2^{hm\log_{2}{n}}$ (45)
where $C$ and $h$ are some positive constants independent of $n$ and $m$. The
first inequality is due to the fact that ${n\choose l}l!\leq n^{l}$ and $l\leq
m$; the second inequality is due to the fact that $\sum_{l=1}^{m}{m\choose
l}=2^{m}-1$. The lemma thus has been proved.
###### Lemma 4.2
Given two random graphs $G=(V,p)$ and $H=(U,q)$, where $p$ and $q$ are
positive constants between 0 and 1, $G$ contains $n$ vertices and $H$ contains
$m$ vertices ($n\geq m$), the probability that $G$ and $H$ contain a common
subgraph of size $n^{\frac{1}{2}}\log_{2}^{\frac{2}{3}}{n}$ is at most
$2^{-\mu n\log_{2}^{\frac{4}{3}}{n}}$, where $\mu$ is a positive constant that
only depends on $p$ and $q$.
Proof. We let $k=n^{\frac{1}{2}}\log_{2}^{\frac{2}{3}}{n}$ and consider two
given subsets of size $k$ in graph $G$ and $H$ respectively. We use
$S_{1}=\\{g_{1},g_{2},\cdots,g_{k}\\}$ and
$S_{2}=\\{h_{1},h_{2},\cdots,h_{k}\\}$ to denote them and $G_{1}$, $H_{1}$ to
denote the subgraphs induced by them in $G$ and $H$ respectively. We assume
that $G_{1}$ is isomorphic to $H_{1}$ under a given one to one mapping $M$,
where vertex $g_{i}$ in $S_{1}$ is mapped to $h_{i}$ in $S_{2}$ for $1\leq
i\leq k$.
We then estimate the probability for $M$ to be such a mapping. If $G_{1}$ is
isomorphic to $H_{1}$ under $M$, for any integer pair $(i,j)$, where $1\leq
i<j\leq k$, either both $(g_{i},g_{j})$ and $(h_{i},h_{j})$ are edges or
neither of them are edges. The probability for the former case is $pq$ and the
probability for the latter case is $(1-p)(1-q)$. Since there are in total
$\frac{k(k-1)}{2}$ such pairs, the probability for $G_{1}$ and $H_{1}$ to be
isomorphic under $M$ is thus $(pq+(1-p)(1-q))^{\frac{k(k-1)}{2}}$.
We use $P(k)$ to denote the probability for $G$ and $H$ to contain a common
subgraph of size $k$. Since the number of vertex subsets of size $k$ in $G$ is
${n\choose k}$ and the number of vertex subsets of size $k$ in $H$ is
${m\choose k}$, we can obtain an upper bound for $P(k)$ using the union bound.
$\displaystyle P(k)$ $\displaystyle\leq$ $\displaystyle{n\choose k}{m\choose
k}\sum_{M}{s^{\frac{k(k-1)}{2}}}$ (46) $\displaystyle\leq$
$\displaystyle{n\choose k}{m\choose k}k!s^{\frac{k(k-1)}{2}}$ (47)
$\displaystyle\leq$ $\displaystyle n^{k}m^{k}s^{\frac{k(k-1)}{2}}$ (48)
$\displaystyle\leq$ $\displaystyle n^{2k}s^{\frac{k(k-1)}{2}}$ (49)
$\displaystyle\leq$ $\displaystyle
2^{2n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n}}s^{\frac{k^{2}}{4}}$ (50)
$\displaystyle=$ $\displaystyle
2^{2n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n}-\frac{n\log_{2}^{\frac{4}{3}}{n}\log_{2}{\frac{1}{s}}}{4}}$
(51) $\displaystyle\leq$ $\displaystyle 2^{-\mu n\log_{2}^{\frac{4}{3}}{n}}$
(52)
where $s=pq+(1-p)(1-q)$ and $\mu$ is some positive constant that depends on
$p$ and $q$ only. The first inequality is due to the union bound. The second
inequality follows from the fact that there are in total $k!$ one to one
mappings between vertices in $S_{1}$ and $S_{2}$. The third inequality is due
to the fact that ${n\choose k}\leq n^{k}$ and ${m\choose k}k!\leq m^{k}$. The
fifth inequality follows from the fact that
$k=n^{\frac{1}{2}}\log_{2}^{\frac{2}{3}}{n}$ and
$\frac{k(k-1)}{2}>\frac{k^{2}}{4}$ when $n$ is sufficiently large. The last
inequality is due to the fact that $s<1$ and
$2n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n}\leq\frac{n\log_{2}^{\frac{4}{3}}{n}\log_{2}{\frac{1}{s}}}{8}$
for sufficiently large $n$.
The proof of the Lemma 4.2 relies on the fact that $p$ and $q$ are both
constants independent of $n$ and $m$, the Lemma does not hold if the values of
$p$ and $q$ depend on $n$ or $m$.
###### Theorem 4.1
Given two random graphs $G=(V,p)$ and $H=(U,q)$, where $p$ and $q$ are
positive numbers between 0 and 1, $G$ contains $n$ vertices and $H$ contains
$m$ vertices ($m\leq n$), a largest common graph of $G$ and $H$ can be
computed in expected time $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$.
Proof. We only need to show that such an algorithm exists when
$m>n^{\frac{1}{2}}\log_{2}^{\frac{2}{3}}{n}$. Since if $m\leq
n^{\frac{1}{2}}\log_{2}^{\frac{2}{3}}{n}$, the algorithm in the proof of Lemma
4.1 can be directly used to find a largest common subgraph of $G$ and $H$ in
time $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$.
Let $k=n^{\frac{1}{2}}\log_{2}^{\frac{2}{3}}{n}$, since $m>k$, we can use the
following algorithm to compute a largest common subgraph in $G$ and $H$.
1. 1.
Enumerate all vertex subsets of size $k$ in $G$. For each such vertex subset
$S_{1}$, enumerate all vertex subsets of size $k$ in $H$;
2. 2.
for each such subset $S_{2}$ in $H$, we enumerate all possible one to one
mappings between $S_{1}$ and $S_{2}$;
3. 3.
for each such mapping $M$, determine whether the subgraph induced by $S_{1}$
in $G$ is isomorphic to the subgraph induced by $S_{2}$ in $H$ under $M$ or
not;
4. 4.
if there exists a mapping that can make the subgraph induced by $S_{1}$ in $G$
isomorphic to the subgraph induced by $S_{2}$ in $H$, call the algorithm in
Lemma 4.1 to compute a largest common subgraph of $G$ and $H$ and return it;
5. 5.
otherwise, for each integer $i$ between $1$ and $k$, use the same approach as
described in steps 1, 2, 3 to determine whether $G$ and $H$ contains a common
subgraph of size $i$ or not;
6. 6.
Based on the result of the exhaustive search performed in step $5$, return a
common subgraph of the largest size.
We then show that the algorithm can compute the largest common subgraph of $G$
and $H$ in expected $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$ time. In
particular, the computation time needed by the exhaustive search performed in
steps 1, 2, and 3 is at most
$\displaystyle C{n\choose k}{m\choose k}k!k^{2}$ $\displaystyle\leq$
$\displaystyle Cn^{k}m^{k}$ (53) $\displaystyle\leq$ $\displaystyle Cn^{2k}$
(54) $\displaystyle\leq$ $\displaystyle C2^{2k\log_{2}{n}}$ (55)
$\displaystyle=$ $\displaystyle
2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$ (56)
where $C$ is some positive constant. The first inequality is due to the fact
that ${m\choose k}k!\leq m^{k}$ and ${n\choose k}k^{2}\leq n^{k}$ for
sufficiently large $n$. From Lemma 4.1, step 4 of the algorithm, if executed,
needs $2^{hm\log_{2}{n}}$ computation time, where $h$ is some positive
constant independent of $n$ and $m$. The computation time needed by step 5 is
at most
$\displaystyle D\sum_{i=1}^{k-1}{{n\choose i}{m\choose i}i!i^{2}}$
$\displaystyle\leq$ $\displaystyle D\sum_{i=1}^{k-1}{n^{i}m^{i}i^{2}}$ (57)
$\displaystyle\leq$ $\displaystyle Dkn^{2k}k^{2}$ (58) $\displaystyle=$
$\displaystyle D2^{3\log_{2}{k}+2k\log_{2}{n}}$ (59) $\displaystyle=$
$\displaystyle 2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$ (60)
where $D$ is some positive constant. The first inequality is due to the fact
that ${m\choose i}i!\leq m^{i}$ and ${n\choose i}\leq n^{i}$. The second
inequality follows from the fact that $1\leq i<k$.
Only one of steps 4 and 5 is executed by the algorithm. From Lemma 4.2, the
probability for step 4 to be executed is at most $2^{-\mu
n\log_{2}^{\frac{4}{3}}{n}}$, where $\mu$ is some constant that depends on $p$
and $q$ only. The expected computation time to execute steps 4 and 5 is thus
at most
$2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}+2^{hm\log_{2}{n}}2^{-\mu
n\log_{2}^{\frac{4}{3}}{n}}.$ (61)
Since $m\leq n$, the second term is bounded by a constant for sufficiently
large $n$. We thus can conclude that the expected computation time for steps 4
and 5 is $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$. Since steps 1, 2,
3 also need $2^{O(n^{\frac{1}{2}}\log_{2}^{\frac{5}{3}}{n})}$ computation
time, the theorem has been proved.
## 5 Approximate Algorithm
As discussed in the introduction, the maximum independent set problem cannot
be approximated within a ratio of $n^{1-\epsilon}$ in polynomial time unless
P=NP, where $\epsilon$ is any positive constant. In [4], it is shown that the
maximum independent set in a graph can be approximated within a ratio of
$O(\frac{n}{\log_{2}^{2}{n}})$. In [15], the approximation ratio is improved
to $O(\frac{n\log_{2}^{2}{\log_{2}{n}}}{\log_{2}^{3}{n}})$. The result so far
remains the best known approximation ratio achieved for this problem in
general graphs. In [19, 22, 29], a polynomial time algorithm that can
approximate the maximum independent set in a random graph within a constant
ratio with high probability is developed and analyzed. However, the
approximation ratio of the algorithm is not guaranteed to be constant for all
graphs. We show that, the maximum independent set in a random graph can be
approximated within a ratio of $\frac{2n}{2^{\sqrt{\log_{2}{n}}}}$ in expected
polynomial time, which is a significant improvement compared with the best
known approximate ratio for this problem in general graphs.
###### Theorem 5.1
Given a random graph $G=(V,p)$ in $n$ vertices where $p$ is a positive
constant between $0$ and $1$, the maximum independent set in $G$ can be
approximated within a ratio of $\frac{2n}{2^{\sqrt{\log_{2}{n}}}}$ in expected
polynomial time.
Proof. We use the following simple algorithm to compute an independent set in
$G$. We let $k=\lfloor 2^{\sqrt{\log_{2}{n}}}\rfloor$ and partition the
vertices in $G$ into $l$ disjoint vertex subsets such that $l-1$ of them
contains $k$ vertices and the remaining one contains at most $k$ vertices. We
use $G_{1},G_{2},\cdots,G_{l}$ to denote the subgraph induced by vertices in
these vertex subsets. It is not difficult to see that
$l\leq\lfloor\frac{n}{k}\rfloor+1$.
We then use the algorithm we have developed in Theorem 2.1 to compute a
maximum independent set in each of $G_{1},G_{2},\cdots,G_{l}$ and return the
one that contains the largest number of vertices.
We first show that the algorithm returns an independent set in expected
polynomial time. $G_{1},G_{2},\cdots,G_{l}$ are disjoint and the expected time
needed to compute a maximum independent set in each of them is at most
$2^{c\log_{2}^{2}{k}}$, where $c$ is some positive constant that only depends
on $p$. Since $k\leq 2^{\sqrt{\log_{2}{n}}}$, the expected computation time
needed to compute the maximum independent set in one subgraph is at most
$2^{c\log_{2}{n}}=n^{c}$. The algorithm thus returns an independent set in
expected time $n^{c+1}$.
We then show that the algorithm can achieve an approximate ratio of
$\frac{2n}{2^{\sqrt{\log_{2}{n}}}}$. We use $APX(G)$ to denote the size of the
independent set returned by the algorithm and $OPT(G)$ to denote the size of a
maximum independent set in $G$. we assume that $I$ is a maximum independent
set in $G$. Since we have partitioned the graph $G$ into $l$ disjoint
subgraphs $G_{0},G_{1},\cdots,G_{l}$, at least one of the $l$ subgraphs
contains at least $\frac{OPT(G)}{l}$ vertices from $I$. These vertices form an
independent set in the subgraph. Since the algorithm computes a maximum
independent set in each subgraph and returns the one with the largest size, we
immediately obtain
$APX(G)\geq\frac{OPT(G)}{l}$ (62)
this suggests that
$\displaystyle\frac{OPT(G)}{APX(G)}$ $\displaystyle\leq$ $\displaystyle l$
(63) $\displaystyle\leq$ $\displaystyle\lfloor\frac{n}{k}\rfloor+1$ (64)
$\displaystyle\leq$ $\displaystyle\frac{n}{k}+1$ (65) $\displaystyle\leq$
$\displaystyle\frac{n}{2^{\sqrt{\log_{2}{n}}}-1}+1$ (66) $\displaystyle\leq$
$\displaystyle\frac{2n}{2^{\sqrt{\log_{2}{n}}}}.$ (67)
The second inequality is due to the fact that
$l\leq\lfloor\frac{n}{k}\rfloor+1$. The fourth inequality is due to the fact
that $k\geq 2^{\sqrt{\log_{2}{n}}}-1$. The last inequality holds for
sufficiently large $n$. The theorem thus has been proved.
## 6 Conclusions
In this paper, we study the independent set problem in random graphs. We show
that a maximum independent set in a random graph can be computed in expected
subexponential time. We also show that the parameterized independent set
problem is fixed parameter tractable with high probability for random graphs.
Using techniques based on enumeration, we show that the largest common
subgraph in two random graphs can be computed in expected subexponential time.
Our work also suggests that the maximum independent set in a random graph can
be approximated within a ratio of $\frac{2n}{2^{\sqrt{\log_{2}{n}}}}$ in
expected polynomial time, which significantly improves on the best known
approximate ratio for this problem in general graphs.
It remains unknown whether the maximum independent set in a random graph can
be computed in expected polynomial time or not. One possible direction of
future work is to study whether there exists such an algorithm. Another
related open question is that if such an algorithm does not exist, whether it
can be approximated within an improved ratio in expected polynomial time.
Further investigations are needed to solve these problems.
## References
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|
arxiv-papers
| 2013-08-07T12:55:41 |
2024-09-04T02:49:49.163453
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yinglei Song",
"submitter": "Yinglei Song",
"url": "https://arxiv.org/abs/1308.1556"
}
|
1308.1665
|
# Protecting quantum states from decoherence of finite temperature using weak
measurement
Shu-Chao Wang1, Zong-Wen Yu2, and Xiang-Bin Wang1,3111Email
Address:[email protected] 1State Key Laboratory of Low Dimensional
Quantum Physics, Tsinghua University, Beijing 100084, People s Republic of
China
2Data Communication Science and Technology Research Institute, Beijing 100191,
China
3 Shandong Academy of Information and Communication Technology, Jinan 250101,
People s Republic of China
###### Abstract
We show how to optimally protect quantum states and quantum entanglement under
non-zero temperature based on measurement reversal from weak measurement. In
particular, we present explicit formulas of the protection.
## I Introduction
The inherit properties of quantum mechanics can be applied to nontrivial tasks
in quantum information processing(QIP) such as the design of fast computation,
the unconditionally secure private communication. However, in practice, the
decoherence can undermine severly the quantum features in QIP. Protecting
quantun states and quantum entanglement under decoherence is crucially
important in effective QIP. Many proposals have been suggested for quantum
coherence protection including passive methods,e.g,decoherence-freefree1 ;
free2 ; free3 subspaces and active methods like quantum error correction
codeqecc1 ; qecc2 ; qecc3 , the technique of dynamical decoupling dd1 ; dd2 ;
dd3 or using quantum Zeno dynamicszeno1 ; zeno2 . When the decoherence is due
to processes with short correlation time scales, it is shown that quantum
reversal scheme has advantagespra ; nature ; enhance1 ; enhance2 .Weak
measurements has also been found useful in entanglement
amplificationamplification .Quantun entanglement plays an essential role in
quantum information processing and gives rise to varieties of interesting
phenomenanielsen . But it is fragile to environmental noises. It is of great
meaning to protect quantum entanglement.
Recently, a novel ideapra ; nature is proposed to protect quantum states and
quantum entanglements from decoherence using weak measurement and measurement
reversal. However, their result is limited to a special class of channel
noise, which corresponds to the zero temperature environmental noise. Most
often, decoherence is caused by the uncontrollable interaction with the
environment. In the case of zero temperature, a type of noise due to
environmental interaction can be modeled as the following amplitude
damping(AD) channelnielsen :
${\varepsilon_{AD}}(\rho)=\sum\limits_{i=0}^{1}{{E_{i}}\rho E_{i}^{\dagger}}$
(1)
with
${E_{0}}=\left({\begin{array}[]{*{20}{c}}1&0\\\
0&{\sqrt{1-r}}\end{array}}\right),{E_{1}}=\left({\begin{array}[]{*{20}{c}}0&{\sqrt{r}}\\\
0&0\end{array}}\right)$ (2)
It has been shownpra ; nature that quantum state and quantum entanglement can
be effectively protected under such a channel. However, in practice,
environmental temperature cannot be zero. In non-zero temperature , the
channel is more complicated than Eq.(2). An important class of dissipation
under finite temperature can be modeled by the following generalized amplitude
(GAD) channelnielsen :
${\varepsilon_{GAD}}(\rho)=\sum\limits_{i=0}^{3}{{E_{i}}\rho E_{i}^{\dagger}}$
(3)
with
$\begin{array}[]{l}{E_{0}}=\sqrt{p}\left({\begin{array}[]{*{20}{l}}1&0\\\
0&{\sqrt{1-r}}\end{array}}\right),{E_{1}}=\sqrt{p}\left({\begin{array}[]{*{20}{l}}0&{\sqrt{r}}\\\
0&0\end{array}}\right)\\\
{E_{2}}=\sqrt{1-p}\left({\begin{array}[]{*{20}{l}}{\sqrt{1-r}}&0\\\
0&1\end{array}}\right),{E_{3}}=\sqrt{1-p}\left({\begin{array}[]{*{20}{l}}0&0\\\
{\sqrt{r}}&0\end{array}}\right)\end{array}.$ (4)
One can see that Eq.(4) reduces to Eq.(4)when $p=1$. In the GAD channel, an
atom can not only transit from the higher energy level to the lower one by
undergoing spontaneous emission, but also can jump from the lower energy state
to the higher energy state by absorbing energy from the finite-temperature
environment. Generalized amplitude damping describes the finite-temperature
relaxation processes due to coupling of spins to their surrounding lattice, a
large system which is in thermal equilibrium at a temperature often much
higher than the spin temperaturenielsen .
In this work, we study how to use weak measurement to battle against the
decoherence in such channels. By performing weak measurements and measurement
reversals, the final fidelity can be optimized by adjusting the measurement
parameters. We have also investigated how to use weak measurements to recover
quantum entanglement at finite temperature environment. Explicit formulas for
optimal results are presented.
This article is organized as follows. In the following section, we show how to
use weak measurement to protect qubit states against decoherence in
generalized amplitude channel. The average fidelity over the initial state is
also studied. The optimal measurement strength is given. In the third section,
we study how to use weak measurements to protect quantum entanglement in GAD
channels, we present an optimal measurement strength for obtaining most
entanglement. The article is ended with a concluding remark.
## II protect quantum qubit through weak measurements
Any pure qubit state can be written as a vector on the Bloch-sphere:
$\rho=\frac{1}{2}(I+\sin\theta\cos\varphi X+\sin\theta\sin\varphi Y+\cos\theta
Z)$. Let us first consider the equatorial states ($\theta=\frac{\pi}{2}$)
which are extensively applied in QKDqkd . In this case, the initial state can
be written as
${\rho_{in}}=\frac{1}{2}\left({\begin{array}[]{*{20}{c}}1&{{e^{-i\varphi}}}\\\
{{e^{i\varphi}}}&1\end{array}}\right).$ (5)
Under the GAD channel as as described by Eq.(4), due to decoherence, the
outcome state is a mixed state,
${\rho_{f}}=\frac{1}{2}\left({\begin{array}[]{*{20}{c}}{1-r+2pr}&{\sqrt{1-r}{e^{-i\varphi}}}\\\
{\sqrt{1-r}{e^{i\varphi}}}&{1+r-2pr}\end{array}}\right)$ (6)
The fidelity of the initial state and this state is,
$F=\frac{1}{2}(1+\sqrt{1-r})$ (7)
In order to improve the fidelity, we should perform two weak measurements $M$
and $N$, before and after the qubit being put into the GAD
channel,respectively. With these weak measurements being implemented, the
final state is,
${\rho_{f}^{(w)}}=N{\varepsilon_{GAD}}(M{\rho_{in}}{M^{\dagger}}){N^{\dagger}}$
(8)
with $\epsilon_{GAD}$ being defined by Eq.(3) and the non-unitary quantum
operations
$M=\left({\begin{array}[]{*{20}{c}}1&0\\\ 0&m\end{array}}\right)$ (9)
and
$N=\left({\begin{array}[]{*{20}{c}}n&0\\\ 0&1\end{array}}\right).$ (10)
It’s easy to see,
${\rho_{f}^{(w)}}=\frac{1}{T}\left({\begin{array}[]{*{20}{c}}{{n^{2}}(pr{m^{2}}+pr-r+1)}&{mn\sqrt{1-r}{e^{-i\varphi}}}\\\
{mn\sqrt{1-r}{e^{i\varphi}}}&{-pr{m^{2}}+{m^{2}}-pr+r}\end{array}}\right),$
(11)
and
$T={n^{2}}(pr{m^{2}}+pr-r+1)-pr{m^{2}}+{m^{2}}-pr+r$ (12)
is the normalization factor. The fidelity of the final state and the initial
state is
$F^{(w)}=\frac{1}{2}+\frac{{mn\sqrt{1-r}}}{{{n^{2}}(pr{m^{2}}+pr-r+1)+{m^{2}}(1-pr)+(1-p)r}}.$
(13)
The overall success probability is
${P_{s}}=\frac{T}{2}\cdot\min\left\\{{1,\frac{1}{m}}\right\\}\cdot\min\left\\{{1,\frac{1}{n}}\right\\}.$
(14)
When
$m=\sqrt[4]{{\frac{{(1-p)(1-r+pr)}}{{p(1-pr)}}}},n=\sqrt[4]{{\frac{{(1-p)(1-pr)}}{{p(1-r+pr)}}}},$
(15)
we obtain the maximal value for $F^{(w)}$ as
${F^{(w)}_{\max}}=\frac{1}{2}\left({1+\frac{{\sqrt{1-r}}}{{\sqrt{(1-pr)(1-r+pr)}+r\sqrt{p(1-p)}}}}\right).$
(16)
Figure 1: The fidelity with varying measurement strength m and n with p=0.8
and r=0.3. One can find that the optimal value of m and n is zero as in the
amplitude channel (p=1) case. But in general when $p\neq 1$,the optimal value
of m and n are not zero.
One can check that ${F_{\max}}\geq{F_{0}}$(see Appendix A), so weak
measurement is useful when we consider the equatorial states, under the
generalized amplitude damping channel. By adjusting the measurement strengths
according to the channel parameters, one can obtain a final state with a
higher fidelity.
In the quantum key distribution, the equatorial states can be used as BB84
statesnielsen . The two basis
$\left\\{{\left|0\right\rangle\equiv\frac{1}{{\sqrt{2}}}\left({\left|0\right\rangle+\left|1\right\rangle}\right),\left|1\right\rangle\equiv\frac{1}{{\sqrt{2}}}\left({\left|0\right\rangle-\left|1\right\rangle}\right)}\right\\}$
and
$\left\\{{\left|0\right\rangle\equiv\frac{1}{{\sqrt{2}}}\left({\left|0\right\rangle+i\left|1\right\rangle}\right),\left|1\right\rangle\equiv\frac{1}{{\sqrt{2}}}\left({\left|0\right\rangle-i\left|1\right\rangle}\right)}\right\\}$
can be used to complete the quantum key distribute processing. The error rate
can be defined as
${R_{E}}={\left\langle{\frac{{\left\langle i\right|{\rho_{i\oplus
1}}\left|i\right\rangle}}{{\left\langle
i\right|{\rho_{i}}\left|i\right\rangle+\left\langle i\right|{\rho_{i\oplus
1}}\left|i\right\rangle}}}\right\rangle_{i}}.$ (17)
Here, $\rho_{i}$ means the obtained density matrix of the qubit after
undergoing the weak measurements and the GAD channel when the initial state is
$|i\rangle$, i=0 or 1 and ${\left\langle\bullet\right\rangle_{i}}$ denotes the
average over the 4 basis states
$\frac{1}{{\sqrt{2}}}\left({\left|0\right\rangle\pm(i)\left|1\right\rangle}\right)$.
By calculating, one can find that ${R_{E}}=1-{F^{(w)}}$, which means that
while maximizing the fidelity, we also minimize the error rate.
We can also maximize the averaged fidelity $\bar{F}$ over six symmetric states
on the Bloch sphere. For experimentally characterizing quantum gates and
channels, it is meaningful to consider the average fidelity $\overline{F}$
over only six initial statespra ; six
:$|0\rangle,|1\rangle,(|0\rangle\pm|1\rangle)/\sqrt{2},(|0\rangle\pm
i|1\rangle)/\sqrt{2}$. Without any weak measurement, one can calculate that,
the final fidelity after passing the channel is $F_{0}=1-r+pr$ for the initial
state $|0\rangle$; $F_{1}=1-pr$, for $|1\rangle$;
$F_{e}=\frac{1}{2}(1+\sqrt{1-r})$, for the equatorial states
$(|0\rangle\pm|1\rangle)/\sqrt{2}$ and $(|0\rangle\pm i|1\rangle)/\sqrt{2}$.
So we have,
$\overline{F}=\frac{1}{6}({F_{0}}+{F_{1}}+4{F_{e}})=\frac{1}{3}+\frac{1}{6}{(1+\sqrt{1-r})^{2}}.$
(18)
With the weak measurements $M$ and $N$ given above, we can get the average
fidelity of these six states,
${\overline{F}^{(w)}}=\frac{1}{3}+\frac{1}{6}\left({\frac{{{n^{2}}(1-r+rp)}}{{r-rp+{n^{2}}(1-r+rp)}}+\frac{{1-rp}}{{1-rp+{n^{2}}rp}}+\frac{{4mn\sqrt{1-r}}}{{{n^{2}}(pr{m^{2}}+pr-r+1)+{m^{2}}(1-pr)+(1-p)r}}}\right).$
(19)
We can show that when Eq.(15) is satisfied, ${\overline{F}^{(w)}}$ has the
maximal value (see Appendix B). Note that when $p=1$, then $m\to 0,n\to 0$,
and the optimal measurements becomes projective measurements. This coincides
with the the previous resultspra .
## III protect entanglement through weak measurements
Quantum entanglement plays an important role in the quantum information
processing. But it is very fragile due to the decoherence. We now study how
the GAD channel affects a two-qubit entangled state. The channel can also be
described as the interaction of the system and the environment with the
initial state ${\left|{00}\right\rangle_{E}}$:
$\begin{array}[]{l}{\left|0\right\rangle_{S}}{\left|{00}\right\rangle_{E}}\to\sqrt{p}{\left|0\right\rangle_{S}}{\left|{00}\right\rangle_{E}}+\sqrt{1-p}\sqrt{1-r}{\left|0\right\rangle_{S}}{\left|{01}\right\rangle_{E}}+\sqrt{1-p}\sqrt{r}{\left|1\right\rangle_{S}}{\left|{11}\right\rangle_{E}}\\\
{\left|1\right\rangle_{S}}{\left|{00}\right\rangle_{E}}\to\sqrt{p}\sqrt{1-r}{\left|1\right\rangle_{S}}{\left|{00}\right\rangle_{E}}+\sqrt{pr}{\left|0\right\rangle_{S}}{\left|{10}\right\rangle_{E}}+\sqrt{1-p}{\left|1\right\rangle_{S}}{\left|{01}\right\rangle_{E}}\end{array}.$
(20)
For simplicity, we call the above channel parameterized by p,r as a GAD
channel of $\\{p,r\\}$.
Figure 2: The scheme for entanglement protection using weak measurement.
Initially, Alice prepare two qubits in an entangled states. Before sending the
two qubits, Alice perform partial collapse weak measurement on the 2 qubits
with the measurement parameter $m_{1}$ and $m_{2}$ respectively. After
obtaining his qubit, Bob (Charlie) does a weak measurement with the strength
$n_{1}$($n_{2}$). The concurrence can be optimized by adjusting the
measurement strength.
Suppose initially, Alice prepare the two qubits in an entangled state:
${\left|\phi\right\rangle_{in}}=\alpha\left|{00}\right\rangle+\beta\left|{11}\right\rangle.$
(21)
Then, Alice sends the two qubits to Bob and Charlie through two CAD channels
characterized by $\\{p_{1},r_{1}\\}$ and $\\{p_{2},r_{2}\\}$. After undergoing
the channels, the density matrix of the two qubits turns to be
${\rho_{C}}=\left({\begin{array}[]{*{20}{c}}a&0&0&e\\\ 0&b&0&0\\\ 0&0&c&0\\\
{{e^{*}}}&0&0&d\end{array}}\right)$ (22)
with
$\begin{split}a&=[1-{r_{1}}-{r_{2}}+{r_{1}}{r_{2}}+{p_{1}}{r_{1}}+{p_{2}}{r_{2}}-({p_{1}}+{p_{2}}){r_{1}}{r_{2}}\\\
&+{p_{1}}{p_{2}}{r_{1}}{r_{2}}]{\left|\alpha\right|^{2}}+{p_{1}}{p_{2}}{r_{1}}{r_{2}}{\left|\beta\right|^{2}},\\\
b&=[{r_{2}}-{r_{2}}{p_{2}}-{r_{1}}{r_{2}}+({p_{1}}+{p_{2}}){r_{1}}{r_{2}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}]{\left|\alpha\right|^{2}}\\\
&+({p_{1}}{r_{1}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}){\left|\beta\right|^{2}},\\\
c&=[{r_{1}}-{p_{1}}{r_{1}}-{r_{1}}{r_{2}}+({p_{1}}+{p_{2}}){r_{1}}{r_{2}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}]{\left|\alpha\right|^{2}}\\\
&+({p_{2}}{r_{2}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}){\left|\beta\right|^{2}},\\\
d&=(1-{p_{1}})(1-{p_{2}}){r_{1}}{r_{2}}{\left|\alpha\right|^{2}}\\\
&+(1-{p_{1}}{r_{1}}-{p_{2}}{r_{2}}+{p_{1}}{p_{2}}{r_{2}}{r_{2}}){\left|\beta\right|^{2}},\\\
e&=\alpha{\beta^{*}}\sqrt{1-{r_{1}}}\sqrt{1-{r_{2}}}\end{split}$ (23)
The concurrenceconcorrence of $\rho_{C}$ is
$\mathcal{C}({\rho_{C}})=\max\left\\{{0,{\Lambda_{1}}\equiv
2(\left|e\right|-\sqrt{bc})}\right\\}.$ (24)
When $\Lambda_{1}>0$, the concurrence is $\Lambda_{1}$, otherwise, the
concurrence is zero. To improve the entanglement Bob and Charlie shared, Alice
chooses weak measurements on both qubits before sending them through the
channel. The two-qubit weak measurement is a non-unitary operation which can
be written as,
$M=\left({\begin{array}[]{*{20}{c}}1&0\\\
0&{{m_{1}}}\end{array}}\right)\otimes\left({\begin{array}[]{*{20}{c}}1&0\\\
0&{{m_{2}}}\end{array}}\right)$ (25)
After obtaining the two qubits, Bob and Charlie does a weak measurement
individually. The second weak measurement can be written as,
$N=\left({\begin{array}[]{*{20}{c}}{{n_{1}}}&0\\\
0&0\end{array}}\right)\otimes\left({\begin{array}[]{*{20}{c}}{{n_{2}}}&0\\\
0&0\end{array}}\right).$ (26)
The final density matrix of the two qubits is
${\rho_{N}}=\frac{1}{P}\left({\begin{array}[]{*{20}{c}}{n_{1}^{2}n_{2}^{2}A}&0&0&{{n_{1}}{n_{2}}E}\\\
0&{n_{1}^{2}B}&0&0\\\ 0&0&{n_{2}^{2}C}&0\\\
{{n_{1}}{n_{2}}{E^{*}}}&0&0&D\end{array}}\right)$ (27)
with
$\begin{split}A&=[1-{r_{1}}-{r_{2}}+{r_{1}}{r_{2}}+{p_{1}}{r_{1}}+{p_{2}}{r_{2}}-({p_{1}}+{p_{2}}){r_{1}}{r_{2}}\\\
&+{p_{1}}{p_{2}}{r_{1}}{r_{2}}]{\left|\alpha\right|^{2}}+m_{1}^{2}m_{2}^{2}{p_{1}}{p_{2}}{r_{1}}{r_{2}}{\left|\beta\right|^{2}}\\\
&\equiv A_{0}+A_{1}m_{1}^{2}m_{2}^{2},\\\
B&=[{r_{2}}-{p_{2}}{r_{2}}-{r_{1}}{r_{2}}+({p_{1}}+{p_{2}}){r_{1}}{r_{2}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}]{\left|\alpha\right|^{2}}\\\
&+m_{1}^{2}m_{2}^{2}({p_{1}}{r_{1}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}){\left|\beta\right|^{2}},\\\
&\equiv B_{0}+B_{1}m_{1}^{2}m_{2}^{2}\\\
C&=[{r_{1}}-{p_{1}}{r_{1}}-{r_{1}}{r_{2}}+({p_{1}}+{p_{2}}){r_{1}}{r_{2}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}]{\left|\alpha\right|^{2}}\\\
&+m_{1}^{2}m_{2}^{2}({p_{2}}{r_{2}}-{p_{1}}{p_{2}}{r_{1}}{r_{2}}){\left|\beta\right|^{2}}\\\
&\equiv C_{0}+C_{1}m_{1}^{2}m_{2}^{2},\\\
D&=(1-{p_{1}})(1-{p_{2}}){r_{1}}{r_{2}}{\left|\alpha\right|^{2}}\\\
&+m_{1}^{2}m_{2}^{2}(1-{p_{1}}{r_{1}}-{p_{2}}{r_{2}}+{p_{1}}{p_{2}}{r_{2}}{r_{2}}){\left|\beta\right|^{2}}\\\
&\equiv D_{0}+D_{1}m_{1}^{2}m_{2}^{2},\\\
E&=\alpha{\beta^{*}}{m_{1}}{m_{2}}\sqrt{1-{r_{1}}}\sqrt{1-{r_{2}}}\end{split}$
(28)
and
$P=n_{1}^{2}n_{2}^{2}A+n_{1}^{2}B+n_{2}^{2}C+D.$ (29)
The overall success probability is
${P_{s}}{\rm{=}}P\prod\limits_{{\rm{c=\\{}}{{\rm{m}}_{1}}{\rm{,}}{{\rm{n}}_{1}}{\rm{,}}{{\rm{m}}_{2}}{\rm{,}}{{\rm{n}}_{2}}{\rm{\\}}}}{\min\\{1,\frac{1}{c^{2}}\\}}$
(30)
The corresponding concurrence is
$\mathcal{C}({\rho_{N}})=\max\left\\{{0,{\Lambda_{2}}\equiv\frac{{2{n_{1}}{n_{2}}(\left|E\right|-\sqrt{BC})}}{{n_{1}^{2}n_{2}^{2}A+n_{1}^{2}B+n_{2}^{2}C+D}}}\right\\}.$
(31)
We can show that $\Lambda_{2}$ gets its maximal value when the following
conditions are met (see Appendix C),
$\displaystyle{n_{1}}$ $\displaystyle=\sqrt[4]{{\frac{{CD}}{{AB}}}},$ (32a)
$\displaystyle{n_{2}}$ $\displaystyle=\sqrt[4]{{\frac{{BD}}{{AC}}}}$ (32b)
One can see that when the above equations are satisfied, the value of
$\Lambda_{2}$ changes only with $m_{1}m_{2}$,we can set $m_{2}=1$,i.e,the weak
measurement on the second qubit is not a necessary and the concurrence can be
optimized by adjusting $m\equiv m_{1}$. Substituting Eqs.(32) into the
expression of $\Lambda_{2}$, we get
$\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{{\left|E\right|-\sqrt{BC}}}{{\sqrt{BC}+\sqrt{AD}}}$ (33)
$\displaystyle=$
$\displaystyle\frac{|\alpha\beta|\sqrt{(1-r_{1})(1-r_{2})}-\sqrt{m^{2}B_{1}C_{1}+\frac{1}{m^{2}}B_{0}C_{0}+B_{1}C_{0}+B_{0}C_{1}}}{\sqrt{m^{2}A_{1}D_{1}+\frac{1}{m^{2}}A_{0}D_{0}+A_{1}D_{0}+A_{0}D_{1}}+\sqrt{m^{2}B_{1}C_{1}+\frac{1}{m^{2}}B_{0}C_{0}+B_{1}C_{0}+B_{0}C_{1}}}.$
In order to maximize the value of $\Lambda_{2}$, we need the following two
inequalities.
$\begin{split}&\sqrt{m^{2}B_{1}C_{1}+\frac{1}{m^{2}}B_{0}C_{0}+B_{1}C_{0}+B_{0}C_{1}}\\\
&\geq\sqrt{2\sqrt{B_{1}C_{1}B_{0}C_{0}}+B_{1}C_{0}+B_{0}C_{1}}\\\
&=\sqrt{B_{1}C_{0}}+\sqrt{B_{0}C_{1}},\end{split}$ (34)
the equality holds when
$m^{4}=\frac{B_{0}C_{0}}{B_{1}C_{1}}.$ (35)
And
$\begin{split}&\sqrt{m^{2}A_{1}D_{1}+\frac{1}{m^{2}}A_{0}D_{0}+A_{1}D_{0}+A_{0}D_{1}}\\\
&\geq\sqrt{2\sqrt{A_{1}D_{1}A_{0}D_{0}}+A_{1}D_{0}+A_{0}D_{1}}\\\
&=\sqrt{A_{1}D_{0}}+\sqrt{A_{0}D_{1}},\end{split}$
the equality holds when
$m^{4}=\frac{A_{0}D_{0}}{A_{1}D_{1}}.$ (36)
Considering the expressions of $A,B,C$ and $D$ presented in Eqs.(28), we can
easily find out that
$\begin{split}m&=\sqrt[4]{{\frac{{{B_{0}}{C_{0}}}}{{{B_{1}}{C_{1}}}}}}=\sqrt[4]{{\frac{{{A_{0}}{D_{0}}}}{{{A_{1}}{D_{1}}}}}}\\\
&=\sqrt[4]{{\frac{{(1-{p_{1}})(1-{p_{2}})(1-{r_{1}}+{r_{1}}{p_{1}})(1-{r_{2}}+{r_{2}}{p_{2}})}}{{{p_{1}}{p_{2}}(1-{r_{1}}{p_{1}})(1-{r_{2}}{p_{2}})}}}}\frac{{|\alpha|}}{{|\beta|}},\end{split}$
(37)
which means that the two inequalities in Eq.(34) and Eq.(III) take the
equality sigh with the same condition
$m^{4}=\frac{B_{0}C_{0}}{B_{1}C_{1}}=\frac{A_{0}D_{0}}{A_{1}D_{1}}$. With the
value of $m$ presented in Eq.(37), we can obtain the maximum value of
$\Lambda_{2}$ such that
$\overline{\Lambda}_{2}=\frac{\sqrt{(1-r_{1})(1-r_{2})}-r_{1}\sqrt{p_{1}(1-p_{1})(1-r_{2}p_{2})(1-r_{2}+r_{2}p_{2})}-r_{2}\sqrt{p_{2}(1-p_{2})(1-r_{1}p_{1})(1-r_{1}+r_{1}p_{1})}}{\left(r_{1}\sqrt{p_{1}(1-p_{1})}+\sqrt{(1-r_{1}p_{1})(1-r_{1}+r_{1}p_{1})}\right)\left(r_{2}\sqrt{p_{2}(1-p_{2})}+\sqrt{(1-r_{2}p_{2})(1-r_{2}+r_{2}p_{2})}\right)}.$
(38)
Then we get the optimal concurrence of the output state $\rho_{N}$
$C(\rho_{N})=\max\\{0,\overline{\Lambda}_{2}\\},$ (39)
with $n_{1},n_{2}$ given by Eqs.(32), $m_{1}=m$ given in Eq.(37), and
$m_{2}=1$. We approach a surprising result: the maximum concurrence does not
depend on the parameters $\alpha$ and $\beta$. Furthermore, under the
condition in Eq.(37), the value of $n_{1}$ and $n_{2}$ can be rewrite into
$\displaystyle n_{1}$ $\displaystyle=$
$\displaystyle\sqrt[4]{\frac{(c_{0}+c_{1}h)(d_{0}+d_{1}h)}{(a_{0}+a_{1}h)(b_{0}+b_{1}h)}},$
(40) $\displaystyle n_{2}$ $\displaystyle=$
$\displaystyle\sqrt[4]{\frac{(b_{0}+b_{1}h)(d_{0}+d_{1}h)}{(a_{0}+a_{1}h)(c_{0}+c_{1}h)}},$
(41)
where $x_{0}|\alpha|^{2}=X_{0},x_{1}|\beta|^{2}=X_{1}$, with $x=a,b,c,d$,
$X=A,B,C,D$, and
$h=\sqrt{\frac{b_{0}c_{0}}{b_{1}c_{1}}}=\sqrt{\frac{a_{0}d_{0}}{a_{1}d_{1}}}$.
This means that $n_{1}$ and $n_{2}$ also do not depend on $\alpha$ and
$\beta$. Then we can optimize the success probability $P_{s}$ by taking
$|\alpha|^{2}=\frac{1}{1+h}.$ (42)
When $\left|\alpha\right|=0$ or $\left|\beta\right|=0$, the success
probability is exactly zero, which means that one can not produce quantum
entanglement only by local operations if there is no entanglement initially.
In Fig.3,we show how the concurrence changes with $m$. We set
$p_{1}=0.9,r_{1}=0.5,p_{2}=0.95,r_{2}=0.3$. One can see that the maximal value
of the concurrence is 0.53, the corresponding measurement parameter is
$m=0.34$, $n_{1}=0.50$ and $n_{2}=0.44$. The success probability is about
$0.06$. Without any weak measurement, the concurrence is about 0.33. The
concurrence can really be enhanced in the pay of success probability. We also
notice that when $p=1$,then the GAD channel reduced to a AD channel, then the
optimal value m and n tends to be zero which means the measurements become
strong measurements. This coincide with the result of Ref.nature .
We have to stress that, in GAD channel, weak measurement can help to
circumvent the ”entanglement sudden death”. In certain conditions,
$\Lambda_{1}$ can be smaller than zero,thus the concurrence become 0, by
choosing proper weak measurement parameters m and n, $\Lambda_{2}$ can be made
non-zero under the same conditions ,e.g,$p_{1}=p_{2}=0.7,r_{1}=r_{2}=0.61$. We
note that, in such a condition, we can not get any entanglement using the
previous schemes in Ref.pra ; nature .
Figure 3: The concurrence with different m when the initial state of the two
qubits is a maximally entangled state
${\left|\psi\right\rangle_{0}}=\frac{1}{{\sqrt{2}}}\left({\left|{00}\right\rangle+\left|{11}\right\rangle}\right)$
and the channel parameters are:$p_{1}=0.9,r_{1}=0.5,p_{2}=0.95,r_{2}=0.3$. One
can see that there is a optimal value for the concurrence at m=0.34. This is
in agrement with Eq.(37)
## IV conclusion
We have studied how weak measurement can be used for quantum state and
entanglement protection exposed to environment with finite temperature. We
found that the pre-channel and post-channel weak measurement are useful to
battle with decoherence in generalized amplitude damping channels. For
equatorial states, we give the optimal measurement strength in analysis
formate. We have also shown that weak measurement are useful in protecting
entanglement in finite temperature environment. When setting $p_{1}=1$ and
$p_{2}=1$, our conclusion coincide with the previous resultspra ; nature .
## ACKNOWLEDGEMENT
We acknowledge the support from the 10000-Plan of Shandong province, the
National High-Tech Program of China Grants No. 2011AA010800 and No.
2011AA010803 and NSFC Grants No. 11174177 and No. 60725416.
## APPENDIX A
To explicate that ${F_{\max}}\geq{F_{0}}$, we have to study the function
$G(p,r)=\sqrt{(1-rp)(1-r+rp)}+r\sqrt{p(1-p)}$. To obtain the maximal value of
$G(p,r)$, we have to solve the equation:
$\left\\{\begin{array}[]{l}{\partial_{p}}G=\frac{1}{2}r(1-2p)\left({\frac{1}{{\sqrt{p(1-p)}}}+\frac{r}{{\sqrt{1-r+(1-p)p{r^{2}}}}}}\right)=0\\\
{\partial_{r}}G=\sqrt{p(1-p)}-\frac{{1-2pr+2{p^{2}}r}}{{2\sqrt{1-r+(1-p)p{r^{2}}}}}=0\end{array}\right.$
(43)
We can find that $G(p,r)$ has the maximal value 1 if and only if $r=0$ or
$p=\frac{1}{2}$,such that ${F_{\max}}\geq{F_{0}}$. When $p=0$ or $p=1$, G has
the minimal value $\sqrt{1-r}$, and $F_{\max}$ can be as large as 1.
## APPENDIX B
In this Appendix, we want to proof that when Eq.(15) is satisfied,
${\overline{F}^{(w)}}$ has the maximal value. In Eq.(20), the variable $m$
only appears in the last term which is just $F_{e}$. We know that,
$F_{e}^{(w)}\leq\frac{1}{2}+\frac{{\sqrt{1-r}}}{{2\sqrt{(1-rp+{n^{2}}rp)[r-rp+{n^{2}}(1-r+rp)}}}$
(44)
and the equality obtained when
$m=\sqrt{\frac{{(1-p)r+{n^{2}}(1-r+pr)}}{{1-pr+{n^{2}}pr}}}.$ (45)
Taking the relation above into the mean fidelity given in Eq.(20), we have,
${\overline{F}^{(w)}}=\frac{1}{3}+\frac{1}{6}\left({\frac{{{n^{2}}(1-r+rp)}}{{r-rp+{n^{2}}(1-r+rp)}}+\frac{{1-rp}}{{1-rp+{n^{2}}rp}}+\frac{{2\sqrt{1-r}}}{{\sqrt{(1-rp+{n^{2}}rp)[r-rp+{n^{2}}(1-r+rp)}}}}\right).$
(46)
One can find that,
${\overline{F}^{(w)}}\leq\frac{1}{3}+\frac{1}{6}{\left[{1+\frac{1}{{\sqrt{{r^{2}}p(1-p)}+\sqrt{(1-rp)(1-r+rp)}}}}\right]^{2}}.$
(47)
and the equality obtained when
$n=\sqrt[4]{{\frac{{(1-p)(1-rp)}}{{p(1-r+rp)}}}}$, Substituting the value of n
into Eq.(36),we have $m=\sqrt[4]{{\frac{{(1-p)(1-r+pr)}}{{p(1-pr)}}}}$.
## APPENDIX C
In this section, we give a proof that when $\Lambda_{2}$ has the maximal
value, then Eq.(33) should be
satisfied.${\Lambda_{2}}=\frac{{2{n_{1}}{n_{2}}(\left|E\right|-\sqrt{BC})}}{{n_{1}^{2}n_{2}^{2}A+n_{1}^{2}B+n_{2}^{2}C+D}}\leq\frac{{2{n_{1}}{n_{2}}(\left|E\right|-\sqrt{BC})}}{{n_{1}^{2}n_{2}^{2}A+2{n_{1}}{n_{2}}\sqrt{BC}+D}}$,and
”=” stands iff $\frac{{{n_{1}}}}{{{n_{2}}}}=\sqrt{\frac{C}{D}}$. And
$\frac{{2{n_{1}}{n_{2}}(\left|E\right|-\sqrt{BC})}}{{n_{1}^{2}n_{2}^{2}A+2{n_{1}}{n_{2}}\sqrt{BC}+D}}=\frac{{2(\left|E\right|-\sqrt{BC})}}{{{n_{1}}{n_{2}}A+2\sqrt{BC}+\frac{D}{{{n_{1}}{n_{2}}}}}}\leq\frac{{\left|E\right|-\sqrt{BC}}}{{\sqrt{BC}+\sqrt{AD}}}$,
the = stands iff ${n_{1}}{n_{2}}=\sqrt{\frac{D}{A}}$. Thus when $\Lambda_{2}$
has the maximal value
$\frac{{\left|E\right|-\sqrt{BC}}}{{\sqrt{BC}+\sqrt{AD}}}$, We should have
Eq.(33).
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|
arxiv-papers
| 2013-08-07T14:08:42 |
2024-09-04T02:49:49.174293
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S.C. Wang, Z.W. Yu, and X.B. Wang",
"submitter": "Xiang-Bin Wang",
"url": "https://arxiv.org/abs/1308.1665"
}
|
1308.1707
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-146 LHCb-PAPER-2013-037 7 August 2013
Measurement of form-factor independent observables in the decay
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
We present a measurement of form-factor independent angular observables in the
decay $B^{0}\rightarrow K^{*}(892)^{0}\mu^{+}\mu^{-}$. The analysis is based
on a data sample corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$, collected by the LHCb experiment in $pp$ collisions at
a center-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$. Four observables
are measured in six bins of the dimuon invariant mass squared, $q^{2}$, in the
range $0.1<q^{2}<19.0$${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$.
Agreement with Standard Model predictions is found for 23 of the 24
measurements. A local discrepancy, corresponding to $3.7$ Gaussian standard
deviations, is observed in one $q^{2}$ bin for one of the observables.
Considering the 24 measurements as independent, the probability to observe
such a discrepancy, or larger, in one is $0.5\%$.
Submitted to Phys. Rev. Lett.
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P.
Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52,
V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A.
Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S. Farry51, D.
Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S.
Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R.
Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph.
Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30,
A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, C. Gotti20, M.
Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35,
G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O.
Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Hess60, A.
Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G.
Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J.
van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S.
Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M.
Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O.
Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U.
Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T. Palczewski27, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B.
Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8,
G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N.
Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina Rios36, E.
Santovetti23,k, M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A. Satta23, M.
Savrie16,e, D. Savrina30,31, P. Schaack52, M. Schiller41, H. Schindler37, M.
Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A. Schopper37, M.-H.
Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba24, M. Seco36, A.
Semennikov30, K. Senderowska26, I. Sepp52, N. Serra39, J. Serrano6, P.
Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T.
Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires9, R.
Silva Coutinho47, M. Sirendi46, T. Skwarnicki58, N.A. Smith51, E. Smith54,48,
J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D.
Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F.
Stagni37, S. Stahl11, O. Steinkamp39, S. Stevenson54, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V.
Vagnoni14, G. Valenti14, A. Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P.
Vazquez Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M.
Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X.
Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A.
Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R.
Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D.
Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D.
Wiedner11, L. Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55,
F.F. Wilson48, J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, S.A.
Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3,
R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L.
Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
The rare decay $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$, where $K^{*0}$
indicates the $K^{*}(892)^{0}\rightarrow K^{+}\pi^{-}$ decay, is a flavor-
changing neutral current process that proceeds via loop and box amplitudes in
the Standard Model (SM). In extensions of the SM, contributions from new
particles can enter in competing amplitudes and modify the angular
distributions of the decay products. This decay has been widely studied from
both theoretical [1, 2, 3] and experimental [4, 5, 6, 7] perspectives. Its
angular distribution is described by three angles ($\theta_{\ell}$,
$\theta_{K}$ and $\phi$) and the dimuon invariant mass squared, $q^{2}$;
$\theta_{\ell}$ is the angle between the flight direction of the $\mu^{+}$
($\mu^{-}$) and the $B^{0}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) meson in the dimuon rest frame;
$\theta_{K}$ is the angle between the flight direction of the charged kaon and
the $B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) meson in the
$K^{*0}$ ($\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) rest frame; and
$\phi$ is the angle between the decay planes of the $K^{*0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$) and the dimuon system in the
$B^{0}$ ($\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) meson rest frame.
A formal definition of the angles can be found in Ref. [7]. Using the
definitions of Ref. [1] and summing over $B^{0}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mesons, the differential angular
distribution can be written as
where the $q^{2}$ dependent observables $F_{\rm L}$ and $S_{i}$ are bilinear
combinations of the $K^{*0}$ decay amplitudes. These in turn are functions of
the Wilson coefficients, which contain information about short distance
effects and are sensitive to physics beyond the SM, and form-factors, which
depend on long distance effects. Combinations of $F_{\rm L}$ and $S_{i}$ with
reduced form-factor uncertainties have been proposed independently by several
authors [8, 9, 2, 3, 10]. In particular, in the large recoil limit
(low-$q^{2}$) the observables denoted as $P_{4}^{\prime}$, $P_{5}^{\prime}$,
$P_{6}^{\prime}$ and $P_{8}^{\prime}$ [11] are largely free from form-factor
uncertainties. These observables are defined as
$P_{i=4,5,6,8}^{\prime}=\frac{S_{j=4,5,7,8}}{\sqrt{F_{\rm L}(1-F_{\rm L})}}.$
(2)
This Letter presents the measurement of the observables $S_{j}$ and the
respective observables $P_{i}^{\prime}$. This is the first measurement of
these quantities by any experiment. Moreover, these observables provide
complementary information about physics beyond the SM with respect to the
angular observables previously measured in this decay [4, 5, 6, 7]. The data
sample analyzed corresponds to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$ of $pp$ collisions at a center-of-mass energy of 7 TeV
collected by the LHCb experiment in 2011. Charged conjugation is implied
throughout this Letter, unless otherwise stated.
The LHCb detector [12] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of approximately $4{\rm\,Tm}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The combined tracking system provides a momentum measurement with
relative uncertainty that varies from 0.4% at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov detectors
[13]. Muons are identified by a system composed of alternating layers of iron
and multiwire proportional chambers [14].
The trigger [15] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, which applies a
full event reconstruction. Candidate events for this analysis are required to
pass a hardware trigger, which selects muons with $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the software trigger, at
least one of the final state particles is required to have both $\mbox{$p_{\rm
T}$}>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter larger than
$100\,\upmu\rm m$ with respect to all of the primary $pp$ interaction vertices
in the event. Finally, the tracks of two or more of the final state particles
are required to form a vertex that is significantly displaced from the primary
vertex.
Simulated events are used in several stages of the analysis, $pp$ collisions
are generated using Pythia 6.4 [16] with a specific LHCb configuration [17].
Decays of hadronic particles are described by EvtGen [18], in which final
state radiation is generated using Photos [19]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [20, *Agostinelli:2002hh] as described in Ref. [22]. This
analysis uses the same selection and acceptance correction technique as
described in Ref. [7].
Signal candidates are required to pass a loose preselection: the $B^{0}$
vertex is required to be well separated from the primary $pp$ interaction
point; the impact parameter with respect to the primary $pp$ interaction point
is required to be small for the $B^{0}$ candidate and large for the final
state particles; and the angle between the $B^{0}$ momentum and the vector
from the primary vertex to the $B^{0}$ decay vertex is required to be small.
Finally, the reconstructed invariant mass of the $K^{*0}$ candidate is
required to be in the range
$792<m_{K\pi}<992$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To further
reject combinatorial background events, a boosted decision tree (BDT) [23]
using the AdaBoost algorithm [24] is applied. The BDT combines kinematic and
geometrical properties of the event.
Several sources of peaking background have been considered. The decays
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{0}\\!\rightarrow\psi{(2S)}K^{*0}$, where the charmonium resonances decay
into a muon pair, are rejected by vetoing events for which the dimuon system
has an invariant mass ($m_{\mu\mu}$) in the range
$2946-3176$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or
$3586-3766$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Both vetoes are
extended downwards by 150${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$B^{0}$ candidates with invariant mass ($m_{K\pi\mu\mu}$) in the range
$5150-5230$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ to account for the
radiative tails of the charmonium resonances. They are also extended upwards
by 25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for candidates with
$5370<m_{K\pi\mu\mu}<5470$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, to
account for non-Gaussian reconstruction effects. Backgrounds from
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
with the kaon or pion from the $K^{*0}$ decay and one of the muons from the
$J/\psi$ meson being misidentified and swapped with each other, are rejected
by assigning the muon mass hypothesis to the $K^{+}$ or $\pi^{-}$ and vetoing
candidates for which the resulting invariant mass is in the range
$3036<m_{\mu\mu}<3156$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Background
from $B^{0}_{s}\rightarrow\phi(\rightarrow K^{+}K^{-})\mu^{+}\mu^{-}$ decays
is removed by assigning the kaon mass hypothesis to the pion candidate and
rejecting events for which the resulting invariant mass $K^{+}K^{-}$ is
consistent with the $\phi$ mass. A similar veto is applied to remove
$\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar
28931\relax(1520)(\rightarrow pK^{-})\mu^{+}\mu^{-}$ events. After these
vetoes, the remaining peaking background is estimated to be negligibly small.
It has been verified with the simulation that these vetos do not bias the
angular observables. In total, 883 signal candidates are observed in the range
$0.1<q^{2}<19.0$${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, with a signal
over background ratio of about 5.
Detector acceptance effects are accounted for by weighting the candidates with
the inverse of their efficiency. The efficiency is determined as a function of
the three angles and $q^{2}$ by using a large sample of simulated events and
assuming factorization in the three angles. Possible non-factorizable
acceptance effects are evaluated and included in the systematic uncertainties.
Several control channels, in particular the decay
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$, which
has the same final state as the signal, are used to verify the agreement
between data and simulation.
Due to the limited number of signal candidates in this dataset, we do not fit
the data to the full differential distribution of Eq. LABEL:eq:masterformula.
In Ref. [7], the data were “folded” at $\phi=0$ ($\phi\rightarrow\phi+\pi$ for
$\phi<0$) to reduce the number of parameters in the fit, while cancelling the
terms containing $\sin{\phi}$ and $\cos{\phi}$. Here, similar folding
techniques are applied to specific regions of the three-dimensional angular
space to exploit the (anti)-symmetries of the differential decay rate with
respect to combinations of angular variables. This simplifies the differential
decay rate without losing experimental sensitivity. This technique is
discussed in more detail in Ref. [25]. The following sets of transformations
are used to determine the observables of interest
$\displaystyle\text{$P_{4}^{\prime}$, $S_{4}$:
}\begin{cases}\phi\rightarrow-\phi&\text{~{}for~{}}\phi<0\\\
\phi\rightarrow\pi-\phi&\text{~{}for~{}}\theta_{\ell}>\pi/2\\\
\theta_{\ell}\rightarrow\pi-\theta_{\ell}&\text{~{}for~{}}\theta_{\ell}>\pi/2,\end{cases}$
(3) $\displaystyle\text{$P_{5}^{\prime}$, $S_{5}$:
}\begin{cases}\phi\rightarrow-\phi&\text{~{}for~{}}\phi<0\\\
\theta_{\ell}\rightarrow\pi-\theta_{\ell}&\text{~{}for~{}}\theta_{\ell}>\pi/2,\end{cases}$
(4) $\displaystyle\text{$P_{6}^{\prime}$, $S_{7}$:
}\begin{cases}\phi\rightarrow\pi-\phi&\text{~{}for~{}}\phi>\pi/2\\\
\phi\rightarrow-\pi-\phi&\text{~{}for~{}}\phi<-\pi/2\\\
\theta_{\ell}\rightarrow\pi-\theta_{\ell}&\text{~{}for~{}}\theta_{\ell}>\pi/2,\end{cases}$
(5) $\displaystyle\text{$P_{8}^{\prime}$, $S_{8}$:
}\begin{cases}\phi\rightarrow\pi-\phi&\text{~{}for~{}}\phi>\pi/2\\\
\phi\rightarrow-\pi-\phi&\text{~{}for~{}}\phi<-\pi/2\\\
\theta_{K}\rightarrow\pi-\theta_{K}&\text{~{}for~{}}\theta_{\ell}>\pi/2\\\
\theta_{\ell}\rightarrow\pi-\theta_{\ell}&\text{~{}for~{}}\theta_{\ell}>\pi/2.\end{cases}$
(6)
Each transformation preserves the first five terms and the corresponding
$S_{i}$ term in Eq. LABEL:eq:masterformula, and cancels the other angular
terms. Thus, the resulting angular distributions depend only on $F_{\rm L}$,
$S_{3}$ and one of the observables $S_{4,5,7,8}$.
Four independent likelihood fits to the $B^{0}$ invariant mass and the
transformed angular distributions are performed to extract the observables
$P_{i}^{\prime}$ and $S_{i}$. The signal invariant mass shape is parametrized
with the sum of two Crystal Ball functions [26], where the parameters are
extracted from the fit to
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ decays
in data. The background invariant mass shape is parametrized with an
exponential function, while its angular distribution is parametrized with the
direct product of three second-order polynomials, dependent on $\phi$,
$\cos{\theta_{K}}$ and $\cos{\theta_{\ell}}$. The angular observables $F_{\rm
L}$ and $S_{3}$ are allowed to vary in the angular fit and are treated as
nuisance parameters in this analysis. Their fit values agree with Ref. [7].
The presence of a $K^{+}\pi^{-}$ system in an S-wave configuration, due to a
non-resonant contribution or to feed-down from $K^{+}\pi^{-}$ scalar
resonances, results in additional terms in the differential angular
distribution. Denoting the right-hand side of Eq. LABEL:eq:masterformula by
$W_{\rm P}$, the differential decay rate takes the form
$\begin{split}(1-F_{\rm S})W_{\text{P}}+\frac{9}{32\pi}\left(W_{\rm S}+W_{\rm
SP}\right),\end{split}$ (7)
where
$\begin{split}W_{\rm S}=\frac{2}{3}F_{\rm S}\sin^{2}\theta_{\ell}\end{split}$
(8)
and $W_{\rm SP}$ is given by
$\begin{split}&\left.\frac{4}{3}A_{\mathrm{S}}\sin^{2}\theta_{\ell}\cos\theta_{K}+A_{\mathrm{S}}^{(4)}\sin\theta_{K}\sin
2\theta_{\ell}\cos\phi+\right.\\\
&\left.A_{\mathrm{S}}^{(5)}\sin\theta_{K}\sin\theta_{\ell}\cos\phi+A_{\mathrm{S}}^{(7)}\sin\theta_{K}\sin\theta_{\ell}\sin\phi\right.\\\
&\left.+A_{\mathrm{S}}^{(8)}\sin\theta_{K}\sin
2\theta_{\ell}\sin\phi~{}.~{}\right.\end{split}$ (9)
The factor $F_{\rm S}$ is the fraction of the S-wave component in the $K^{*0}$
mass window, and $W_{\rm SP}$ contains all the interference terms,
$A_{\mathrm{S}}^{(i)}$, of the S-wave with the $K^{*0}$ transversity
amplitudes as defined in Ref. [27]. In Ref. [7], $F_{\rm S}$ was measured to
be less than $0.07$ at $68\%$ confidence level. The maximum value that the
quantities $A_{\mathrm{S}}^{(i)}$ can assume is a function of $F_{\rm S}$ and
$F_{\rm L}$ [11]. The S-wave contribution is neglected in the fit to data, but
its effect is evaluated and assigned as a systematic uncertainty using pseudo-
experiments. A large number of pseudo-experiments with $F_{\rm S}=0.07$ and
with the interference terms set to their maximum allowed values are generated.
All other parameters, including the angular observables, are set to their
measured values in data. The pseudo-experiments are fitted ignoring S-wave and
interference contributions. The corresponding bias in the measurement of the
angular observables is assigned as a systematic uncertainty.
Table 1: Measurement of the observables $P_{4,5,6,8}^{\prime}$ and $S_{4,5,7,8}$ in the six $q^{2}$ bins of the analysis. For the observables $P_{i}^{\prime}$ the measurement in the $q^{2}$-bin $1.0<q^{2}<6.0$ ${\mathrm{\,Ge\kern-0.90005ptV^{2}\\!/}c^{4}}$, which is the theoretically preferred region at large recoil, is also reported. The first uncertainty is statistical and the second is systematic. $q^{2}$[${\mathrm{\,Ge\kern-0.90005ptV^{2}\\!/}c^{4}}$ ] | $P_{4}^{\prime}$ | $P_{5}^{\prime}$ | $P_{6}^{\prime}$ | $P_{8}^{\prime}$
---|---|---|---|---
$\phantom{0}0.10\phantom{0}-\phantom{0}2.00$ | $\phantom{0}\;0.00^{+0.26}_{-0.26}\pm 0.03$ | $\phantom{0}\;0.45^{+0.19}_{-0.22}\pm 0.09$ | $-0.24^{+0.19}_{-0.22}\pm 0.05$ | $-0.06^{+0.28}_{-0.28}\pm 0.02$
$\phantom{0}2.00\phantom{0}-\phantom{0}4.30$ | $-0.37^{+0.29}_{-0.26}\pm 0.08$ | $\phantom{0}\;0.29^{+0.39}_{-0.38}\pm 0.07$ | $\phantom{0}\;0.15^{+0.36}_{-0.38}\pm 0.05$ | $-0.15^{+0.29}_{-0.28}\pm 0.07$
$\phantom{0}4.30\phantom{0}-\phantom{0}8.68$ | $-0.59^{+0.15}_{-0.12}\pm 0.05$ | $-0.19^{+0.16}_{-0.16}\pm 0.03$ | $-0.04^{+0.15}_{-0.15}\pm 0.05$ | $\phantom{0}\;0.29^{+0.17}_{-0.19}\pm 0.03$
$10.09\phantom{0}-12.90$ | $-0.46^{+0.20}_{-0.17}\pm 0.03$ | $-0.79^{+0.16}_{-0.19}\pm 0.19$ | $-0.31^{+0.23}_{-0.22}\pm 0.05$ | $-0.06^{+0.23}_{-0.22}\pm 0.02$
$14.18\phantom{0}-16.00$ | $\phantom{0}\;0.09^{+0.35}_{-0.27}\pm 0.04$ | $-0.79^{+0.20}_{-0.13}\pm 0.18$ | $-0.18^{+0.25}_{-0.24}\pm 0.03$ | $-0.20^{+0.30}_{-0.25}\pm 0.03$
$16.00\phantom{0}-19.00$ | $-0.35^{+0.26}_{-0.22}\pm 0.03$ | $-0.60^{+0.19}_{-0.16}\pm 0.09$ | $\phantom{0}\;0.31^{+0.38}_{-0.37}\pm 0.10$ | $\phantom{0}\;0.06^{+0.26}_{-0.27}\pm 0.03$
$\phantom{0}1.00\phantom{0}-\phantom{0}6.00$ | $-0.29^{+0.18}_{-0.16}\pm 0.03$ | $\phantom{0}\;0.21^{+0.20}_{-0.21}\pm 0.03$ | $-0.18^{+0.21}_{-0.21}\pm 0.03$ | $\phantom{0}\;0.23^{+0.18}_{-0.19}\pm 0.02$
$q^{2}$[${\mathrm{\,Ge\kern-0.90005ptV^{2}\\!/}c^{4}}$ ] | $S_{4}$ | $S_{5}$ | $S_{7}$ | $S_{8}$
$\phantom{0}0.10\phantom{0}-\phantom{0}2.00$ | $\phantom{0}\;0.00^{+0.12}_{-0.12}\pm 0.03$ | $\phantom{0}\;0.22^{+0.09}_{-0.10}\pm 0.04$ | $-0.11^{+0.11}_{-0.11}\pm 0.03$ | $-0.03^{+0.13}_{-0.12}\pm 0.01$
$\phantom{0}2.00\phantom{0}-\phantom{0}4.30$ | $-0.14^{+0.13}_{-0.12}\pm 0.03$ | $\phantom{0}\;0.11^{+0.14}_{-0.13}\pm 0.03$ | $\phantom{0}\;0.06^{+0.15}_{-0.15}\pm 0.02$ | $-0.06^{+0.12}_{-0.12}\pm 0.02$
$\phantom{0}4.30\phantom{0}-\phantom{0}8.68$ | $-0.29^{+0.06}_{-0.06}\pm 0.02$ | $-0.09^{+0.08}_{-0.08}\pm 0.01$ | $-0.02^{+0.07}_{-0.08}\pm 0.04$ | $\phantom{0}\;0.15^{+0.08}_{-0.08}\pm 0.01$
$10.09\phantom{0}-12.90$ | $-0.23^{+0.09}_{-0.08}\pm 0.02$ | $-0.40^{+0.08}_{-0.10}\pm 0.10$ | $-0.16^{+0.11}_{-0.12}\pm 0.03$ | $-0.03^{+0.10}_{-0.10}\pm 0.01$
$14.18\phantom{0}-16.00$ | $\phantom{0}\;0.04^{+0.14}_{-0.08}\pm 0.01$ | $-0.38^{+0.10}_{-0.09}\pm 0.09$ | $-0.09^{+0.13}_{-0.14}\pm 0.01$ | $-0.10^{+0.13}_{-0.12}\pm 0.02$
$16.00\phantom{0}-19.00$ | $-0.17^{+0.11}_{-0.09}\pm 0.01$ | $-0.29^{+0.09}_{-0.08}\pm 0.04$ | $\phantom{0}\;0.15^{+0.16}_{-0.15}\pm 0.03$ | $\phantom{0}\;0.03^{+0.12}_{-0.12}\pm 0.02$
Figure 1: Measured values of $P_{4}^{\prime}$ and $P_{5}^{\prime}$ (black
points) compared with SM predictions from Ref. [11] (blue bands).
The results of the angular fits to the data are presented in Table 1. The
statistical uncertainties are determined using the Feldman-Cousins method
[28]. The systematic uncertainty takes into account the limited knowledge of
the angular acceptance, uncertainties in the signal and background invariant
mass models, the angular model for the background, and the impact of a
possible S-wave amplitude. Effects due to $B^{0}$/$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ production asymmetry have been
considered and found negligibly small. The comparison between the measurements
and the theoretical predictions from Ref. [11] are shown in Fig. 1 for the
observables $P_{4}^{\prime}$ and $P_{5}^{\prime}$. The observables
$P_{6}^{\prime}$ and $P_{8}^{\prime}$ (as well as $S_{7}$ and $S_{8}$) are
suppressed by the small size of the strong phase difference between the decay
amplitudes, and therefore are expected to be close to zero across the whole
$q^{2}$ region.
In general, the measurements agree with SM expectations [11], apart from a
sizeable discrepancy in the interval
$4.30<q^{2}<8.68$${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for the
observable $P_{5}^{\prime}$. The $p$-value, calculated using pseudo-
experiments, with respect to the upper bound of the theoretical predictions
given in Ref. [11], for the observed deviation is $0.02\%$, corresponding to
$3.7$ Gaussian standard deviations ($\sigma$). If we consider the 24
measurements as independent, the probability that at least one varies from the
expected value by $3.7\,\sigma$ or more is approximately $0.5\%$. A
discrepancy of $2.5\,\sigma$ is observed integrating over the region
$1.0<q^{2}<6.0$${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ (see Table 1),
which is considered the most robust region for theoretical predictions at
large recoil. The discrepancy is also observed in the observable $S_{5}$. The
value of $S_{5}$ quantifies the asymmetry between decays with positive and
negative value of $\cos{\theta_{K}}$ for $|\phi|<\pi/2$, averaged with the
opposite asymmetry of events with $|\phi|>\pi/2$ [1]. As a cross check, this
asymmetry was also determined from a counting analysis. The result is
consistent with the value for $S_{5}$ determined from the fit. It is worth
noting that the predictions for the first two $q^{2}$-bins and for the region
$1.0<q^{2}<6.0$${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ are also
calculated in Ref. [29], where power corrections to the QCD factorization
framework and resonance contributions are considered. However, there is not
yet in the literature unanimous consensus about the best approach to treat
these power corrections. The technique used in Ref. [29] leads to a larger
theoretical uncertainty with respect to Ref. [11].
In conclusion, we measure for the first time the angular observables $S_{4}$,
$S_{5}$, $S_{7}$, $S_{8}$ and the corresponding form-factor independent
observables $P_{4}^{\prime}$, $P_{5}^{\prime}$, $P_{6}^{\prime}$ and
$P_{8}^{\prime}$ in the decay $B^{0}\\!\rightarrow K^{*0}\mu^{+}\mu^{-}$.
These measurements have been performed in six $q^{2}$ bins for each of the
four observables. Agreement with SM predictions [11] is observed for 23 of the
24 measurements, while a local discrepancy of $3.7\,\sigma$ is observed in the
interval $4.30<q^{2}<8.68$${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for
the observable $P_{5}^{\prime}$. Integrating over the region
$1.0<q^{2}<6.0$${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, the observed
discrepancy in $P_{5}^{\prime}$ is $2.5\,\sigma$. The observed discrepancy in
the angular observable $P_{5}^{\prime}$ could be caused by a smaller value of
the Wilson coefficient $C_{9}$ with respect to the SM, as has been suggested
to explain some other small inconsistencies between the $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ data [7] and SM predictions [30]. Measurements with more
data and further theoretical studies will be important to draw more definitive
conclusions about this discrepancy.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-08-07T21:59:34 |
2024-09-04T02:49:49.182845
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana,\n F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S.\n Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, C. Gotti, M. Grabalosa G\\'andara, R. Graciani Diaz,\n L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando\n Morata, E. van Herwijnen, M. Hess, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N.\n Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R.\n Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M.\n Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S.\n Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R.\n Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N.\n Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J.\n Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A.\n Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins\n Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E.\n Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J.\n Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, S.\n Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. 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"submitter": "Nicola Serra",
"url": "https://arxiv.org/abs/1308.1707"
}
|
1308.1799
|
# Coupling of exciton states as the origin of their biexponential decay
dynamics in GaN nanowires
Christian Hauswald [email protected] Timur Flissikowski Tobias Gotschke
Raffaella Calarco Lutz Geelhaar Holger T. Grahn Oliver Brandt Paul-Drude-
Institut für Festkörperelektronik, Hausvogteiplatz 5–7, 10117 Berlin, Germany
###### Abstract
Using time-resolved photoluminescence spectroscopy, we explore the transient
behavior of bound and free excitons in GaN nanowire ensembles. We investigate
samples with distinct diameter distributions and show that the pronounced
biexponential decay of the donor-bound exciton observed in each case is not
caused by the nanowire surface. At long times, the individual exciton
transitions decay with a common lifetime, which suggests a strong coupling
between the corresponding exciton states. A system of non-linear rate-
equations taking into account this coupling directly reproduces the
experimentally observed biexponential decay.
###### pacs:
71.35.-y, 78.67.Uh,78.55.Cr,78.47.jd
Spontaneously formed GaN nanowires (NWs) exhibit a high structural perfection
regardless of the substrate used.Geelhaar _et al._ (2011) Their geometry
inhibits the propagation of dislocations along the NW axis, and the material
is thus indeed virtually free of threading dislocations, which plague
epitaxial GaN films.Bennett (2010) Hence, it is expected that the exciton
lifetimes of GaN NWs rival those measured for the highest quality epitaxial
GaN layers available to date.Morkoç (2001); *Scajev2012b However,
photoluminescence (PL) transients obtained for GaN NWs in time-resolved
experiments do not generally exhibit a monoexponential decay as expected for a
single excitonic transition. Instead, bi- and nonexponential transients were
obtained,Yoo _et al._ (2006); Corfdir _et al._ (2009); Korona _et al._
(2012); Gorgis _et al._ (2012) which impede the extraction of a single
lifetime. Analogous observations were made for ZnO NWs.Wischmeier _et al._
(2006); Zhao _et al._ (2008) This nonexponential decay was attributed to
surface-related effects by different groups.Wischmeier _et al._ (2006); Zhao
_et al._ (2008); Corfdir _et al._ (2009); Park _et al._ (2009); Gorgis _et
al._ (2012) In fact, single GaN NWs with a very high surface-to-volume ratio
were recently shown to exhibit individual single exponential decays,Gorgis
_et al._ (2012) and their superposition in ensemble measurements thus
inevitably results in a nonexponential transient.
In the present article, we investigate the exciton decay dynamics in GaN NWs
of larger diameter. We focus on two different ordered NW arrays having narrow
diameter distributions and on one spontaneously formed NW ensemble with a
broad diameter distribution. The dominant radiative transition decays
biexponentially for each of these samples. Neither a spectral superposition of
different states nor the NW surface are responsible for these biexponential
transients. Instead, we show that it is the coupling of all exciton states
participating in recombination which determines their temporal evolution. This
insight allows us to extract the actual lifetime of the donor-bound exciton
from our experimental results. For low excitation, the values obtained are
much below the radiative lifetimes of at least 1 ns measured in free-standing
GaN layersMonemar _et al._ (2008, 2010) and are thus governed by a
nonradiative decay channel that is not related to the NW surface.
Figure 1: Low-temperature PL spectrum and TRPL transient of sample A
$\left[\text{(a) and (d)}\right]$, sample B $\left[\text{(b) and (e)}\right]$,
and sample C $\left[\text{(c) and (f)}\right]$, respectively. The spectra in
(a)–(c) are dominated by transitions due to donor-bound excitons
[($D^{0},X_{\text{A}}$)], but also acceptor-bound [($A^{0},X_{\text{A}}$)] and
free ($X_{\text{A}}$) exciton transitions are observed. The shaded areas
indicate the spectral range of integration used for obtaining the TRPL
transients displayed in (d)–(f). The decay times given next to the transients
have been extracted by a fit (solid line) of the experimental data with a
phenomenological biexponential decay convoluted with the system response
function. The insets show the diameter distribution of the respective NW
ensemble. The mean NW diameter and the full width at half maximum of the
respective histogram (grey bars) are obtained by fits (solid lines) with a
normal distribution for samples A and B and a shifted Gamma distribution for
sample C.
The three GaN NW ensembles under investigation were synthesized by plasma-
assisted molecular-beam epitaxy on Si(111) substrates. Samples A and B were
obtained by selective area growth (see Ref. Schumann _et al._ , 2011 for
details regarding substrate and mask preparation) and contain spatially
ordered arrays of GaN NWs with a pitch of 360 nm and well-defined diameters of
120 and 175 nm, respectively. Sample C is a representative example of a self-
induced GaN NW ensemble (see Ref. Geelhaar _et al._ , 2011 for details
regarding growth) characterized by a high density of NWs with random position
and a broad diameter distribution with a mean of 100 nm.
For PL spectroscopy, the samples were cooled in a microscope cryostat to a
temperature of 10 K. In all cases, the excited area was several µm in diameter
and thus spanned over at least 100 NWs. Continuous-wave PL was excited by the
325 nm (3.814 eV) line of a He-Cd laser focused onto the samples with an
excitation density of less than 1 W/cm2. The PL intensity was spectrally
dispersed by a 80 cm monochromator providing a spectral resolution of 0.25 meV
and detected with a cooled charge-coupled device array. Time-resolved (TR) PL
measurements were performed by exciting the samples with the second harmonic
(325 nm) of fs pulses from an optical parametric oscillator synchronously
pumped by a femtosecond Ti:sapphire laser, which itself was pumped by a
frequency-doubled Nd:YVO4 laser. The energy fluence per pulse was set to 0.2
µJ/cm2 for samples A and B and 0.8 µJ/cm2 for sample C. Assuming that all
incident light is absorbed by the NWs, the upper limit of the photogenerated
carrier density in all samples is estimated to be $5\times 10^{16}$ cm-3 (the
higher fluence used for sample C is compensated by the higher NW density). The
transient PL signal was dispersed by a monochromator providing a spectral
resolution of 4 meV and detected by a streak camera with a temporal resolution
of 50 ps.
Figure 2: (Color online) Streak camera image and transient PL spectra of
sample A $\left[\text{(a) and (d)}\right]$, sample B $\left[\text{(b) and
(e)}\right]$, and sample C $\left[\text{(c) and (f)}\right]$, respectively.
The intensity in the streak camera images (a)–(c) is displayed on a
logarithmic scale from blue (low intensity) to red (high intensity). The
spectra [grey lines in (d)–(f)] are extracted from these images at times
$t_{1}$ = 0.18, $t_{2}$ = 0.5, and $t_{3}$ = 1.35 ns after excitation and are
also displayed on a logarithmic intensity scale. Lineshape fits (black lines)
to the experimental data allow us to perform a spectral deconvolution of the
transitions (the $X_{\text{A}}$ transition can be reliably fit only for sample
C, for which its intensity is comparatively high). The vertical lines
represent the spectral positions of the individual transitions determined from
the PL measurements presented in Fig. 1.
Figures 1(a)–1(c) show the PL spectra of the three samples on a logarithmic
intensity scale. The dominant transitions in all spectra originate from the
recombination of A excitons bound to neutral O and Si donors at $(3.4713\pm
0.0001)$ [(O${}^{0},X_{\text{A}}$)] and $(3.4721\pm 0.0001)$ eV
[(Si${}^{0},X_{\text{A}}$)], respectively. These values are essentially equal
to those obtained in free-standing GaN layers within our experimental
uncertainty.Freitas Jr. _et al._ (2002); Monemar _et al._ (2008) As expected
for the comparatively large NW diameters, we do not observe a contribution
from excitons bound to surface donors.Brandt _et al._ (2010) The observed
linewidth of about 1 meV for both transitions is thus determined by the
residual microstrain within the GaN NWs.Kaganer _et al._ (2012) In addition
to these dominant ($D^{0},X_{\text{A}}$) transitions, all three samples
exhibit a narrow line at 3.467 eV stemming from the recombination of A
excitons bound to neutral acceptors [($A^{0}_{1},X_{\text{A}}$)].Monemar _et
al._ (2008); Morkoç (2008) Samples A and B exhibit an extra set of lines
between 3.455 and 3.463 eV [($A^{0}_{2},X_{\text{A}}$)], which we attribute to
the deeper acceptor states identified recently.Monemar _et al._ (2006, 2008)
Finally, a transition due to the recombination of B excitons bound to neutral
donors [($D^{0},X_{\text{B}}$)] at 3.475 eV and from free A excitons
($X_{\text{A}}$) at 3.478 eV is observed in all samples.
Figure 1(d)–1(f) displays the PL transients of the three samples integrated
over a spectral window of 5 meV width centered at the ($D^{0},X_{\text{A}}$)
transition energy. The decay is biexponential and remains virtually unchanged
when varying the width of the spectral window between 2 and 20 meV. The two
components of the transients differ significantly in their decay time,
particularly for samples A and B. The integrated intensity is dominated by the
short component, accounting for 85%, 90%, and 85% for samples A, B, and C,
respectively. The biexponential decay thus cannot be caused by the integration
over the two transitions related to excitons bound to O and Si, since the
intensity of these transitions is comparable [cf. Fig. 1(a)–1(c)]. Moreover,
the lifetimes of excitons bound to O and Si were reported to be
similar.Monemar _et al._ (2008, 2010)
The biexponential decay can neither be attributed to nonradiative
recombination of bound excitons in close proximity to the surface.Gorgis _et
al._ (2012) Following Ref. Gorgis _et al._ , 2012 and assuming surface
recombination to be the dominant nonradiative decay channel for donor-bound
excitons situated close to the surface, the short decay time of 90 ps measured
for samples A and B would correspond to an average NW diameter of 23 nm, in
blatant disagreement with the actual diameter distribution of the NW arrays
under investigation [cf. insets of Fig. 1(d)–1(e)]. Moreover, to explain the
amplitude of the short component would require 85% to 90% of all donors to be
in close proximity to the surface and even with the exact same distance.
Besides the fact that this situation is of course entirely unlikely, it would
manifest itself also in an energy shift of the transition,Brandt _et al._
(2010) which we do not observe in the PL spectra shown in Figs. 1(a)–1(c).
Having ruled out the two most obvious possibilities for a biexponential decay,
we next examine the transient PL spectra of the samples. Figures 2(a)–2(c)
show the raw streak camera images obtained after pulsed excitation.
Immediately after excitation and up to a time of about 0.5 ns, the
($D^{0},X_{\text{A}}$) transition clearly dominates the spectra. For longer
times, the ($A^{0},X_{\text{A}}$) transition takes over as the dominant line
in the spectra, i. e., its decay is significantly slower than that of the
($D^{0},X_{\text{A}}$) transition.
This result can be inspected more closely in Figs. 2(d)–2(f), which display
transient spectra extracted from the streak camera images at three different
times after excitation, namely, at $t_{1}$ = 180, $t_{2}$ = 500, and $t_{3}$ =
1350 ps. Between $t_{1}$ and $t_{2}$, the ($D^{0},X_{\text{A}}$) transition
for samples A and B decreases in intensity by an order of magnitude with
respect to the ($A^{0}_{2},X_{\text{A}}$) transition. Between $t_{2}$ and
$t_{3}$, however, the intensity ratio between these two transitions stays the
same, i. e., they decay with a common time constant for long times. For sample
C [Fig. 2(f)], we observe a qualitatively similar behavior, but the
($A^{0}_{1},X_{\text{A}}$) transition becomes comparable in intensity with the
($D^{0},X_{\text{A}}$) transition only at longer time ($>3$ ns).
The transient spectra shown in Figs. 2(d)–2(f) reveal a significant spectral
overlap of the ($D^{0},X_{\text{A}}$) and ($A^{0},X_{\text{A}}$) lines. Even
with the narrow spectral window used to obtain the transients shown in Figs.
1(d)–1(f), it is inevitable that we monitor a superposition of the
corresponding transitions. Since the ($A^{0},X_{\text{A}}$) transitions have a
longer decay time than the ($D^{0},X_{\text{A}}$) transition as seen in Fig.
2, the biexponential decay may thus be interpreted as being simply due to the
spectral overlap of these lines. The decay times of the two components of the
transient would then correspond to the lifetime of the transition dominating
the spectrum in a given time interval.
To examine this interpretation, we extract a series of transient spectra from
the streak camera images and fit them by a sum of Voigt functions (three for
samples A and B, four for sample C) as shown by the black lines in Figs.
2(d)–2(f). This spectral deconvolution allows us to explore the decay dynamics
of each radiative recombination channel separately. Figure 3 shows the time-
dependent intensities of each transition as obtained by the deconvolution.
While the ($A^{0}_{1},X_{\text{A}}$) and ($A^{0}_{2},X_{\text{A}}$) transients
are monoexponential, the ($D^{0},X_{\text{A}}$) transient is still clearly
biexponential. This behavior is thus _not_ caused by the spectral overlap, and
the above naive interpretation of the decay times of the two components of
this transient is incorrect.
Figure 3: (Color online) PL transients for the ($D^{0},X_{\text{A}}$)
(triangles), ($A^{0}_{1},X_{\text{A}}$) and ($A^{0}_{2},X_{\text{A}}$)
(circles), and $X_{\text{A}}$ [(squares, only in (c)] transitions obtained by
the spectral deconvolution of the transient spectra [Fig. 2(a)–2(c)] for (a)
sample A, (b) sample B, and (c) sample C. The solid lines represent the decay
of these transitions as obtained by Eqs. (1)–(3). The fast initial decay ($50$
ps) of the free exciton is caused by its capture by neutral donors and
acceptors. Note the common decay time of all transitions at longer times which
is a signature of their strong coupling.
The key for the understanding of this result is the observation that the
($D^{0},X_{\text{A}}$) and ($A^{0},X_{\text{A}}$) transients are strictly
parallel at long times. In addition, the $X_{\text{A}}$ and
($D^{0},X_{\text{A}}$) transients for sample C are found to evolve in
parallel, very similar to the results reported by Korona Korona (2002) for
bulk GaN and Corfdir et al. Corfdir _et al._ (2009) for GaN NWs. These
transitions thus exhibit a common decay time, suggesting a strong coupling
between all states participating in radiative recombination.Brandt _et al._
(1998); Korona (2002); Corfdir _et al._ (2009) To facilitate a quantitative
analysis of our data and to extract the actual lifetimes of these states, we
model the time-dependent densities of the $X_{\text{A}}$ ($n_{\text{F}}$),
($D^{0},X_{\text{A}}$) ($n_{\text{D}}$), and ($A^{0},X_{\text{A}}$)
($n_{\text{A}}$) states by the following set of coupled rate-equations:
$\displaystyle\frac{dn_{\text{F}}}{dt}$
$\displaystyle=-b_{\text{D}}n_{\text{F}}\left(N_{\text{D}}-n_{\text{D}}\right)-b_{\text{A}}n_{\text{F}}\left(N_{\text{A}}-n_{\text{A}}\right)$
(1) $\displaystyle\quad+\hat{\gamma}_{\hskip
0.85358pt\text{D}}n_{\text{D}}+\hat{\gamma}_{\text{A}}n_{\text{A}}-\gamma_{\hskip
0.85358pt\text{F}}n_{\text{F}},$ $\displaystyle\frac{dn_{\text{D}}}{dt}$
$\displaystyle=b_{\text{D}}n_{\text{F}}\left(N_{\text{D}}-n_{\text{D}}\right)-\hat{\gamma}_{\hskip
0.85358pt\text{D}}n_{\text{D}}-\gamma_{\hskip 0.85358pt\text{D}}n_{\text{D}},$
(2) $\displaystyle\frac{dn_{\text{A}}}{dt}$
$\displaystyle=b_{\text{A}}n_{\text{F}}\left(N_{\text{A}}-n_{\text{A}}\right)-\hat{\gamma}_{\text{A}}n_{\text{A}}-\gamma_{\text{A}}n_{\text{A}},$
(3)
with the initial densities $n_{\text{F}}(0)=n_{\text{F}}^{0}$, and
$n_{\text{D}}(0)=n_{\text{A}}(0)=0$.
Figure 4: Schematic energy diagram visualizing Eqs. (1)–(3). The involved
states are denoted by $|n_{i}\rangle$, and the crystal groundstate is
represented by $|0\rangle$.
The first terms of Eqs. (1)–(3), which are illustrated in the scheme displayed
in Fig. 4, describe the capture of free excitons by neutral donors and
acceptors with a total density $N_{\text{D}}$ and $N_{\text{A}}$ and the rate
coefficients $b_{\text{D}}$ and $b_{\text{A}}$, respectively. The second terms
account for the dissociation of the bound excitons with the rate constants
$\hat{\gamma}_{\hskip 0.85358pt\text{D}}$ and $\hat{\gamma}_{\hskip
0.85358pt\text{A}}$ and the third ones for the recombination of free and bound
excitons with the rate constants $\gamma_{\hskip 0.85358pt\text{F}}$,
$\gamma_{\hskip 0.85358pt\text{D}}$, and $\gamma_{\hskip 0.85358pt\text{A}}$.
These rate constants are the inverse of the effective decay times measured
experimentally and implicitly contain radiative ($\gamma_{i,\text{r}}$) and
nonradiative ($\gamma_{i,\text{nr}}$) contributions. The PL intensity of each
transition is then given by
$\gamma_{i,\text{r}}\,n_{i}$.*[Theradiativerateconstant$γ_i;
\text{r}$directlydeterminesthepeakintensityofthetransient[see][].Toreproducetheexperimentallyobservedpeakintensitiesofthedifferenttransitionsforeachsample;
weassumearadiativelifetimeforthe($D^0; X_\text{A}$)of$γ^-1_i;
\text{r}=1$ns(seeRefs.~11and12)whichinturnsetstheradiativelifetimesforthe$X_\text{A}$;
($A^0_1; X_\text{A}$); and($A^0_2; X_\text{A}$)transitionsto10; 7.7; and5.5ns;
respectively.]Brandt2002 The free parameters of our model are the rate
constants $\gamma_{i}$ for recombination, $\hat{\gamma}_{i}$ for the
dissociation of bound excitons, and the rate constants for the capture of free
excitons ($b_{i}N_{i}$).111Due to our low excitation density, we remain in the
small-signal regime such that $N_{i}\gg n_{i}$. Thus, the product $b_{i}N_{i}$
approximates the capture dynamics of free excitons very well. The solid lines
in Fig. 3 depict the simulated PL transients based on a numerical solution of
Eqs. (1)–(3) using values for the free parameters as summarized in Tab. 1. The
obtained capture rate constants are consistent with the experimentally
observed rise times of the respective PL lines (not shown here).
Table 1: Summary of the free parameters, all in units of ns-1, of the rate-equation model [Eqs. (1)–(3)] used for computing the PL transients shown in Fig. 3. Sample | $\gamma_{\hskip 0.85358pt\text{F}}$ | $\gamma_{\hskip 0.85358pt\text{D}}$ | $\gamma_{\hskip 0.85358pt\text{A}}$ | $\hat{\gamma}_{\hskip 0.85358pt\text{D}}$ | $\hat{\gamma}_{\text{A}}$ | $b_{\text{D}}N_{\text{D}}$ | $b_{\text{A}}N_{\text{A}}$
---|---|---|---|---|---|---|---
A | 8 | 11 | 0.5 | 10 | 0.65 | 20 | 2.8
B | 8 | 11 | 0.6 | 10 | 0.60 | 20 | 2.0
C | 3 | 7.5 | 0.4 | 10 | 1.3 | 26 | 2.8
The excitonic states can be depopulated not only by recombination, but also by
dissociation as depicted in Fig. 4. The experimentally observed decay times
are thus not necessarily equal to the actual lifetimes of these states. In
this respect, our simulations provide a valuable guide for the interpretation
of the experimentally observed transients. With the parameters listed in Tab.
1, the fast component of the biexponential decay of the ($D^{0},X_{\text{A}}$)
transition is essentially given by its effective lifetime $1/\gamma_{\hskip
0.85358pt\text{D}}$ and is thus governed by nonradiative recombination of the
($D^{0},X_{\text{A}}$) complex. In contrast, the slow component is caused by a
re-population of the ($D^{0},X_{\text{A}}$) state due to its coupling with the
deeper acceptor-bound excitons. In this particular case, its decay rate is
approximately given by $\gamma_{\text{A}}+\hat{\gamma}_{\text{A}}$ and thus
results from the simultaneous dissociation and recombination of the
($A^{0},X_{\text{A}}$) complex.
At first glance, the strong coupling of the exciton states suggested by our
results is surprising given the low measurement temperature of 10 K. Corfdir
et al. Corfdir _et al._ (2009) attributed the parallel temporal evolution of
the $X_{\text{A}}$ and ($D^{0},X_{\text{A}}$) states at a lattice temperature
of 8 K to an enhanced thermal dissociation of bound excitons due to an
electronic (carrier) temperature of 35 K deduced from the high-energy tail of
the transient spectra. Despite the low excitation density used in the present
experiments, we obtain similar values from the exponential high-energy tail of
the transient PL spectra immediately after excitation. However, for an
electronic temperature of 35 K and an exciton binding energy of 6–7 meV,
detailed balance arguments would predict a significantly smaller ratio of
dissociation and capture rate constants than that obtained from the
fits.Brandt _et al._ (1998)
We propose that the enhanced dissociation rate of bound excitons evident from
our experiments is non-thermal in nature and related to the presence of
electric fields within the GaN NWs.Calarco _et al._ (2011) The strength of
these fields, which arise from the pinning of the Fermi level at the NW
sidewall _M_ -plane surfaces,Calarco _et al._ (2005); Van de Walle and Segev
(2007) amounts to 10 to 17 kV/cm for a moderate doping density of $2\times
10^{16}$ cm-3 and the present range of NW diameters.Pfüller _et al._ (2010)
Fields of this magnitude are theoretically expected to directly ionize the
($D^{0},X_{\text{A}}$) complexBlossey (1971); Yamabe _et al._ (1977); Pedrós
_et al._ (2007) and have been experimentally found to quench the
($D^{0},X_{\text{A}}$) line in GaN layers due to the dissociation of donor-
bound excitons by impact ionization.Pedrós _et al._ (2007) Note that the
magnitude of these fields depends linearly on NW diameter and doping
concentration for the characteristic dimensions of GaN NWs. For the same
doping level, these fields are thus significantly weaker in thin GaN NWs such
as investigated in Ref. Gorgis _et al._ , 2012. However, since they are an
inherent property of GaN NWs of small to medium diameter, their effect on the
exciton dynamics in these NWs must not be ignored.
Finally, our results imply that the lifetime of the ($D^{0},X_{\text{A}}$)
complex in thick GaN NWs is short and governed by a nonradiative decay channel
not related to the NW surface. The actual origin of this decay channel is
currently under investigation and will be the subject of a forthcoming
publication. At present, we can firmly state that the nonradiative process is
not intrinsic to GaN NWs in that it is neither related to the free surface nor
to an excitonic AugerKharchenko _et al._ (1990) process. In particular with
regard to the latter, an increase of the fluence of the excitation by one
order of magnitude results in a clear increase of the decay time, i. e., the
nonradiative process can be saturated. Schlager _et al._ Schlager _et al._
(2011) even observed lifetimes up to 1 ns (i. e., close to the radiative one)
by exciting very thick GaN NWs with a fluence two orders of magnitude larger
than that used in the present work. For small-signal excitation as in the
present work, however, the internal quantum efficiency of the GaN NWs under
investigation is not larger than 20% even at 10 K. Whether higher values can
be achieved in a different growth regime remains to be seen.
The authors would like to thank Vladimir Kaganer for providing a robust and
fast batch fitting routine, the AMO GmbH for the preparation of the pre-
patterned substrates and Manfred Ramsteiner for a critical reading of the
manuscript. The work was partially funded by the German BMBF joint research
project MONALISA (Contract No. 01BL0810).
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|
arxiv-papers
| 2013-08-08T09:36:28 |
2024-09-04T02:49:49.196170
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christian Hauswald, Timur Flissikowski, Tobias Gotschke, Raffaella\n Calarco, Lutz Geelhaar, Holger T. Grahn, Oliver Brandt",
"submitter": "Christian Hauswald",
"url": "https://arxiv.org/abs/1308.1799"
}
|
1308.1857
|
# PANAS-t: A Pychometric Scale for Measuring Sentiments on Twitter
Pollyanna Gonçalves⋆ Fabrício Benevenuto⋆† Meeyoung Cha‡
†Computer Science Department, Federal University of Ouro Preto, Brazil
⋆Computer Science Department, Federal University of Minas Gerais, Brazil
‡Graduate School of Culture Technology, KAIST, Korea
###### Abstract
Online social networks have become a major communication platform, where
people share their thoughts and opinions about any topic real-time. The short
text updates people post in these network contain emotions and moods, which
when measured collectively can unveil the public mood at population level and
have exciting implications for businesses, governments, and societies.
Therefore, there is an urgent need for developing solid methods for accurately
measuring moods from large-scale social media data. In this paper, we propose
PANAS-t, which measures sentiments from short text updates in Twitter based on
a well-established psychometric scale, PANAS (Positive and Negative Affect
Schedule). We test the efficacy of PANAS-t over 10 real notable events drawn
from 1.8 billion tweets and demonstrate that it can efficiently capture the
expected sentiments of a wide variety of issues spanning tragedies, technology
releases, political debates, and healthcare.
###### Index Terms:
Twitter, sentiment analysis, public emotion, public mood, psychometric scales,
PANAS.
## I Introduction
Online social networks (OSNs) like Facebook and Twitter have become an
important communication platform, where people share their thoughts and
opinions about any topic in a collaborative manner in real-time. As of 2012,
Facebook has over one billion active users, which is one-seventh of the world
population, and Twitter similarly has over 400 million registered users each
producing hundreds of millions of status updates every day [36]. Given the
scale and the richness of these networks, the potential for mining the data
within OSNs and utilizing observations from such data is tremendous, and OSN
data have been a gold mine for scholars in fields like linguistics, sociology,
and psychology who are looking for real-time language data to analyze [25].
The massive-scale detailed human lifelog data found in OSNs have important
implications for businesses, governments, and societies. The following areas
of research demonstrate directly how useful observations from mining OSN data
could be. First, social media data can be used to find resonance of important
real-time debates and breaking news. As more people are seamlessly connected
to the Web and OSN sites by mobile devices, people participate in delivering
and propagating prominent and urgent information like political uprising [21,
9], natural disasters [32], and the upheaval of epidemics [15]. Second, social
media data can be used to not only understand the current trends but also
predict future trends such as movie sales [2], political elections [33, 12,
28], as well as stock market [7].
The key features of OSN data that allow for the above implications is at their
immediacy and immensity, for which the development of new methods on large-
scale and real-time collection and analysis of OSN data are crucial. One such
important development is at inferring sentiments in OSNs. A recent work has
showed that real-time moods of people can be gauged on a global level, instead
of relying on questionnaires and other laborious and time-consuming methods of
data collection [14]. Measuring sentiments from unstructured OSN data can not
only broaden our understanding of the human nature, but also comprehend how,
when, and why individuals’ feelings fluctuate according to various social and
economic events.
While sentiment analysis in OSNs is getting great attention, existing work on
measuring sentiments from OSN data has focused on extracting opinions (not
feelings) for marketing purposes [30] and on finding correlation of moods with
some other factor such as happiness [13] and stock price [7]. Most research on
inferring moods from social media texts have directly employed existing
natural language processing tools like LIWC (Linguistic Inquiry and Word
Count) [31], PANAS (Positive and Negative Affect Schedule) [35, 34], ANEW
(Affective Norms for English Words) [23], and Profile of Mood States (POMS)
[7] that have been developed to suit more traditional style writing, such as
formal articles that uses proper language (but not for unstructured and less-
formal OSN data). However, relatively little attention has been paid
developing solid methods for adjusting existing natural language processing
tools for specific types of OSN data.
In this paper, we use well-established psychometric scales, PANAS, to measure
sentiments from short text updates in Twitter and propose PANAS-t, which is an
eleven-sentiment psychometric scale adapted to the context of Twitter. PANAS-t
contains positive and negative mood states and is suitable to measure
sentiments about any sort of event in Twitter. To establish PANAS-t, we used
empirical data from a unique dataset containing 1.8 billion tweets. We used
such data to compute normalization scores for each sentiment, so that any
increase or decrease in positive or negative moods over time can be measured
relatively to the presence of the overall sentiments in this dataset. This
approach makes PANAS-t very simple and practical to be used for large amounts
of data and even for real-time analysis.
To validate our approach, we extracted 10 real notable events that span a wide
variety of issues spanning tragedies, technology releases, political debates,
and healthcare from the 3.5 years worth of Twitter data, and demonstrated that
PANAS-t can effectively capture the mood fluctuations during these events. The
10 events studied include the 2009 Presidential election in the US, death of
the singer Michael Jackson, as well as the natural disasters like the 2010
Earthquake in Haiti. Our qualitative evaluation offers strong evidences that
PANAS-t correctly captured expected sentiments for the analyzed events.
The remainder of this paper is organized as follows. Section 2 surveys
existing approaches to measure sentiments from text. Section 3 details how
PANAS-t works and Section 4 describes the Twitter dataset. Section 5 provides
experimental evidences that our approach is able to capture public mood from
tweets associated to noteworthy events. Finally, Section 6 concludes the paper
and offers directions for future work.
## II Related Work
With the growth of social networking on web, sentiment analysis and opinion
mining have become a subject of study for many researches. In this section, we
survey different techniques used to measure sentiments from online text and
describe related work that studied sentiments in Twitter.
Several methodologies have being used by researchers to extract sentiment from
online text. An overview of a number of these approaches was well-presented in
Pang and Lee’s survey [30], which covers several methods that use Natural
Language Processing (NLP) techniques for sentiment analysis—techniques by
which subjective properties of text are inferred using statistical methods.
Those methods are usually suitable for constructing sentiment-aware and
opinion mining Web applications, which analyze feedback of consumers or users
about a particular product or service [3, 1].
Chesley et al. [10] utilized verbs and adjectives extracted from Wikipedia to
classify text from blogs into three categories: objective, subjective-
positive, or subjective-negative. The verb classes used in the paper can
express objectivity and polarity (i.e., a positive or negative opinion), and
the polarity of adjectives can be drawn from their entries in the online
dictionary, with high accuracy rates of two verb classes demonstrating
polarity near 90%. More recently, Pak and Paroubek [29] utilized strategies of
grammatical structures’ recognition to define if a tweet written by a user is
a subjective phrase or not. They demonstrated that superlative adjectives,
verbs in first person, and personal pronouns are often used for expressing
emotions and opinions as opposed to comparative adjectives, common, and proper
nouns that are a strong indicator of an objective text.
Other approaches that extract sentiment from online text rely on machine
learning, a technique in which algorithms learn a classification model from a
set of previously labeled data, and then apply the acquired knowledge to
classify text new into sentiment categories. In [6], the authors use Support
Vector Machine (SVM) and Multinomial Naive Bayes (MNB) classifiers to test
whether brevity in microblog posts give any advantage in classifying sentiment
and in fact find that short document length suggests a more compact and
explicit sentiment than long document length. In [16], the authors use Random
Walk (RW)-based model and compare it with SVM to predict bias in user
opinions. Although these approaches are applicable for several scenarios,
supervised learning techniques require manual intervention for pre-classifying
training data, which may be infeasible for massive-scale social media data.
Another line of research on extracting sentiments from online text is at
measuring a happiness index from text [14]. Dodds and Danforth [13] proposed a
method that computes the level of happiness of an unstructured text. They
showed that while the happiness index inferred from song lyrics trends
downward from the 1960s to the mid 1990s remained stable within genres, that
of blogs has steadily increased from 2005 to 2009. While providing new
insights, one drawback of this approach is that the happiness index proposed
has a single scale and do not provide any other categorization of rich
sentiments, which is the focus of this work.
Miyoshi, T. [27] et al. propose a method to estimate the semantic orientation
of Japanese reviews about some target products. Authors selected words that
possible change the semantic orientation of a text and then concluded if the
review of a product can be considered desirable or not. In order to evaluat
their approach, authors analyzed 1,400 Japanese reviews of eletric products
such as LCD and MP3 Players in order to separated it in positive and negative
reviews.
There are two studies that are more closely related to our goals. Kim et al.
[23] proposed a method for detecting emotions using Affective Norms for
English Words (ANEW), which is a dataset that contains normative emotional
ratings for 1034 English words. Each word in the ANEW dataset is associated
with a rating of 1–9 along each of three dimensions: valence, arousal, and
dominance. Based on these scales, the authors examined sample tweets about
celebrity deaths and found ANEW to be a promising tool mine Twitter data.
Another study [7] utilized Profile of Mood States (POMS), which is a
psychological rating scale that measures certain mood states consisting of 65
adjectives that qualify 6 negative feelings: tension, depression, anger,
vigor, fatigue and confusion. The authors applied this scale to identify
sentiments on a sample of tweets and evaluate the mood of users related to
market fluctuations and events like political elections in the United States.
This paper builds upon the above efforts and adopt a different psychometric
scale called PANAS (Positive and Negative Affect Schedule) [35, 34] to achieve
new contributions. First, compared to the machine learning-based or other
dictionary-based approaches, PANAS contains a well-balanced set of both
positive and negative affects. This makes PANAS suitable to analyze reactions
of people not only on crisis events such as celebrity deaths and natural
disasters, but also amusing events that incur positive emotions. Second,
compared to existing work that tested sentiment extraction on sample data, we
use the complete data gathered from Twitter to test the idea, which allows us
to perform appropriate normalization to adjust PANAS for Twitter.
## III PANAS-t: Affect Measure for Twitter
Our approach to measure sentiments in Twitter is rooted on a well-known
psychometric scale, namely PANAS. We begin by describing PANAS-x, a popular
expanded version of PANAS, which we utilize and then describe the
normalization steps that we take to adapt the psychometric scale for Twitter.
### III-A The PANAS and PANAS-x Scales
The original PANAS consists of two 10-item mood scales and was developed by
Watson and Clark [35] to provide brief measures of PA (Positive Affect) and NA
(Negative Affect). Respondents are asked to rate the extent to which they have
experienced each particular emotion within a specified time period (typically
during the past week), with reference to a 5-point scale. Ever since the
development of the test, the words appearing in the checklist broadly tapped
the affective lexicon. Later, the same authors developed an expanded version
by including 60 items. The expanded version, called PANAS-x, not only measures
the two original higher order scales (PA and NA), but also 11 specific
affects: Fear, Sadness, Guilt, Hostility, Shyness, Fatigue, Surprise,
Joviality, Self-Assurance, Attentiveness, and Serenity.
Table I summarizes the word composition of the PANAS-x scale [34]. The
negative affect includes words like “afraid,” “scared,” and “nervous,” while
the fatigue affect state includes words like “sleepy,” “tired,” and
“sluggish.” The items in PANAS-x has been validated extensively and also is
known to have strongly relationship with POMS categories, with convergent
correlations ranging above 0.85. In addition, PANAS-x has been demonstrated
with its excellence over POMS, because the items in PANAS-x tend to be less
highly correlated with one another, and thus show better discriminant
validity. For instance, the mean correlation among the PANAS-x Fear,
Hostility, Sadness, and Fatigue scales was 0.45, which is significantly lower
than the mean correlation (0.60) among the corresponding POMS scales.
The authors also validated that individual trait scores on the PANAS-X scales
(a) are stable over time, (b) show significant convergent and discriminant
validity when correlated with peer-judgments, (c) are highly correlated with
corresponding measures of aggregated state affect, and (d) are strongly and
systematically related to measures of personality and emotionality [34]. Due
to this excellence, we choose to adopt PANAS-x for analyzing short text
updates from online social media.
General Dimension Scales |
---|---
Negative Affect (10) | afraid, scared, nervous, jittery, irritable, hostile, guilty, ashamed, upset, distressed.
Positive Affet (10) | active, alert, attentive, determined, enthusiastic, excited, inspired, interested, pround, strong.
Basic Negative Emotions Scales |
Fear (6) | afraid, scared, frightened, nervous, jittery, shaky.
Hostility (6) | angry, hostile, irritable, scornful, disgusted, loathing.
Guilt (6) | guilty, ashamed, blameworthy, angry at self, disgusted with self, dissatisfied with self.
Sadness (5) | sad, blue, downhearted, alone, lonely.
Basic Positive Emotions Scales |
Joviality (8) | happy, joyful, delighted, cheerful, excited, enthusiastic, lively, energetic.
Self-assurance (6) | proud, strong, confident, bold, daring, fearless.
Attentiveness (4) | alert, attentiveness, concentrating, determined.
Other Affective States |
Shyness (4) | shy, bashful, sheepish, timid.
Fatigue (4) | sleepy, tired, sluggish, drowsy.
Serenity (3) | calm, relaxed, at ease.
Surprise (3) | amazed, surprised, astonished.
| Note. The number of terms comprising each scale is shown in parentheses.
Table I: Item composition of the PANAS-x scales.
### III-B Adjusting PANAS-x for Twitter
Tweets expressing certain sentiments may appear more frequently than others,
leading to a bias or dominance of a small set of sentiments in OSN data. Thus,
in order to tell if tweets expressing a specific type of sentiment has
increased or decreased for a given event (e.g., celebrity death or natural
disasters), we first need to know what kinds of sentiments appear during
“typical” or non-event periods. Unfortunately, it is hard or impossible to
determine which dates would be classified as such. One natural baseline would
be to aggregate sentiments over a long period of time and consider the
proportion of each type of sentiment as the baseline. Therefore, by comparing
the proportion of tweets that contain a specific sentiment during a given
event against the entire baseline, one can know how sentiments have changed
related to the presence of a given event in the entire dataset.
We describe the methods to compute the baselines for comparison. We assume
each normalized tweet can be mapped to a single sentiment. When a tweet
contains any of the adjectives in Table I, we associate the corresponding
sentiment $s$ as the main sentiment of the tweet. In case none of the
sentiment words in Table I appear in a tweet, we cannot infer the sentiment
for that tweet. This limitation is common to most other sentiment tools
described in the related work. In case there is a tie and more than two
sentiments can be found in a single tweet, we choose the first sentiment that
appears in the tweet (based on the locatio of the adjectives) as the major
sentiment of that tweet, although such ties are very rare and hence are
negligible for analysis.
The baseline sentiment can be then calculated as follows. Let $T$ be the
entire set of normalized tweets and $T_{s}$ the subset of these tweets related
to sentiment $s$. The baseline value for each sentiment, $\alpha_{s}$, is
defined as the proportion that divides the number of occurrences of tweets of
each type of sentiment by the total number of normalized tweets in our
dataset:
$\alpha_{s}=\frac{|T_{s}|}{|T|}$ (1)
Table II shows the baseline values for all 11 sentiments in PANAS-x from the
3.5 years worth of Twitter data, which we will describe in detail in the next
section. Some sentiments occur orders of magnitude more frequently than
others. Tweets expressing fatigue occurs nearly 32 more frequently than tweets
expressing shyness. This skew in frequency indicates that normalization is
needed to comprehend the effective change of a given sentiment, because
treating the any increase in the number of fatigue and shyness tweets equally
will result in under-estimation and over-estimation of these sentiments,
respectively. Therefore, the inherent skew in sentiments reinforces that a
proper normalization specific to the OSN is necessary.
Sentiment ($s$) | Baseline ( $\alpha_{s}$)
---|---
Fear | 0.0063791
Sadness | 0.0086279
Guilt | 0.0021756
Hostility | 0.0018225
Shyness | 0.0007608
Fatigue | 0.0240757
Surprise | 0.0084612
Joviality | 0.0182421
Self-assurance | 0.0036012
Attentiveness | 0.0008997
Serenity | 0.0022914
Table II: Fraction of tweets for each sentiment in the entire dataset.
Given the baseline sentiment values in Table II, we can now compute the
relative increase or decrease in sentiments for a particular sample of tweets
as follows. Let $S$ be the set of tweets (e.g., natural disaster) and $S_{s}$
the subset of these tweets related to sentiment $s$. We define $\beta_{s}$ as
the relative occurrence of sentiment $s$ for the event $S$ and compute it as
follows:
$\beta_{s}=\frac{|S_{s}|}{|S|}$ (2)
Finally, we define the PANAS-t score as an eleven-dimensional sentiment
vector, where the PANAS-t score function $P(s)$ for sentiment $s$ is computed
as bellow:
$P(s)=\begin{cases}\frac{(\alpha_{s}-\beta_{s})}{\alpha_{s}}&\text{if
}\beta_{s}\leq{\alpha_{s}}\\\
-\frac{(\beta_{s}-\alpha_{s})}{\beta_{s}}&\text{otherwise}\end{cases}$ (3)
The value of $P(s)$ varies between -1 and 1 for each sentiment $s$. An event
with $P(fear)$ = 0 means that the event has no increase or decrease for the
sentiment fear in comparison with the entire dataset of tweets posted as of
2009. A positive value of 0.3 would mean an increase of 30%, and so on. Our
strategy to compute the PANAS-t score is simple and suitable for allowing the
comparison of both the increase and decrease for each type of sentiment
relatively to a non-bias dataset. More importantly, Table II provides a
baseline for comparison against any kinds of sample tweets. For instance, one
could easily crawl tweet samples using the Twitter API and normalize the
sentiment scores found with our baselines.
### III-C Most popular words of PANAS-t
Having seen that the level of baseline sentiments in tweets are skewed, we
quantify which words of the PANAS-t scales appear most frequently in the
dataset. Table III shows the frequency of each adjective based on the entire
Twitter data. Even within a given sentiment, certain adjectives are used more
frequently to express feelings. The most popular adjectives are “sleepy” in
the fatigue category (appearing over 8.0 million times), followed by “happy”
in the joviality category (appearing over 3.8 million times). Other popular
words include “tired”, “excited”, “sad”, “amazed”, “alone”, and “surprised”,
which all appear more than 1 million times.
However, certain words in the PANAS-x scales are rarely used in Twitter to
express the moods, such as “downherted” in the sadness category and
“blameworth” in the guilt category. We may expect that not all words in the
PANAS-x will appear frequently in OSNs, because the PANAS-x scale was
originally designed to be used in a different environment (i.e., intrusive
surveys). A patient submitted to PANAS test needs to mark in a scale from 1 to
5 how much each of these words tell about her mood state. Despite of this
difference between PANAS-x and PANAS-t, the next section presents a number of
situations in which PANAS-t can capture the expected mood states of
populations about a number of noteworthy events accurately.
Self-assurance | Attentiveness | Fatigue
---|---|---
proud: 762,990 | alert: 209,062 | sleepy: 8,043,591
strong: 596,376 | concentrating: 123,725 | tired: 3,486,574
daring: 295,047 | determined: 96,616 | sluggish: 19,938
confident: 95,858 | attentive: 5,456 | drowsy: 18,435
bold: 90,101 | |
fearless: 20,084 | |
Guilt | Fear | Sadness
ashamed: 492,371 | scare: 1,649,193 | sad: 2,765,458
guilty: 324,446 | nervous: 668,867 | alone: 1,096,592
angry at self: 7,873 | afraid: 515,224 | lonely: 15,858
disgusted with self: 2,853 | shaky: 173,142 | blue: 987
dissatisfied with self: 61 | frightened: 75,260 | downhearted: 286
blameworthy: 19 | jittery: 12,791 |
Hostility | Joviality | Serenity
angry: 483,937 | happy: 3,802,662 | at ease: 1,030,236
irritable: 268,546 | excited: 3,170,837 | relaxed: 737,668
disgusted: 220,470 | delighted: 117,074 | calm: 258,576
loathing: 72,330 | lively: 43,552 |
hostile: 12,614 | enthusiastic: 34,323 |
scornful: 7,516 | energic: 22,159 |
| joyful: 21,663 |
| cheerful: 19,178 |
Surprise | Shyness | -
amazed: 2,758,114 | shy: 320,611 |
surprised: 1,050,164 | timid: 13,521 |
astonished: 19,047 | bashful: 2,556 |
| sheepish: 6,850 |
Table III: Frequency of each term of PANAS-t in the total database.
## IV Twitter dataset
The dataset used in this work includes extensive data from a previous
measurement study that included a complete snapshot of the Twitter social
network and the complete history of tweets posted by all users as of August
2009 [8]. More specifically, the dataset contains 54,981,152 users who had
1,963,263,821 follow links among themselves and posted 1,755,925,520 tweets
(as of August 2009). Out of all users, nearly 8% of the accounts were set as
private, which implies that only their friends could view their links and
tweets. We ignore these users in our analysis.
This dataset is appropriate for the purpose of this work for the following
reasons. First, the dataset contains all users with accounts created before
August 2009. Thus, it is not based on sampling techniques that can introduce
bias towards some characteristics of the users. Second, this dataset contains
all tweets of these users, which is essential for measuring the increase or
decrease of a certain sentiment related to tweets of a specific event. Thus,
this dataset uniquely allows us to normalize the presence of sentiments of a
sample of tweets relatively to the inherit sentiments in Twitter.
### IV-A Data cleaning steps
In order to analyze only those tweets that possibly express individuals’
feelings, we only into account tweets that contain explicit statements of
their author’s mood states by matching the following expressions in tweets:
“I’m”, “I am”, “I”, “am”, “feeling”, “me” and “myself”. A similar approach has
been used in [7] in finding correlations of Twitter moods and stock price. In
total, we found 479,356,536 tweets that match these patterns, which correspond
to about 27% of the entire dataset of tweets.
Once we found a set of candidate tweets that contain emotions and moods, we
further cleaned the data as follows. We first applied common language
processing approaches such as case-folding, stemming, and removal of stop
words, URLs, and common verb-forms. We then separated individual terms using
white-space as delimiters and also removed commas, dashes, and others non-
alphanumeric characters. For example, a tweet “I am so scared about swine flu”
terns into the following set of terms, [I, am, scare, swine, flu]. In the
remainder of this paper, we use the above described normalization and analyze
a total of 479,356,536 normalized tweets.
## V Evaluation of PANAS-t
In order to evaluate the extent to which PANAS-t can accurately measure
sentiments of Twitter users, we need ground truth data to compare the results
with our methods. Such ground truth data is difficult to obtain because
sentiments are subjective by nature. In this paper, we consider a few number
of strategies to perform this evaluation. First we evaluate a set of popular
events, for which the sentiments associated with them are expected or easy to
be verified. Second, we compare our results obtained using PANAS-t with an
analysis performed using common emoticons most used by users for express their
feeling on social networkings. Third, we show that the baseline values
computed for PANAS-t were useful to measure sentiments from a dataset of
tweets collected in a different period.
Event | Duration | Description (Example keywords) | # Tweets
---|---|---|---
H1N1 | Mar 1 – Jul 31, 2009 | Disease outbreak (tamiflu, outbreak, influenza, pandemia, pandemic, h1n1, swine, world health organization) | 335,969
AirFrance | Jun 1–6, 2009 | A plane crash (victims, passengers, A330, 447, crash, airplane, airfrance) | 29,765
US-Elec | Nov 2–6, 2008 | US presidential election (clinton, biden, palin, vote, mccain, democrat, republican, obama) | 185,477
Obama | Jan 18–22, 2009 | Presidential inauguration speech (barack obama, white house, presidential, inauguration) | 43,015
Michael-Jackson | Jun 25–30, 2009 | Death of celebrity (rip, mj, michael jackson, death, king of pop, overdose, drugs, heart attack, conrad murray) | 56,259
Susan-Boyle | Apr 11–16, 2009 | Appearance of a new celebrity (susan boyle, I dreamed a dream, britain’s got talent) | 7,142
Harry-Potter | Jul 13–17, 2009 | Release of a movie (harry potter, half-blood prince, rowling) | 194,356
Olympics | Aug 6–26, 2008 | Beijing Olympics (olympics, medals, china, beijing, sports, peking, sponsor) | 12,815
Samoa | Sep 28 – Oct 4, 2009 | Natural disaster (tsunami, samoa islands, tonga, earthquake) | 23,881
Haiti | Jan 11–17, 2010 | Natural disaster (haiti, earthquake, richter, port au prince, jacmel, leogane) | 236,096
Table IV: Summary of events that were analyzed.
### V-A Testing across popular real-world events
We picked nine events that were widely reported to have been covered by
Twitter111Top Twitter trends http://tinyurl.com/yb4965e. These events,
summarized in Table IV, span topics related to tragedies, products and movie
releases, politics, health, as well as sport events. To extract tweets
relevant to the these events, we first identified a set of keywords describing
each topic by consulting news websites, blogs, wikipedia, and informed
individuals. Given the selected list of keywords, we identified the topics by
searching for keywords in the tweet dataset. We limited the duration of each
event because popular keywords are typically hijacked by spammers after
certain time [5, 11]. Table IV also displays the keywords used and the total
number of tweets for each topic.
In order to test how accurately PANAS-t can measure sentiment fluctuations, we
calculated the PANAS-t scales for all events and present them in Kiviat
representations. In each Kiviat graph, radial lines starting at the central
point -1 represents each sentiment with the maximum value of 1 [22]. In Figure
1, we plot the eleven sentiments in each figure so that each figure represents
the corresponding event.
(a) H1N1
(b) AirFrance
(c) US-Elec
(d) Obama
(e) MJ-death
(f) Susan-Boyle
(g) Harry-Potter
(h) Olympics-begin
(i) Olympics-end
Figure 1: Events and feelings associated with them using PANAS-t.
The first event we examine is H1N1, which represents the worldwide disease
outbreak of the H1N1 influenza. The marking date, March 1st of 2009 was the
day, where the influenza was declared by World Health Organization (WHO) as
the global pandemic. To identify the event, we searched for a number of
keywords including “pandemic” and “swine” and found a total of 335,969
relevant tweets during the five months period. Figure 1(a) shows the sentiment
scores of this event based on PANAS-t scales. It demonstrates that the
emotional state of Twitter users increased in attentiveness (P(s) = 0.8774)
and fear (P(s) = 0.6768) in the days just after the announcement. Indeed,
these two feelings correspond to the most likely feelings to expect from this
event as people were both attentive to the precautions as well as afraid of a
global pandemic.
The second event is AirFrance, which describes the tragic crash of an airplane
on July 1st, 2009, which caused a big commotion in Twitter. The AirFrance
Flight 447 was a scheduled as commercial flight from Rio de Janeiro to Paris,
but crashed in Ocean and killed all the 216 passengers. As expected, the crash
caused sad emotions towards those who died and also fear that a something
similar might happen again. Figure 1(b) shows the Kiviat representation for
this event. As expected, fear (P(s) = 0.72914) and sadness (P(s) = 0.6992)
were the two most predominated feelings in the tweets associated to this
event.
The third event is US-Elec, which describes the presidential election related
tweets in the US. With the election, many voters might feel apprehensive and
even excited about the power of choice that is given to them. Our results show
sentiments on this direction. Figure 1(c) shows that users had the feeling for
self-assurance (P(s) = 0.6741), joviality (P(s) = 0.4277) and fear (P(s) =
0.3072) increased, when the election results came out.
The fourth event, Obama, describes the president Barack Obama’s inauguration
speech, which received wide attention in Twitter. As reported in reference
[17], the majority of Americans were more confident in the improvement of the
country after viewing President Barack Obama’s inauguration speech. Our
analysis of the mood of Twitter’s users performed on the day of Obama’s speech
shows a particularly large increase in self-assurance’s (P(s) = 0.7980),
followed by surprise (P(s) = 0.5802), and joviality (P(s) = 0.5227). But
despite all the positive manifestation regarding the election of Obama, we can
also see a positive, but not so high value for sadness (P(s) = 0.1789), which
might naturally represent tweets from Barack Obama’s oppositors. Figure 1(d)
shows that the feelings measured with PANAS-t are agreement with the ones
reported in reference [17].
The fifth Kiviat chart, Michael-Jackson, is about the death of singer Michael
Jackson. According to DailyMail [4], nine of the ten most popular topics in
Twitter were dedicated to the event the day after his death. In Figure 1(e),
we can see an increase in sadness (P(s) = 0.4055), fear (P(s) = 0.5676),
shyness (P(s) = 0.4055), guilt (P(s) = 0.1616), and surprise (P(s) = 0.0810).
It is interesting to perceive that, in addition to the expected feelings
associated with a sudden death like sadness and fear, we could see increase in
guilt. This may be explained by the fact that many speculated about who or
what killed Michael Jackson and fans and critics blamed the high stress caused
by paparazzi and media for the death of celebrity. Therefore, some Twitter
users felt guilt for his death and expressed such feeling in their tweets.
The next event we analyze is Susan-Boyle, who’s appearance as a contestant the
TV show, Britain’s Got Talent, had an incredible repercussion in the media.
Global interest was triggered by the contrast between her powerful voice
singing “I Dreamed a Dream” from the musical Les Miserables and her plain
appearance on stage. The contrast of the audience’s first impression of her,
with the standing ovation she received during and after her performance, led
to an immediate viral spread over the social networks and a huge attention of
the global media. Figure 1(f) shows that the sentiments expressed in Twitter
associated with Susan Boyle’s first appearance are surprise (P(s) = 0.9066),
followed by self-assurance (P(s) = 0.4751), and guilt (P(s) = 0.1367). The
high surprise factor could also explain why Susan Boyle’s video went viral on
the Internet. People also felt self-assured as it is encouraging to see a
woman successfully facing an audience that is laughing at her. Finally, guilt
is also expected as the event is based on wrong prejudice based on appearance.
The seventh event we studied is Harry-Potter, which describes the release of
the movie “Harry Potter and the Half-Blood Prince”. Figure 1(g) shows that the
main feelings associated are joviality (P(s) = 0.6355), surprise (P(s) =
0.4926), and sadness (P(s) = 0.2056), which also is described by many other
critics that say the movie will leave the audience “pleased, amused, excited,
scared, infuriated, delighted, sad, surprised, and thoughtful.”
The last two charts shown in the figure are related to the Olympics games that
were held in the summer of 2008 in Beijing, China. For this event, we show two
Kiviat charts: one drawn based on the beginning sentiments and the other based
on the ending sentiments of the event. Figure 1(h) is based on sentiments from
the day of opening ceremony on August 08, where people felt surprise (P(s) =
0.7024), attentiveness (P(s) = 0.4621), and joviality (P(s) = 0.3298).
However, in the end of the event, on August 24th, we can see that these
feelings had a decrease, whereas sadness increased from P(s)= 0.1222 in this
day and to P(s) = 0.5245 in the next day, as we can see in Figure 1(i).
### V-B Testing across different geographical regions
In order to evaluate whether PNAS-t can effectively capture the subtle
sentiment differences across different geographical areas, we take the example
of the popular H1N1 event and examine how sentiments on the event fluctuate
over time in two different regions: USA and Europe.
To give further context of the H1N1 event, we start by describing its impact
on society. The H1N1 influenza, or also known as the “swine flu” by the
public, has killed as many as half a million people in 2009. The World Health
Organization (WHO) declare it as the first global pandemic since the 1968 Hong
Kong flu, which caused a large concern in the world population. Later, WHO
launched several warnings and precautions that should be taken by governments
and by public, taking the entire population to a state of world alert against
the disease.
In this section we compare the fluctuations of the mood of users about H1N1 in
two different locations. More specifically, we want to verify how USA and
European Twitter users felt about the event and quantify differences in public
mood according to geographic regions.
In examining the difference in sentiments across North America and Europe, we
focus on only English tweets. Therefore sentiments in Europe are limited to
those tweets residing from Europe written in English. To be consistent in
language representativeness, we limited our focus to tweets residing from the
following regions in Europe: Ireland, Kingdom of the Netherlands, Malta, and
United Kingdom. To do this we used a database collected and used in reference
[24]. In this paper, authors used an expressive database from Twitter to
separate unique ids, that represents users, by location.
The sparkline charts shown in Figures 2(a) and 2(b) present the fluctuations
of four of the major sentiments related to the event in PANAS-t scales for
Europe and USA, respectively. The charts are marked with five dates that
indicate the day of important announcements made by WHO. In March, Mexican
authorities begin picking up cases of what WHO called an “influenza-like-
illness.” This event led European users to have an increase in the feeling of
surprise (P(s) = 0.8730) but the same did not happen with users in the US.
(a) H1N1-Europe
(b) H1N1-US
Figure 2: Public mood for H1N1 over 2009 in Europe and U.S.
In April, the first case of H1N1 in the United States was confirmed and WHO
issued a health advisory on the outbreak of “influenza like illness in the
United States and Mexico”, and the charts shown a similar increase in fear in
both locations, P(s) = 0.7401 for Europe and P(s) = 0.6154 for the US. We also
see an increase in attentiveness, but this trend is only for Europe (P(s) =
0.4423).
In June, WHO declared the new strain of swine-origin H1N1 as a pandemic,
causing an increase of fear (P(s) = 0.5385) but also in attentiveness in the
US (P(s) = 0.3491) and in users from Europe (P(s) = 0.3174). In July, 26,089
new cases of H1N1 were confirmed in Europe by WHO, which leads to a further
increase in sentiment of fear (P(s) = 0.4887), mainly among the European
users.
On the last marked date in August, the most affected countries and deaths were
announced as being located in Europe and America [19]. In this period,
European users had an increase in feeling of hostility (P(s) = 0.2542),
whereas users in the US increased the feeling of fear (P(s) = 0.4112). These
variations in the degree of sentiments expressed over time can effectively
capture the dynamics in people’s moods across different geographical regions.
### V-C Testing across different time periods
(a) Samoa
(b) Haiti
Figure 3: Feeling expressed by Twitter’s users for Tsunami, in Samoa Islands,
and Earthquake, in Haiti.
The baseline values computed for PANAS-t in Table II is based on longitudinal
data, based on 3.5 years worth of tweets between 2006 and until mid 2009, and
represent a rather stable base sentiment of Twitter users. Therefore, these
baseline values can be used to detect feelings of Twitter users from much
later time periods (beyond mid 2009). Here, we use a different Twitter dataset
that contains tweets posted between the end of 2009 to the end of 2010 that
was collected by [26] and have extracted tweets associated with two last
events in Table IV: Samoa and Haiti.
The 2009 Samoa Islands Tsunami was caused by a submarine earthquake that took
place in the Samoan Islands on September 29th with a magnitude of 8.1, which
was the largest earthquake of 2009. A tsunami was generated causing
substantial damage and loss of life in Samoa, American Samoa, and Tonga. More
than 189 people were killed including children, which caused a large commotion
around the world and generated a state of alert in neighboring coastal
countries [18]. Figure 3(a) shows the Kiviat chart for mood of users on the
day of tsunami and the day after, which shows dominance in feelings of fear
(P(s) = 0.9280), attentiveness (P(s) = 0.9932), hostility (P(s) = 0.8451),
surprise (P(s) = 0.6528), and sadness (P(s) = 0.6483).
A similar tragic event happened in three months later in another part of the
world. The 2010 Haiti earthquake was a catastrophic natural disaster, which
caused severe damage in Port-au-Prince and the nearby region killing at least
250,000 people. Figure 3(b) shows that feelings of hostility (P(s) = 0.9280),
attentiveness (P(s) = 0.3678), surprise (P(s) = 0.4576) and sadness (P(s) =
0.3975) had an increase. We also see an increase in shyness and guilt. After
this event the world’s eyes were focused on the disaster and people around the
world offered help to Haiti [20]. As the poverty and precarious situation of
the Haiti people was unveiled in the news, it is possible that this situation
has generated an increase of these two feelings among the Twitter users. This
finding demonstrates that PNAS-t is stable and can effectively represent
sentiments of tweets gathered much later in time.
## VI Conclusions
In this paper, we present PANAS-t an eleven-sentiment psychometric scale
adapted to the context of Twitter. PANAS-t is based on the expanded version of
the well known Positive Affect Negative Affect Scale (PANAS-x). Using
empirical data from a unique Twitter dataset containing 1.8 billion tweets, we
were able to compute the normalization scores for each sentiment. We conducted
a three-step evaluation. We first applied PANAS-t to 11 notable events that
were widely discussed in Twitter. We next compared PANAS-t with a method using
most common emoticons that are used for users in Web. We finally showed that
our method can be used in other database and also in other periods. These
results provide strong evidences that PANAS-t can accurately capture the
positive and negative sentiments about events in Twitter.
The normalized scores of sentiments provided in this paper allow anyone to
easily use PANAS-t, making it very simple and practical to be used for large
amounts of data and even for real-time analysis. We hope that this
psychometric scale can be used by any researches with the purpose of create
tools that can be used for government agencies or companies that might be
interested in improving their products using social networks. From the
researcher perspective our method would allow one to comprehend how, when, and
why individuals feel and their feelings fluctuate according to social and
economic events.
Despite the new opportunities our work brings, there are several limitations.
First, the tweets we examined do no represent everyone who expressed
sentiments in Twitter. We only focused on those tweets that explicitly
contained “I am feeling” kinds of tags, although other tweets may contain
emotions as well. Nonetheless, classifying emotional content from
informational content remains an important challenge in social media analysis.
Second, one criticism of sentiment analysis is that it takes a naive view of
emotional states, assuming that personal moods can simply be divined from word
selection. This might seem particularly perilous on a medium like Twitter,
where sarcasm and other playful uses of language may subvert the surface
meaning of a tweet. Deeper linguistic analysis should be explored to provide
“a richer and a more nuanced view” of how people present themselves to the
world.
We expect that in the future more applications will utilize sentiment analysis
for specific vocabularies especially in a dynamic environment like Twitter to
understand people’s moods. Thus, we plan to combine other techniques such as
machine learning to dynamically incorporate sentiments to PANAS-t according to
the context.
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|
arxiv-papers
| 2013-08-08T14:06:51 |
2024-09-04T02:49:49.205004
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pollyanna Gon\\c{c}alves, Fabr\\'icio Benevenuto, Meeyoung Cha",
"submitter": "Pollyanna Gon\\c{c}alves Ms",
"url": "https://arxiv.org/abs/1308.1857"
}
|
1308.1878
|
# n-fold filters in residuated lattices
Albert Kadji, Celestin Lele, Marcel Tonga Department of Mathematics,
University of Yde 1 [email protected] Department of Mathematics,
University of Dschang [email protected] Department of Mathematics,
University of Yde 1 [email protected]
###### Abstract.
Residuated lattices play an important role in the study of fuzzy logic based
of t-norm. In this paper, we introduced the notions of n-fold implicative
filters, n-fold positive implicative filters, n-fold boolean filters, n-fold
fantastic filters, n-fold normal filters and n-fold obstinate filters in
residuated lattices and study the relations among them.
This generalized the similar existing results in BL-algebra with the
connection of the work of Kerre and all in [14], Kondo and all in [7], [11]
and Motamed and all in [9].
At the end of this paper, we draw two diagrams; the first one describe the
relations between some type of n-fold filters in residuated lattices and the
second one describe the relations between some type of n-fold residuated
lattices.
Key words: residuated lattices, filters, n-fold filters, n-fold residuated
lattices.
2000 Mathematics Subject Classification. Primary 06D99, 08A30
## 1\. Introduction
Since Hájek introduced his Basic Fuzzy logics, (BL-logics) in short in 1998
[1], as logics of continuous t-norms, a multitude research papers related to
algebraic counterparts of BL-logics, has been published. In [2], [3],[9] and
[13] the authors defined the notion of n-fold implicative filters, n-fold
positive implicative filters, n-fold boolean filters, n-fold fantastic
filters, n-fold obstinate filters, n-fold normal filters in BL-algebras and
studied the relation among many type of n-fold filters in BL-algebra.
The aim of this paper is to extend this research to residuated lattices with
the connection of the results obtaining in [14], [11], [7].
## 2\. Preliminaries
A residuated lattice is a nonempty set $L$ with four binary operations
$\wedge,\vee,\otimes,\rightarrow$, and two constants $0,1$ satisfying:
L-1: $\mathbb{L}(L):=(L,\wedge,\vee,0,1)$ is a bounded lattice;
L-2: $(L,\otimes,1)$ is a commutative monoid;
L-3: $x\otimes y\leq z$ iff $x\leq y\rightarrow z$ (Residuation);
A MTL-algebra is a residuated lattice $L$ which satisfies the following
condition:
L-4: $(x\rightarrow y)\vee(y\rightarrow x)=1$ (Prelinearity);
A BL-algebra is a MTL-algebra $L$ which satisfies the following condition:
L-5: $x\wedge y=x\otimes(x\rightarrow y)$ (Divisibility).
A MV-algebra is a BL-algebra $L$ which satisfies the following condition:
L-6: $\overline{\overline{x}}=x$ where $\overline{x}:=x\rightarrow 0$.
In this work, unless mentioned otherwise,
$(L,\wedge,\vee,\otimes,\rightarrow,0,1)$ will be a residuated lattice, which
will often be referred by its support set $L$.
###### Proposition 2.1.
[4],[6],[7],[11]For all $x,y,z\in L$
(1) $\displaystyle x\leq y\;\text{iff}\;x\rightarrow y=1;x\otimes y\leq
x\wedge y;$ (2) $\displaystyle x\rightarrow(y\rightarrow z)=(x\otimes
y)\rightarrow z;$ (3) $\displaystyle x\rightarrow(y\rightarrow
z)=y\rightarrow(x\rightarrow z);$ (4) $\displaystyle\text{If}\;x\leq
y,\;\text{then}\;y\rightarrow z\leq x\rightarrow z\;\text{and}\;z\rightarrow
x\leq z\rightarrow y;$ (5) $\displaystyle x\leq y\rightarrow(x\otimes
y);x\otimes(x\rightarrow y)\leq y;$ (6) $\displaystyle 1\rightarrow
x=x;x\rightarrow x=1;x\rightarrow 1=1;x\leq y\rightarrow
x,x\leq\bar{\bar{x}},\bar{\bar{\bar{x}}}=\bar{x};$ (7) $\displaystyle
x\otimes\bar{x}=0;x\otimes y=0\;\text{iff}\;x\leq\bar{y};$ (8) $\displaystyle
x\leq y\;\text{implies}\;x\otimes z\leq y\otimes z,z\rightarrow x\leq
z\rightarrow y,y\rightarrow z\leq x\rightarrow z,\bar{y}\leq\bar{x};$ (9)
$\displaystyle\overline{x\otimes y}=x\rightarrow\bar{y};$ (10) $\displaystyle
x\vee y=1\;\text{implies}\;x\otimes y=x\wedge y\;\text{and}\ x^{n}\vee
y^{n}=1\;\text{for every }\;n\geq 1;$ (11) $\displaystyle x\otimes(y\vee
z)=(x\otimes y)\vee(x\otimes z);$ (12) $\displaystyle(x\vee y)\rightarrow
z=(x\rightarrow z)\wedge(y\rightarrow z);(x\rightarrow z)\vee(y\rightarrow
z)\leq(x\wedge y)\rightarrow z;$ (13) $\displaystyle(x\vee y)\otimes(x\vee
z)\leq x\vee(y\otimes z),\;\text{hence}\;(x\vee y)^{mn}\leq x^{n}\vee y^{m};$
(14) $\displaystyle x\vee y\leq((x\rightarrow y)\rightarrow
y)\wedge((y\rightarrow x)\rightarrow x);$ (15) $\displaystyle x\rightarrow
y\leq(y\rightarrow z)\rightarrow(x\rightarrow z);$ (16) $\displaystyle
y\rightarrow x\leq(z\rightarrow y)\rightarrow(z\rightarrow x);$ (17)
$\displaystyle((x\rightarrow y)\rightarrow y)\rightarrow y=x\rightarrow y.$
Besides equations (1)-(17), we will use the following results.
Fact 1
A nonempty subset $F$ of a residuated lattice
$(L,\wedge,\vee,\otimes,\rightarrow,0,1)$ is called a filter if it satisfies:
(F1): For every $x,y\in F$, $x\otimes y\in F$;
(F2): For every $x,y\in L$, if $x\leq y$ and $x\in F$, then $y\in F$.
A deductive system of a residuated lattices $L$ is a subset $F$ containing $1$
such that for all $x,y\in L$; $x\rightarrow y\in F\ \ \text{and}\ \ x\in F\ \
\text{imply}\ \ y\in F.$
It is known that in a residuated lattices, filters and deductive systems
coincide [4].
Fact 2
The following Examples will be use as a residuated lattices which are not BL-
algebra.
###### Example 2.2.
[12] Let $L=\\{0,a,b,c,d,1\\}$ be a lattice such that $0<a<c$, $0<b<c<d<1$,
$a$ and $b$ are incomparable. Define the operations $\otimes$ and
$\rightarrow$ by the two tables. Then $L$ is a residuated lattice which is not
a BL-algebra since $(a\longrightarrow b)\vee(b\longrightarrow a)=c\neq 1$.
$\otimes$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
---|---|---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$a$ | $0$ | $a$ | $0$ | $a$ | $a$ | $a$
$b$ | $0$ | $0$ | $b$ | $b$ | $b$ | $b$
$c$ | $0$ | $a$ | $b$ | $c$ | $c$ | $c$
$d$ | $0$ | $a$ | $b$ | $c$ | $c$ | $d$
$1$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
$\longrightarrow$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
---|---|---|---|---|---|---
$0$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$a$ | $b$ | $1$ | $b$ | $1$ | $1$ | $1$
$b$ | $a$ | $a$ | $1$ | $1$ | $1$ | $1$
$c$ | $0$ | $a$ | $b$ | $1$ | $1$ | $1$
$d$ | $0$ | $a$ | $b$ | $d$ | $1$ | $1$
$1$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
$F=\\{1,b,c,d\\}$; $F_{1}=\\{1,a,c,d\\}$; $F_{2}=\\{1,c,d\\}$ are proper
filters of $L$.
###### Example 2.3.
[12] Let $L=\\{0,a,b,c,d,1\\}$ be a lattice such that $0<a,b,d,c<1$, $a,b,c,d$
are pairwise incomparable. Define the operations $\otimes$ and $\rightarrow$
by the two tables. Then $L$ is a residuated lattice which is not a BL-algebra
since $a\otimes(a\longrightarrow b)=b\neq 0=a\wedge b$.
$\otimes$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
---|---|---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$a$ | $0$ | $a$ | $b$ | $d$ | $d$ | $a$
$b$ | $0$ | $b$ | $b$ | $0$ | $0$ | $b$
$c$ | $0$ | $d$ | $0$ | $d$ | $d$ | $c$
$d$ | $0$ | $d$ | $0$ | $d$ | $d$ | $d$
$1$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
$\longrightarrow$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
---|---|---|---|---|---|---
$0$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$a$ | $a$ | $1$ | $b$ | $c$ | $c$ | $1$
$b$ | $c$ | $a$ | $1$ | $c$ | $c$ | $1$
$c$ | $b$ | $a$ | $b$ | $1$ | $a$ | $1$
$d$ | $b$ | $a$ | $b$ | $a$ | $1$ | $1$
$1$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
$F=\\{1,c,d\\}$ is a proper filter of $L$.
###### Example 2.4.
[4] Let $L=\\{0,a,b,c,d,1\\}$ be a lattice such that $0<a<c<d<1$, $0<b<c<d<1$,
$a$ and $b$ are incomparable. Define the operations $\otimes$ and
$\rightarrow$ by the two tables. Then
$(L,\wedge,\vee,\otimes,\rightarrow,0,1)$ is a residuated lattices which is
not a BL-algebra since $(a\longrightarrow b)\vee(b\longrightarrow a)=c\vee
c=c\neq 1$.
$\otimes$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
---|---|---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$a$ | $0$ | $0$ | $0$ | $0$ | $a$ | $a$
$b$ | $0$ | $0$ | $0$ | $0$ | $b$ | $b$
$c$ | $0$ | $0$ | $0$ | $0$ | $c$ | $c$
$d$ | $0$ | $a$ | $b$ | $c$ | $d$ | $d$
$1$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
$\longrightarrow$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
---|---|---|---|---|---|---
$0$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$a$ | $c$ | $1$ | $c$ | $1$ | $1$ | $1$
$b$ | $c$ | $c$ | $1$ | $1$ | $1$ | $1$
$c$ | $c$ | $c$ | $c$ | $1$ | $1$ | $1$
$d$ | $0$ | $a$ | $b$ | $c$ | $1$ | $1$
$1$ | $0$ | $a$ | $b$ | $c$ | $d$ | $1$
$F=\\{1,d\\}$ is a proper filter of $L$.
###### Example 2.5.
[6] Let $L=\\{0,a,b,c,1\\}$ be a lattice such that $0<c<a,b<1$, $a,b$ are
incomparable. Define the operations $\otimes$ and $\rightarrow$ by the two
tables. Then $(L,\wedge,\vee,\otimes,\rightarrow,0,1)$ is a residuated lattice
which is not BL-algebra since $a\otimes(a\longrightarrow b)=c\neq 0=a\wedge
b$.
$\otimes$ | $0$ | $c$ | $a$ | $b$ | $1$
---|---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$ | $0$
$c$ | $0$ | $c$ | $c$ | $c$ | $c$
$a$ | $0$ | $c$ | $a$ | $c$ | $a$
$b$ | $0$ | $c$ | $c$ | $b$ | $b$
$1$ | $0$ | $c$ | $a$ | $b$ | $1$
$\longrightarrow$ | $0$ | $c$ | $a$ | $b$ | $1$
---|---|---|---|---|---
$0$ | $1$ | $1$ | $1$ | $1$ | $1$
$c$ | $0$ | $1$ | $1$ | $1$ | $1$
$a$ | $0$ | $b$ | $1$ | $b$ | $1$
$b$ | $0$ | $a$ | $a$ | $1$ | $1$
$1$ | $0$ | $c$ | $a$ | $b$ | $1$
$F_{1}=\\{1,a\\},F_{2}=\\{1,b\\},F_{3}=\\{1,a,b,c\\}$ are proper filters of
$L$.
###### Definition 2.6.
[11] A residuated lattice $L$ is said to be locally finite if for every $x\neq
1$, there exists an integer $n\geq 1$ such that $x^{n}:=\underbrace{x\otimes
x\cdots\otimes x}_{n\;times}=0$.
###### Definition 2.7.
[7] Let $F$ be a filter of a residuated lattice $L$. For $x,y\in L$, a
relation $\equiv_{F}$ on $L$, define by
$x\equiv_{F}y\Longleftrightarrow(x\longrightarrow y,y\longrightarrow x)\in F$,
is a congruence on $L$ and a quotient structure $L/F$ is also a residuated
lattice where : $x/F\wedge y/F=(x\wedge y)/F$; $x/F\vee/F=(x\vee y)/F$;
$x/F\otimes y/F=(x\otimes y)/F$; $x/F\longrightarrow y/F=(x\longrightarrow
y)/F$.
###### Definition 2.8.
[14] [11] [4] A Proper filter $F$ is said to be:
* (i)
prime if it satisfies the following condition:
For all $x,y\in L$, $x\longrightarrow y\in F$ or $y\longrightarrow x\in F$.
* (ii)
prime of the second kind if it satisfies the following condition:
For all $x,y\in L$, $x\vee y\in F$ implies $x\in F$ or $y\in F$.
* (iii)
prime of the third kind if it satisfies the following condition:
For all $x,y\in L$, $(x\longrightarrow y)\vee(y\longrightarrow x)\in F$.
* (iv)
boolean if it satisfies the following condition:
For all $x\in L$, $x\vee\overline{x}\in F$.
* (v)
boolean filter in the second kind if it satisfies the following condition:
For all $x\in L$, $x\in F$ or $\overline{x}\in F$.
###### Remark 2.9.
[14][11] [6]
* (i)
Prime filters are prime filters in the second kind. The converse is true if
$L$ is a MTL-algebra.
* (ii)
Prime filters are prime filters in the third kind. The converse is true if $L$
is a MTL-algebra.
* (iii)
Boolean filters in the second kind are boolean filters.
* (iv)
Maximal filters are prime filters in the second kind.
* (v)
If $L$ is a MTL-algebra, then maximal filters are prime filters.
We have the following results.
###### Proposition 2.10.
[6] For any filter $F$ of a residuated lattices $L$, the following conditions
are equivalent:
* (i)
$F$ is a maximal filter of $L$.
* (ii)
For any $x\in L$, $x\notin F$ if and only if $\overline{x^{n}}\in F$ for some
$n\geq 1$
* (iii)
For any $x\notin F$, there is $f\in F$ and $n\geq 1$ such that $f\otimes
x^{n}=0$
Follows from Prop.2.10, we have the following lemma:
###### Lemma 2.11.
$F$ is a maximal filter of $L$ if and only if $L/F$ is a locally finite
residuated lattice.
Now, unless mentioned otherwise, $n\geq 1$ will be an integer and $F\subseteq
L$.
## 3\. SEMI MAXIMAL FILTER IN RESIDUATED LATTICES
###### Definition 3.1.
[5] Let $F$ be a proper filter of $L$. The intersection of all maximal filters
of $L$ which contain $F$ is called the radical of $F$ and it is denoted by
Rad($F$).
###### Definition 3.2.
A proper filter $F$ of $L$ is said to be a semi maximal filter of $L$ if
Rad($F$)= $F$.
The following example shows that the notion of semi maximal filters in
residuated lattices exist and semi maximal filter may not be maximal filter.
###### Example 3.3.
Let $L$ be a residuated lattice from Example 2.2. It is easy to check that
Rad($\\{1,c,d\\}$)= $\\{1,c,d\\}$. Hence $\\{1,c,d\\}$ is a semi maximal
filter of $L$.
But $\\{1,c,d\\}\subseteq\\{1,a,c,d\\}$ and $\\{1,c,a,d\\}$ is a filter of
$L$, hence $\\{1,c,d\\}$ is a semi maximal filter which is not a maximal
filter of $L$.
###### Remark 3.4.
It is clear that maximal filters are semi maximal filters.
## 4\. N-FOLD IMPLICATIVE FILTER IN RESIDUATED LATTICES
###### Definition 4.1.
An n-fold implicative residuated lattice $L$ is a residuated lattices which
satifies the following condition:
$x^{n+1}=x^{n}$ for all $x,y\in L$.
The following examples shows that n-fold implicative residuated lattices exist
and that residuated lattice is not in general n-fold implicative residuated
lattice.
###### Example 4.2.
Let $L$ be a residuated lattice from Example 2.4. We have:
* (i)
$a^{1+1}=0\neq a^{1}$ so $L$ is not an 1-fold implicative residuated lattice.
* (ii)
For all $n\geq 2$, $x^{n+1}=x^{n}$ for all $x,y\in L$. So $L$ is an n-fold
implicative residuated lattice for all $n\geq 2$.
###### Definition 4.3.
$F$ is an n-fold implicative filter if it satisfies the following conditions:
* (i)
$1\in F$
* (ii)
For all $x,y,z\in L$, if $x^{n}\longrightarrow(y\longrightarrow z)\in F$ and
$x^{n}\longrightarrow y\in F$, then $x^{n}\longrightarrow z\in F$.
In particular 1-fold implicative filters are implicative filters.[7]
###### Example 4.4.
Let $n\geq 1$ and $L$ be a residuated lattice from Example 2.5. Simple
computations proves that $F_{1}=\\{1,a\\},F_{2}=\\{1,b\\},F_{3}=\\{1,a,b,c\\}$
are n-fold implicative filters.
The following lemma gives a characterization of n-fold implicative filters.
###### Lemma 4.5.
Let $a\in L$. Let $F$ be a filter of $L$. Then $L_{a}=\\{b\in
L:a^{n}\longrightarrow b\in F\\}$ is a filter of $L$ if and only if $F$ is an
n-fold implicative filter of $L$.
###### Proof.
Let $F$ be an n-fold implicative filter of $L$. Since $a^{n}\longrightarrow
1=1\in F$, we have $1\in L_{a}$. Let $x,y\in L$ be such that
$x,x\longrightarrow y\in L_{a}$, then $a^{n}\longrightarrow x\in F$ and
$a^{n}\longrightarrow(x\longrightarrow y)\in F$. Since $F$ is an n-fold
implicative filter of $L$, by Definition 4.3, $a^{n}\longrightarrow y\in F$,
hence $y\in L_{a}$. Therefore $L_{a}$ is a filter of $L$.
Conversely suppose that $L_{a}$ is a filter of $L$ for all $a\in L$. Let
$x,y,z\in L$ be such that $x^{n}\longrightarrow(y\longrightarrow z)\in F$ and
$x^{n}\longrightarrow y\in F$. We have $y,y\longrightarrow z\in L_{x}$, by the
hypothesis $L_{x}$ is a filter of $L$, so $z\in L_{x}$ and hence
$x^{n}\longrightarrow z\in F$. ∎
The following proposition gives another characterization of n-fold implicative
filters in residuated lattices.
###### Proposition 4.6.
Let $F$ be a filter of $L$. Then for all $x\in L$, the following conditions
are equivalent:
* (i)
$F$ is an n-fold implicative filter of $L$.
* (ii)
$x^{n}\longrightarrow x^{2n}\in F$.
###### Proof.
$(i)\longrightarrow(ii)$: Let $x\in L$, by Prop. 2.1 we have:
$x^{n}\longrightarrow(x^{n}\longrightarrow x^{2n})=x^{2n}\longrightarrow
x^{2n}=1\in F$ and $x^{n}\longrightarrow x^{n}=1\in F$. Since $F$ is an n-fold
implicative filter of $L$, we get $x^{n}\longrightarrow x^{2n}\in F$.
$(ii)\longrightarrow(i)$: Let $x,y,z\in L$ be such that
$x^{n}\longrightarrow(y\longrightarrow z)\in F$ and $x^{n}\longrightarrow y\in
F$. By Prop. 2.1 we have the following:
* (1)
$x^{n}\otimes[x^{n}\longrightarrow(y\longrightarrow z)]\leq y\longrightarrow
z$.
* (2)
$x^{n}\otimes(x^{n}\longrightarrow y)\leq y$.
* (3)
By (1) and (2) we have : $[x^{n}\otimes[x^{n}\longrightarrow(y\longrightarrow
z)]]\otimes[x^{n}\otimes(x^{n}\longrightarrow y)]\leq
y\otimes(y\longrightarrow z)\leq z$.
* (4)
By (3) we have : $([x^{n}\longrightarrow(y\longrightarrow
z)]\otimes(x^{n}\longrightarrow y))\otimes x^{2n}\leq z$.
* (5)
By (4), we have :$([x^{n}\longrightarrow(y\longrightarrow
z)]\otimes(x^{n}\longrightarrow y))\leq x^{2n}\longrightarrow z$.
* (6)
Since $x^{n}\longrightarrow(y\longrightarrow z)\in F$ and
$x^{n}\longrightarrow y\in F$, by the fact that $F$ is a filter, we get
$[x^{n}\longrightarrow(y\longrightarrow z)]\otimes(x^{n}\longrightarrow y)\in
F$
* (7)
By (5),(6) and the fact that $F$ is a filter, we get $x^{2n}\longrightarrow
z\in F$.
* (8)
$x^{n}\longrightarrow x^{2n}\leq(x^{2n}\longrightarrow
z)\longrightarrow(x^{n}\longrightarrow z)$
* (9)
By (7), (8) and the fact that $x^{n}\longrightarrow x^{2n}\in F$, we obtain
$x^{n}\longrightarrow z\in F$.
Hence $F$ is an n-fold implicative filter of $L$. ∎
###### Proposition 4.7.
Let $F$ be a filter of $L$. Then for all $x,y\in L$, the following conditions
are equivalent:
* (i)
$x^{n}\longrightarrow x^{2n}\in F$.
* (ii)
If $x^{n+1}\longrightarrow y\in F$, then $x^{n}\longrightarrow y\in F$.
###### Proof.
$(i)\longrightarrow(ii)$: Since $(i)$ holds, by Prop. 4.6 $F$ is an n-fold
implicative filter of $L$. On the other hand by Prop. 2.1 we have :
$x^{n+1}\longrightarrow y=x^{n}\longrightarrow(x\longrightarrow y)\in F$ and
$x^{n}\longrightarrow x=1\in F$, by the fact that $F$ is an n-fold implicative
filter of $L$ we get $x^{n}\longrightarrow y\in F$.
$(ii)\longrightarrow(i)$: We have :
$x^{n+1}\longrightarrow(x^{n-1}\longrightarrow x^{2n})=x^{2n}\longrightarrow
x^{2n}=1\in F$. From this and the fact that (ii) holds, we also have :
$x^{n}\longrightarrow(x^{n-1}\longrightarrow x^{2n})\in F$.
But $x^{n+1}\longrightarrow(x^{n-2}\longrightarrow
x^{2n})=x^{n}\longrightarrow(x^{n-1}\longrightarrow x^{2n})\in F$.
From this and the fact that (ii) holds, we also have :
$x^{n}\longrightarrow(x^{n-2}\longrightarrow x^{2n})\in F$.
By repeating the process n times, we get
$x^{n}\longrightarrow(x^{n-n}\longrightarrow x^{2n})=x^{n}\longrightarrow
x^{2n}\in F$. ∎
###### Proposition 4.8.
Let $F$ be a filter of $L$. Then for all $x,y,z\in L$, the following
conditions are equivalent:
* (i)
$x^{n}\longrightarrow x^{2n}\in F$.
* (ii)
If $x^{n}\longrightarrow(y\longrightarrow z)\in F$, then
$(x^{n}\longrightarrow y)\longrightarrow(x^{n}\longrightarrow z)\in F$.
###### Proof.
$(i)\longrightarrow(ii)$: Assume that $x^{n}\longrightarrow(y\longrightarrow
z)\in F$. By Prop. 2.1 we have the following equations:
* (1)
$y\longrightarrow z\leq(x^{n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow z)$.
* (2)
$x^{n}\longrightarrow(y\longrightarrow z)\leq
x^{n}\longrightarrow[(x^{n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow z)]$.
* (3)
$x^{n}\longrightarrow[(x^{n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow
z)]=x^{n}\longrightarrow[x^{n}\longrightarrow((x^{n}\longrightarrow
y)\longrightarrow z)]=x^{2n}\longrightarrow[(x^{n}\longrightarrow
y)\longrightarrow z]$.
* (4)
By (2) and (3) we have : $x^{n}\longrightarrow(y\longrightarrow z)\leq
x^{2n}\longrightarrow[(x^{n}\longrightarrow y)\longrightarrow z]$.
* (5)
Since $F$ is a filter, by (4) and the fact that
$x^{n}\longrightarrow(y\longrightarrow z)\in F$, we have :
$x^{2n}\longrightarrow[(x^{n}\longrightarrow y)\longrightarrow z]\in F$
* (6)
$x^{2n}\longrightarrow[(x^{n}\longrightarrow y)\longrightarrow
z]\leq(x^{n}\longrightarrow
x^{2n})\longrightarrow(x^{n}\longrightarrow[(x^{n}\longrightarrow
y)\longrightarrow z])=(x^{n}\longrightarrow
x^{2n})\longrightarrow((x^{n}\longrightarrow
y)\longrightarrow[(x^{n}\longrightarrow z])$.
* (7)
Since $F$ is a filter, by (6) and the fact that
$x^{2n}\longrightarrow[(x^{n}\longrightarrow y)\longrightarrow z]\in F$, we
have : $(x^{n}\longrightarrow x^{2n})\longrightarrow((x^{n}\longrightarrow
y)\longrightarrow[(x^{n}\longrightarrow z])\in F$.
* (8)
Since $F$ is a filter, by (7) and the fact that $x^{n}\longrightarrow
x^{2n}\in F$, we obtain $(x^{n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow z)\in F$.
$(ii)\longrightarrow(i)$: Since $x^{n}\longrightarrow(x^{n}\longrightarrow
x^{2n})=x^{2n}\longrightarrow x^{2n}=1\in F$, by (ii) we have :
$(x^{n}\longrightarrow x^{n})\longrightarrow(x^{n}\longrightarrow x^{2n})\in
F$, hence $x^{n}\longrightarrow x^{2n}\in F$. ∎
By Prop. 4.6, Prop. 4.7 and Prop. 4.8, we have the following result:
###### Proposition 4.9.
Let $F$ be a filter of $L$. Then for all $x,y,z\in L$, the following
conditions are equivalent:
* (i)
$F$ is an n-fold implicative filter of $L$.
* (ii)
$x^{n}\longrightarrow x^{2n}\in F$.
* (iii)
If $x^{n+1}\longrightarrow y\in F$, then $x^{n}\longrightarrow y\in F$.
* (iv)
If $x^{n}\longrightarrow(y\longrightarrow z)\in F$, then
$(x^{n}\longrightarrow y)\longrightarrow(x^{n}\longrightarrow z)\in F$.
###### Proposition 4.10.
If a filter $F$ is an n-fold implicative filter, then $F$ is an (n+1)-fold
implicative filter.
###### Proof.
Let $F$ be a filter. Assume that $F$ is an n-fold implicative filter. Let
$x,y\in L$ such that $x^{n+2}\longrightarrow y\in F$, by Prop. 2.1,
$x^{n+1}\longrightarrow(x\longrightarrow y)=x^{n+2}\longrightarrow y\in F$.
Since $F$ is an n-fold implicative filter, apply Prop. 4.9(iii),
we obtain $x^{n}\longrightarrow(x\longrightarrow y)\in F$. Hence
$x^{n+1}\longrightarrow y\in F$ and by Prop. 4.9, $F$ is an (n+1)-fold
implicative filter. ∎
By the following example, we show that the converse of Prop. 4.10 is not true
in general.
###### Example 4.11.
[10]
Let $L=\\{0,a,b,1\\}$ be a lattice such that $0<a<b<1$. Define the operations
$\otimes$ and $\rightarrow$ by the two tables. Then
$(L,\wedge,\vee,\otimes,\rightarrow,0,1)$ is a residuated lattice.
$\otimes$ | $0$ | $a$ | $b$ | $1$
---|---|---|---|---
$0$ | $0$ | $0$ | $0$ | $0$
$a$ | $0$ | $0$ | $0$ | $a$
$b$ | $0$ | $0$ | $a$ | $b$
$1$ | $0$ | $a$ | $b$ | $1$
$\longrightarrow$ | $0$ | $a$ | $b$ | $1$
---|---|---|---|---
$0$ | $1$ | $1$ | $1$ | $1$
$a$ | $b$ | $1$ | $1$ | $1$
$b$ | $a$ | $b$ | $1$ | $1$
$1$ | $0$ | $a$ | $b$ | $1$
$\\{1\\}$ is an 3-fold implicative filter but $\\{1\\}$ is not an 2-fold
implicative filter, since $b^{2}\longrightarrow a\in\\{1\\}$ but
$b^{1}\longrightarrow a=b\notin\\{1\\}$
###### Proposition 4.12.
n-fold implicative filters are filters.
###### Proof.
Suppose that $F$ is an n-fold implicative filter of $L$. Let $z,y\in L$ such
that $y,y\longrightarrow z\in F$. We have $1^{n}\longrightarrow
y,1^{n}\longrightarrow(y\longrightarrow z)\in F$, this implies
$z=1^{n}\longrightarrow z\in F$. Hence $F$ is a deductive system of $L$ and
the thesis follows from the fact1. ∎
By the following example, we show that the converse of Prop. 4.12 is not true
in general.
###### Example 4.13.
Let $L$ be a residuated lattice from Example 4.11. $\\{1\\}$ is a filter but
$\\{1\\}$ is not an 2-fold implicative filter, since $b^{2}\longrightarrow
a\in\\{1\\}$ but $b^{1}\longrightarrow a=b\notin\\{1\\}$.
Using Prop.4.9, it is easy to show the following results:
###### Corollary 4.14.
If $L$ is an n-fold implicative residuated lattice then, the concepts of
n-fold implicative filters and filters coincide.
###### Theorem 4.15.
Let $F_{1}$ and $F_{2}$ two filters of $L$ such that $F_{1}\subseteq F_{2}$.
If $F_{1}$ is an n-fold implicative filter, then $F_{2}$ is an n-fold
implicative filter.
The following theorem gives the relation between n-fold implicative residuated
lattice and n-fold implicative filter.
###### Proposition 4.16.
Let $F$ be a filter of $L$. The following conditions are equivalent:
* (i)
$L$ is an n-fold implicative residuated lattice.
* (ii)
Every filter of $L$ is an n-fold implicative filter of $L$.
* (iii)
{1} is an n-fold implicative filter of $L$.
* (iv)
$x^{n}=x^{2n}$ for all $x\in L$.
###### Proof.
$(i)\longrightarrow(ii)$ : follows from Corollary. 4.14
$(ii)\longrightarrow(iii)$ : follows from the fact that {1} is a filter of
$L$.
$(iii)\longrightarrow(iv)$ : Assume that {1} is an n-fold implicative filter
of $L$. From Prop 4.9, we have $x^{n}\longrightarrow x^{2n}=1$ for all $x\in
L$. So $x^{n}\leq x^{2n}$ for all $x\in L$. Since $x^{2n}\leq x^{n}$ for all
$x\in L$, we obtain $x^{n}=x^{2n}$ for all $x\in L$.
$(iv)\longrightarrow(i)$ : If $x^{n}=x^{2n}$ for all $x\in L$, we have
$x^{n}\longrightarrow x^{2n}=1\in\\{1\\}$ for all $x\in L$, by Prop. 4.9,
$\\{1\\}$ is an n-fold implicative filter of $L$. Since
$x^{n}\longrightarrow(x^{n}\longrightarrow x^{n+1})=1\in\\{1\\}$ and
$x^{n}\longrightarrow x^{n}=1\in\\{1\\}$, we get $x^{n}\longrightarrow
x^{n+1}\in\\{1\\}$, that is $x^{n+1}=x^{n}$ for all $x\in L$. ∎
###### Corollary 4.17.
A filter $F$ of a residuated lattice $L$ is an n-fold implicative filter if
and only if $L/F$ is an n-fold implicative residuated lattice.
###### Proof.
Let $F$ be a filter.
Suppose that $F$ is an n-fold implicative filter. By Prop. 4.9(ii), we have
$x^{n}\longrightarrow x^{2n}\in F$ for all $x\in L$, that is
$(x^{n}\longrightarrow x^{2n})/F=1/F$ for all $x\in L$. So
$(x/F)^{n}\longrightarrow(x/F)^{2n}=(x^{n}/F)\longrightarrow(x^{2n}/F)=(x^{n}\longrightarrow
x^{2n})/F=1/F$ for all $x/F\in L/F$, by Prop. 4.16(iv), $L/F$ is an n-fold
implicative residuated lattice.
Suppose conversely that $L/F$ is an n-fold implicative residuated lattice. By
Prop. 4.16(iv), we get $(x/F)^{n}=(x/F)^{2n}$ for all $x/F\in L/F$ or
equivalently $(x^{n}/F)=(x^{2n}/F)$ for all $x\in L$. That is
$(x^{n}\longrightarrow x^{2n})/F=1/F$ for all $x\in L$. Hence
$x^{n}\longrightarrow x^{2n}\in F$ for all $x\in L$, we obtain the result by
apply Prop. 4.9(ii). ∎
By (3)[4] and Corollary 4.17, we have the following result.
###### Corollary 4.18.
A filter $F$ of a residuated lattice $L$ is an 1-fold implicative filter if
and only if $L/F$ is a Heyting algebra. As a consequence, it is easy to
observe that, a residuated lattice $L$ is a Heyting algebra if and only if
$\\{1\\}$ is an 1-fold implicative filter of $L$ if and only if $L$ is an
1-fold implicative residuated lattice.
## 5\. N-FOLD POSITIVE IMPLICATIVE FILTERS OF RESIDUATED LATTICES
###### Definition 5.1.
$F$ is an n-fold positive implicative filter if it satisfies the following
conditions:
* (i)
$1\in F$
* (ii)
For all $x,y,z\in L$, if $x\longrightarrow((y^{n}\longrightarrow
z)\longrightarrow y)\in F$ and $x\in F$, then $y\in F$.
In particular 1-fold positive implicative filters are positive implicative
filters.[7]
###### Example 5.2.
Let $n\geq 1$. Let $L$ be a residuated lattice from Example 2.5. Simple
computations proves that $F_{3}=\\{1,a,b,c\\}$ is an n-fold positive
implicative filter.
###### Proposition 5.3.
Every n-fold positive implicative filter is a filter.
###### Proof.
Let $F$ be an n-fold positive implicative filter of $L$, it is clear that
$1\in F$. Since for any $y\in F$, $y^{n}\longrightarrow 1=1$, by setting $z=1$
in the definition of n-fold positive implicative filter, we obtain the result.
∎
The following Example shows that filters are not n-fold positive implicative
filters in general.
###### Example 5.4.
Let $L$ be a residuated lattice from Example 2.5 and $n\geq 1$.
$F_{1}=\\{1,a\\},F_{2}=\\{1,b\\}$ are filters but not n-fold positive
implicative filters since $1\longrightarrow((b^{n}\longrightarrow
0)\longrightarrow b)\in F_{1}$ and $1\in F_{1}$, but $b\notin F_{1}$;
$1\longrightarrow((a^{n}\longrightarrow 0)\longrightarrow a)\in F_{2}$ and
$1\in F_{2}$, but $a\notin F_{2}$.
The following proposition gives a characterization of n-fold positive
implicative filter for any $n\geq 1$ .
###### Proposition 5.5.
The following conditions are equivalent for any filter $F$ and any $n\geq 1$ :
* (i)
$F$ is an n-fold positive implicative filter
* (ii)
For all $x,y\in L$, $(x^{n}\longrightarrow y)\longrightarrow x\in F$ implies
$x\in F$.
* (iii)
For all $x\in L$, $\overline{x^{n}}\longrightarrow x\in F$ implies $x\in F$.
###### Proof.
$(i)\longrightarrow(ii)$ : Suppose that $F$ is n-fold positive implicative
filter of $L$ and $(x^{n}\longrightarrow y)\longrightarrow x\in F$, since
$1\longrightarrow((x^{n}\longrightarrow y)\longrightarrow
x)=(x^{n}\longrightarrow y)\longrightarrow x\in F$ and $1\in F$, we apply the
fact that $F$ is n-fold positive implicative filter of $L$ and obtain the
result.
$(ii)\longrightarrow(iii)$ : We obtain the result by setting $y=0$ in the
equation $(ii)$.
$(iii)\longrightarrow(i)$ : Suppose that
$x\longrightarrow((y^{n}\longrightarrow z)\longrightarrow y)\in F$ and $x\in
F$, from the fact that $F$ is filter, we obtain $(y^{n}\longrightarrow
z)\longrightarrow y\in F$.
On the other hand, from Prop. 2.1(4), we have : $(y^{n}\longrightarrow
z)\longrightarrow y\leq(y^{n}\longrightarrow 0)\longrightarrow y$, from the
fact that $F$ is filter, we obtain $(y^{n}\longrightarrow 0)\longrightarrow
y\in F$, we apply the hypothesis and obtain $y\in F$. ∎
###### Corollary 5.6.
A proper filter $F$ is an n-fold positive implicative filter if an only if for
all $x\in L$, $x\vee\overline{x^{n}}\in F$.
###### Proof.
Assume that for all $x\in L$, $\overline{x^{n}}\longrightarrow x\in F$ and
$x\vee\overline{x^{n}}\in F$. By Prop. 5.5, we must show that $x\in F$. Since
by (14)Prop. 2.1, $x\vee\overline{x^{n}}\leq(\overline{x^{n}}\longrightarrow
x)\longrightarrow x$, we have $(\overline{x^{n}}\longrightarrow
x)\longrightarrow x\in F$. Using the fact that
$\overline{x^{n}}\longrightarrow x\in F$, we have $x\in F$.
Conversely suppose that $F$ is an n-fold positive implicative filter. Let
$x\in L$. Let $t=x\vee\overline{x^{n}}$, we must show that $t\in F$. Since
$x\leq t$, we have $x^{n}\leq t^{n}$ and then
$\overline{t^{n}}\leq\overline{x^{n}}\leq\overline{x^{n}}\vee x=t$. Hence
$\overline{t^{n}}\leq t$ or equivalently $\overline{t^{n}}\longrightarrow
t=1$. So $\overline{t^{n}}\longrightarrow t\in F$. From this and the fact that
$F$ is an n-fold positive implicative filter, by Prop. 5.5, we get that $t\in
F$. ∎
###### Definition 5.7.
$F$ is an n-fold boolean filter if it satisfies the following conditions:
$x\vee\overline{x^{n}}\in F$ for all $x\in L$. In particular 1-fold boolean
filters are boolean filters.[4].
The extension theorem of n-fold positive implicative filters is obtained from
the following result:
###### Theorem 5.8.
Let $n\geq 1$. Let $F_{1}$ and $F_{2}$ two filters of $L$ such that
$F_{1}\subseteq F_{2}$. If $F_{1}$ is an n-fold positive implicative filter,
then so is $F_{2}$.
###### Proof.
If $F_{1}$ is an n-fold positive implicative filter, then by Corollary 5.6, we
get $\overline{x^{n}}\vee x\in F_{1}$ for all $x\in L$. Since $F_{1}\subseteq
F_{2}$, we have $\overline{x^{n}}\vee x\in F_{2}$ for all $x\in L$ and by
Corollary 5.6, $F_{2}$ is an n-fold positive implicative filter. ∎
The following theorem gives the relation between n-fold positive implicative
filters and n-fold implicative filters in residuated lattices.
###### Theorem 5.9.
Every n-fold positive implicative filter of $L$ is an n-fold implicative
filter of $L$.
###### Proof.
Let $F$ be an n-fold positive implicative filter of $L$. Let $x,y\in L$ be
such that $x^{n+1}\longrightarrow y\in F$. By Prop. 2.1 we have the following:
* (1)
$(x^{n+1}\longrightarrow y)^{n}\longrightarrow(x^{n}\longrightarrow y)$=
$[(x^{n+1}\longrightarrow y)^{n-1}\otimes(x^{n+1}\longrightarrow
y)]\longrightarrow(x^{n}\longrightarrow y)$=
$[(x^{n+1}\longrightarrow y)^{n-1}]\longrightarrow[(x^{n+1}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow y)$=
$[(x^{n+1}\longrightarrow y)^{n-1}]\longrightarrow[(x^{n+1}\longrightarrow
y)\longrightarrow[x^{n-1}\longrightarrow(x\longrightarrow y)]$=
$[(x^{n+1}\longrightarrow
y)^{n-1}]\longrightarrow[x^{n-1}\longrightarrow[(x^{n+1}\longrightarrow
y)\longrightarrow(x\longrightarrow y)]]$=
$[(x^{n+1}\longrightarrow
y)^{n-1}]\longrightarrow[x^{n-1}\longrightarrow[(x\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x\longrightarrow y)]]$
* (2)
So by (1) we have :$(x^{n+1}\longrightarrow
y)^{n}\longrightarrow(x^{n}\longrightarrow y)$=
$[(x^{n+1}\longrightarrow
y)^{n-1}]\longrightarrow[x^{n-1}\longrightarrow[(x\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x\longrightarrow y)]]$
* (3)
We have : $(x^{n}\longrightarrow y)\longrightarrow
y\leq[(x\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x\longrightarrow y)]$
* (4)
By (3) we have :$[(x^{n+1}\longrightarrow
y)^{n-1}]\longrightarrow[x^{n-1}\longrightarrow[(x^{n}\longrightarrow
y)\longrightarrow y]]\leq[(x^{n+1}\longrightarrow
y)^{n-1}]\longrightarrow[x^{n-1}\longrightarrow[[(x\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x\longrightarrow y)]]]$
* (5)
By (4) and (2) , we have : $((x^{n+1}\longrightarrow
y)^{n-1})\longrightarrow(x^{n-1}\longrightarrow((x^{n}\longrightarrow
y)\longrightarrow y))\leq(x^{n+1}\longrightarrow
y)^{n}\longrightarrow(x^{n}\longrightarrow y)$.
* (6)
We have: $[(x^{n+1}\longrightarrow
y)^{n-1}]\longrightarrow[x^{n-1}\longrightarrow[(x^{n}\longrightarrow
y)\longrightarrow y]]=[(x^{n+1}\longrightarrow
y)^{n-1}]\longrightarrow[(x^{n}\longrightarrow
y)\longrightarrow[x^{n-1}\longrightarrow y]]=[(x^{n}\longrightarrow
y)]\longrightarrow[(x^{n+1}\longrightarrow
y)^{n-1}\longrightarrow[x^{n-1}\longrightarrow y]]$
* (7)
By (6) and (5) we get : $(x^{n}\longrightarrow
y)\longrightarrow[(x^{n+1}\longrightarrow
y)^{n-1}\longrightarrow(x^{n-1}\longrightarrow y)]\leq(x^{n+1}\longrightarrow
y)^{n}\longrightarrow(x^{n}\longrightarrow y)$
* (8)
We have : $(x^{n}\longrightarrow y)\otimes(x^{n+1}\longrightarrow
y)^{n-1}\otimes x^{n-1}=(x^{n}\longrightarrow y)\otimes(x^{n+1}\longrightarrow
y)^{n-2}\otimes(x^{n+1}\longrightarrow y)\otimes x^{n-2}\otimes
x=(x^{n}\longrightarrow y)\otimes(x^{n+1}\longrightarrow y)^{n-2}\otimes
x^{n-2}\otimes x\otimes(x^{n+1}\longrightarrow y)$
* (9)
We also have : $x\otimes(x^{n+1}\longrightarrow
y)=x\otimes[x\longrightarrow(x^{n}\longrightarrow y)]\leq x^{n}\longrightarrow
y$
* (10)
Then : $(x^{n}\longrightarrow y)\otimes(x^{n+1}\longrightarrow y)^{n-2}\otimes
x^{n-2}\otimes x\otimes(x^{n+1}\longrightarrow y)\leq(x^{n}\longrightarrow
y)\otimes(x^{n+1}\longrightarrow y)^{n-2}\otimes
x^{n-2}\otimes(x^{n}\longrightarrow y)$
* (11)
So: $(x^{n}\longrightarrow y)\otimes(x^{n+1}\longrightarrow y)^{n-1}\otimes
x^{n-1}\leq(x^{n}\longrightarrow y)^{2}\otimes(x^{n+1}\longrightarrow
y)^{n-2}\otimes x^{n-2}$
* (12)
By (11) we get: $[(x^{n}\longrightarrow y)^{2}\otimes(x^{n+1}\longrightarrow
y)^{n-2}\otimes x^{n-2}]\longrightarrow y\leq[(x^{n}\longrightarrow
y)\otimes(x^{n+1}\longrightarrow y)^{n-1}\otimes x^{n-1}]\longrightarrow y$
* (13)
By (12), we have: $((x^{n}\longrightarrow y)^{2}\otimes(x^{n+1}\longrightarrow
y)^{n-2})\longrightarrow(x^{n-2}\longrightarrow y)\leq((x^{n}\longrightarrow
y)\otimes(x^{n+1}\longrightarrow
y)^{n-1})\longrightarrow(x^{n-1}\longrightarrow y)$
* (14)
So : $(x^{n}\longrightarrow y)^{2}\longrightarrow((x^{n+1}\longrightarrow
y)^{n-2}\longrightarrow(x^{n-2}\longrightarrow y))\leq(x^{n}\longrightarrow
y)\longrightarrow((x^{n+1}\longrightarrow
y)^{n-1}\longrightarrow(x^{n-1}\longrightarrow y))$
* (15)
By (14) and (7), we have : $(x^{n}\longrightarrow
y)^{2}\longrightarrow((x^{n+1}\longrightarrow
y)^{n-2}\longrightarrow(x^{n-2}\longrightarrow y))\leq(x^{n+1}\longrightarrow
y)^{n}\longrightarrow(x^{n}\longrightarrow y)$
By repeating (15) n times, we obtain: $(x^{n}\longrightarrow
y)^{n}\longrightarrow((x^{n+1}\longrightarrow
y)^{0}\longrightarrow(x^{0}\longrightarrow y))\leq(x^{n+1}\longrightarrow
y)^{n}\longrightarrow(x^{n}\longrightarrow y)$. This implies
$(x^{n}\longrightarrow y)^{n}\longrightarrow(1\longrightarrow(1\longrightarrow
y))\leq(x^{n+1}\longrightarrow y)^{n}\longrightarrow(x^{n}\longrightarrow y)$.
Hence $(x^{n}\longrightarrow y)^{n}\longrightarrow
y\leq(x^{n+1}\longrightarrow y)^{n}\longrightarrow(x^{n}\longrightarrow y)$.
Then $((x^{n}\longrightarrow y)^{n}\longrightarrow
y)\longrightarrow((x^{n+1}\longrightarrow
y)^{n}\longrightarrow(x^{n}\longrightarrow y))=1$. Hence by Prop. 2.1 we have
: $(x^{n+1}\longrightarrow y)^{n}\longrightarrow(((x^{n}\longrightarrow
y)^{n}\longrightarrow y)\longrightarrow(x^{n}\longrightarrow y))=1\in F$.
Since $(x^{n+1}\longrightarrow y)\in F$ and $F$ is a filter, we have
$(x^{n+1}\longrightarrow y)^{n}\in F$. By the fact that $F$ is a filter and
$(x^{n+1}\longrightarrow y)^{n}\longrightarrow(((x^{n}\longrightarrow
y)^{n}\longrightarrow y)\longrightarrow(x^{n}\longrightarrow y))\in F$, we
have $((x^{n}\longrightarrow y)^{n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow y)\in F$. Since $F$ is an n-fold
positive implicative filter, by Prop. 5.5 we have: $x^{n}\longrightarrow y\in
F$.
By Prop. 4.9, $F$ is an n-fold implicative filter.
∎
The following Example shows that n-fold implicative filters may not be n-fold
positive implicative filters.
###### Example 5.10.
Let $L$ be a residuated lattice from Example 2.5 and $n\geq 1$.
$F_{1}=\\{1,a\\},F_{2}=\\{1,b\\}$ are n-fold implicative filters but not
n-fold positive implicative filters since
$1\longrightarrow((b^{n}\longrightarrow 0)\longrightarrow b)\in F_{1}$ and
$1\in F_{1}$, but $b\notin F_{1}$; $1\longrightarrow((a^{n}\longrightarrow
0)\longrightarrow a)\in F_{2}$ and $1\in F_{2}$, but $a\notin F_{2}$.
###### Proposition 5.11.
Every n-fold positive implicative filter is an (n+1)-fold positive implicative
filter.
###### Proof.
Let $F$ be an n-fold positive implicative filter of $L$. Let $x\in L$ such
that $\overline{x^{n+1}}\longrightarrow x\in F$. We show that $x\in F$. Since
$\overline{x^{n+1}}\longrightarrow x\leq\overline{x^{n}}\longrightarrow x$, by
the fact that $F$ is a filter, we get $\overline{x^{n}}\longrightarrow x\in
F$. Since $F$ is an n-fold positive implicative filter of $L$, by Prop. 5.5,
we obtain $x\in F$. By Prop. 5.5, $F$ is an (n+1)-fold positive implicative
filter of $L$. ∎
By the following example, we show that the converse of Prop. 5.11 is not true
in general.
###### Example 5.12.
Let $L$ be a residuated lattice from Example 4.11.It is clear that $\\{1\\}$
is an 3-fold positive implicative filter but $\\{1\\}$ is not an 2-fold
positive implicative filter, since $\overline{b^{2}}\longrightarrow
b\in\\{1\\}$ and $b\notin\\{1\\}$.
###### Definition 5.13.
A residuated lattice $L$ is called n-fold positive implicative residuated
lattice if it satisfies $\overline{y^{n}}\longrightarrow y=y$ for each $y\in
L$.
###### Remark 5.14.
Since $\overline{x^{n}}\longrightarrow x\leq\overline{x}\longrightarrow x$ for
all $x\in L$, it is clear that 1-fold residuated lattices are n-fold
residuated lattices. It is also clear that n-fold residuated lattices are
(n+1)-fold residuated lattices since $\overline{x^{n+1}}\longrightarrow
x\leq\overline{x^{n}}\longrightarrow x$ for all $x\in L$
The following example shows that residuated lattices are not in general n-fold
positive implicative residuated lattices.
###### Example 5.15.
Let $L$ be a residuated lattice from Example 2.5 and $n\geq 2$. $L$ is not an
n-fold positive implicative residuated lattice since $(b^{n}\longrightarrow
0)\longrightarrow b=(b\longrightarrow 0)\longrightarrow b=0\longrightarrow
b=1\neq b$
It is follows from Prop.5.5 and Prop.5.3 the following proposition:
###### Proposition 5.16.
If $L$ is an n-fold positive implicative residuated lattice then the notion of
n-fold positive implicative filter and filter coincide.
###### Proposition 5.17.
The following conditions are equivalent :
* (i)
$L$ is an n-fold positive implicative residuated lattice.
* (ii)
Every filter $F$ of $L$ is an n-fold positive implicative filter of $L$
* (iii)
$\\{1\\}$ is an n-fold positive implicative filter
###### Proof.
$(i)\longrightarrow(ii)$ : Follows from Prop.5.16
$(ii)\longrightarrow(iii)$ : Follows from the fact that $\\{1\\}$ is a filter
of $L$.
$(iii)\longrightarrow(i)$: Assume that $\\{1\\}$ is an n-fold positive
implicative filter. Let $x\in L$. By Corollary 5.6, for all $x\in L$ holds
$x\vee\overline{x^{n}}=1$. By (14)Prop. 2.1,
$x\vee\overline{x^{n}}\leq(\overline{x^{n}}\longrightarrow x)\longrightarrow
x$, Hence $\overline{x^{n}}\longrightarrow x\longrightarrow x=1$ or
equivalently $\overline{x^{n}}\longrightarrow x\leq x$, and by the fact that
$x\leq\overline{x^{n}}\longrightarrow x$, we have
$\overline{x^{n}}\longrightarrow x=x$. ∎
The following result which follows from Prop. 5.17 and Prop. 4.16, gives the
relation between n-fold positive implicative residuated lattices and n-fold
implicative residuated lattices.
###### Proposition 5.18.
n-fold positive implicative residuated lattices are n-fold implicative
residuated lattices.
By the following example, we show that the converse of Prop. 5.18 is not true
in general.
###### Example 5.19.
A residuated lattice $L$ from Example 2.5 is an n-fold implicative residuated
lattice but by Example 5.15, it is not an fold positive implicative residuated
lattice.
The following result which follows from Prop. 5.17 and Corollary 5.6, gives
new a characterization of n-fold positive implicative residuated lattices.
###### Corollary 5.20.
A residuated lattice $L$ is an n-fold positive implicative residuated lattice
if and only if it satisfies $\overline{y^{n}}\vee y=1$ for each $y\in L$.
###### Proposition 5.21.
If $L$ is a totaly ordered residuated lattice, then any n-fold positive
implicative filter $F$ is maximal filter of $L$ and $L/F$ is a locally finite
residuated lattice.
###### Proof.
Let $L$ be a totaly ordered residuated lattice. Assume that $F$ is n-fold
positive implicative filter and let $x\in L$ be such an element that $x\notin
F$. From Prop. 5.5 an assumption $\overline{x^{n}}\leq x$, or equivalently
$\overline{x^{n}}\ \longrightarrow x=1\in F$ leads to a contradiction $x\in
F$, so we necessarily have $x<\overline{x^{n}}$. Therefore $x^{n+1}=0\in F$
and so $\overline{x^{n+1}}=1\in F$ . The thesis follows from Prop. 2.10. ∎
At consequence of Prop. 5.21, we have the following results:
###### Corollary 5.22.
A totaly ordered residuated lattice is a locally finite if $\\{1\\}$ is an
n-fold positive implicative filter. A totaly ordered n-fold positive
implicative residuated lattice is a locally finite.
###### Proposition 5.23.
A filter $F$ of $L$ is an n-fold positive implicative filter if and only $L/F$
is an n-fold positive implicative residuated lattice.
###### Proof.
Suppose that $F$ is an n-fold positive implicative filter. Let $x\in L$ be
such that $\overline{(x/F)^{n}}\longrightarrow x/F\in\\{1/F\\}$, then
$(\overline{x^{n}}\longrightarrow x)/F=\overline{(x/F)^{n}}\longrightarrow
x/F=1/F$. So $(\overline{x^{n}}\longrightarrow x)\in F$, by the fact that $F$
is an n-fold positive implicative filter, we have $x\in F$. Hence
$x/F\in\\{1/F\\}$, by Prop. 5.5, $\\{1/F\\}$ is an n-fold positive implicative
filter of $L/F$, by Prop. 5.17, $L/F$ is an n-fold positive implicative
residuated lattice.
Suppose conversely that $L/F$ is an n-fold positive implicative residuated
lattice. Let $x\in L$ be such that $\overline{x^{n}}\longrightarrow x\in F$.
We have $(\overline{x^{n}}\longrightarrow x)/F=1/F$, this implies
$\overline{(x/F)^{n}}\longrightarrow x/F\in\\{1/F\\}$. Since $L/F$ is an
n-fold positive implicative residuated lattice, by Prop. 5.17, $\\{1/F\\}$ is
an n-fold positive implicative filter of $L/F$. Hence $x/F\in\\{1/F\\}$ or
equivalently $x\in F$. By Prop. 5.5, $F$ is an n-fold positive implicative
filter of $L$. ∎
The following examples shows that the notion of n-fold positive implicative
residuated lattices exist.
###### Example 5.24.
Let $L$ be a residuated lattice from Example 2.5 and $n\geq 2$. Since
$F_{3}=\\{1,a,b,c\\}$ is an n-fold positive implicative filter, by Prop. 5.23,
$L/F_{3}$ is an n-fold positive implicative residuated lattice.
Follows from Corollary 5.6, we have the following proposition.
###### Proposition 5.25.
In any residuated lattices, the concepts of n-fold boolean filters and n-fold
positive implicative filters coincide.
###### Definition 5.26.
$L$ is an n-fold boolean residuated lattice if it satisfies the following
conditions:
$x\vee\overline{x^{n}}=1$ for all $x\in L$. In particular 1-fold boolean
residuated lattices are boolean algebra.
It is easy to observe that:
###### Remark 5.27.
A residuated lattice $L$ is an n-fold boolean residuated lattice if and only
if $\\{1\\}$ is an n-fold boolean filter of $L$ if and only if $L$ is an
n-fold positive implicative residuated lattice.
## 6\. N-Fold Normal Filter in Residuated Lattices
In [13], the authors introduce the notion of n-fold normal filter in BL-
algebra. This motives us to introduce the notion of n-fold normal filter in
residuated lattices.
###### Proposition 6.1.
Let $n\geq 1$. The following conditions are equivalent for any filter $F$:
* (i)
For all $x,y,z\in L$, if $z\longrightarrow((y^{n}\longrightarrow
x)\longrightarrow x)\in F$ and $z\in F$, then $(x\longrightarrow
y)\longrightarrow y\in F$.
* (ii)
For all $x,y\in L$, $(y^{n}\longrightarrow x)\longrightarrow x\in F$ implies
$(x\longrightarrow y)\longrightarrow y\in F$.
###### Proof.
Let $F$ be a filter which satisfying (i) . Assume that $(y^{n}\longrightarrow
x)\longrightarrow x\in F$. We have $(y^{n}\longrightarrow x)\longrightarrow
x=1\longrightarrow((y^{n}\longrightarrow x)\longrightarrow x)\in F$ and $1\in
F$, by the fact that $F$ satisfying (i), we obtain $(x\longrightarrow
y)\longrightarrow y\in F$. Conversely, let
$z\longrightarrow((y^{n}\longrightarrow x)\longrightarrow x)\in F$ and $z\in
F$, Since $F$ is a filter, we have $(y^{n}\longrightarrow x)\longrightarrow
x\in F$. By hypothesis, we obtain $(x\longrightarrow y)\longrightarrow y\in
F$. ∎
###### Definition 6.2.
A filter $F$ is an n-fold normal filter if it satisfies one of the conditions
of Prop. 6.1. In particular 1-fold normal filters are normal filters.
###### Example 6.3.
Let $n\geq 1$. Let $L$ be a residuated lattice from Example 2.4. Simple
computations proves that $F_{3}=\\{1,a,b,c\\}$ is an n-fold normal filter.
The following example shows that filters may not be n-fold normal in general.
###### Example 6.4.
Let $L$ be a residuated lattice from Example 2.4. $F_{2}=\\{1,b\\}$ is not an
n-fold normal filter since $(a^{n}\longrightarrow 0)\longrightarrow
0=1\in\\{1,b\\}$ but $[(0\longrightarrow a)\longrightarrow
a]=a\notin\\{1,b\\}$.
###### Proposition 6.5.
n-fold positive implicative filters are n-fold normal filters.
###### Proof.
Assume that $F$ is an n-fold positive implicative filter an let
$(x^{n}\longrightarrow y)\longrightarrow y\in F$.
We must show that $(y\longrightarrow x)\longrightarrow x\in F$.
Since $y\leq(y\longrightarrow x)\longrightarrow x$, by (4) Prop.2.1, we obtain
$(x^{n}\longrightarrow y)\longrightarrow y\leq(x^{n}\longrightarrow
y)\longrightarrow((y\longrightarrow x)\longrightarrow x)$.
From this and the fact that $F$ is a filter, we obtain $(x^{n}\longrightarrow
y)\longrightarrow((y\longrightarrow x)\longrightarrow x)\in F$.
Since $x\leq(y\longrightarrow x)\longrightarrow x$, by (4)Prop.2.1 we have
$x^{n}\leq((y\longrightarrow x)\longrightarrow x)^{n}$, hence
$(x^{n}\longrightarrow y)\longrightarrow[(y\longrightarrow x)\longrightarrow
x]\leq([(y\longrightarrow x)\longrightarrow x]^{n}\longrightarrow
y)\longrightarrow[(y\longrightarrow x)\longrightarrow x]$.
Since $(x^{n}\longrightarrow y)\longrightarrow[(y\longrightarrow
x)\longrightarrow x]\in F$, by the fact that $F$ is a filter, we obtain
$([(y\longrightarrow x)\longrightarrow x]^{n}\longrightarrow
y)\longrightarrow[(y\longrightarrow x)\longrightarrow x]\in F$. From this and
the fact that $F$ is an n-fold positive implicative filter, we obtain the
result by apply Prop.5.5. ∎
By the following example, we show that the converse of Prop. 6.5 is not true
in general.
###### Example 6.6.
Let $L$ be a lattice from Example 4.11 $\\{1\\}$ is an 1-fold normal filter
but $\\{1\\}$ is not an 1-fold positive implicative filter.
## 7\. n-fold fantastic Filter in Residuated Lattices
###### Definition 7.1.
Let $n\geq 1$. $F$ is an n-fold fantastic filter $L$ if it satisfies the
following conditions:
* (i)
$1\in F$
* (ii)
For all $x,y\in L$, $y\longrightarrow x\in F$ implies $[(x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow x\in F$.
In particular 1-fold fantastic filters are fantastic filters.[7]
###### Example 7.2.
Let $n\geq 1$. Let $L$ be a residuated lattice from Example 2.3. It is easy to
check that $\\{1\\}$ is an n-fold fantastic filter.
The following example shows that filters may not be n-fold fantastic in
general.
###### Example 7.3.
Let $L$ be a residuated lattice from Example 2.2. $\\{1\\}$ is not an n-fold
fantastic filter since $a\longrightarrow c=1\in\\{1\\}$ but
$[(c^{n}\longrightarrow a)\longrightarrow a]\longrightarrow c=c\notin\\{1\\}$.
###### Proposition 7.4.
Let $n\geq 1$. n-fold positive implicative filters are n-fold fantastic
filters.
###### Proof.
Assume that $F$ is an n-fold positive implicative filter. Let $x,y\in L$ be
such that $y\longrightarrow x\in F$.
By Prop. 2.1, we have:
$x\leq[((x^{n}\longrightarrow y)\longrightarrow y)\longrightarrow x]$. (a)
Then by Prop. 2.1, we also have:
$x^{n}\leq[((x^{n}\longrightarrow y)\longrightarrow y)\longrightarrow x]^{n}$.
(b)
By (b) and Prop. 2.1, we get, $(x^{n}\longrightarrow
y)\geq[((x^{n}\longrightarrow y)\longrightarrow y)\longrightarrow
x]^{n}\longrightarrow y$. (c)
By Prop. 2.1, we get, $y\longrightarrow x\leq((x^{n}\longrightarrow
y)\longrightarrow y)\longrightarrow((x^{n}\longrightarrow y)\longrightarrow
x)$. (d)
We also have :
$((x^{n}\longrightarrow y)\longrightarrow
y)\longrightarrow((x^{n}\longrightarrow y)\longrightarrow
x)=((x^{n}\longrightarrow y)\longrightarrow(((x^{n}\longrightarrow
y)\longrightarrow y)\longrightarrow x)$. (e)
So, by (d)and(e), we get, $y\longrightarrow x\leq((x^{n}\longrightarrow
y)\longrightarrow(((x^{n}\longrightarrow y)\longrightarrow y)\longrightarrow
x)$. (f)
By (c) and Prop. 2.1, we get,
$[((x^{n}\longrightarrow y)\longrightarrow[((x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow x]\leq[[((x^{n}\longrightarrow
y)\longrightarrow y)\longrightarrow x]^{n}\longrightarrow
y]\longrightarrow[[((x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow
x]]$ (g)
By (f) and (g) , we obtain,
$y\longrightarrow x\leq[[((x^{n}\longrightarrow y)\longrightarrow
y)\longrightarrow x]^{n}\longrightarrow
y]\longrightarrow[[((x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow
x]]$ (h)
Since $F$ is a filter (see Prop. 5.3), by (h) and the fact that
$y\longrightarrow x\in F$, we get :
$[[((x^{n}\longrightarrow y)\longrightarrow y)\longrightarrow
x]^{n}\longrightarrow y]\longrightarrow[[((x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow x]]\in F$. (w)
By (w), Prop. 5.5 and the fact that $F$ is an n-fold positive implicative
filter, we obtain $[[((x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow x]]\in F$.
Hence $F$ is an n-fold fantastic filter.
∎
The following example shows that n-fold fantastic filters may not be n-fold
positive implicative in general.
###### Example 7.5.
Let $n\geq 1$. Let $L$ be a residuated lattice from Example 2.3. It is easy to
check that $\\{1\\}$ is an n-fold fantastic filter, but not n-fold positive
implicative filter since $\overline{a^{n}}\longrightarrow a\in F$ and $a\notin
F$.
###### Proposition 7.6.
Let $n\geq 1$. n-fold fantastic filters are n-fold normal filters.
###### Proof.
Assume that $F$ is an n-fold fantastic filter. Let $x,y\in L$ be such that
$(x^{n}\longrightarrow y)\longrightarrow y\in F$ and $t=(y\longrightarrow
x)\longrightarrow x$. We must show that $t\in F$.
By Prop. 2.1, we have:
$y\leq(y\longrightarrow x)\longrightarrow x$, so $(x^{n}\longrightarrow
y)\longrightarrow y\leq[(x^{n}\longrightarrow
y)\longrightarrow[(y\longrightarrow x)\longrightarrow x]]$, that is
$(x^{n}\longrightarrow y)\longrightarrow y\leq[(x^{n}\longrightarrow
y)\longrightarrow t]$. (a)
Since $(x^{n}\longrightarrow y)\longrightarrow y\in F$, by (a) an the fact
that $F$ is a filter, it follows that $(x^{n}\longrightarrow y)\longrightarrow
t\in F$. (b)
By (b) and the fact that $F$ is an n-fold fantastic filter, we get that
$[(t^{n}\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow y)]\longrightarrow t\in F$. (c)
By Prop. 2.1, we also have $t^{n}\longrightarrow(x^{n}\longrightarrow
y)=x^{n}\longrightarrow(t^{n}\longrightarrow y)$, so
$(t^{n}\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow
y)=(x^{n}\longrightarrow(t^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow y)$ and then
$[(x^{n}\longrightarrow(t^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow y)]\longrightarrow t\in F$. (d)
On the other hand, by Prop. 2.1, we also have $(t^{n}\longrightarrow
y)\longrightarrow y\leq(x^{n}\longrightarrow(t^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow y)$. (e)
Since $x\leq t$, it follows that $(x^{n}\longrightarrow y)\longrightarrow
y\leq(t^{n}\longrightarrow y)\longrightarrow y$. (f)
Since $F$ is a filter and $(x^{n}\longrightarrow y)\longrightarrow y\in F$, by
(f) we obtain $(t^{n}\longrightarrow y)\longrightarrow y\in F$. (g)
Since $F$ is a filter, by (g) and (e) it follows that
$(x^{n}\longrightarrow(t^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow y)\in F$. (h)
Since $F$ is a filter, by (h) and (d) it follows that $t\in F$.
Hence $F$ is an n-fold normal filter. ∎
###### Lemma 7.7.
For all $x,y\in L$, we have:
$[(x^{n}\longrightarrow x^{2n})\otimes(x^{2n}\longrightarrow y)]\leq
x^{n}\longrightarrow y$.
###### Proof.
Let $x,y\in L$, by Prop.2.1 we have the following:
* (1)
$x^{n}\longrightarrow x^{2n}\leq[(x^{2n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow y)]$
* (2)
By (1) we have : $[(x^{2n}\longrightarrow y)\otimes(x^{n}\longrightarrow
x^{2n})]\leq[(x^{2n}\longrightarrow y)\otimes((x^{2n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow y))]$
* (3)
$[(x^{2n}\longrightarrow y)\otimes((x^{2n}\longrightarrow
y)\longrightarrow(x^{n}\longrightarrow y))]\leq x^{n}\longrightarrow y$
By (2) and (3) we have : $[(x^{n}\longrightarrow
x^{2n})\otimes(x^{2n}\longrightarrow y)]\leq x^{n}\longrightarrow y$. ∎
###### Proposition 7.8.
Let $n\geq 1$. Let $F$ a filter of $L$. If $F$ is n-fold fantastic and n-fold
implicative filter, then $F$ is an n-fold positive implicative filter.
###### Proof.
Let $x,y\in L$ be such that $(x^{n}\longrightarrow y)\longrightarrow x\in F$.
Assume that $F$ is both n-fold fantastic filter and n-fold implicative filter.
Since $F$ is a n-fold fantastic filter, by the fact that
$(x^{n}\longrightarrow y)\longrightarrow x\in F$, we have :
$[(x^{n}\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow y)]\longrightarrow x\in F$.
By Lemma 7.7 and residuation we get : $(x^{n}\longrightarrow
x^{2n})\leq(x^{2n}\longrightarrow y)\longrightarrow(x^{n}\longrightarrow y)$.
So, $(x^{n}\longrightarrow
x^{2n})\leq[x^{n}\longrightarrow(x^{n}\longrightarrow
y)]\longrightarrow(x^{n}\longrightarrow y)$. Hence, $[(x^{n}\longrightarrow
x^{2n})\longrightarrow x]\geq[[x^{n}\longrightarrow(x^{n}\longrightarrow
y)]\longrightarrow(x^{n}\longrightarrow y)]\longrightarrow x$. Since $F$ is a
filter, by the fact that $[(x^{n}\longrightarrow(x^{n}\longrightarrow
y))\longrightarrow(x^{n}\longrightarrow y)]\longrightarrow x\in F$, we have :
$(x^{n}\longrightarrow x^{2n})\longrightarrow x\in F$. Since $F$ is and n-fold
implicative filter, by Prop. 4.9, we get, $x\in F$. By Prop. 5.5, $F$ is an
n-fold positive implicative filter.
∎
Follows from Prop. 7.4, Prop. 7.8, Prop. 5.9, it is easy to show the following
theorem.
###### Theorem 7.9.
Let $n\geq 1$. Let $F$ a filter. $F$ is an n-fold positive implicative filter
of $L$ if and only if $F$ is n-fold fantastic and n-fold implicative filter of
$L$.
###### Definition 7.10.
$L$ is said to be n-fold fantastic residuated lattice if for all $x,y\in L$,
$y\longrightarrow x=[(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow
x$.
The following example shows that the notion of n-fold fantastic residuated
lattice exist.
###### Example 7.11.
Let $n\geq 1$. Let $L$ be a residuated lattice from Example 2.3. It is easy to
check that $L$ is an n-fold fantastic residuated lattice.
The following example shows that residuated lattices may not be n-fold
fantastic in general.
###### Example 7.12.
Let $L$ be a residuated lattice from Example 2.2. $L$ is not an n-fold
fantastic residuated lattice since $a\longrightarrow c=1\neq
c=[(c^{n}\longrightarrow a)\longrightarrow a]\longrightarrow c$.
The following proposition gives a characterization of n-fold fantastic
residuated lattice.
###### Proposition 7.13.
$L$ is an n-fold fantastic residuated lattice if and only if the inequality
$(x^{n}\longrightarrow y)\longrightarrow y\leq(y\longrightarrow
x)\longrightarrow x$ holds for all $x,y\in L$.
###### Proof.
Assume that $L$ is an n-fold fantastic residuated lattice. Let $x,y\in L$. We
have $[(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow[(y\longrightarrow x)\longrightarrow x]=(y\longrightarrow
x)\longrightarrow[[(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow
x]$. (a)
By hypothesis $y\longrightarrow x=[[(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow x]$. Hence $(y\longrightarrow
x)\longrightarrow[[(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow
x]=1$. (b)
By (a) and (b), we get $[(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow[(y\longrightarrow x)\longrightarrow x]=1$ or equivalently
$[(x^{n}\longrightarrow y)\longrightarrow y]\leq[(y\longrightarrow
x)\longrightarrow x]$.
Suppose conversely that the inequality $(x^{n}\longrightarrow
y)\longrightarrow y\leq(y\longrightarrow x)\longrightarrow x$ holds for all
$x,y\in L$. Then $(y\longrightarrow x)\longrightarrow[[(x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow x]=[(x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow[(y\longrightarrow x)\longrightarrow x]$.
(e)
Since $(x^{n}\longrightarrow y)\longrightarrow y\leq(y\longrightarrow
x)\longrightarrow x$, by Prop. 2.1, we get $[(x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow[(x^{n}\longrightarrow y)\longrightarrow
y]\leq[(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow[(y\longrightarrow x)\longrightarrow x]$, that is
$1\leq[(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow[(y\longrightarrow x)\longrightarrow x]$ or equivalently
$[(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow[(y\longrightarrow
x)\longrightarrow x]=1$. (f)
By (e) and (f), its follows that $(y\longrightarrow
x)\leq[[(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow x]$. (g)
Since $y\leq(x^{n}\longrightarrow y)\longrightarrow y$, we also get by Prop.
2.1, $y\longrightarrow x\geq[[(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow x]$. (h)
From (h) and (g), we obtain $y\longrightarrow x=[[(x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow x]$. Hence $L$ is an n-fold fantastic
residuated lattice. ∎
###### Proposition 7.14.
The following conditions are equivalent for any filter $F$:
* (i)
$L$ is an n-fold fantastic residuated lattice.
* (ii)
Every filter $F$ of $L$ is an n-fold fantastic filter of $L$
* (iii)
$\\{1\\}$ is an n-fold fantastic filter of $L$.
###### Proof.
$(i)\longrightarrow(ii)$ : Follows from Definition 7.10
$(ii)\longrightarrow(iii)$ : Follows from the fact that $\\{1\\}$ is a filter
of $L$.
$(iii)\longrightarrow(i)$: Assume that $\\{1\\}$ is an n-fold fantastic
filter.
Let $x,y\in L$ and $t=(y\longrightarrow x)\longrightarrow x$. By Prop. 2.1,
$y\leq t$. So $y\longrightarrow t=1$ and by the hypothesis, we have
$[(t^{n}\longrightarrow y)\longrightarrow y]\longrightarrow t=1$,
that is $[(t^{n}\longrightarrow y)\longrightarrow y]\leq t$. (w)
On the other hand, $x\leq t$ implies $x^{n}\leq t^{n}$, hence
$[(x^{n}\longrightarrow y)\longrightarrow y]\leq(t^{n}\longrightarrow
y)\longrightarrow y$. (z)
By (z) and (w), it follows that $[(x^{n}\longrightarrow y)\longrightarrow
y]\leq t=(y\longrightarrow x)\longrightarrow x$. Hence by Prop. 7.13, $L$ is
an n-fold fantastic residuated lattice. ∎
Combine Prop.7.14, Prop.5.17, Prop.4.16 and Theorem 7.9, we have the following
result:
###### Corollary 7.15.
Let $n\geq 1$. $L$ is an n-fold positive implicative residuated lattice if and
only if $L$ is n-fold fantastic residuated lattice and n-fold implicative
residuated lattice.
The following corollary gives a characterization of n-fold fantastic filter in
residuated lattice.
###### Corollary 7.16.
Let $n\geq 1$. Let $F$ be a filter of $L$. Then $F$ is an n-fold fantastic
filter if and only $L/F$ is is an n-fold fantastic residuated lattice.
###### Proof.
Let $F$ be a filter of $L$. Assume that $F$ is an n-fold fantastic filter. We
show that $L/F$ is is an n-fold fantastic residuated lattice. Let $x,y\in L$
be such that $y/F\longrightarrow x/F\in\\{1/F\\}$, then $(y\longrightarrow
x)/F=1/F$ or equivalently $y\longrightarrow x\in F$. Since $F$ is an n-fold
fantastic filter, we get $[(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow x\in F$ or equivalently $([(x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow x)/F=1/F$, so $([((x/F)^{n}\longrightarrow
y/F)\longrightarrow y/F]\longrightarrow x/F)\in\\{1/F\\}$. Hence $\\{1/F\\}$
is an n-fold fantastic filter of $L/F$, therefore by Prop. 7.14, $L/F$ is is
an n-fold fantastic residuated lattice.
Conversely, assume that $L/F$ is is an n-fold fantastic residuated lattice.
Let $x,y\in L$ be such that $y\longrightarrow x\in F$ then $(y\longrightarrow
x)/F=1/F$ or equivalently $y/F\longrightarrow x/F\in\\{1/F\\}$. Since $L/F$ is
is an n-fold fantastic residuated lattice, by Prop. 7.14, $\\{1/F\\}$ is an
n-fold fantastic filter of $L/F$. From this and the fact that
$y/F\longrightarrow x/F\in\\{1/F\\}$, we have: $([((x/F)^{n}\longrightarrow
y/F)\longrightarrow y/F]\longrightarrow x/F)\in\\{1/F\\}$ or equivalently
$([(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow x)/F=1/F$, so
$[(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow x\in F$. Hence $F$
is an n-fold fantastic filter. ∎
The extension theorem of n-fold fantastic filters is obtained from the
following result:
###### Theorem 7.17.
Let $n\geq 1$. Let $F_{1}$ and $F_{2}$ two filters of $L$ such that
$F_{1}\subseteq F_{2}$. If $F_{1}$ is an n-fold fantastic filter, then so is
$F_{2}$.
###### Proof.
Let $x,y\in L$ be such that $y\longrightarrow x\in F_{2}$. Since $F_{1}$ is an
n-fold fantastic filter, by Corollary. 7.16, $L/F_{1}$ is an n-fold positive
implicative residuated lattice. So $([((x/F_{1})^{n}\longrightarrow
y/F_{1})\longrightarrow y/F_{1}]\longrightarrow
x/F_{1})=y/F_{1}\longrightarrow x/F_{1}$, so $(y\longrightarrow
x)\longrightarrow([(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow
x)\in F_{1}$, so $(y\longrightarrow x)\longrightarrow([(x^{n}\longrightarrow
y)\longrightarrow y]\longrightarrow x)\in F_{2}$. Since $F_{2}$ is a filter of
$L$, by the fact that $y\longrightarrow x\in F_{2}$ and $(y\longrightarrow
x)\longrightarrow([(x^{n}\longrightarrow y)\longrightarrow y]\longrightarrow
x)\in F_{2}$, we get $([(x^{n}\longrightarrow y)\longrightarrow
y]\longrightarrow x)\in F_{2}$. Hence $F_{2}$ is an n-fold fantastic filter. ∎
## 8\. n-fold obstinate Filters in Residuated Lattices
###### Definition 8.1.
A filter $F$ is an n-fold obstinate filter of $L$ if it satisfies the
following conditions for any $n\geq 1$:
* (i)
$0\notin F$
* (ii)
For all $x,y\in L$, $x,y\notin F$ imply $x^{n}\longrightarrow y\in F$ and
$y^{n}\longrightarrow x\in F$
In particular 1-fold obstinate filters are obstinate filters.[8]
The following proposition gives a characterization of n-fold obstinate filter
of $L$.
###### Proposition 8.2.
For any $n\geq 1$, the following conditions are equivalent for any filter $F$:
* (i)
$F$ is an n-fold obstinate filter
* (ii)
For all $x\in L$, if $x\notin F$ then there exist $m\geq 1$ such that
$(\overline{x^{n}})^{m}\in F$
###### Proof.
$(i)\longrightarrow(ii)$ : Suppose that $F$ is n-fold obstinate filter of $L$.
Let $x\in L$ be such that $x\notin F$. By setting $y=0$ in Definition 8.1, we
get: $x^{n}\longrightarrow 0\in F$. Hence for $m=1$, we have:
$(\overline{x^{n}})^{m}\in F$
$(ii)\longrightarrow(i)$: Conversely, let $x,y\notin F$, we show that
$x^{n}\longrightarrow y\in F$ and $y^{n}\longrightarrow x\in F$. By the
hypothesis, there are $m_{1},m_{2}\geq 1$ such that
$(\overline{x^{n}})^{m_{1}},(\overline{y^{n}})^{m_{2}}\in F$.
By Prop. 2.1(8), we have:
$(\overline{x^{n}})^{m_{1}}\leq\overline{x^{n}}\leq x^{n}\longrightarrow y$
(a)
and $(\overline{y^{n}})^{m_{2}}\leq\overline{x}^{n}\leq y^{n}\longrightarrow
x$ (b)
Since $F$ is a filter, by (a) and (b), we get $x^{n}\longrightarrow y\in F$
and $y^{n}\longrightarrow x\in F$. ∎
###### Example 8.3.
Let $L$ be a lattice from Example 2.2. $F=\\{1,b,c,d\\}$ is a proper filter of
$L$. Using Prop. 8.2, for any $n\geq 1$, it is easy to check that $F$ is an
n-fold obstinate filter of $L$.
The following example shows that any filters may not be n-fold obstinate
filter.
###### Example 8.4.
Let $L$ be a lattice from Example 2.3. $F=\\{1,c,d\\}$ is a proper filter of
$L$. For any $n\geq 1$, we have: $a,b\notin F$ but $a^{n}\longrightarrow
b=b\notin F$. Hence $F$ is not an n-fold obstinate filter of $L$.
###### Proposition 8.5.
Every n-fold obstinate filter of $L$ is a (n+1)-fold obstinate filter of $L$.
###### Proof.
Let $F$ be an n-fold obstinate filter of $L$ and $x,y\notin F$. We show that
$x^{n+1}\longrightarrow y\in F$ and $y^{n+1}\longrightarrow x\in F$. By
hypothesis, $x^{n}\longrightarrow y\in F$ and $y^{n}\longrightarrow x\in F$.
(c)
By Prop. 2.1(8), $x^{n+1}\leq x^{n}$ and $y^{n+1}\longrightarrow y^{n}$.
Then, $x^{n}\longrightarrow y\leq x^{n+1}\longrightarrow y$ and
$y^{n}\longrightarrow x\leq y^{n+1}\longrightarrow x$. (d)
By (c) and (d) and the fact that $F$ is a filter, we obtain
$x^{n+1}\longrightarrow y\in F$ and $y^{n+1}\longrightarrow x\in F$. Hence $F$
is a (n+1)-fold obstinate filter of $L$. ∎
The extension theorem of n-fold obstinate filters is obtained from the
following result and any $n\geq 1$:
###### Theorem 8.6.
Let $F_{1},F_{2}$ two filter of $L$ be such that $F_{1}\subseteq F_{2}$. If
$F_{1}$ is an n-fold obstinate filter of $L$ then so is $F_{2}$.
###### Proof.
Let $F_{1},F_{2}$ two filter of $L$ be such that $F_{1}\subseteq F_{2}$.
Assume that $F_{1}$ is an n-fold obstinate filter of $L$, and let $x\notin
F_{2}$. Since $F_{1}\subseteq F_{2}$, we have $x\notin F_{1}$. Since $F_{1}$
is an n-fold obstinate filter of $L$, by Prop. 8.2, there exist $m\geq 1$ such
that $(\overline{x^{n}})^{m}\in F_{1}$. Since $F_{1}\subseteq F_{2}$, we have
$(\overline{x^{n}})^{m}\in F_{2}$. Therefore there exist $m\geq 1$ such that
$(\overline{x^{n}})^{m}\in F_{2}$. Hence by Prop. 8.2, $F_{2}$ is an n-fold
obstinate filter of $L$. ∎
From Theorem 8.6, it is easy to shows the following result for any $n\geq 1$:
###### Corollary 8.7.
$\\{1\\}$ is an n-fold obstinate filter of $L$ if and only if every filter of
$L$ is an n-fold obstinate filter of $L$.
###### Proposition 8.8.
The following conditions are equivalent for any filter $F$ and any $n\geq 1$:
* (i)
$F$ is an n-fold obstinate filter
* (ii)
$F$ is a maximal and n-fold positive implicative filter
* (iii)
$F$ is a maximal and n-fold implicative filter
###### Proof.
$(i)\longrightarrow(ii)$: Assume that $F$ is an n-fold obstinate filter. We
show that $F$ is a maximal and n-fold positive implicative filter. At first we
show that $F$ is a maximal. Let $x\notin F$, since $F$ is an n-fold obstinate
filter, by Prop. 8.2, there exist $m\geq 1$ such that
$(\overline{x^{n}})^{m}\in F$. Since
$(\overline{x^{n}})^{m}\leq\overline{x^{n}}$, by the fact that $F$ is a
filter, we get $\overline{x^{n}}\in F$, by Prop. 2.10, $F$ is a maximal
filter.
On the other hand, assume in the contrary that there exist $x\in L$ such that
$\overline{x^{n}}\longrightarrow x\in F$ and $x\notin F$. Since $F$ is an
n-fold obstinate filter, by Prop. 8.2, there exist $m\geq 1$ such that
$(\overline{x^{n}})^{m}\in F$. Since
$(\overline{x^{n}})^{m}\leq\overline{x^{n}}$, by the fact that $F$ is a
filter, we get $\overline{x^{n}}\in F$. Since $F$ is a filter, by the fact
that $\overline{x^{n}}\longrightarrow x\in F$, we obtain $x\in F$ which is a
contradiction. Hence for all $x\in L$, $\overline{x^{n}}\longrightarrow x\in
F$ implies $x\in F$.
By Prop. 5.5, $F$ is an n-fold positive implicative filter.
$(ii)\longrightarrow(iii)$: follows from Prop. 5.9
$(iii)\longrightarrow(i)$: Assume that $F$ is a maximal and n-fold implicative
filter of $L$. Let $x,yL$ be such that $x,y\notin F$. By Lemma 4.5,
$L_{x}=\\{b\in L:x^{n}\longrightarrow b\in F\\}$ is a filter of $L$.
$L_{y}=\\{b\in L:y^{n}\longrightarrow b\in F\\}$ is a filter of $L$.
Let $z\in F$, since $z\leq x^{n}\longrightarrow z$, we have
$x^{n}\longrightarrow z\in F$, hence $x^{n}\longrightarrow z\in F$, so $z\in
L_{x}$ and we obtain $F\subseteq L_{x}$. On the other hand,
$x^{n}\longrightarrow x=1\in F$ since $x^{n}\subseteq x$, hence $x\in L_{x}$.
By hypothesis, $x\notin F$, So $F\varsubsetneq L_{x}\subseteq L$. Since $F$ is
a maximal filter of $L$, we get $L_{x}=L$. Therefore $y\in L_{x}$ or
equivalently $x^{n}\longrightarrow y\in F$. Similarly, we get
$y^{n}\longrightarrow x\in F$. Hence $F$ is an n-fold obstinate filter of $L$.
∎
###### Proposition 8.9.
The following conditions are equivalent for any filter $F$ and any $n\geq 1$:
* (i)
$F$ is an n-fold obstinate filter
* (ii)
$F$ is a maximal and n-fold boolean filter
* (iii)
$F$ is a prime of second kind and n-fold boolean filter
###### Proof.
$(i)\longrightarrow(ii)$: Assume that $F$ is an n-fold obstinate filter. First
observe that by Prop. 8.8, $F$ is a maximal filter. Let $x\in L$. We have two
cases: $x\in F$ or $x\notin F$
case 1: $x\in F$. Since $x\leq x\vee\overline{x^{n}}$, by the fact that $F$ is
a filter, we have $x\vee\overline{x^{n}}\in F$.
case 2: $x\notin F$. Since $F$ is an n-fold obstinate filter, by Prop. 8.2,
there exist $m\geq 1$ such that $(\overline{x^{n}})^{m}\in F$. Since
$(\overline{x^{n}})^{m}\leq\overline{x^{n}}\leq x\vee\overline{x^{n}}$, we
have $(\overline{x^{n}})^{m}\leq x\vee\overline{x^{n}}$. By the fact that $F$
is a filter, we have $x\vee\overline{x^{n}}\in F$.
Since in both the two cases $x\vee\overline{x^{n}}\in F$, it is clear that for
all $x\in L$, $x\vee\overline{x^{n}}\in F$, hence $F$ is an n-fold boolean
filter.
$(ii)\longrightarrow(iii)$: Follows in the fact that a maximal filter of $L$
is a prime filter of second kind of $L$.
$(iii)\longrightarrow(i)$: Assume that $F$ is a prime filter of the second
kind and n-fold boolean filter. Let $x\in L$ be such that $x\notin F$. $F$ is
an n-fold boolean filter, we have $x\vee\overline{x^{n}}\in F$. Since $F$ is a
prime filter of the second kind , by the fact $x\notin F$, we have
$\overline{x^{n}}^{1}\in F$. by Prop. 8.2, $F$ is an n-fold obstinate filter.
∎
Combine Prop. 8.8 and Prop. 7.4, we have the following proposition.
###### Proposition 8.10.
Any n-fold obstinate filter $F$ is an n-fold fantastic filter. The converse is
not true in general.
The following example shows that the converse of the Proposition 8.10 is not
true in general.
###### Example 8.11.
Let $L$ be a residuated lattice from Example 2.3. It is easy to check that
$\\{1\\}$ is an n-fold fantastic filter but not n-fold obstinate filter.
Follows from Prop.8.9 and Prop.5.25, we have the following result:
###### Corollary 8.12.
The following conditions are equivalent for any filter $F$ and any $n\geq 1$:
* (i)
$F$ is an n-fold obstinate filter
* (ii)
$F$ is a prime filter in the second kind and n-fold positive implicative
filter
###### Proposition 8.13.
for any $n\geq 1$, a filter $F$ is an n-fold obstinate filter if and only if
every filter of $L/F$ is an n-fold obstinate filter of $L/F$.
###### Proof.
Assume that $F$ is an n-fold obstinate filter. Let $x,y\in L$ be such that
$x/F,y/F\notin\\{1/F\\}$, then $x,y\notin F$. Since $F$ is an n-fold obstinate
filter, we have $x^{n}\longrightarrow y,y^{n}\longrightarrow x\notin F$. From
this we have, $(x^{n}\longrightarrow y)/F,(y^{n}\longrightarrow x)/F\notin
1/F$ or equivalently $(x/F)^{n}\longrightarrow y/F,(y/F)^{n}\longrightarrow
x/F\notin 1/F$. Hence $\\{1/F\\}$ is an n-fold obstinate filter of $L/F$ and
by Corollary , every filter of $L/F$ is an n-fold obstinate filter of $L/F$.
Conversely, let $x,y\notin F$. Then $x/F,y/F\notin\\{1/F\\}$. Since
$\\{1/F\\}$ is an n-fold obstinate filter of $L/F$, we have
$(x/F)^{n}\longrightarrow y/F,(y/F)^{n}\longrightarrow x/F\notin 1/F$ or
equivalently $(x^{n}\longrightarrow y)/F,(y^{n}\longrightarrow x)/F\notin
1/F$. So $x^{n}\longrightarrow y,y^{n}\longrightarrow x\notin F$ and $F$ is an
n-fold obstinate filter. ∎
###### Definition 8.14.
A residuated lattice $L$ is said to be an n-fold obstinate residuated it
satisfies the following condition :
For all $x,y\in L$, $x,y\neq 1$ implies $x^{n}\longrightarrow y=1$ and
$y^{n}\longrightarrow x=1$.
###### Proposition 8.15.
The following conditions are equivalent for any $n\geq 1$:
* (i)
$L$ is an n-fold obstinate residuated lattice
* (ii)
$\\{1\\}$ is an n-fold obstinate filter of $L$.
* (iii)
Every filter of $L$ is n-fold obstinate
###### Proof.
$(i)\longrightarrow(ii)$: Obvious
$(ii)\longrightarrow(iii)$:Follows from Theorem 8.6
$(iii)\longrightarrow(i)$:Assume that every filter of $L$ is n-fold obstinate,
then $\\{1\\}$ is an n-fold obstinate filter of $L$ since $\\{1\\}$ is a
filter of $L$. The thesis follows by setting $F=\\{1\\}$ in Definition 8.1. ∎
The following example shows that the notion of n-fold obstinate residuated
lattice exist.
###### Example 8.16.
Let $L$ be a lattice from Example 2.2. $F$ is an n-fold obstinate filter of
$L$, then by Prop. 8.13 and Prop. 8.15, $L/F$ is an n-fold obstinate
residuated lattice, for any $n\geq 1$.
The following example shows that any residuated lattice may not be n-fold
obstinate residuated lattice.
###### Example 8.17.
Let $L$ be a lattice from Example 2.4. For any $n\geq 1$, $F=\\{1,d\\}$ is not
an n-fold obstinate filter of $L$, then by Prop. 8.13 and Prop. 8.15, $L$ is
not an n-fold obstinate residuated lattice.
Follows from Prop. 8.15, Prop. 8.9 and Prop. 5.27, we have the following
proposition.
###### Proposition 8.18.
n-fold obstinate residuated lattices are n-fold boolean residuated lattices
## 9\. Diagram among type of n-fold filters in Residuated Lattices
## References
* [1] Hájek, P.:Metamathematics of fuzzy logic. trends in logic, studia logica library, vol. 4. Kluwer, Dordrecht(1998).
* [2] Haveshki, M.,Eslami, E.:n-fold filters in BL-algebra. Math. Log. Quart.54, 176-186(2008)
* [3] C.Lele, Algorithms and computations in BL-algebra. International journal of Artificial Life Research, 1(4),29-47(2010)
* [4] L. Lianzhen, L. Kaitai, Boolean Filters and Positive Implicative Filters of Residuated lattices. Information Sciences and International Journal, 177(2007)5725-5738
* [5] Shokoofeh Ghorbani and Lida Torkzadeh, Nilpotent Elements of Residuated Lattices, International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 763428, 9 pages, 2012. doi:10.1155/2012/763428
* [6] C.Busneag, D. Piciu, The Stable Topology for residuated Lattices,soft computing, Springler Verlag (2012)1639-1655.
* [7] M.Kondo, Classification of Residuated Lattices by Filters. School of Information Environment (2011)33-38
* [8] A. Borumand and M. Pourkhatoun, obstinate filters in residuated lattices. Bull. Math. Soc . Sci. Math . Roumanie. Tome 55(103) No. 4, 2012, 413-422
* [9] S. Motamed, A. Borumand, : n-fold obstinate filters in BL-algebras. Neural Computing and Applications 20, 461-472(2011)
* [10] M. Haveshki and M. Mohamadhasani, Annals of the University of Graiova, Mathematics and Computer Science Series Volume 37(4), 2010, Pages 9-17 ISSN: 1223- 6934
* [11] M.Kondo, E.Turunen, Prime Filters on residuated Lattices,(2012)IEEE 42 nd International Symposium on multiple-value logic 89-91.
* [12] A.B.Saeid, M.Pourkhatoun, Obstinate Filters in Residuated Lattices, Bull. Math. Soc. Sci. Math. Roumanie. Tome55(103)No,4,(2012), 413-422
* [13] E.Turenen, N.Tchikapa, C.Lele: A New Characterization for n-fold Positive Implicative BL-Logics. S. Greco et al. (Eds.): IPMU 2012, PartI, CCIS 297, 552-560, 2012. ©Springler-Verlag Berlin Heidelberg 2012
* [14] B.V. Gasse, G.Deschrijver, G. Cornelis, E.E. Kerre, Filters of residuated lattices and triangle algebras. Information Sciences 180(2010)3006-3020
|
arxiv-papers
| 2013-08-08T15:36:49 |
2024-09-04T02:49:49.221477
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A, Kadji, C.Lele, M.Tonga",
"submitter": "Celestin Lele",
"url": "https://arxiv.org/abs/1308.1878"
}
|
1308.1898
|
# The non-Abelian Weyl-Yang-Kaluza-Klein gravity model
Halil Kuyrukcu [email protected] Physics Department, Bülent Ecevit
University, 67100, Zonguldak, Turkey
###### Abstract
The Weyl-Yang gravitation gauge theory is investigated in the framework of a
pure higher-dimensional non-Abelian Kaluza-Klein background. We construct the
dimensionally reduced field equations and energy-momentum tensors as well as
the four dimensional modified Weyl-Yang+Yang-Mills theory from an arbitrary
curved $internal$ space which is a extension of our previous model. In
particular, the coset space case is considered to obtain explicitly the
interactions between the gravitational and the gauge fields. The results not
only appear to be generalization of the well-established equations of non-
Abelian theory but also contain intrinsically the generalized gravitational
source term and the Lorentz force density.
PACS numbers: 04.50.Cd, 04.50.Kd, 04.50.-h, 11.25.Mj
## I Introduction
The Kaluza-Klein (KK) theories Kaluza ; Klein ; Mandel ; KleinN base on the
idea of a unification of well-known gravitation and $U(1)$ Abelian gauge
interactions on a circle bundle with purely geometrical point of view in the
four-dimensional ($4$D) effective theory, when extra space dimension is
compactified (i.e. invisible through Kaluza’s cylinder condition). This idea
can naturally be generalized to the case of a non-Abelian gauge fields
Einstein ; Witt ; Rayski ; Kerner ; Trautman ; Cho ; Choo ; Chooo ; Chang
describe the strong and weak interactions leads to most natural generalization
of Einstein-Maxwell theory on a principal fibre bundle structure in the
$(4+N)$-dimensional spacetime, i.e. the usual $4$D spacetime ($external$) plus
$N$-dimensional compact sub-space ($internal$), usually preferred to be the
coset space Helgason ; Kobayashi ; Gilmore ; witten77 ; forgacs ; witten ;
kapetanakis or the group manifold scherk79 (and references therein). The
gauge symmetry group of $external$ space comes remarkably from the isometry
group of $internal$ space in the KK-type multi-dimensional unified field
theories, and therefore they have been extensively discussed from many
different angles over the years by other unification theories (for a complete
overview see Appelquist ; Bailin ; Overduin ) such that the supergravity Duff
and superstring Schwarz theories.
To obtain equations of motions of Abelian or non-Abelian KK theories (most of
gravitational theories) we conventionally prefer to employ well-established
Hilbert-Einstein action as a gravitational Lagrangian. However, there is an
alternatively Yang-Mills-style gravitational formulations (also known as gauge
theories of gravitation) which had already been considered very early by Weyl
weyl1 , Lanczos lanczos , Lichnerowicz lichnerowicz , Stephenson stephenson ,
Higgs higgs , Kilmister and Newman kilmister , later improved by Yang yang
and further investigated by Thompson thompson , Pavelle pavelle and Fairchild
fairchild .
The Weyl-Yang gravitational Lagrangian is a quadratic in Riemann-Christofell
tensor, and two distinct Einsteinian field equations are obtained by
considering the metric and the affine connection are independent dynamical
field variables without torsion and with torsion Szczyrba ; mashhoon88 ;
mashhoon91 . The theory may be called Yang-Mills approach to gravity because
the equations of motions are similarly achieved as those of the Yang-Mills
yangmills or the well-known classical electromagnetic theory. Actually, these
Maxwellian field equations may appear a special limit of those are given by
Hehl and $et~{}al$ hehl78 considering nonzero torsion is also independent
variable within the framework of the theory so-called Poincaré Gauge theory of
Gravity hehl76 , and references given there. Although the initial appeal for
quadratic-type of alternative gravitational theory is faded as a gauge
formulation of gravitational physics, it is interesting in its own right, it
is still prevailing as effective theories of modified gravity, specifically in
quantum gravity lausher2002 and loop quantum cosmology cognola2013 .
Recently, that simple quadratic gravitational Lagrangian model is shown to be
very useful to overcome some cosmological problems in the real four space-time
dimensions gerard ; cook ; gonzalez ; chen ; chen1 ; yeung ; wang2013 . The
physical plane-wave solutions of the theory in the four and five dimensions is
also developed and discussed by Başkal baskal and Kuyrukcu kuyrukcu ,
respectively.
In the present paper, we completely generalize methods and results from our
previous work Başkal and Kuyrukcu sibelhalil to the non-Abelian case without
including any scalar fields by taking into account spinless and torsionless
Weyl-Yang gravity model in the context of the more than five dimensional pure
KK theories. As is well-known in KK theories, the $4$D matter is induced from
geometry in higher dimensions that the space-time is empty. In our approach,
the $4$D matter-spin source term is induced from those that matter carrying
energy-momentum but not possessing any spin tensor. We also extend the reduced
equations to more physical forms by taking the compact $internal$ space as a
homogeneous coset spaces background needed in KK theories. We can achieve this
construction the following outline of our article. In Sec. II and III, we
begin a brief review of both of the non-Abelian KK theory and Weyl-Yang
gravity model, respectively, to remind the reader some basic elements of those
theories for convenient reading, and to introduce the relations which we
employ all along in the work. In Sec. IV, we perform a popular dimensional KK
reduction procedure to obtain the modified $4$D Weyl-Yang+Yang-Mills action
from a pure $(4+N)$-dimensional Weyl-Yang gravitational Lagrangian without
supplementary matter fields. In that respect, the dimensionally reduced field
equations which contain naturally the generalized gravitational source term
and the Lorentz force density and stress-energy tensors are simultaneously
investigated, and they are compared with the our previous gravity model and
standard higher-dimensional KK theories in Sec. V and Sec. VI, respectively.
As a by-product, the coset space case of the theory is also included which
leads to more physical results and some plain solution of the field equations
in some detail. Finally, the last section demonstrates a brief discussion and
conclusions obtained from our analysis.
## II The non-Abelian Kaluza-Klein theory
We briefly review here the major steps of non-Abelian KK framework unifying
gravitation and Yang-Mills theories in more than five dimensions. First, let
us remember some basic notions of that theory in the usual way for convenient
reading. In what follows, the Greek indices $\mu,\nu,...=0,...,3$ refer to the
$external$ $4D$ flat Minkowski (Ricci flat) space-time $M_{4}$, admitting the
metric ${g}_{\mu\nu}(x)$ with usual signature and collectively coordinates
$x\in M_{4}$. The Latin indices $i,j,...=5,...,4+N$ refer to the curved
$internal$ $N$-dimensional compact subspace $M_{N}$ such as simply the
hypersphere or hypertorus, admitting the metric ${g}_{ij}(y)$ with Euclidean
signature and collectively coordinates $y\in M_{N}$, whereas the Latin capital
indices $A,B,...=0,...,3,5,...,4+N$ refer to the whole $(4+N)$-dimensional
Minkowski space $M_{4+N}$, with the metric $\hat{g}_{AB}(x,y)$ and associated
with the collectively event $z\in M_{4+N}$, $z=(x,y)$. The quantities
with/without the hat symbol demonstrate the ones in the $(4+N)$-dimensional
entire space/in the usual $4$D $external$ space,respectively. The stable
ground state of the generalized 5D KK theory is assumed to be, at least
locally, a direct space product of the form $M_{4+N}=M_{4}\times M_{N}$ which
is a trivial principal bundle over a $M_{4}$ with fibres $M_{N}$, instead of
assuming to be only $M_{4+N}$. Finally, the Greek indices $\alpha,\beta,...$
refer to any compact isometry (Lie) group $G$ of $M_{N}$, running over the
rank of $G$, i.e. $\alpha,\beta,...=1,...,dim(G)$. The isometries of the
$internal$ manifold produce linearly independent space-like Killing vectors
fields $\xi^{i}_{\alpha}(y)$ each corresponding to a metric symmetry in an
elegant way. The symmetries of $M_{N}$ appear to be gauge group in the real 4D
world for the effective observer as to be the 5D KK approach. The Killing
vectors fields $\xi^{i}_{\alpha}(y)$ satisfy the Lie’s equation
$\displaystyle[\xi_{\alpha},\xi_{\beta}]^{i}\equiv\xi^{j}_{\alpha}\partial_{j}\xi^{i}_{\beta}-\xi^{j}_{\beta}\partial_{j}\xi^{i}_{\alpha}$
$\displaystyle=$ $\displaystyle-f_{\alpha\beta}\,^{\gamma}\xi^{i}_{\gamma},$
(1)
corresponding to the Lie algebra by the Lie bracket and the following isometry
condition
$\displaystyle\mathcal{L}_{\xi}g_{ij}\equiv\xi^{\alpha
k}\partial_{k}g_{ij}+g_{ik}\partial_{j}\xi^{\alpha
k}+g_{jk}\partial_{i}\xi^{\alpha k}=0,$ (2)
which gives Killing’s equation $D_{(i}\xi^{\alpha}_{j)}=0$, respectively.
Here, $f_{\alpha\beta}\,^{\gamma}$ are the real antisymmetric
$f_{\alpha\beta}\,^{\gamma}=-f_{\beta\alpha}\,^{\gamma}$ structure constants
of $G$. The components of the $(4+N)$-dimensional metric $\hat{g}_{AB}(x,y)$
can be written in terms of the massless gauge fields (Yang-Mills vector
bosons) $A^{\alpha}_{\mu}(x)$ of the general group $G$ and the Killing vectors
fields $\xi^{i}_{\alpha}(y)$ in the higher-dimensional space-time $M_{4}\times
M_{N}$ as follow
$\displaystyle\hat{g}_{\mu\nu}(x,y)={g}_{\mu\nu}(x)+{g}_{ij}(y)\xi^{\alpha
i}(y)\xi^{\beta j}(y)A^{\alpha}_{\mu}(x)A^{\beta}_{\nu}(x),$
$\displaystyle\hat{g}_{\mu j}(x,y)={g}_{ij}(y)\xi^{\alpha
i}(y)A^{\alpha}_{\mu}(x),$ (3) $\displaystyle\hat{g}_{ij}(x,y)={g}_{ij}(y).$
It is very useful and convenient to make metric $\hat{g}_{AB}(x,y)$ block
diagonal for calculations. It can achieve choosing the basis in the so-called
noncoordinate (anholonomic, horizontal lift) basis Cho with
$\displaystyle\hat{E}^{\mu}(x,y)=dx^{\mu},$
$\displaystyle\hat{E}^{i}(x,y)=dy^{i}+\xi^{\alpha
i}(y)A^{\alpha}_{\mu}(x)dx^{\mu}.$ (4)
The dual basis can be found by help of the identity
$\hat{E}^{A}\hat{\iota}_{B}=\hat{\delta}^{A}\,_{B}$ in the following forms
$\displaystyle\hat{\iota}_{\mu}(x,y)=\partial_{\mu}-\xi^{\alpha
i}(y)A^{\alpha}_{\mu}(x)\partial_{i},$ (5)
$\displaystyle\hat{\iota}_{i}(x,y)=\partial_{i}.$
Hence, the metric components in equation (II) reduce a simple forms so that
the $(4+N)$-dimensional metric $\hat{g}_{AB}(x,y)$ becomes
$\displaystyle\hat{g}_{AB}=\left(\begin{array}[]{cc}{g}_{\mu\nu}(x)&0\\\
0&{g}_{ij}(y)\\\ \end{array}\right).$ (8)
It is very easy to raise and lower indices by taking into account the form of
$\hat{g}_{AB}(x,y)$ in equation (8). By employing necessary relations can be
found in misner for this basis, the non-zero components of the
$(4+N)$-dimensional Riemann tensor $\hat{R}^{A}\,_{BCD}$ decomposes into
$\displaystyle\hat{R}^{\mu}\,_{\nu\lambda\sigma}={R}^{\mu}\,_{\nu\lambda\sigma}-l_{\alpha\beta}(F^{\alpha\mu}\,_{\nu}F^{\beta}_{\lambda\sigma}+F^{\alpha\mu}\,_{[\lambda}F^{\beta}_{|\nu|\sigma]}),$
$\displaystyle\hat{R}^{i}\,_{\nu\lambda\sigma}=\frac{1}{2}\xi^{\alpha
i}\mathcal{D}_{\nu}F^{\alpha}_{\lambda\sigma},$
$\displaystyle\hat{R}^{i}\,_{\nu j\sigma}=\frac{1}{2}D_{j}\xi^{\alpha
i}F^{\alpha}_{\nu\sigma}-\frac{1}{2}l_{\alpha\beta}F^{\alpha}_{\sigma\tau}F^{\beta\tau}\,_{\nu},$
(9) $\displaystyle\hat{R}^{\mu}\,_{\nu
ij}=D_{j}\xi^{\alpha}_{i}F^{\alpha\mu}\,_{\nu}+\frac{1}{4}\xi^{\alpha}_{i}\xi^{\beta}_{j}(F^{\alpha\mu\tau}F^{\beta}_{\tau\nu}-F^{\beta\mu\tau}F^{\alpha}_{\tau\nu}),$
$\displaystyle\hat{R}^{i}\,_{jkl}={R}^{i}\,_{jkl},$
where the following abbreviation are introduced
$\displaystyle l_{\alpha\beta}=\frac{1}{2}{g}_{ij}\xi^{\alpha i}\xi^{\beta
j}.$ (10)
We give these expressions, since they are the basis of what follows. For
completeness, the electromagnetic field strength tensor
$F^{\alpha}_{\mu\nu}(x)$ and the Yang-Mills total covariant derivative in
equation (II) are explicitly given by
$\displaystyle F^{\alpha}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}A^{\alpha}_{\nu}-\partial_{\nu}A^{\alpha}_{\nu}+f_{\beta\gamma}\,^{\alpha}A^{\beta}_{\mu}A^{\gamma}_{\nu},$
(11) $\displaystyle\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}$ $\displaystyle=$
$\displaystyle
D_{\mu}F^{\alpha}_{\nu\lambda}+f_{\beta\gamma}\,^{\alpha}A^{\beta}_{\mu}F^{\gamma}_{\nu\lambda}.$
(12)
The reduced forms of Ricci tensor $\hat{R}_{AB}$ are, on the other hand,
expressed as
$\displaystyle\hat{R}_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\mathcal{P}_{\mu\nu}=R_{\mu\nu}-l_{\alpha\beta}F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda},$
(13) $\displaystyle\hat{R}_{i\nu}$ $\displaystyle\equiv$
$\displaystyle\mathcal{Q}_{i\nu}=\xi^{\alpha}_{i}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu},$
(14) $\displaystyle\hat{R}_{ij}$ $\displaystyle\equiv$
$\displaystyle\mathcal{U}_{ij}=R_{ij}+\frac{1}{4}\xi^{\alpha}_{i}\xi^{\beta}_{j}F^{\alpha}_{\lambda\tau}F^{\beta\lambda\tau}.$
(15)
Here, $R_{\mu\nu}$ and $R_{ij}$ are the $4$D Ricci tensors of $external$ and
$internal$ spaces, respectively. In the usual manner, we assume that a matter-
free Hilbert-Einstein Lagrangian which is linear in curvature components
proportional to the dimensional constant $1/{2\hat{\kappa}^{2}}$ leads to
$\hat{S}_{HE}(x,y)=\frac{1}{2\hat{\kappa}^{2}}\int_{M_{4+N}}\hat{R}\,\sqrt{-\hat{g}}\,d^{4}x\,d^{N}y,$
(16)
where $\hat{R}$ is the $(4+N)$-dimensional curvature scalar of the Riemann
space and the $\hat{\kappa}$ is coupling constant which satisfy
$\hat{\kappa}^{2}\equiv 8\pi\hat{\mathcal{G}}/c^{4}$ together with the
universal gravitational constant $\hat{\mathcal{G}}$. The Ricci tensor
contracting to get Ricci scalar, we find the curvature invariant corresponding
to the non-Abelian ansatz is given by
$\hat{R}(x,y)=R(x)+R(y)-\frac{1}{2}l_{\alpha\beta}(y)F^{\alpha}_{\lambda\tau}(x)F^{\beta\lambda\tau}(x),$
(17)
where $R(x)$ and $R(y)$ are scalar curvatures in four and $N$ dimensions,
respectively. As is well-known, to obtain conventional form of the $4$D gauge
fields in equation (17), i.e. to construct a desired $4$D field theory which
only includes the graviton and massless Yang-Mills fields, we must select and
normalize the Killing vectors fields such that
$2l_{\alpha\beta}\equiv{g}_{ij}(y)\xi^{\alpha i}(y)\xi^{\beta
j}(y)=c\delta^{\alpha\beta},$ (18)
for some constant $c$ so that the right-hand side of equation (17) is
independent of the $internal$ coordinates Duff84 ; Duff85 . We also add an
appropriate cosmological constant term $\hat{\Lambda}$ to the action of the
theory (16) to avoid contributions from non-zero $R(y)$ term.
We can obtain an appropriate vacuum solution looking for the equations of
motion $\hat{R}_{AB}=0$ as classical KK framework. However, the last equation
(15) is not vanish $\hat{R}_{ij}\neq 0$ (not Ricci flat) because of the
$internal$ space has to be curved for any non-Abelian group. This is well-
known consistency problem of the non-Abelian KK theories and the supergravity
theories as well. However, by adding suitable matter fields to the action
(16), we can obtain acceptable vacuum solution of the form $M_{4}\times M_{N}$
which is called $spontaneous~{}compactification$ by Cremmer and Scherk Cremmer
. Finally, one also can consider generalize the KK ansatz (II) including
massless scalar fields $\phi$ Awada ; Appequist which play an important role
in supergravity but it will not be discussed in this work.
## III The Weyl-Yang gravity model in a (4+N) dimensions
The dynamics of Weyl-Yang theory of gravity is determined by the curvature-
squared gravitational action on $M_{4+N}$ (or may be alternatively called
Stephenson-Kilmister-Yang action)
$\hat{S}_{WY}(x,y)=\hat{k}\int_{M_{4+N}}\hat{R}_{ABCD}^{2}\,\sqrt{-\hat{g}}\,d^{4}x\,d^{N}y,$
(19)
for matter-free gravity in the $(4+N)$-dimensional space-time with the
shorthand notation $\hat{R}_{ABCD}^{2}\equiv\hat{R}_{ABCD}\hat{R}^{ABCD}$ and
a real coupling constant $\hat{k}$. This Riemannian model is introduced by
Yang yang as an alternative approach to the Einstein’s theory of General
Relativity by employing perfect analogy with Yang-Mills theories. The Yang’s
theory is developed by Fairchild fairchild applying à la Palatini variation
(P-variation) principle palatini to obtain Euler-Lagrange equations of the
theory in a natural manner. Thus, the independent variation of the action (19)
with respect to the $(4+N)$-dimensional connection (gauge potential)
$\hat{\Gamma}^{A}\,_{BC}$ and the metric $\hat{g}_{AB}$ gives the desired
Yang-Mills style field equations. If spin and torsion vanish, from the
connection variation, symbolically as
$\delta\hat{S}_{\hat{\Gamma}}[\hat{g},\hat{\Gamma}]=0$, we obtain third-order
Yang’s pure gravitational field equation
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{BCD}=0,$ (20)
which can also equivalently be written by consecutive contractions and
applying the Bianchi identities as follows
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{BCD}\equiv\hat{D}_{C}\hat{R}_{BD}-\hat{D}_{D}\hat{R}_{BC}=0.$
(21)
Obviously, Yang’s equation (20) generalize Einstein’s field equations and
contains naturally vacuum Einsteinian solutions as well as non-Einsteinian
ones. The matter-source term $\hat{S}_{BCD}$, antisymmetric in $C$ and $D$,
may also be added to the right hand side of the equation (20), namely,
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{BCD}=\hat{\lambda}\hat{S}_{BCD},$ (22)
together with some suitable coupling constant $\hat{\lambda}$ kilmister ;
camenzind75 ; pavelle76 ; fairchild77 ; tseytlin . This gravitational current
term is interpreted as the covariant derivative of the Einstein’s matter
stress-energy tensor by Kilmister kilmister66 later Camenzind camenzind75
without a variational principle or more conveniently the spin tensor of the
matter fields by Fairchild fairchild . We can also rewrite sourceless field
equations (20) in many other ways, see baekler ; garecki .
From the metric variation
$\delta\hat{S}_{\hat{g}}[\hat{g},\hat{\Gamma}]=\hat{T}_{AB}$, we arrive at the
symmetric second-rank gravitational energy-momentum tensor together with
choosing appropriate $\hat{k}$ in the following form
$\hat{T}_{AB}\equiv\hat{R}_{ACDE}\hat{R}_{B}\,^{CDE}-\frac{1}{4}\hat{g}_{AB}\hat{R}^{2}_{CDEF}.$
(23)
It is called Stephenson equation for the case $\hat{T}_{AB}=0$ stephenson
which is divergence-free $\hat{D}_{A}\hat{T}^{A}\,_{B}=0$ but not traceless in
higher dimensions. It is actually $\hat{T}=-(N/4)\hat{R}^{2}_{ABCD}$ and
totally traceless only $N=0$ case, i.e. for the usual $4$D space-time. Let us
finally mention that, the equation (23) is indeed the contraction form of the
well-known Bel-Robinson superenergy tensor bel ; robinson , and it appear
perfectly analogous to stress-energy of classical electromagnetic theories as
well.
## IV The Reduction of the Quadratic Curvature
The method of dimensional reduction, from higher-dimensional theory to the
actual $4$D space-time, bring into the clear the types of gravitational and
gauge fields together with the forms of interactions between the constituent
fields. The reduced form of the $(4+N)$-dimensional quadratic curvature after
performing dimensional reduction procedure is given by
$\displaystyle\hat{R}_{ABCD}^{2}$ $\displaystyle=$
$\displaystyle\hat{R}_{\mu\nu\lambda\sigma}^{2}+4(\hat{R}_{i\nu\lambda\sigma}^{2}+\hat{R}_{i\nu
k\sigma}^{2}+\hat{R}_{ijk\sigma}^{2})$ (24)
$\displaystyle+2\hat{R}_{ij\lambda\sigma}^{2}+\hat{R}_{ijkl}^{2}.$
The substitutions of Riemann’s components in equation (II) into above with a
straightforward calculation and help of the relation
$\displaystyle
R_{\mu\nu\lambda\sigma}F^{\alpha\mu\lambda}F^{\beta\nu\sigma}=\frac{1}{2}R_{\mu\nu\lambda\sigma}F^{\alpha\mu\nu}F^{\beta\lambda\sigma},$
(25)
gives the dimensionally reduced invariant in the following form
$\displaystyle\hat{R}_{ABCD}^{2}$
$\displaystyle=R_{\mu\nu\lambda\rho}^{2}-3l_{\alpha\beta}R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\beta\lambda\rho}$
(26)
$\displaystyle+\frac{1}{2}l_{\alpha\beta}l_{\gamma\eta}\left(\mathcal{F}^{4}_{\alpha\gamma\beta\eta}+3\mathcal{F}^{2}_{\alpha\gamma}\mathcal{F}^{2}_{\beta\eta}+4\mathcal{F}^{4}_{\alpha\gamma\eta\beta}\right)$
$\displaystyle+3(m_{\alpha\beta\gamma}\mathcal{F}^{3}_{\alpha\beta\gamma}+k_{\alpha\beta}\mathcal{F}^{2}_{\alpha\beta})$
$\displaystyle+2l_{\alpha\beta}\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}+R_{ijkl}^{2}.$
Here, we have shortly defined the notations
$\displaystyle\mathcal{F}_{\alpha\beta}^{2}=F^{\alpha}_{\mu\nu}F^{\beta\mu\nu},$
$\displaystyle\mathcal{F}_{\alpha\beta\gamma}^{3}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma}\,{}^{\mu},$
$\displaystyle\mathcal{F}^{3~{}\alpha\beta\gamma}_{\mu\nu}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma\nu},$
(27)
$\displaystyle\mathcal{F}_{\alpha\beta\gamma\eta}^{4}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma\nu}F^{\eta\nu\mu},$
$\displaystyle\mathcal{F}^{4~{}\alpha\beta\gamma\eta}_{\mu\nu}=F^{\alpha}_{\mu\lambda}F^{\beta\lambda\sigma}F^{\gamma}_{\sigma\rho}F^{\eta\rho}\,_{\nu},$
and, the invariant coefficients are
$\displaystyle k_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
D_{i}\xi^{\alpha}_{j}D^{i}\xi^{\beta j},$ (28) $\displaystyle
m_{\alpha\beta\gamma}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{j}D^{i}\xi^{\gamma j}.$ (29)
Furthermore, in equation (26) and what follows, the gauge
$\mathcal{D}_{[\mu}F^{\alpha}_{\nu\lambda]}=0$ and the gravitational
$D_{[\tau}R_{\mu\nu]\lambda\sigma}=0$ Bianchi identities are used, whenever we
need them. We also recognize the fact that, the equation (26) is a common
result for all the basis, because $\hat{R}_{ABCD}^{2}$ is clearly an
invariant. We can overcome the term $D_{i}\xi^{\alpha}_{j}$ by taking into
account Wu and Zee’s assumption Yong-shi , then we get simple formula for
covariant derivative of Killing vector fields as
$D_{i}\xi^{\alpha}_{j}=\frac{1}{2}f^{\alpha\beta\gamma}\xi^{\beta}_{i}\xi^{\gamma}_{j}.$
(30)
However, in this case the $internal$ manifold is reduced to be homogeneous
space which is first discussed by Luciani Luciani in KK theories. Thus, the
extra space can now be written as the symmetric coset space $M_{N}=G/H$. $H$
is defined the isotropy subgroup of $G$, $H\subset G$ with $N=dim(G)-dim(H)$.
All the Killing vector terms of equation (26) are then disappeared, just as
expected, by employing equations (18) and (30) with $c=1$
$\displaystyle\hat{R}_{ABCD}^{2}$
$\displaystyle=R_{\mu\nu\lambda\rho}^{2}-\frac{3}{2}R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\alpha\lambda\rho}+\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\alpha\nu\lambda}$
$\displaystyle+\frac{3}{4}(f^{\alpha\gamma\eta}f^{\beta\gamma\eta}\mathcal{F}^{2}_{\alpha\beta}+2f^{\alpha\beta\gamma}\mathcal{F}^{3}_{\alpha\beta\gamma})$
$\displaystyle+\frac{1}{8}[\mathcal{F}^{4}_{\alpha\beta\alpha\beta}+3(\mathcal{F}^{2}_{\alpha\beta})^{2}+4\mathcal{F}^{4}_{\alpha\beta\beta\alpha}]+R_{ijkl}^{2}.$
The equation (IV) defines a new Lagrangian formalism in a background Yang-
Mills fields. It is easy see that, in addition to the standard $4$D Weyl-Yang
Lagrangian term $R_{\mu\nu\lambda\rho}^{2}$, the effective $4$D action (IV)
also contains the well-known $RF^{2}$-type term ( liu and references therein)
which describes a non-minimal coupling between curvature and the non-Abelian
gauge field and the self-interacting Yang-Mills field invariants which are the
cubic term $F^{3}$ and the quartic terms $F^{4}$ as well. The invariant form
$F^{3}$ appears only non-Abelian theories, and it is first studied by Alekseev
and Arbuzov Alekseev82 ; Alekseev84 . The ordinary Yang-Mills Lagrangian term
$\mathcal{F}^{2}_{\alpha\alpha}$ can be obtained from the
$f^{\alpha\gamma\eta}f^{\beta\gamma\eta}\mathcal{F}^{2}_{\alpha\beta}$ term,
if we normalize the structure constants such that
$f^{\alpha\gamma\eta}f^{\beta\gamma\eta}=(N-1)\delta^{\alpha\beta}$. By virtue
of Leibniz rule, the term with covariant derivative of gauge field tensor may
be rewritten symbolically as
$(\mathcal{D}F)^{2}=\mathcal{D}(F\mathcal{D}F)-F\mathcal{D}^{2}F$ of which
first term is total derivative so it can be dropped from the action, thus we
just have the $F\mathcal{D}^{2}F$-type interactions. Finally, the last term
$R_{ijkl}^{2}$ can be interpreted as the cosmological constant term as well as
the $4$D Ricci scalar of $internal$ space $R(y)$ or it can be just ignored so
that it does not contain physical fields. As a conclusion, the equation (IV)
leads to a modified $4$D Weyl-Yang+Yang-Mills action, and in fact it is some
part of second-order Euler-Poincaré (Gauss-Bonnet) curvature invariant Huang ;
m ller ; dereli ; kimm . We should also emphasise that, although the
dimensional reduction of quadratic Lagrangian from $4+N$ to $4$ dimensions is
discussed very early by Cho and $et~{}al$ Choo , they not have obtained proper
Yang-Mills term $\mathcal{F}^{2}_{\alpha\alpha}$ but rather discussed only
$(\mathcal{D}F)^{2}$ term.
## V The Reduction of the Field Equations
Now, we ready to work out the reduced components of $(4+N)$-dimensional
source-free field equations $\hat{D}_{A}\hat{R}^{A}\,_{BCD}=0$ in (20) by
taking into account the KK reduction scheme, as traditional way. Thus, we have
the six different equations of motions (one $4$D part and five $N$-dimensional
parts) are complementary each other
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{\nu\lambda\sigma}=0,$ (32)
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{i\lambda\sigma}=0,\qquad\qquad\hat{D}_{A}\hat{R}^{A}\,_{\nu
j\sigma}=0,$ (33) $\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{i\lambda
k}=0,\qquad\qquad\hat{D}_{A}\hat{R}^{A}\,_{\nu jk}=0,$ (34)
$\displaystyle\hat{D}_{A}\hat{R}^{A}\,_{ijk}=0.$ (35)
After some lengthy but careful manipulations, on account of mainly equation
(II) together with the Killing vector relations (equations (1) and (2)) and
the dual basis equation (5), one obtains the $4$D part of the
$(4+N)$-dimensional Yang’s equation in (32) as the $4$D Yang’s equation with
current term
$\displaystyle D_{\mu}R^{\mu}\,_{\nu\lambda\sigma}=S_{\nu\lambda\sigma},$ (36)
where
$\displaystyle
S_{\nu\lambda\sigma}(x,y)=l_{\alpha\beta}[J^{\alpha}_{\nu}F^{\beta}_{\lambda\sigma}+J^{\alpha}_{[\lambda}F^{\beta}_{|\nu|\sigma]}+2\mathcal{D}_{[\sigma}(F^{\alpha}_{\lambda]\mu}F^{\beta\mu}\,_{\nu})].$
Here, $J^{\alpha}_{\nu}$ is usual non-zero four current term of Yang-Mills
theory, i.e. $J^{\alpha}_{\nu}=\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu}$. The
term $S_{\nu\lambda\sigma}(x,y)$ may be interpreted as the gravitational
source like-term of the $4$D matter, and it consists of various combinations
of $F$ and $DF$ terms. It is no difficult to conjecture that, the equation (V)
satisfies the cyclic symmetry $S_{[\nu\lambda\sigma]}=0$ and covariant
conservation $\mathcal{D}_{\nu}S^{\nu}\,_{\lambda\sigma}=0$ properties as
expected from a source term. F. Öktem comments that “Those identities then
reveal that matter is endowed with angular momentum” oktem . This result (36)
very valuable from the physical point of view in case the $4$D matter-spin
tensor term is induced from in higher dimensions that matter carrying energy-
momentum but not possessing any spin tensor. In this sense our approach is
formally similar to the KK theories. Besides, we can remark that the first
reduced field equation (36) governs mainly the gravitational fields, since the
$DR$ term have the highest order derivative of the $external$ metric
$g_{\mu\nu}(x)$.
The reduced form of field equations (33) however give more complicated
relations
$\displaystyle\xi^{\alpha}_{i}\mathcal{D}_{\mu}\mathcal{D}^{\mu}F^{\alpha}_{\lambda\sigma}$
$\displaystyle=$
$\displaystyle-\xi^{\alpha}_{i}R_{\mu\nu\lambda\sigma}F^{\alpha\mu\nu}+4\xi^{\alpha
j}D_{j}\xi^{\beta}_{i}F^{[\alpha}_{\lambda\mu}F^{\beta]\mu}\,_{\sigma}$
$\displaystyle+\xi^{\alpha}_{i}l_{\beta\gamma}(\mathcal{F}^{2}_{\alpha\beta}F^{\gamma}_{\lambda\sigma}+2\mathcal{F}^{3~{}\alpha\beta\gamma}_{[\sigma\lambda]})$
$\displaystyle-2F^{\alpha}_{\lambda\sigma}D_{j}D^{j}\xi^{\alpha}_{i},$
$\displaystyle\xi^{\alpha}_{i}\mathcal{D}_{\mu}\mathcal{D}_{\sigma}F^{\alpha\mu}\,_{\nu}$
$\displaystyle=$
$\displaystyle-\xi^{\alpha}_{i}R_{\mu\nu\tau\sigma}F^{\alpha\tau\mu}+2\xi^{\alpha
j}D_{j}\xi^{\beta}_{i}F^{\alpha}_{\sigma\mu}F^{\beta\mu}\,_{\nu}$
$\displaystyle-\frac{1}{2}\xi^{\alpha}_{i}l_{\beta\gamma}(\mathcal{F}^{2}_{\alpha\beta}F^{\gamma}_{\nu\sigma}-2\mathcal{F}^{3~{}\alpha\beta\gamma}_{\nu\sigma})$
$\displaystyle+F^{\alpha}_{\nu\sigma}D_{j}D^{j}\xi^{\alpha}_{i}.$
The equations (V) and (V) basically govern the Yang-Mills gauge fields in view
of $\mathcal{D}\mathcal{D}\mathcal{F}$, and they include non-minimal $RF$ type
couplings as well as the cubic terms $F^{3}$. Additionally, we can obtain the
covariant derivative of the non-Abelian current
$\mathcal{D}_{\mu}J^{\alpha}_{\nu}$ by summing above two equations ((V) and
(V)) as well as employing Bianchi identities with following relation
$\displaystyle\xi^{\alpha}_{i}[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]F^{\alpha\lambda}\,_{\sigma}$
$\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}(R^{\lambda}\,_{\tau\mu\nu}F^{\alpha\tau}\,_{\sigma}-R^{\tau}\,_{\sigma\mu\nu}F^{\alpha\lambda}\,_{\tau})$
(40) $\displaystyle+2\xi^{\beta
j}D_{j}\xi^{\alpha}_{i}F^{\alpha}_{\mu\nu}F^{\beta\lambda}\,_{\sigma}.$
Hence, the result is
$\displaystyle\xi^{\alpha}_{i}\mathcal{D}_{\mu}J^{\alpha}_{\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\xi^{\alpha}_{i}(\mathcal{F}^{2}_{\alpha\beta}\mathcal{F}^{\beta}_{\mu\nu}-\mathcal{F}^{3~{}\alpha\beta\beta}_{\mu\nu}+2\mathcal{F}^{3~{}\alpha\beta\beta}_{\nu\mu})$
(41)
$\displaystyle+\xi^{\alpha}_{i}F^{\alpha}_{\nu\lambda}R^{\lambda}\,_{\mu}+\xi^{\alpha}_{j}F^{\alpha}_{\mu\nu}R_{i}\,^{j}.$
We can prove that, the contraction of free indices gives precisely the
covariant conservation law for external current such that
$\mathcal{D}_{\mu}J^{\alpha\mu}=0$.
In an equivalent way, the field equations with two $internal$ space indices
(34) are respectively reduced as
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}(\mathcal{D}_{\lambda}\mathcal{F}^{2}_{\alpha\beta}+F^{\alpha}_{\lambda\tau}J^{\beta\tau})+2J^{\alpha}_{\lambda}D_{k}\xi^{\alpha}_{i}=0,$
(42)
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}F^{[\alpha}_{\lambda\tau}J^{\beta]\tau}+2J^{\alpha}_{\lambda}D_{k}\xi^{\alpha}_{i}=0.$
(43)
Another interesting result is obtained, if we eliminate the common term
$J^{\alpha}_{\lambda}D_{k}\xi^{\alpha}_{i}$ of equations (42) and (43) by
adding the equations together. Then, we read
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}[\mathcal{D}_{\lambda}\mathcal{F}^{2}_{\alpha\beta}+F^{(\alpha}_{\lambda\tau}J^{\beta)\tau}]=0,$
(44)
or, alternatively
$\displaystyle
f_{\lambda}^{\alpha\beta}+f_{\lambda}^{\beta\alpha}=-2\mathcal{D}_{\lambda}\mathcal{F}^{2}_{\alpha\beta}.$
(45)
Here, $f_{\lambda}^{\alpha\alpha}=F^{\alpha}_{\lambda\tau}J^{\alpha\tau}$ is
confidentially interpreted as a Lorentz-like force density of the non-Abelian
theory. Although such a terms generally emerge from the higher-dimensional
geodesic equations Kerner , they naturally appears in the reduced field
equations of our approach. It is difficult to understand physical meaning of
this circumstance (45), however we can, at least theoretically, say that the
Yang-Mills invariant $\mathcal{F}^{2}_{\alpha\beta}$ is defined as a field
whose gradient is equal and opposite to the generalized Lorentz force density.
Finally, from the last field equation (35), we again obtain the Yang’s
equation with source term but in this case for the $internal$ space
$D_{l}R^{l}\,_{ijk}=S_{ijk},$ (46)
where the unphysical source-like term is found to be
$S_{ijk}(x,y)=-\frac{1}{2}\mathcal{F}^{2}_{\alpha\beta}(\xi^{\alpha}_{[k}D_{j]}\xi^{\beta}_{i}-\xi^{\alpha}_{i}D_{k}\xi^{\beta}_{j}).$
(47)
which provides the same conservation and symmetry conditions as the term
$S_{\nu\lambda\sigma}$ in (V). In the absence of Yang-Mills fields we only
have source-free Yang’s equation for $external$
$D_{\mu}R^{\mu}\,_{\nu\lambda\sigma}=0$ and $internal$ $D_{l}R^{l}\,_{ijk}=0$
spaces respectively.
Attention will now be turned to investigate the couplings between the
components of $(4+N)$-dimensional Ricci tensor $\mathcal{P}_{\mu\nu}$,
$\mathcal{Q}_{i\nu}$, $\mathcal{U}_{ij}$ in equations (13)-(15) and the gauge
fields $F$ together with Killing fields $\xi$. By virtue of alternative field
equation (21), one can recognize the fact that
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set are
embedded in the Weyl-Yang field equations of non-Abelian theory (32)-(35).
Hence, from the reduced equations ((36) with (V), (V), (V), (42), (43) and
(46) with (47)) we deduce the following compact embedded equations
$\displaystyle\mathcal{D}_{[\lambda}\mathcal{P}_{|\nu|\sigma]}+\frac{1}{4}\xi^{\alpha
i}(F^{\alpha}_{\nu[\lambda}\mathcal{Q}_{|i|\sigma]}-F^{\alpha}_{\lambda\sigma}\mathcal{Q}_{i\nu})=0,$
(48)
$\displaystyle\mathcal{D}_{[\lambda}\mathcal{Q}_{|i|\sigma]}+\xi^{\alpha}_{i}F^{\alpha\mu}\,_{[\sigma}\mathcal{P}_{|\mu|\lambda]}+\xi^{\alpha
j}F^{\alpha}_{\lambda\sigma}\mathcal{U}_{ij}=0,$ (49)
$\displaystyle\mathcal{D}_{\sigma}\mathcal{Q}_{i\nu}+\xi^{\alpha}_{i}F^{\alpha\mu}\,_{\nu}\mathcal{P}_{\mu\sigma}-\xi^{\alpha
j}F^{\alpha}_{\nu\sigma}\mathcal{U}_{ij}=0,$ (50)
$\displaystyle\mathcal{D}_{\lambda}\mathcal{U}_{ik}+\frac{1}{4}(2D_{k}\mathcal{Q}_{i\lambda}+\xi^{\alpha}_{i}F^{\alpha}_{\lambda}\,{}^{\tau}\mathcal{Q}_{k\tau})=0,$
(51) $\displaystyle
D_{k}\mathcal{Q}_{j\nu}+\frac{1}{2}F^{\alpha}_{\nu}\,{}^{\tau}\xi^{\alpha}_{[j}\mathcal{Q}_{k]\tau}=0,$
(52) $\displaystyle D_{[i}\mathcal{U}_{|k|j]}=0,$ (53)
on account of identities as mentioned before and the relation
$D_{j}D_{k}\xi^{\alpha}_{l}=\xi^{\alpha}_{i}R^{i}\,_{jkl}$ weinberg . The
equation (49) can be obtain from the equation (50), thus it is not essential.
The equations (48)-(53) contain the non-Abelian covariant derivative of
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set, as well
as the ordinary derivative of $\\{\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set
and various $\mathcal{P}F$, $\mathcal{Q}F$-type coupling terms. Above
generalized field equations are welcome from another point of view that any
solution to the
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{i\nu},\mathcal{U}_{ij}\\}$-set solves
the embedded equations (48)-(53) naturally.
There is a mismatch between the left-hand side of the field equation (36)
which depends only on $external$ coordinates $x$ and the right-hand side which
depends both on $external$ and $internal$ coordinates $x,y$. Therefore, to not
only avoid this problem but also to reduce equations (48)-(53) to the more
simpler and physical forms, we restrict our considerations such that the
$internal$ space is taken to be the homogeneous coset space the type $G/H$. It
is easy to see from equation (18) with $c=1$ that the source term (V) yields
$\displaystyle
S_{\nu\lambda\sigma}(x)=\frac{1}{2}[J^{\alpha}_{\nu}F^{\alpha}_{\lambda\sigma}+J^{\alpha}_{[\lambda}F^{\alpha}_{|\nu|\sigma]}+2\mathcal{D}_{[\sigma}(F^{\alpha}_{\lambda]\mu}F^{\alpha\mu}\,_{\nu})],$
which depend only on $external$ coordinates $x$ as it should be. Next by using
equation (30) and reduce form of equation (40)
$\displaystyle[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]F^{\alpha\lambda}\,_{\sigma}$
$\displaystyle=$ $\displaystyle
R^{\lambda}\,_{\tau\mu\nu}F^{\alpha\tau}\,_{\sigma}-R^{\tau}\,_{\sigma\mu\nu}F^{\alpha\lambda}\,_{\tau}$
(55)
$\displaystyle+f^{\beta\gamma\alpha}F^{\beta}_{\mu\nu}F^{\gamma\lambda}\,_{\sigma}.$
the first embedded equation (48) becomes
$\begin{array}[]{l}\mathcal{D}_{\lambda}(R_{\sigma\nu}-\frac{1}{2}F^{\alpha}_{\sigma\mu}F^{\alpha}_{\nu}\,{}^{\mu})-\mathcal{D}_{\sigma}(R_{\lambda\nu}-\frac{1}{2}F^{\alpha}_{\lambda\mu}F^{\alpha}_{\nu}\,{}^{\mu})\\\\[8.61108pt]
+\frac{1}{4}\left[F^{\alpha}_{\nu\lambda}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\sigma}-F^{\alpha}_{\nu\sigma}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\lambda}-2F^{\alpha}_{\lambda\sigma}\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu}\right]=0.\end{array}$
(56)
Second one (49) simultaneously yields
$\displaystyle\xi^{\alpha}_{i}[\mathcal{D}_{\lambda}(\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\sigma})-\mathcal{D}_{\sigma}(\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\lambda})$
$\displaystyle+F^{\alpha\mu}\,_{\sigma}(R_{\mu\lambda}-\frac{1}{2}F^{\beta}_{\mu\tau}F^{\beta}_{\lambda}\,{}^{\tau})-F^{\alpha\mu}\,_{\lambda}(R_{\mu\sigma}-\frac{1}{2}F^{\beta}_{\mu\tau}F^{\beta}_{\sigma}\,{}^{\tau})$
$\displaystyle+\frac{1}{2}F^{\beta}_{\lambda\sigma}(\mathcal{F}^{2}_{\alpha\beta}-f^{\eta\gamma\alpha}f^{\beta\gamma\eta})]=0.$
(57)
The third equation (50) implies that
$\begin{array}[]{l}\xi^{\alpha}_{i}[\mathcal{D}_{\sigma}(\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu})+F^{\alpha\mu}\,_{\nu}(R_{\mu\sigma}-\frac{1}{2}F^{\beta}_{\mu\tau}F^{\beta}_{\sigma}\,{}^{\tau})\\\\[8.61108pt]
+\frac{1}{4}F^{\beta}_{\nu\sigma}(\mathcal{F}^{2}_{\alpha\beta}-f^{\eta\gamma\alpha}f^{\beta\gamma\eta})]=0.\end{array}$
(58)
and the equations (51), (52) are similarly reorganized as
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{k}[\mathcal{D}_{\lambda}(\mathcal{F}^{2}_{\alpha\beta}-f^{\eta\gamma\alpha}f^{\beta\gamma\eta})+F^{\alpha}_{\lambda\tau}\mathcal{D}_{\mu}F^{\beta\mu\tau}$
$\displaystyle+f^{\gamma\beta\alpha}\mathcal{D}_{\mu}F^{\gamma\mu}\,_{\lambda}]=0,$
(59)
$\displaystyle\xi^{\alpha}_{j}\xi^{\beta}_{k}[F^{\alpha}_{\nu\tau}\mathcal{D}_{\mu}F^{\beta\mu\tau}-F^{\beta}_{\nu\tau}\mathcal{D}_{\mu}F^{\alpha\mu\tau}+2f^{\gamma\beta\alpha}\mathcal{D}_{\mu}F^{\gamma\mu}\,_{\nu}]=0,$
respectively. Finally, the last equation (53), on the other hand, can be
expressed in the following form
$\displaystyle\xi_{i}^{\alpha}\xi_{j}^{\beta}\xi_{k}^{\gamma}[f_{\eta\alpha\gamma}(\mathcal{F}^{2}_{\beta\eta}-f^{\zeta\pi\beta}f^{\eta\pi\zeta})-f_{\eta\beta\gamma}(\mathcal{F}^{2}_{\alpha\eta}-f^{\zeta\pi\alpha}f^{\eta\pi\zeta})$
$\displaystyle+2f_{\eta\alpha\beta}(\mathcal{F}^{2}_{\eta\gamma}-f^{\zeta\pi\eta}f^{\gamma\pi\zeta})]=0.$
(61)
If we remove all the Killing vector terms from embedded equations (78)-(V),
then everything would be fine. Furthermore, it is possible to recognize the
following plain solutions in the reduced equations
$\displaystyle R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F^{\alpha}_{\mu\tau}F^{\alpha}_{\nu}\,{}^{\tau},$
$\displaystyle\mathcal{D}_{\mu}F^{\alpha\mu}\,_{\nu}$ $\displaystyle=$
$\displaystyle 0,$ (62) $\displaystyle F^{\alpha}_{\mu\nu}F^{\beta\mu\nu}$
$\displaystyle=$ $\displaystyle f^{\eta\gamma\alpha}f^{\beta\gamma\eta}.$
For $N=1$ case i.e. $G=U(1)$ and $G/H=S^{1}$ which gives the usual $5$D KK
ground state $M_{4}\times S^{1}$, we dramatically obtain well-known KK field
equations in the case of scalar field $\phi(x)=1$ where $F^{\alpha}_{\mu\nu}$
reduces to $F_{\mu\nu}$, $\mathcal{D}_{\mu}$ to ${D}_{\mu}$ and
$f^{\alpha\beta\gamma}$ to $0$. That is
$\displaystyle R_{\mu\nu}=\frac{1}{2}F_{\mu\tau}F_{\nu}\,^{\tau},\qquad
D_{\mu}F^{\mu}\,_{\nu}=0,\qquad F_{\mu\nu}F^{\mu\nu}=0.$
In that respect, the equations (V) and (V) vanish identically out, and the
remaining four reduced field equations (56)-(V) (with the current term (V))
precisely gives those of 5D Weyl-Yang-Kaluza-Klein model sibelhalil , as
expected. Besides, the equation (45) is reduced to be more interesting
expression $f_{\lambda}=-D_{\lambda}\mathcal{F}^{2}$ (also see sibelhalil ).
## VI The Reduction of the Stress-Energy Tensor
In this section, we obtain the components of energy-momentum tensor of our
model $\hat{T}_{AB}$ (23) in the $(4+N)$-dimensional entire space. The
computations are more straightforward by taking into account non-trivial
Riemann tensors (II) and reduced form of the quadratic curvature term
$\hat{R}^{2}_{ABCD}$ (26), however we should recall that the noncoordinate
components of the full metric (8) are also used here. It is useful to
decompose bak (separate) the $\hat{T}_{\mu\nu}$ as well as $\hat{T}_{ij}$
component of the stress-energy tensor into trace-free
$\hat{T}_{\mu\nu}^{(tf)}$ and trace $\hat{T}_{\mu\nu}^{(t)}$ parts
$\displaystyle\hat{T}_{\mu\nu}=\hat{T}_{\mu\nu}^{(tf)}+\hat{T}_{\mu\nu}^{(t)}$
(64)
because not only the former look nice but also the later can be employed to
find the trace of $\hat{T}_{AB}$ more easily. Hence, after performing some
manipulations, the traceless parts are found to be
$\displaystyle\hat{T}^{(tf)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle{T}^{(g)}_{\mu\nu}+3(k_{\alpha\beta}+\frac{1}{2}l_{\alpha\gamma}l_{\beta\eta}\mathcal{F}^{2}_{\gamma\eta})T^{(em)\alpha\beta}_{\mu\nu}$
$\displaystyle+3l_{\alpha\beta}{T}^{(c)\alpha\beta}_{\mu\nu}+3m_{\alpha\beta\gamma}{T}^{(1)\alpha\beta\gamma}_{\mu\nu}$
$\displaystyle+\frac{1}{2}(l_{\alpha\beta}l_{\gamma\eta}+4l_{\alpha\eta}l_{\gamma\beta}){T}^{(2)\alpha\gamma\beta\eta}_{\mu\nu}+2l_{\alpha\beta}{T}^{(3)\alpha\beta}_{\mu\nu}.$
The first term of equation (VI) is labelled as ${T}^{(g)}_{\mu\nu}$ for the
reason that it exactly gives pure gravitational stress-energy tensor of Weyl-
Yang gauge theory in the actual $4$D $external$ space-time
$\displaystyle{T}^{(g)}_{\mu\nu}=R_{\mu\lambda\sigma\rho}R_{\nu}\,^{\lambda\sigma\rho}-\frac{1}{4}g_{\mu\nu}R_{\tau\lambda\sigma\rho}R^{\tau\lambda\sigma\rho}.$
(66)
Another well-known term $T^{(em)\alpha\beta}_{\mu\nu}$ is also welcome from
the equation (VI). That is gauge invariant energy-momentum tensor of the non-
Abelian Yang-Mills fields which can be written
$\displaystyle{T}^{(em)\alpha\beta}_{\mu\nu}=F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda}-\frac{1}{4}g_{\mu\nu}F^{\alpha}_{\tau\lambda}F^{\beta\tau\lambda}.$
(67)
The resting non-trivial terms of equation (VI) have more complicated
structures because they come, as might be expected, from higher-order
quadratic action. We respectively identify these terms as the stress-energy
tensor of the $RF^{2}$-type non-minimal coupling fields
$\displaystyle{T}^{(c)\alpha\beta}_{\mu\nu}=F^{\beta\rho\sigma}F^{\alpha}_{(\mu}\,{}^{\lambda}R_{\nu)\lambda\sigma\rho}-\frac{1}{4}g_{\mu\nu}F^{\alpha\tau\lambda}F^{\beta\rho\sigma}R_{\tau\lambda\sigma\rho},$
that of the cubic fields
$\displaystyle{T}^{(1)\alpha\beta\gamma}_{\mu\nu}=\mathcal{F}^{3~{}\alpha\beta\gamma}_{(\mu\nu)}-\frac{1}{4}g_{\mu\nu}\mathcal{F}^{3}_{\alpha\beta\gamma},$
(69)
of the quartic constituent fields
$\displaystyle{T}^{(2)\alpha\gamma\beta\eta}_{\mu\nu}=\mathcal{F}_{\mu\nu}^{4~{}\alpha\gamma\beta\eta}-\frac{1}{4}g_{\mu\nu}\mathcal{F}^{4}_{\alpha\gamma\beta\eta},$
(70)
and of the $(\mathcal{D}F)^{2}$-type interactions
$\displaystyle{T}^{(3)\alpha\beta}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\mathcal{D}_{\mu}F^{\alpha}_{\sigma\rho}\mathcal{D}_{\nu}F^{\beta\sigma\rho}-\frac{1}{4}g_{\mu\nu}\mathcal{D}_{\sigma}F^{\alpha}_{\tau\rho}\mathcal{D}^{\sigma}F^{\beta\tau\rho}.$
(71)
The trace part which consists of the remaining terms of $\hat{T}_{\mu\nu}$
(64) is become
$\displaystyle\hat{T}_{\mu\nu}^{(t)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}g_{\mu\nu}R^{2}_{ijkl}-\frac{3}{2}m_{\alpha\beta\gamma}\mathcal{F}^{3~{}\alpha\beta\gamma}_{(\mu\nu)}-\frac{3}{2}k_{\alpha\beta}F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda}$
(72)
$\displaystyle+\frac{1}{2}l_{\alpha\beta}[l_{\gamma\eta}(\mathcal{F}_{\mu\nu}^{4~{}\alpha\gamma\beta\eta}-2\mathcal{F}_{\mu\nu}^{4~{}\alpha\gamma\eta\beta})$
$\displaystyle+2\mathcal{D}_{\sigma}F^{\alpha}_{\mu\rho}\mathcal{D}^{\sigma}F^{\beta}_{\nu}\,{}^{\rho}-3\mathcal{D}_{\mu}F^{\alpha}_{\sigma\rho}\mathcal{D}_{\nu}F^{\beta\sigma\rho}].$
Another reduced component of the energy-momentum tensor is the $\hat{T}_{\mu
i}$ which turns out that
$\displaystyle\hat{T}_{\mu i}$
$\displaystyle=\frac{1}{2}\xi^{\alpha}_{i}R_{\mu\lambda\sigma\rho}\mathcal{D}^{\lambda}F^{\alpha\sigma\rho}+\xi^{\alpha}_{j}D^{j}\xi^{\beta}_{i}F^{\beta\sigma\rho}\mathcal{D}_{(\mu}F^{\alpha}_{\rho)\sigma}$
$\displaystyle+\xi^{\alpha}_{i}l_{\beta\gamma}[\frac{3}{2}F^{\beta}_{\mu}\,{}^{\lambda}F^{\sigma\rho}_{\gamma}\mathcal{D}_{\sigma}F^{\alpha}_{\rho\lambda}-F^{\alpha\sigma\lambda}F^{\beta}_{\lambda}\,{}^{\rho}\mathcal{D}_{(\mu}F^{\gamma}_{\sigma)\rho}].$
Similarly, the $i-j$ component of equation 23) can be split into a trace-free
and a trace part as follows
$\displaystyle\hat{T}_{ij}=\hat{T}_{ij}^{(tf)}+\hat{T}_{ij}^{(t)},$ (74)
where
$\displaystyle\hat{T}_{ij}^{(tf)}$ $\displaystyle=$
$\displaystyle\frac{N}{4}{T}^{(g)}_{ij}+\frac{N}{16}[l_{\gamma\eta}(\mathcal{F}^{4}_{\alpha\gamma\beta\eta}+4\mathcal{F}_{\alpha\gamma\eta\beta}^{4})$
$\displaystyle+4\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}]{T}^{(1)\alpha\beta}_{ij}$
$\displaystyle+\frac{3N}{4}[\mathcal{F}^{2}_{\alpha\beta}{T}^{(2)\alpha\beta}_{ij}+\mathcal{F}^{3}_{\alpha\beta\gamma}{T}^{(3)\alpha\beta\gamma}_{ij}].$
The ${T}^{(g)}_{ij}$ appear formally to be $4$D stress-energy tensor of Weyl-
Yang theory but in this case for the $internal$ space. That is,
${T}^{(g)}_{ij}$ is written in terms of the Riemann tensors of $internal$
space
$\displaystyle{T}^{(g)}_{ij}=R_{inkl}R_{j}\,^{nkl}-\frac{1}{N}g_{ij}R_{mnkl}R^{mnkl}.$
(76)
Other unphysical energy-momentum tensors correspondingly are
$\displaystyle{T}^{(1)\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{j}-\frac{1}{N}g_{ij}\mathcal{\xi}^{\alpha}_{k}\xi^{\beta
k},$ (77) $\displaystyle{T}^{(2)\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle
D_{n}\xi^{\alpha}_{i}D^{n}\xi^{\beta}_{j}-\frac{1}{N}g_{ij}D_{n}\xi^{\alpha}_{m}D^{n}\xi^{\beta
m},$ (78) $\displaystyle{T}^{(3)\alpha\beta\gamma}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{n}\xi^{\beta}_{(i}D^{n}\xi^{\gamma}_{j)}-\frac{1}{N}g_{ij}\xi^{\alpha}_{n}\xi^{\beta}_{m}D^{n}\xi^{\gamma
m},$ (79)
together with more complicated trace part
$\displaystyle\hat{T}_{ij}^{(t)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}g_{ij}[R^{2}_{\mu\nu\lambda\rho}-\frac{3}{2}l_{\alpha\beta}(2R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\beta\lambda\rho}$
(80) $\displaystyle-
l_{\gamma\eta}\mathcal{F}^{2}_{\alpha\gamma}\mathcal{F}^{2}_{\beta\eta})]-(\frac{N-4}{4})R_{inkl}R_{j}\,^{nkl}$
$\displaystyle-(\frac{3N-6}{4})[D_{n}\xi^{\alpha}_{i}D^{n}\xi^{\beta}_{j}\mathcal{F}^{2}_{\alpha\beta}+\xi^{\alpha}_{n}\xi^{\beta}_{(i}D^{n}\xi^{\gamma}_{j)}\mathcal{F}^{3}_{\alpha\beta\gamma}]$
$\displaystyle-\frac{1}{16}\xi^{\alpha}_{i}\xi^{\beta}_{j}\big{\\{}l_{\gamma\eta}[(N+4)\mathcal{F}^{4}_{\alpha\gamma\beta\eta}+2(N-2)\mathcal{F}^{4}_{\alpha\gamma\eta\beta}]$
$\displaystyle+4(N-1)\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}\big{\\}}.$
In $5$D KK theories, $\hat{T}_{\mu 5}$ may be interpreted as the current
density by help of conservation law $\hat{D}_{\mu}\hat{T}^{\mu}\,_{5}=0$. In
our approach, it is not only easy to find physical roles or consequences of
components $\hat{T}_{\mu i}$ and $\hat{T}_{ij}$ but also to obtain an
analogues of those in the $4$D Einstein’s theory of gravity. Now, we can
calculate the trace of the full $\hat{T}_{AB}$ (i.e. $\hat{T}$) by using
$\displaystyle\hat{T}\equiv\hat{g}^{AB}\hat{T}_{AB}=\hat{g}^{\mu\nu}\hat{T}^{(t)}_{\mu\nu}+\hat{g}^{ij}\hat{T}^{(t)}_{ij}.$
(81)
The trace is easily obtained by employing in the manner of equations (72) and
(80). The result will not here be considered further, however the resulting
form can be checked on due to the fact that $\hat{T}=-(N/4)\hat{R}^{2}_{ABCD}$
(from equation (23)) with equation (26). It is no difficult to conjecture
that, the absence of non-Abelian gauge fields $A_{\mu}^{\alpha}=0$ we only get
pure gravitational stress-energy tensor $\hat{T}_{\mu\nu}={T}^{(g)}_{\mu\nu}$
with $\hat{T}_{\mu i}=\hat{T}_{ij}=0$.
Attention will now be turned to investigate the reduced energy-momentum
tensors which are already obtained below for the case where the $internal$
space is the homogeneous coset space. On account of all necessarily
assumptions which are mentioned previous sections, the traceless part
$\hat{T}_{\mu\nu}^{(tf)}$ in (VI) is become
$\displaystyle\hat{T}_{\mu\nu}^{(tf)}$ $\displaystyle=$
$\displaystyle{T}^{(g)}_{\mu\nu}+\frac{3}{8}(2f^{\alpha\gamma\eta}f^{\beta\gamma\eta}+\mathcal{F}^{2}_{\alpha\beta}){T}^{(em)\alpha\beta}_{\mu\nu}+\frac{3}{2}{T}^{(c)\alpha\alpha}_{\mu\nu}$
(82)
$\displaystyle+\frac{3}{2}f^{\alpha\beta\gamma}{T}^{(1)\alpha\beta\gamma}_{\mu\nu}+\frac{1}{8}{T}^{(2)\alpha\beta\alpha\beta}_{\mu\nu}+\frac{1}{2}{T}^{(2)\alpha\beta\beta\alpha}_{\mu\nu}$
$\displaystyle+{T}^{(3)\alpha\alpha}_{\mu\nu},$
and the trace part in (72) is
$\displaystyle\hat{T}_{\mu\nu}^{(t)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}g_{\mu\nu}R^{2}_{ijkl}-\frac{3}{8}f^{\alpha\gamma\eta}f^{\beta\gamma\eta}F^{\alpha}_{\mu\lambda}F^{\beta}_{\nu}\,{}^{\lambda}$
(83)
$\displaystyle-\frac{3}{4}f^{\alpha\beta\gamma}\mathcal{F}^{3~{}\alpha\beta\gamma}_{(\mu\nu)}\
+\frac{1}{8}(\mathcal{F}_{\mu\nu}^{4~{}\alpha\beta\alpha\beta}-2\mathcal{F}_{\mu\nu}^{4~{}\alpha\beta\beta\alpha})$
$\displaystyle+\frac{1}{2}\mathcal{D}_{\sigma}F^{\alpha}_{\mu\rho}\mathcal{D}^{\sigma}F^{\beta}_{\nu}\,{}^{\rho}-\frac{3}{4}\mathcal{D}_{\mu}F^{\alpha}_{\sigma\rho}\mathcal{D}_{\nu}F^{\alpha\sigma\rho}.$
Hence, we recognize the fact that the component $\hat{T}_{\mu\nu}$ (the
summation of equation (82) and equation (83)) is Killing term-free equation,
and the resulting expression is more convenient to bring out type of
interactions between constituent fields.
Next, the $\hat{T}_{\mu i}$ (VI) yields
$\displaystyle\hat{T}_{\mu i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\xi^{\alpha}_{i}[R_{\mu\lambda\sigma\rho}\mathcal{D}^{\lambda}F^{\alpha\sigma\rho}+f^{\alpha\beta\gamma}F^{\beta\sigma\rho}\mathcal{D}_{(\mu}F^{\gamma}_{\rho)\sigma}$
(84)
$\displaystyle+\frac{3}{2}F^{\beta}_{\mu}\,{}^{\lambda}F^{\beta\sigma\rho}\mathcal{D}_{\sigma}F^{\alpha}_{\rho\lambda}-F^{\alpha\sigma\lambda}F^{\beta}_{\lambda}\,{}^{\rho}\mathcal{D}_{(\mu}F^{\beta}_{\sigma)\rho}].$
Let us finally evaluate the last component $\hat{T}_{ij}$ of the stress-energy
tensor $\hat{T}_{AB}$. The equation (VI) is reduced to
$\displaystyle\hat{T}_{ij}^{(tf)}$ $\displaystyle=$
$\displaystyle\frac{N}{4}{T}^{(g)}_{ij}+\frac{N}{16}(\mathcal{F}^{4}_{\alpha\gamma\beta\gamma}+4\mathcal{F}^{4}_{\alpha\gamma\gamma\beta}+4\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}$
(85)
$\displaystyle+3\mathcal{F}^{2}_{\eta\zeta}f^{\eta\gamma\alpha}f^{\zeta\gamma\beta}){T}^{\alpha\beta}_{ij}+\frac{3N}{8}\mathcal{F}^{3}_{\eta\alpha\gamma}f^{\gamma\eta\beta}{\widetilde{T}}^{\alpha\beta}_{ij}.$
Here, we write to show that
$\displaystyle{T}^{\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{i}\xi^{\beta}_{j}-\frac{1}{N}g_{ij}\delta^{\alpha\beta},$
(86) $\displaystyle{\widetilde{T}}^{\alpha\beta}_{ij}$ $\displaystyle=$
$\displaystyle\xi^{\alpha}_{(i}\xi^{\beta}_{j)}-\frac{1}{N}g_{ij}\delta^{\alpha\beta}.$
(87)
It means that, the ${T}^{(1)\alpha\beta}_{ij}$ in (77) and the
${T}^{(2)\alpha\beta}_{ij}$ in (78) reduced to the ${T}^{\alpha\beta}_{ij}$
equation (86) , the ${T}^{(3)\alpha\beta\gamma}_{ij}$ in (79) to the
${\widetilde{T}}^{\alpha\beta}_{ij}$ equation (87), respectively. As another
point of view, the equation (80) which is trace part of $\hat{T}_{ij}$ is
equivalent to
$\displaystyle\hat{T}_{ij}^{(t)}$
$\displaystyle=-\frac{1}{4}g_{ij}[R^{2}_{\mu\nu\lambda\rho}-\frac{3}{2}R_{\mu\nu\lambda\rho}F^{\alpha\mu\nu}F^{\alpha\lambda\rho}+\frac{3}{8}(\mathcal{F}^{2}_{\alpha\beta})^{2}]$
$\displaystyle-(\frac{N-4}{4})R_{inkl}R_{j}\,^{nkl}-(\frac{3N-6}{8})f^{\gamma\eta\beta}\xi^{\alpha}_{(i}\xi^{\beta}_{j)}\mathcal{F}^{3}_{\eta\alpha\gamma}$
$\displaystyle-\frac{1}{32}\xi^{\alpha}_{i}\xi^{\beta}_{j}[(N+4)\mathcal{F}^{4}_{\alpha\gamma\beta\gamma}+4(N-2)\mathcal{F}^{4}_{\alpha\gamma\gamma\beta}$
$\displaystyle+8(N-1)\mathcal{D}_{\mu}F^{\alpha}_{\nu\lambda}\mathcal{D}^{\mu}F^{\beta\nu\lambda}+6(N-2)f^{\eta\gamma\alpha}f^{\zeta\gamma\beta}\mathcal{F}^{2}_{\eta\zeta}].$
The compatibility is also welcome here for the $5$D KK picture where
$\xi^{\alpha}_{i}=1$ and $R_{ijkl}=0$. The equations (82), (83), (84) and (VI)
exactly reduce to the those of our previous model sibelhalil , and the
equation (85) goes explicitly to $0$. It should be emphasised that, we use in
(84) well-known tensorial rule, the inner product of a symmetric and an
antisymmetric tensor vanishes.
## VII Conclusions
In this paper, we have completely generalized methods and results from our
previous work Başkal and Kuyrukcu sibelhalil to the non-Abelian case without
including any scalar fields by taking into account spinless and torsionless
Weyl-Yang gravity model in the context of the more than five dimensional pure
KK theories. We have firstly given a brief review of both of the non-Abelian
KK theory and Weyl-Yang gravity model, respectively. Next, we have performed a
popular dimensional KK reduction procedure to obtain the modified $4$D Weyl-
Yang+Yang-Mills action from a pure $(4+N)$-dimensional Weyl-Yang gravitational
Lagrangian without supplementary matter fields. In that respect, the
dimensionally reduced field equations which contain naturally the generalized
gravitational source term and the Lorentz force density and stress-energy
tensors are simultaneously investigated, and they are compared with the our
previous gravity model and standard higher-dimensional KK theories,
respectively. In our approach, the $4$D matter-spin source term is induced
from those that matter carrying energy-momentum but not possessing any spin
tensor. We also extend the reduced equations to more physical forms by taking
the compact $internal$ space as a homogeneous coset spaces background needed
in KK theories.
S. Başkal comments that “The $5$D Weyl-Yang theory appears to be more complete
compared to that of the standard KK model by accommodating terms that can be
identified as the Lorentz force density.” sibelhalil . Our theory which is
extension of $5$D Weyl-Yang model to the case of a higher dimensional non-
Abelian KK theory has effectively completed this remark with the generalized
gravitational source term which is not exist in the literature.
###### Acknowledgements.
I would like to thank S. Başkal for showing me ref. dereli .
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|
arxiv-papers
| 2013-08-08T16:26:16 |
2024-09-04T02:49:49.232491
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Halil Kuyrukcu",
"submitter": "Halil Kuyrukcu",
"url": "https://arxiv.org/abs/1308.1898"
}
|
1308.1986
|
0
# A DETERMINISTIC PSEUDORANDOM PERTURBATION SCHEME FOR ARBITRARY POLYNOMIAL
PREDICATES
Geoffrey Irving Forrest Green11footnotemark: 1 , Otherlab
{irving,forrest}@otherlab.com
###### Abstract
We present a symbolic perturbation scheme for arbitrary polynomial geometric
predicates which combines the benefits of Emiris and Canny’s simple randomized
linear perturbation scheme with Yap’s multiple infinitesimal scheme for
general predicates. Like the randomized scheme, our method accepts black box
polynomial functions as input. For nonmaliciously chosen predicates, our
method is as fast as the linear scheme, scaling reasonably with the degree of
the polynomial even for fully degenerate input. Like Yap’s scheme, the
computed sign is deterministic, never requiring an algorithmic restart
(assuming a high quality pseudorandom generator), and works for arbitrary
predicates with no knowledge of their structure. We also apply our technique
to exactly or nearly exactly rounded constructions that work correctly for
degenerate input, using l’Hôpital’s rule to compute the necessary singular
limits. We provide an open source prototype implementation including example
algorithms for Delaunay triangulation and Boolean operations on polygons and
circular arcs in the plane.
## 1 Introduction
Symbolic perturbation is a standard technique in computational geometry for
avoiding degeneracies by adding an infinitesimally small perturbation to the
inputs of a geometric algorithm. The technique was introduced by [6], with
refinements in [19], [7], [8], and [17]. Consider a geometric function
$G:\mathbb{R}^{N}\to S$ mapping input coordinates $x\in\mathbb{R}^{N}$ into
some discrete set $S$. Examples of $G(x)$ include Delaunay triangulation,
arrangements of lines or circles, and Boolean operations on shapes. We will
assume $G(x)$ can be computed using an algorithm that queries its input $x$
only through the signs of various polynomials $f(x)$ with integer
coefficients, each representing a geometric predicate such as “is this
triangle counterclockwise?” or “do two circles intersect inside a third
circle?”. If $f(x)=0$, the algorithm either fails due to ambiguity or requires
special logic to handle the degeneracy.
We describe symbolic perturbation in the framework of nonstandard analysis;
see [19], [8], and [17] for the geometric meaning of this approach. To extend
$G(x)$ to degenerate inputs, we introduce one or more positive infinitesimal
quantities $\epsilon_{1},\epsilon_{2},\ldots$, with $0<\epsilon_{i}<1/n$ for
all $i,n>0$. If we introduce more than one infinitesimal, we define a relative
ordering of the different monomials
$\epsilon_{1}^{p_{1}}\epsilon_{2}^{p_{2}}\cdots$; the simplest is
lexicographic ordering where $\epsilon_{i}^{p}>\epsilon_{i+1}$ for all
$i,p>0$. We then form an infinitesimal perturbation
$\delta\in\mathbb{R}[\epsilon_{1},\epsilon_{2},\ldots]^{N}$ from linear
combinations of the infinitesimals (here $\mathbb{R}[\epsilon_{i}]$ is the
ring of multivariate polynomials over $\mathbb{R}$ generated by
$\epsilon_{i}$), and evaluate
$\displaystyle G^{\prime}(x)=G(x+\delta).$
In detail, whenever the algorithm asks for the sign of $f(x)$ for some integer
coefficient polynomial $f$, we instead compute $f(x+\delta)$, which is a
multivariate polynomial in the infinitesimals. The sign of $f(x+\delta)$ is
the sign of the “least infinitesimal” nonzero monomial coefficient of this
polynomial. We distinguish between three existing symbolic perturbation
schemes that can be expressed in this framework and discuss their advantages
and disadvantages.
Yap’s deterministic scheme [19] introduces one infinitesimal $\epsilon_{i}$
per input coordinate $x_{i}$, and lets $\delta_{i}=\epsilon_{i}$. This
corresponds to evaluating $f(x_{1}+\epsilon_{1},x_{2}+\epsilon_{2},\ldots)$.
Since each coordinate has its own infinitesimal, $f(x+\delta)$ has at least
one nonzero monomial unless $f$ is identically zero, so the scheme produces a
nonzero sign for all nonzero polynomials. Unfortunately, a degree $d$
polynomial $f$ results in an
$f(x+\delta)\in\mathbb{R}[\epsilon_{1},\epsilon_{2},\ldots]$ with up to
$\binom{n+d}{n}$ monomial terms where $n$ is the number of input coordinates
used by $f$, which is worst case exponential in the degree of the predicate.
For extremely degenerate input, we may need to evaluate a large number of
coefficients before finding a nonzero.
Emiris and Canny’s deterministic linear scheme [7] arranges the input
coordinates into $n$ $k$-vectors based on the dimension $k$ of the geometric
space as $x_{a,b}$, $1\leq a\leq n$, $1\leq b\leq k$. They introduce a single
infinitesimal $\epsilon$ and write
$\delta_{a,b}=\epsilon\cdot(a^{b}\operatorname{mod}p)$
where $p>n$ is a prime. They show that this scheme produces a nonzero sign for
simplex orientation tests up to dimension $k$ and for the incircle tests used
in Delaunay triangulation. However, as discussed in [17], extending this
technique to other predicates is difficult.
In addition, as noted in [3], a fixed deterministic perturbation may turn
highly degenerate input into worst case behavior for algorithms like convex
hull: ignoring the $\operatorname{mod}p$, the deterministic linear scheme
produces a convex hull of size $n^{\lceil d/2\rceil}$ when all input points
are at the origin. We believe this also applies to Yap’s scheme and may arise
with the modular deterministic linear scheme.
Emiris and Canny’s randomized linear scheme [8] again introduces a single
infinitesimal $\epsilon$, but now sets $\delta_{i}=\epsilon y_{i}$ using
random coefficients $y_{i}$ chosen from some space $Y$. By the Schwartz-Zippel
lemma [16], $f(x+\delta)$ will be nonvanishing as a polynomial in $\epsilon$
with probability at least $1-d/|Y|$, where $d$ is the degree of the
polynomial. Unfortunately, what we actually need is for _all_ polynomials
evaluated during the algorithm to not vanish, which reduces the probability of
success to $(1-d/|Y|)^{T}$ where $T$ is the number of branches required.
Emiris and Canny show that their randomized scheme is very efficient in the
algebraic computation model, but suffers from a worst case cubic slowdown in
the bit computation model due to the large $|Y|$ required. For some algorithms
it is possible to reduce this slowdown by restarting only part of the
algorithm, but this adds significant complexity (in the authors’ experience).
To summarize: Yap’s deterministic scheme and the randomized linear scheme work
for arbitrary polynomial predicates, but suffer from unfortunate performance
penalties. The randomized linear scheme occasionally requires a restart of all
or part of the computation, adding extra complexity to the surrounding
algorithm especially if multiple computations are chained together (possibly
with user interaction in between). The deterministic linear scheme is ideal
when it works but requires special analysis to verify correctness for each
predicate.
Our contribution is to combine the advantages of each of the above methods.
## 2 A deterministic pseudorandom perturbation
Our approach is to introduce an infinite sequence of infinitesimals
$\epsilon_{1},\epsilon_{2},\ldots$, choose deterministic pseudorandom vectors
$y_{1},y_{2},\ldots$ with $y_{k,i}=\operatorname{rand}(k,i)$ for $1\leq
k<\infty,1\leq i\leq n$, and set
$\delta=\epsilon_{1}y_{1}+\epsilon_{2}y_{2}+\cdots.$
Here $\operatorname{rand}$ is a deterministic pseudorandom generator with
random access capability. Our implementation uses the Threefry generator of
[15], with
$\operatorname{rand}:[0,2^{128})\times[0,2^{128})\to[0,2^{32}).$
We order the infinitesimals largest first, so that
$\epsilon_{i}^{p}>\epsilon_{i+1}$ for all $p>0$. As in Yap’s scheme, this
ordering lets us add one term of the perturbation series at a time, evaluating
$\displaystyle f_{0}$ $\displaystyle=f(x)$ $\displaystyle f_{1}$
$\displaystyle=f(x+\epsilon_{1}y_{1})$ $\displaystyle f_{2}$
$\displaystyle=f(x+\epsilon_{1}y_{1}+\epsilon_{2}y_{2})$ $\displaystyle f_{3}$
$\displaystyle=f(x+\epsilon_{1}y_{1}+\epsilon_{2}y_{2}+\epsilon_{3}y_{3})$
$\displaystyle\vdots$
and stopping as soon as we arrive at a nonzero polynomial
$f_{k}(\epsilon_{1}.\ldots,\epsilon_{k})$. To compute the coefficients of a
given $f_{k}$, we temporarily view the infinitesimals $\epsilon_{i}$ as
integer variables and use a black box function for $f(x)$ to evaluate
$f_{k}(\epsilon_{1},\ldots,\epsilon_{k})$ with
$(\epsilon_{1},\ldots,\epsilon_{k})$ replaced with all $\binom{k+d}{k}$
nonnegative integer tuples satisfying $\epsilon_{1}+\cdots+\epsilon_{k}\leq d$
as discussed in [12] and Appendix A. If any values are nonzero, we use
multivariate polynomial interpolation to recover the $\binom{k+d}{k}$
coefficients of $f_{k}$ and return the sign of the least infinitesimal nonzero
term. Note that we have replaced the $\binom{n+d}{n}$ coefficients of Yap’s
scheme with $\binom{k+d}{k}$ coefficients.
We show that the computational cost is dominated by the first perturbation
term even for arbitrarily degenerate input, as long as the range $Y$ of the
random generator satisfies $d^{3}\ll|Y|$. In other words, our scheme has the
same cost as the simple linear scheme. To see this, note that if $f_{k}$ is
zero, setting one $\epsilon_{j}$ to one and the others to zero shows that
$f(x+y_{1}),\ldots,f(x+y_{k})$ are zero. Thus, if the polynomial predicate
$f(x)$ is not identically zero, the Schwartz-Zippel lemma gives
$\Pr(f_{k}=0)\leq\frac{d^{k}}{|Y|^{k}}.$
The sizes of the lattice points on which we evaluate $f$ grow slowly with $k$,
so the cost of a single polynomial evaluation is effectively $O(1)$ where the
constant depends on the polynomial. Similarly, the sizes of the numbers used
for multivariate interpolation also grow slowly with $k$, so the cost of
multivariate interpolation at level $k$ is $O\left(d\binom{d+k}{k}^{2}\right)$
(see Appendix A). Thus, the expected cost of the perturbation scheme is
$\displaystyle\sum_{k=0}^{\infty}\Pr(f_{k}=0)O\left(d\binom{d+k+1}{k+1}^{2}\right)\leq\sum_{k=0}^{\infty}\frac{d^{k}}{|Y|^{k}}O(d^{2k+3})=O\left(d^{3}\sum_{k=0}^{\infty}\frac{d^{3k}}{|Y|^{k}}\right)=O(d^{3})$
where we need $d^{3}<|Y|$ to guarantee a convergent geometric series. In
practice, $d^{3}\ll|Y|$; for $|Y|=2^{32}$ terms with $k\geq 2$ contribute less
than $1/4000$th of the expected cost for polynomials up to degree $100$. We
emphasize that this bound is independent of the input $x$, and therefore holds
even for maliciously chosen input data. However, we do assume that
$\operatorname{rand}$ behaves as a strong random source and, in particular,
that the polynomials $f(x)$ are not chosen with knowledge of
$\operatorname{rand}$.111Though maliciously choosing $f(x)$ so that
$f_{1}=f_{2}=0$ is quite useful for unit testing purposes.
Thus, our method has the same complexity as the deterministic linear scheme,
but like Yap’s scheme and the randomized linear scheme it works on arbitrary
polynomials. As in the randomized scheme, the perturbation does not create any
worst case behavior not already present in the input data. Since the
occasional random fallbacks occur one polynomial at a time, the outer
structure of a geometric algorithm is blissfully unaware that randomness is
used internally, and in particular we avoid poor bit complexity scaling when
evaluating many predicates over the course of an algorithm.
In practice, the dominant cost of the algorithm is black box predicate
evaluation. Even a single multiplication of two degree $d/2$ terms has
complexity $O(d^{2})$ using naive quadratic multiplication (which is typically
the fastest algorithm for small degrees). The linear perturbation phase
performs $d$ polynomial evaluations, for a total complexity of $O(d^{3})$, and
the constant is typically higher than for interpolation since most polynomials
involve several such multiplications. An $O(d^{3})$ slowdown for degenerate
cases is faster than previous general approaches but still a significant
drawback (see section 5 for benchmarks). Fortunately, a tiny amount of finite
perturbation applied to the input can minimize both the $O(d^{3})$ slowdown of
perturbation and the $O(d^{2})$ slowdown of unperturbed exact evaluation,
relying on symbolic perturbation to unconditionally correctly handle the few
remaining degeneracies.
## 3 Other approaches
Since the original introduction of the symbolic perturbation method several
alternative schemes have been introduced for treating degeneracies in
numerical algorithms. All of these approaches seem to require some algorithm
or predicate specific treatment, which complicates the process of developing
and especially testing new algorithms. However, the algorithm specific
approaches may be superior to a general approach such as ours when they apply,
either by avoiding the slowdown of occasional exact arithmetic entirely by
treating degenerate cases faster (our approach introduces a slowdown of $O(d)$
for the first perturbation level over exact evaluation), or by computing the
true exact answer rather than a perturbed answer.
Perhaps the most natural approach to treating degeneracies is to manually
extend the definition of $G(x)$ to degenerate cases and write algorithms which
treat these cases directly. For example, in an arrangement of lines,
intersections of three or more lines can be detected and represented as higher
degree vertices in the arrangement graph. Burnikel et al. [3] argue that
perturbation is slower and more complicated to implement than simply handling
degeneracies directly and present two degeneracy-aware algorithms as evidence.
We believe our method reduces the implementation complexity of symbolic
perturbation, but agree that a tailored algorithm is faster on highly
degenerate input. Unlike the deterministic symbolic perturbation schemes, an
algorithm built on our method will treat fully degenerate data as purely
random data, in particular avoiding the worst case behavior of convex hull
discussed in [3].
The _controlled perturbation_ approach of [11] applies a small finite
perturbation to the input points to avoid degeneracies, allowing the rest of
the algorithm to run with inexact floating point arithmetic. Input points
(spheres in their case) are processed one at a time, perturbing each new input
to avoid degeneracies against all previous inputs. Controlled perturbation
requires a careful enumeration of the possible degeneracies that may arise,
and a careful choice of the finite tolerance required for the algorithm to run
safely. A good tolerance bound may be computed with numerical analysis
techniques as in [10], at the cost of significant algorithm-specific analysis.
The main advantage of their approach over ours is speed: the majority of their
algorithms avoid all exact arithmetic and even all interval arithmetic or
other filters. As noted above, if degeneracies are pervasive and a slowdown of
$O(d^{3})$ is too large, an input to a symbolically perturbed algorithm can be
randomly jittered by a small amount, reducing the practical overhead to the
cost of interval analysis filtering without affecting correctness. Unlike
controlled perturbation, this requires no algorithm specific analysis.
Devillers et al. [5] present _qualitative symbolic perturbation_ , which
replaces the algebraic perturbations used in previous perturbation schemes
(and ours) by a sequence of carefully chosen, geometrically meaningful
perturbations. Their approach replaces the $O(d)$ slowdown of the first
perturbation level with a predicate dependent slowdown and may be faster than
our method when it applies. However, the geometric perturbations and the
analysis of their effect on the predicates must be performed separately for
each predicate, which complicates the design of algorithms and is a likely
source of complexity during implementation and debugging. Moreover, since the
perturbations depend on the algorithm, chaining two algorithms together
requires adjusting the perturbations to be compatible. Their approach shares
with ours (and indeed with Yap’s) the idea of a sequence of increasingly small
perturbations, applied one at a time until a nonsingular result is obtained.
Finally, we address a common complaint against symbolic perturbation (e.g.,
[3]), namely that a complicated postprocessing step is required to obtain the
exact answer from the perturbed result. We argue that the input to a typical
geometric algorithm already contains some degree of noise or numerical
inaccuracy, and therefore that classes of errors arising from infinitesimal
symbolic perturbation already arise in practice for exact algorithms run on
slightly bad input data. For example, consider the Boolean union of two
squares which touch exactly along one edge. An exact algorithm run on this
ideal input would merge the two squares into one rectangle, while symbolic
perturbation may leave the squares separate or even join them only partway
along the edge. However, if the input is already slightly shifted, both
algorithms produce exactly the same result. The solution in both cases is to
offset the squares slightly outwards prior to union, which resolves both
infinitesimal and finite errors.
## 4 Implementation
A C++ implementation of our symbolic perturbation technique is available under
a BSD license at https://github.com/otherlab/core/tree/exact222See
https://github.com/otherlab/core/commit/dc0f10918d17507d for the version
benchmarked below.. The code includes three algorithms built on top of the
perturbation core: Delaunay triangulation, Boolean operations on polygons, and
Boolean operations on polygons built from circular arcs. We plan to expand the
set of implemented algorithms and use them for various tasks in CAD/CAM such
as shape decomposition for manufacturing and motion planning. Benchmarks and
plotting scripts are available along with the paper source at
https://github.com/otherlab/perturb.
For simplicity and speed, our implementation quantizes all input coordinates
to the integer range $[-2^{53},2^{53}]$, the largest range of integers exactly
representable in double precision. This allows use of fast interval arithmetic
filters [2], falling back to exact integer evaluation using GMP if the filter
fails [9], and falling back to symbolic perturbation if the exact answer is
zero. The polynomial is provided as a black box evaluation routine (see
`exact/perturb.h` in the code). For multivariate interpolation we evaluate
$f_{k}(\epsilon_{1},\ldots,\epsilon_{k})$ on our fixed set of
$(\epsilon_{1},\ldots,\epsilon_{k})$ tuples, use the algorithm of [12] to map
into the Newton basis, then expand into the monomial basis. It is possible to
perform all computations required for polynomial interpolation using integers
only; see Appendix A. To avoid a significant slowdown due to memory allocation
inside GMP, the final version was written using manual memory allocation and
the low level interface to GMP.
In addition to computing the perturbed signs of polynomial predicates, we use
our scheme to compute exactly rounded perturbed constructions. Given a
rational function $f(x)/g(x)$ with $g(x)=0$, we compute the perturbation
series $g_{1},g_{2},\ldots$ until we find a nonzero $g_{k}$, compute the
perturbed numerator $f_{k}$, then evaluate the perturbed result as the ratio
of the matching least infinitesimal nonzero term in $f_{k}$ and $g_{k}$. In a
correct algorithm this ratio will always be finite, in that $f_{k}$ will never
contain a nonzero term larger than $g_{k}$, but it is easy to detect this case
and throw an exception as an aid to debugging. Note that the ratio of matching
least infinitesimal terms is exactly l’Hôpital’s rule for computing limits.
Finally, the ratio is rounded to the nearest integer. We can similarly compute
$\sqrt{f(x)/g(x)}$ by evaluating the limit of the ratio as a rational and
taking an exactly rounded square root.
We emphasize that these perturbed constructions are guaranteed to be within
$L_{0}$ distance $1/2$ of the true answer, where the true answer is consistent
with the rest of the algorithm and obeys any geometric invariants that apply
in the exact case. For example, a constructed union of a convex polygon with
itself will be within $L_{0}$ distance $1/2$ of the input, and in particular
will avoid all but extremely tiny foldovers that might result from performing
constructions with floating point arithmetic when an algorithm completes.
Moreover, since the maximum error is known, they can be fed back into the same
algorithm as tight interval bounds without fear of introducing
inconsistencies. Our circular arc Boolean code makes use of this to perform
more accurate interval-based filtering. For example, when comparing $y$
coordinates of different intersections of circles, we precompute the rounded
intersections and avoid costly polynomial evaluation if the rounded
coordinates differ.
Debugging and testing the symbolically perturbed algorithms we have
implemented so far has been a quite pleasant experience. Once the perturbation
core itself is trusted, bugs in the surrounding algorithm necessarily manifest
on a set of positive measure, since any taken branching path through the code
is described by algebraic inequalities which give rise to open sets. Thus, all
bugs are likely to be found by running the algorithm on random input. In
contrast, an algorithm which handles degeneracies specially or tailors the
perturbation to the predicates involved must actually test each kind of
degeneracy when debugging the algorithm. Any speedup logic such as interval
filtering can be easily checked by including a compile time flag to
unconditionally evaluate both fast and slow paths. This tests both the
correctness of the filter and the correctness of the predicate, which is
important for complicated predicates.
Although our currently implemented algorithms are serial, our symbolic
perturbation scheme can easily be used in parallel algorithms since each
predicate evaluation is deterministic. However, the dramatic slowdown between
interval filtering and perturbed exact evaluation might interfere with load
balancing at very high levels of parallelism, such as on a GPU.
In a correct geometric algorithm, no polynomial passed to symbolic
perturbation will be identically zero; this would correspond to a
fundamentally degenerate question such as “Is the triangle
$(x_{7},x_{7},x_{7})$ counterclockwise?”. However, it is convenient for
debugging to detect these cases and produce useful output. Therefore, if both
$f_{1}$ and $f_{2}$ are identically zero, our code pauses to run a randomized
polynomial identity check [16] and throws an exception if a nonzero is not
found. The identity test evaluates the polynomial on 20 random points; this
produces a false positive with probability under $10^{-171}$ (sufficient for
the lifetime of the code) and always reports failure for a truly zero
polynomial. The check has negligible effect on overall cost, since usually
$f_{1}\neq 0$.
For Delaunay triangulation, we use the partially randomized incremental
construction of [1]. Our implementation is $O(n\log n)$ for arbitrarily
degenerate input, and happily computes a random but valid Delaunay
triangulation if all points are at the origin. For Boolean operations, we find
intersections using axis-aligned bounding box hierarchies and find winding
numbers for each contour by tracing rays along horizontal lines (horizontal
lines are safe due to symbolic perturbation). Our current Boolean operation
algorithms degrade to $O(n^{2}\log n)$ for fully degenerate input since they
compute an arrangement of curves as the first step; this slowdown is
independent of the perturbation technique used, and also occurs for badly
formed nondegenerate input. Compared to [4], which used degree 12 predicates
for circular arc arrangements, our implementation uses predicates of degree at
most 8 via a combination of polynomial factoring and algorithmic changes (see
Appendix B). Even degree 8 is problematic for Yap’s scheme due to the worst
case exponential blowup in the number of terms. Other work on circle
arrangements in CGAL was done by [18]; this is orthogonal to our contribution.
## 5 Results
slope $1.071$slope $1.070$
Figure 1: Left: Delaunay triangulation of 2000 normally distributed points.
Right: computation time for Delaunay triangulation of (green, lower) $n$
normally distributed points and (blue, upper) $n$ copies of the origin. The
fully degenerate case ranges from $13.1$ to $15.5$ times as slow as the random
case due to falling back from interval arithmetic filters to integer
computation and symbolic perturbation. To reproduce these figures, run
``examples delaunay –count 2000 –plot 1+ and ``examples delaunay –count
1000000+.
Results for Delaunay triangulation are shown in Figure 1. Since our algorithm
is worst case $O(n\log n)$ independent of degeneracies, the slowdown ratio
from random input to fully degenerate input (all points at the origin) is
constant: between $13$ and $15.5$ due to falling back from interval arithmetic
filters to exact integer computation and symbolic perturbation. We note that
our current Delaunay triangulation algorithm is not state of the art, though
this is orthogonal to our contributions: CGAL’s routine is 4.3 times faster on
$10^{6}$ normally distributed points ($0.704$ s vs. $3.05$ s). It is also
dramatically faster for all points at the origin ($0.11$ s vs. $43$ s), though
only because CGAL prunes duplicate points as a preprocess. To reproduce our
CGAL benchmarks, run `examples delaunay --count 1000000 --cgal 1`.
slope $2.275$slope $2.069$slope $1.918$
Figure 2: Left: Boolean union of 1000 randomly chosen circular arc 4-gons.
Right: computation time for union of different numbers of (red, lower)
randomly distributed 4-gons, (green, middle) nearly but not exactly degenerate
4-gons, and (blue, top) exactly degenerate 4-gons. The exactly degenerate case
ranges from 65 to 252 times slower than the nearly degenerate case, which is
as expected since most of the cost is in degree 6 or 8 predicates
($6^{3}=216$, $8^{3}=512$). Both random and nearly degenerate cases use almost
entirely interval arithmetic; the latter is slower since it is closer to the
quadratic worst case. To reproduce these figures, run ``examples circles –plot
1 –count 1000+ and ``examples –mode circles –count 1000 –min-count 10+.
Results for circular arc Booleans are shown in Figure 2. Log-log slopes near 2
are expected because of the $O(n^{2})$ complexity of general arrangements of
circles. The slowdown for the exactly vs. nearly degenerate case is much
greater than for Delaunay triangulation because of the higher degree and
increased complexity of the predicates. Further optimizations to the
degenerate case are possible, in particular inlining GMP calls for small
arguments and caching certain repeated predicate evaluations, but these are of
questionable importance in practice since a tiny amount of finite jittering
removes the vast majority of degeneracies.
## 6 Conclusion
We have presented a deterministic pseudorandom symbolic perturbation scheme
which combines the advantages of several existing techniques. Given a
polynomial $f(x)$, we evaluate the sign of
$f(x+\epsilon_{1}y_{1}+\epsilon_{2}y_{2}+\cdots)$ where $y_{k}$ are
deterministic pseudorandom and $\epsilon_{k}$ are infinitesimals in decreasing
order of size. Typically only the first infinitesimal in this series need be
considered, so our method is as fast as the linear symbolic perturbation
schemes, but works for arbitrary polynomials and appears deterministic to the
caller.
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## Appendix A Polynomial interpolation
We found several useful papers discussing different aspects of univariate and
multivariate polynomial interpolation, and collect these results for
convenience. The algorithms discussed here perform $O(N^{2})$ linear
operations to convert $N$ samples to $N$ coefficients. Adds and multiply-by-
constants for degree $d$ integers require time $O(d)$, so the total complexity
is $O(dN^{2})$. Asymptotically faster algorithms using spectral methods exist,
but we do not consider them here.
In order to recover the coefficients of
$f_{k}(\epsilon_{1},\ldots,\epsilon_{k})$ we must perform multivariate
interpolation given the values of $f_{k}$ at our chosen set of tuples. In the
univariate case, this amounts to the classical divided difference algorithm.
As discussed in [14] and [13], the divided difference algorithm can be
beautifully expressed as the following factorization of the Vandermonde matrix
into bidiagonal matrices, shown here for the degree 3 case:
$\displaystyle\left(\begin{matrix}1&x_{0}&x_{0}^{2}&x_{0}^{3}\\\
1&x_{1}&x_{1}^{2}&x_{1}^{3}\\\ 1&x_{2}&x_{2}^{2}&x_{2}^{3}\\\
1&x_{3}&x_{3}^{2}&x_{3}^{3}\end{matrix}\right)=$
$\displaystyle\left(\begin{matrix}1&0&0&0\\\
\frac{1}{x_{0}-x_{1}}&\frac{1}{x_{1}-x_{0}}&0&0\\\
0&\frac{1}{x_{1}-x_{2}}&\frac{1}{x_{2}-x_{1}}&0\\\
0&0&\frac{1}{x_{2}-x_{3}}&\frac{1}{x_{3}-x_{2}}\end{matrix}\right)^{-1}$ (1)
$\displaystyle\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\
0&\frac{1}{x_{0}-x_{2}}&\frac{1}{x_{2}-x_{0}}&0\\\
0&0&\frac{1}{x_{1}-x_{3}}&\frac{1}{x_{3}-x_{1}}\end{matrix}\right)^{-1}$
$\displaystyle\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&\frac{1}{x_{0}-x_{3}}&\frac{1}{x_{3}-x_{0}}\end{matrix}\right)^{-1}$
$\displaystyle\left(\begin{matrix}1&x_{0}&0&0\\\ 0&1&x_{1}&0\\\ 0&0&1&x_{2}\\\
0&0&0&1\end{matrix}\right)$ $\displaystyle\left(\begin{matrix}1&0&0&0\\\
0&1&x_{0}&0\\\ 0&0&1&x_{1}\\\ 0&0&0&1\end{matrix}\right)$
$\displaystyle\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&x_{0}\\\
0&0&0&1\end{matrix}\right)$
This factorization was given in [14], though in a somewhat less elegant form
due to placing ones along the diagonal of $L$ instead of $U$ in the $LU$
factorization. The clean $LU$ factorization was given in [13], though without
the further bidiagonal factorization.
The first half of this factorization is the classical divided difference
algorithm to convert values $f(x_{0}),\ldots,f(x_{k})$ into the coefficients
of $f$ in the Newton basis $x(x-1)\cdots(x-n+1)$. The second half expands from
the Newton basis down to monomials. In our case, we have $x_{k}=k$, so all of
the ratios in each bidiagonal matrix have the same denominator. In particular,
we can clear fractions by multiplying the inverse by $d!$ where $d$ is the
degree of $f$, after which all computations can be performed in integers.
Alternatively, we can use the fact that while the inverse of the Vandermonde
matrix is not integral, both our polynomial values and the coefficients of the
polynomials in both Newton and monomial basis are integers. It turns out that
in this case all intermediate results in the divided difference algorithm are
integers as well. To show this, we must prove that the $k$th forward
difference $\Delta^{k}f(x)$ of an integer polynomial is divisible by $k!$. We
use the following argument due to Qiaochu
Yuan333http://math.stackexchange.com/questions/413600. Since the
transformation to and from the monomial basis to Newton basis (the second half
of (1)) is integral, it suffices to check $k!\mid\Delta^{k}f(x)$ for an
element of the Newton basis
$f(x)=x(x-1)\cdots(x-n+1)=n!\binom{x}{n}.$
Since $\Delta\binom{x}{n}=\binom{x}{n-1}$ we have
$\displaystyle\Delta^{k}x(x-1)\cdots(x-(n-1))$
$\displaystyle=n!\binom{x}{n-k}$
$\displaystyle=\frac{n!}{(n-k)!}x(x-1)\cdots(x-(n-k-1))$
$\displaystyle=k!\binom{n}{n-k}x(x-1)\cdots(x-(n-k-1))$
For the multivariate case, Neidinger [12] provides an elegant generalization
of the univariate divided difference algorithm when the polynomial is
evaluated on an “easy corner” of points, which includes the
$0\leq\epsilon_{i}$, $\epsilon_{1}+\cdots+\epsilon_{k}\leq d$ set that we use.
All intermediate results in their algorithm are multivariate divided
differences and are therefore integral by the above argument. They discuss
only interpolation into the multivariate Newton basis consisting of
polynomials such as
$\prod_{i}x_{i}(x_{i}-1)\cdots(x_{i}-(n_{i}-1))$
which corresponds to the first half of Equation 1. The multivariate
generalization of the second half of Equation 1 is easy, since the
multivariate Newton to monomial basis transformation matrix factors into
commuting matrices each expanding one variable, and these matrices are block
diagonal with respect to the other variables.
## Appendix B Degree 8 circular arc predicates
The critical predicate required for circular arc arrangements, determining
whether one intersection of two arcs is above another intersection, can be
reduced to degree 12 using resultant techniques [4]. This holds for the
general case of two unrelated intersections between pairs of circles
$C_{0},C_{1}$ and $C_{2},C_{3}$. However, to compute a circular arc
arrangement it suffices to consider the case where $C_{0}=C_{2}$; that is,
comparing the $y$ coordinates of the intersections of one circle with two
others. In this case, the polynomials can be factored into terms of degree
$\leq 8$. One significant algorithmic change is required, since we can no
longer fire a horizontal or vertical ray from the intersection of
$C_{0},C_{1}$ and detect intersections against unrelated circle arcs. Instead,
we must fire rays along exactly known (degree 1) $y$ coordinates, which is
sufficient to determine the winding number of a given circular arc polygon (or
connected component of an arrangement) as long as the bounding box touches at
least one ray. For most applications, polygons smaller than this may be safely
discarded.
We derived the degree 8 version of the predicate by starting with an
inequality involving square roots, then iteratively checking polynomial signs
and squaring to eliminate square roots until a fully polynomial inequality is
reached. All polynomials to be tested were then factored in Mathematica down
to their minimal degree, then manually simplified down to the more compact
expressions shown below (Mathematica’s `FullSimplify` was insufficient for
this purpose), using Mathematica to check each stage of the simplification.
The resultant techniques used in [4] would have also found the degree 8
solution had they been applied to the three circle special case. It should be
possible to automate the entire process from algebraic inequality to optimized
minimum degree polynomial expressions, but we have not yet done so.
The derivations below make several simplifications, for example assuming that
squaring does not reverse the direction of inequalities. For full details,
refer to
https://github.com/otherlab/core/blob/b186ab68303/exact/circle_predicates.cpp#L289
or `circles.nb` in https://github.com/otherlab/perturb.
### B.1 The intersection of two circles
Let circle $C_{i}$ have center $c_{i}$ and radius $r_{i}$, and define
$c_{ij}=c_{j}-c_{i}$. Assuming $C_{0}$ and $C_{1}$ intersect, parameterize one
of their intersections by
$\displaystyle p_{01}$ $\displaystyle=c_{0}+\alpha c_{01}+\beta
c_{01}^{\perp}.$
where $v^{\perp}$ is $v$ rotated left by $90^{\circ}$. We have
$\displaystyle(p_{01}-c_{i})^{2}$ $\displaystyle=r_{i}^{2}$ $\displaystyle
p_{01}^{2}-2p_{01}\cdot c_{i}+c_{i}^{2}$ $\displaystyle=r_{i}^{2}.$
Subtracting the two circle equations gives
$\displaystyle-2p_{01}\cdot c_{01}+c_{1}^{2}-c_{0}^{2}$
$\displaystyle=r_{1}^{2}-r_{0}^{2}$ $\displaystyle-2c_{0}\cdot c_{01}-2\alpha
c_{01}^{2}+(c_{0}+c_{1})\cdot c_{01}$ $\displaystyle=r_{1}^{2}-r_{0}^{2}$
$\displaystyle(1-2\alpha)c_{01}^{2}$ $\displaystyle=r_{1}^{2}-r_{0}^{2}$
$\displaystyle 1-2\alpha$
$\displaystyle=\frac{r_{1}^{2}-r_{0}^{2}}{c_{01}^{2}}$
$\displaystyle\hat{\alpha}=2c_{01}^{2}\alpha$
$\displaystyle=c_{01}^{2}+r_{0}^{2}-r_{1}^{2}$
Substituting into $C_{0}$’s equation gives
$\displaystyle(p_{01}-c_{0})^{2}$ $\displaystyle=r_{0}^{2}$
$\displaystyle\left(\alpha c_{01}+\beta c_{01}^{\perp}\right)^{2}$
$\displaystyle=r_{0}^{2}$
$\displaystyle\alpha^{2}c_{01}^{2}+\beta^{2}c_{01}^{2}$
$\displaystyle=r_{0}^{2}$ $\displaystyle\beta^{2}$
$\displaystyle=\frac{r_{0}^{2}}{c_{01}^{2}}-\alpha^{2}$
$\displaystyle\hat{\beta}^{2}=\left(2c_{01}^{2}\beta\right)^{2}$
$\displaystyle=4r_{0}^{2}c_{01}^{2}-\hat{\alpha}^{2}.$
To summarize, the intersection between circles $C_{0}$ and $C_{1}$ is
described by
$\displaystyle p_{01}$ $\displaystyle=c_{0}+\alpha c_{01}+\beta
c_{01}^{\perp}$ $\displaystyle\hat{\alpha}=2\alpha c_{01}^{2}$
$\displaystyle=c_{01}^{2}-r_{1}^{2}+r_{0}^{2}$
$\displaystyle\hat{\beta}^{2}=(2c_{01}^{2}\beta)^{2}$
$\displaystyle=4r_{0}^{2}c_{01}^{2}-\hat{\alpha}^{2}$
where we choose the positive or negative square root for $\beta$ depending on
which intersection is desired.
### B.2 Is one circle intersection above another?
Given three circles $C_{0},C_{1},C_{2}$, is $p_{01}$ below $p_{02}$? This
predicate has the form
$\displaystyle p_{01y}$ $\displaystyle<p_{02y}$ $\displaystyle
c_{0y}+\alpha_{01}c_{01y}+\beta_{01}c_{01x}$
$\displaystyle<c_{0y}+\alpha_{02}c_{02y}+\beta_{02}c_{02x}$ $\displaystyle 0$
$\displaystyle<\alpha_{02}c_{02y}-\alpha_{01}c_{01y}-\beta_{01}c_{01x}+\beta_{02}c_{02x}$
$\displaystyle 0$
$\displaystyle<\hat{\alpha}_{02}c_{02y}c_{01}^{2}-\hat{\alpha}_{01}c_{01y}c_{02}^{2}-\hat{\beta}_{01}c_{01x}c_{02}^{2}+\hat{\beta}_{02}c_{02x}c_{01}^{2}$
$\displaystyle 0$ $\displaystyle<A+B_{1}\sqrt{C_{1}}+B_{2}\sqrt{C_{2}}$
where $A,B_{1},B_{2},C_{1},C_{2}$ are polynomials and $C_{1},C_{2}>0$ since
the two intersections are assumed to exist. To reduce this equality to purely
polynomial equalities, we first compute the signs of $A,B_{1},B_{2}$. If these
all match, we are done. Otherwise we move the square root terms that differ
from $A$ in sign to the RHS and square. Assuming $A>0$, this gives either
$\displaystyle A+B_{1}\sqrt{C_{1}}$ $\displaystyle>-B_{2}\sqrt{C_{2}}$
$\displaystyle A^{2}+B_{1}^{2}C_{1}+2AB_{1}\sqrt{C_{1}}$
$\displaystyle>B_{2}^{2}C_{2}$ $\displaystyle
A^{2}+B_{1}^{2}C_{1}-B_{2}^{2}C_{2}$ $\displaystyle>-2AB_{1}\sqrt{C_{1}}$ (2)
or
$\displaystyle A$ $\displaystyle>-B_{1}\sqrt{C_{1}}-B_{2}\sqrt{C_{2}}$
$\displaystyle A^{2}$
$\displaystyle>B_{1}^{2}C_{1}+B_{2}^{2}C_{2}+2B_{1}B_{2}\sqrt{C_{1}C_{2}}$
$\displaystyle A^{2}-B_{1}^{2}C_{1}-B_{2}^{2}C_{2}$
$\displaystyle>2B_{1}B_{2}\sqrt{C_{1}C_{2}}$ (3)
The signs of the RHS’s of (2) and (3) are known. The polynomial LHS’s are
degree 10, but factor as
$\displaystyle\begin{array}[]{@{}r}A^{2}+B_{1}^{2}C_{1}-B_{2}^{2}C_{2}\phantom{\bigg{(}}=\\\
\phantom{\bigg{(}}\end{array}$
$\displaystyle\begin{array}[]{@{}r@{}l@{}}c_{02}^{2}\bigg{(}&c_{01}^{2}\left(\hat{\alpha}_{02}\left(\hat{\alpha}_{02}c_{01}^{2}-2\hat{\alpha}_{01}c_{01y}c_{02y}\right)+4r_{0}^{2}(c_{01x}^{2}c_{02y}^{2}-c_{01y}^{2}c_{02x}^{2})\right)\\\
&-\hat{\alpha}_{01}^{2}\left(c_{01x}^{2}-c_{01y}^{2}\right)c_{02}^{2}\bigg{)}\end{array}$
$\displaystyle\begin{array}[]{@{}r}A^{2}-B_{1}^{2}C_{1}-B_{2}^{2}C_{2}\phantom{\bigg{(}}=\\\
\phantom{\bigg{(}}\end{array}$
$\displaystyle\begin{array}[]{@{}r@{}l@{}}c_{01}^{2}c_{02}^{2}\bigg{(}&c_{02}^{2}\hat{\alpha}_{01}^{2}+c_{01}^{2}\hat{\alpha}_{02}^{2}-2c_{01y}c_{02y}\hat{\alpha}_{01}\hat{\alpha}_{02}\\\
&-4r_{0}^{2}(c_{01y}^{2}c_{02x}^{2}+c_{01x}^{2}c_{02y}^{2}+2c_{01x}^{2}c_{02x}^{2})\bigg{)}\end{array}$
and therefore reduce to degree 8 and 6, respectively. If the LHS and RHS of
(2) or (3) have the same sign, we square once more to eliminate the final
square root. Assuming positive LHS, squaring (2) gives
$\displaystyle(A^{2}+B_{1}^{2}C_{1}-B_{2}^{2}C_{2})^{2}$
$\displaystyle>4A^{2}B_{1}^{2}C_{1}$ $\displaystyle
A^{4}-2A^{2}B_{1}^{2}C_{1}+B_{1}^{4}C_{1}^{2}-2A^{2}B_{2}^{2}C_{2}-2B_{1}^{2}B_{2}^{2}C_{1}C_{2}+B_{2}^{4}C_{2}^{2}$
$\displaystyle>0$ $\displaystyle E$ $\displaystyle>0$
and squaring (3) gives
$\displaystyle(A^{2}-B_{1}^{2}C_{1}-B_{2}^{2}C_{2})^{2}$
$\displaystyle>4B_{1}^{2}B_{2}^{2}C_{1}C_{2}$ $\displaystyle
A^{4}-2A^{2}B_{1}^{2}C_{1}+B_{1}^{4}C_{1}^{2}-2A^{2}B_{2}^{2}C_{2}-2B_{1}^{2}B_{2}^{2}C_{1}C_{2}+B_{2}^{4}C_{2}^{2}$
$\displaystyle>0$ $\displaystyle E$ $\displaystyle>0.$
That is, the two inequalities square into the same degree 20 polynomial $E$,
which factors into degree $\leq 6$ terms as
$\displaystyle E$ $\displaystyle=c_{01}^{4}c_{02}^{4}E_{+}E_{-}$
$\displaystyle E_{\pm}$
$\displaystyle=c_{02}^{2}\hat{\alpha}_{01}^{2}+c_{01}^{2}\hat{\alpha}_{02}^{2}-2\hat{\alpha}_{01}\hat{\alpha}_{02}(c_{01y}c_{02y}\pm
c_{01x}c_{02x})-4r_{0}^{2}(c_{01x}c_{02y}\mp c_{01y}c_{02x})^{2}$
If intersections between four circles are compared, the analog to $E$ is still
divisible by $c_{01}^{4}c_{02}^{4}$, but the remaining degree $12$ polynomial
is irreducible as expected from [4].
As might be expected, performing these calculations only semiautomatically
resulted in a large number of typos and copying errors. The fact that the
final result is automatically checked against interval filters in the code was
critical to making the debugging process practical.
|
arxiv-papers
| 2013-08-08T21:44:36 |
2024-09-04T02:49:49.247251
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Geoffrey Irving and Forrest Green",
"submitter": "Geoffrey Irving",
"url": "https://arxiv.org/abs/1308.1986"
}
|
1308.1996
|
Article 15 in eConf C1304143
X - Ray Flares and Their Connection With Prompt Emission in GRBs
E. Sonbas1,2, G. A. MacLachlan3, A. Shenoy3, K.S. Dhuga3,
W. C. Parke3
1University of Adiyaman, Department of Physics, 02040 Adiyaman, Turkey
2NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
3Department of Physics, The George Washington University, Washington, DC
20052, USA
> We use a wavelet technique to investigate the time variations in the light
> curves from a sample of GRBs detected by Fermi and Swift. We focus primarily
> on the behavior of the flaring region of Swift-XRT light curves in order to
> explore connections between variability time scales and pulse parameters
> (such as rise and decay times, widths, strengths, and separation
> distributions) and spectral lags. Tight correlations between some of these
> temporal features suggest a common origin for the production of X-ray flares
> and the prompt emission.
> PRESENTED AT
>
>
>
>
> GRB 2013
> the Seventh Huntsville Gamma-Ray Burst Symposium
> Nashville, Tennessee, 14–18 April 2013
## 1 Introduction
In addition to the prompt emission in Gamma-ray Bursts (GRBs), rich and
diverse X-ray afterglow components have been identified by a number of studies
[2, 12, 13, 18]. Often embedded within the X-ray lightcurves are X-ray flares
(XRFs) in a large percentage of the GRBs [2, 14, 5, 3].
A number of studies suggest a connection between the prompt emission and the
X-ray flaring activity. For example, the lag-luminosity relation for XRFs has
been investigated by [9] and was found to be consistent with the existing
relation for the prompt emission [17, 11]. A very similar study [16] makes a
connection between the prompt emission data and the late afterglow X-ray data
and suggests that the lag-luminosity relation is valid over a time scale well
beyond the early steep-declining phase of the X-ray light curve. [10] present
a summary of the salient properties of XRFs and also show, using an internal
shell collision model, that the main time histories of XRFs can be explained
by the late activity of the central engine. Another study that hints at a
connection between the prompt emission and the X-ray afterglow is that of [6]
in which the authors examined the evolution of pulse widths of the flares and
found that the correlation between the widths of the pulses and time is
consistent with the effects of internal shocks at ever increasing collision
radii. In this work we focus on several temporal properties of the prompt
emission and flaring emissions as seen especially in long bursts.
## 2 Data And Methodology
Following the work of [8], we have used a technique based on wavelets to
extract a minimum time scale (MTS) for a sample of GRBs. The MTS determines
the time scale at which scaling processes dominate over random noise
processes. For the extraction of X-ray light curves, we used the method
developed by [4]. Using the available software, we extracted X- ray-flare
light curves with different time bins. By constructing log-scale diagrams (log
(variance) of signal vs. inverse frequency in octaves) for the sample, we have
determined the minimum time scale. An example of a log-scale diagram (in
addition to the light curve) for an X-ray flare is shown in Figure 1. We also
studied the extent to which the extraction of the MTS is sensitive to detector
thresholds. Shown in Figure 2 is a result of a simulation of the extraction of
the MTS for a number of bright GRBs. The figure shows MTS in octaves (inverse
frequency) vs. Brightness (cast as signal-to- noise ratio, and defined as
$\xi$ in [8]). The input octaves are indicated as horizontal colored lines
corresponding to the two selected octaves (6 and 7). The prompt emission
sample lies in the brightness range of 0.3 $<\xi<$ 0.74, corresponding roughly
to the range indicated by the horizontal double-headed black arrow. The
extracted (octave) values, shown as blue squares and red circles, match well
with the input values, indicating little dependence on brightness. Black
triangles correspond to XRF data. The majority of the XRFs lie in a region of
higher brightness (with a typical value of $\xi$ $\sim$ 3.0) compared to the
prompt emission (with $\xi$ $\sim$ 0.5). This shows that the signal-to-noise
ratio is significantly different for XRFs compared to that of the prompt
emission, and in addition, further illustrates that the extracted MTS varies
little with brightness.
Figure 1: Logscale diagram (and light curve) for the bright X-ray flare in
GRB070520B: Log(Variance) of signal as a function of octave (inverse
frequency). Plateau region is white noise and the sloped region is red noise.
Figure 2: MTS vs. Brightness for a sample of GRB prompt emission and XRFs.
Input octaves shown as colored lines and extracted ones indicated by colored
squares and circles. Triangles are XRFs.
## 3 Results
Using the extracted MTS and pulse-fit parameters (taken from literature), we
show that a correlation exists between the MTS and the pulse-rise times. The
correlation extends several decades of variability and includes the XRFs. This
result is depicted in Figure 3, which shows pulse rise-times vs. MTS. Black
data points indicate the prompt emission (with the pulse-fit parameters from
[1]; the blue and green points depict the XRF data with pulse-fit parameters
taken from [6, 9] respectively. Also shown in the figure is a line depicting
the equality of time scales. The best-fit line (not shown) leads to a slope of
1.26 $\pm$ 0.05. The Spearman correlation is 0.96 $\pm$ 0.02 and the Kendall
correlation is 0.79 $\pm$ 0.02.
Figure 3: Observer frame Pulse rise-times vs. MTS: Black points (prompt
emission); green and blue points (XRF data). The red line indicates the
equality of the respective temporal scales.
Figure 4: Observer frame Spectral lags vs. MTS: Black points (prompt emission
for long bursts); magenta point prompt emission for short burst); and blue
points (XRF data). The red line indicates the best-fit to the data.
This result extends the work of [7], who examined prompt emission only, to the
temporal domain covered by XRFs and reinforces their main conclusion that the
two techniques, wavelets and pulse-fitting, can be used independently to
extract a minimum time scale for physical processes of interest as long as
close attention is paid to time binning and the proper identification of
distinct pulses.
In order to pursue the apparent connection between the temporal properties of
prompt emission and the XRFs further, we explore below the possible link
between another temporal property, that of spectral lags, and the MTS. For the
prompt emission data, we extracted spectral lags for various observer-frame
energy bands using the CCF method described in detail by [17]. Some of these
results have been presented by [15]. Using the flare peak times reported by
[9], we have also extracted the spectral lags for the XRFs between the energy
bands 0.3-1 keV and the 3-10 keV respectively. A plot of the spectral lags vs.
the MTS is shown in Figure 4. Black and magenta data points depict the prompt
emission for long and short bursts; the blue points represent the XRF data.
The red line indicates the best-fit (a slope of 1.44 $\pm$ 0.07) through the
combined data set. The result clearly indicates a strong positive correlation
(a Spearman correlation of 0.96 $\pm$ 0.05 and a Kendall correlation of 0.86
$\pm$ 0.05) between the two temporal features, spectral lag and the MTS.
The two correlations taken together i.e., the pulse-rise times vs. MTS and the
spectral lag vs. MTS, are suggestive of more than a trivial connection between
the prompt emission and the XRFs.
## 4 Conclusions
For a sample of long-duration GRBs detected by Fermi/GBM and Swift, we have
extracted the minimum variability time scale (MTS) and spectral lags for both
prompt emission and XRF light curves. We compare the MTS, extracted through a
technique based on wavelets, both with the pulse rise times extracted through
a fitting procedure, and spectral lags extracted via the CCF method. Our main
results are summarized as follows;
* •
The prompt emission and the XRFs exhibit a significant positive correlation
between pulse rise times and the MTS, with time scales ranging from several
milliseconds to a few seconds respectively, and
* •
The spectral lag for both the prompt emission and the XRFs shows a strong
positive correlation with the MTS.
These results suggest a direct link between the mechanisms that lead to the
production of XRFs and prompt emission in GRBs.
## References
* [1] Bhat, P. N. et al. 2012, ApJ, 744, 141.
* [2] Burrows, D. N. et al. 2005a, SSRv, 120, 165
* [3] Chincarini, G. et al. 2007, ApJ, 671, 1903
* [4] Evans, P. A. et al., 2009, MNRAS, 397, 1177-1201
* [5] Falcone, A. D. et al. 2006, ApJ, 641, 1010
* [6] Kocevski D., et al., 2007, ApJ, 667, 1024-1032
* [7] MacLachlan G. A. et al., 2012, MNRAS, 425, L32-L35
* [8] MacLachlan G. A. et al., 2013, MNRAS, 432, Issue 2, p.857-865
* [9] Margutti, R. et al., 2010, MNRAS, 406, 2149-2167
* [10] Maxham, A. & Zhang, B. 2009, ApJ, 707, 1623-1633
* [11] Norris J. P. 2002, ApJ, 579, 386-403
* [12] Nousek, J. A. et al. 2006, ApJ, 642, 389
* [13] O′Brien, P. T. et al. 2006, ApJ, 647, 1213
* [14] Romano, P. et al. 2006, A&A, 450, 59
* [15] Sonbas, E. et al., 2012, Proceedings of the Gamma-Ray Bursts 2012 Conference May 7-11, 2012. Munich, Germany
* [16] Sultana, J. et al., 2012 ApJ, 758, 32
* [17] Ukwatta, T. N. et al., 2012, MNRAS, 419, 614-623
* [18] Willingale, R. et al. 2007, The Astrophysical Journal, 662:1093-1110
|
arxiv-papers
| 2013-08-08T22:56:29 |
2024-09-04T02:49:49.256105
|
{
"license": "Public Domain",
"authors": "E. Sonbas, G. A. MacLachlan, A. Shenoy, K.S. Dhuga, W. C. Parke",
"submitter": "Eda Sonbas",
"url": "https://arxiv.org/abs/1308.1996"
}
|
1308.2185
|
# The RHIC Beam Energy Scan Program: Results from the PHENIX Experiment
Brookhaven National Laboratory
E-mail
###### Abstract:
The PHENIX Experiment at RHIC has conducted a beam energy scan at several
collision energies in order to search for signatures of the QCD critical point
and the onset of deconfinement. PHENIX has conducted measurements of
transverse energy production, muliplicity fluctuations, the skewness and
kurtosis of net charge distributions, Hanbury-Brown Twiss correlations,
charged hadron flow, and energy loss. The data analyzed to date show no
significant indications of the presence of the critical point.
## 1 Introduction
Recent lattice QCD calculations predict that there is a first order phase
transition from hadronic matter to a Quark-Gluon Plasma that ends in a
critical point. There is a continuous phase transition on the other side of
the critical point. The Relativistic Heavy Ion Collider (RHIC) has conducted a
program to probe different regions of the QCD phase diagram in the vicinity of
the possible critical point with a beam energy scan. During 2010 and 2011,
RHIC provided Au+Au collisions to PHENIX at $\sqrt{s_{NN}}$ = 200 GeV, 62.4
GeV, 39 GeV, 27 GeV, 19.6 GeV, and 7.7 GeV. The strategy of the data analysis
focuses on looking for signs of the onset of deconfinement by comparing to
results at the top RHIC energy, and searching for direct signatures of a
critical point. Results from PHENIX covering charged particle multiplicity and
transverse energy production, multipicity and net charge fluctuations,
Hanbury-Brown Twiss correlations (HBT), charged hadron flow, and energy loss
will be discussed.
## 2 Multiplicity and Transverse Energy Production
PHENIX has measured charged particle multiplicity and transverse energy
($E_{T}$) production in Au+Au collisions at the following collision energies:
200, 62.4, 39, 19.6, and 7.7 GeV. These observables are closely related to the
geometry of the system and are fundamental measurements necessary to
understand the global properties of the collision. This work extends the
previous PHENIX measurements in 200, 130, and 19.6 GeV Au+Au collisions [1].
The charged particle multiplicity expressed as $dN_{ch}/d\eta$ normalized by
the number of participant pairs is shown in Figure 1. Included are
measurements from other experiments including ALICE and ATLAS. The red line is
a straight line fit to all of the points excluding the points at LHC energies.
Charged particle production at LHC energies exceeds the trend established at
lower energies.
Total $E_{T}$ production results are summarized in Figure 2, which shows the
excitation function of the estimated value of the Bjorken energy density [2]
expressed as
$\epsilon_{BJ}=\frac{1}{A_{\perp}\tau}\frac{dE_{T}}{dy},$ (1)
where $\tau$ is the formation time and $A_{\perp}$ is the transverse overlap
area of the nuclei. The Bjorken energy density increases monotonically over
the range of the RHIC beam energy scan. Also shown is the estimate for 200 GeV
U+U collisions taken during the 2012 running period.
Although $N_{ch}$ and $E_{T}$ production dramatically increases at LHC
energies compared to RHIC energies, the shape of the distributions as a
function of the number of participants, $N_{part}$, is independent of the
collision energy. This is illustrated in Figures 3 and 4, which each show an
overlay of the distributions for 7.7 GeV, 200 GeV, and 2.76 TeV Au+Au
collisions. The 200 GeV and 7.7 GeV distributions have been scaled up to match
the 2.76 TeV distributions. The shape of the distributions as a function of
$N_{part}$ appears to be driven by the collision geometry.
Figure 1: The value of $dN_{ch}/d\eta$ at mid-rapidity normalized by the
number of participant pairs as a function of $\sqrt{s_{NN}}$ for Au+Au
collisions. The red line is an exponential fit to all of the data points
excluding the ALICE and ATLAS points. Figure 2: The estimated value of the
Bjorken energy density, $\epsilon_{BJ}$, multiplied by the formation time in
central Au+Au collisions at mid-rapidity as a function of $\sqrt{s_{NN}}$. The
open circle represents the estimate for 200 GeV U+U collisions. Figure 3:
$dN_{ch}/d\eta$ normalized by the number of participant pairs as a function
$N_{part}$. Overlayed are the distributions from 7.7 GeV, 200 GeV, and 2.76
TeV Au+Au collisions. The PHENIX data has been scaled up to overlay the ATLAS
data [3]. Figure 4: $dE_{T}/d\eta$ normalized by the number of participant
pairs as a function $N_{part}$. Overlayed are the distributions from 7.7 GeV,
200 GeV, and 2.76 TeV Au+Au collisions. The PHENIX data has been scaled up to
overlay the ALICE data [4].
## 3 Multiplicity and Net Charge Fluctuations
Near the QCD critical point, it is expected that fluctuations in the charged
particle multiplicity will increase [5]. PHENIX has extended the previous
analysis of multiplicity fluctuations in 200 and 62.4 GeV Au+Au collisions [6]
to 39 and 7.7 GeV Au+Au collisions. Charged particle multiplicity fluctuations
are measured using the scaled variance, $\omega_{ch}=\sigma_{ch}/\mu_{ch}$,
which is the standard deviation scaled by the mean of the distribution. The
scaled variance is corrected for contributions due to non-dynamic impact
parameter fluctuations using the method described in [6]. Figure 5 shows the
PHENIX results for central collisions as a function of $\sqrt{s_{NN}}$. There
is no indication of the presence of a critical point from the PHENIX results
alone.
Figure 5: Charged particle multiplicity fluctuations in central Au+Au
collisions expressed in terms of the scaled variance as a function of
$\sqrt{s_{NN}}$.
The shapes of the distributions of the event-by-event net charge are expected
to be sensitive to the presence of the critical point [7]. PHENIX has measured
the skewness ($S=\langle(N-\langle N\rangle)^{3}\rangle/\sigma^{3}$) and the
kurtosis ($\kappa=\langle(N-\langle N\rangle)^{4}\rangle/\sigma^{4}-3$) of net
charge distributions in Au+Au collisions at 200, 62.4, 39, and 7.7 GeV. These
values are expressed in terms that can be associated with the quark number
susceptibilities, $\chi$: $S\sigma\approx\chi^{(3)}/\chi^{(2)}$ and
$\kappa\sigma^{2}\approx\chi^{(4)}/\chi^{(2)}$ [8]. The skewness and kurtosis
for central collisions are shown in Figure 6 as a function of $\sqrt{s_{NN}}$.
The data are compared to URQMD and HIJING simulation results processed through
the PHENIX acceptance and detector response. There is no excess above the
simulation results observed in the data at these four collision energies. More
details on this analysis are available in these proceedings [9].
Figure 6: The skewness multiplied by the standard deviation and the kurtosis
multiplied by the variance from net charge distributions from central Au+Au
collisions. The circles represent the data. The grey error bars represent the
systematic errors. Also shown are URQMD and HIJING simulation results
processed through the PHENIX acceptance. The increase in the kurtosis from
URQMD and HIJING may be due to an increase in resonance production at 200 GeV.
## 4 Hanbury-Brown Twiss Correlations
HBT measurements provide information about the space-time evolution of the
particle emitting source in the collision. An emitting system which undergoes
a strong first order phase transition is expected to demonstrate a much larger
space-time extent than would be expected if the system had remained in the
hadronic phase throughout the collision process [10]. The shape of the
emission source function can also provide signals for a second order phase
transition or proximity to the QCD critical point [11].
PHENIX has measured the 3-dimensional source radii ($R_{side},R_{out}$, and
$R_{long}$) for charged pions in 200 GeV, 62.4 GeV, and 39 GeV Au+Au
collisions. The measurements have been made for $0.2<k_{T}<2.0$ GeV/c. The
results are summarized in Figure 7, which shows the excitation function for
the radii for central collisions at $<k_{T}>$ = 0.3 GeV/c. There is no
significant variation in the radii $R_{out}$ and $R_{side}$ over this energy
range while $R_{long}$ follows an increasing trend as collision energy
increases. The freeze-out volume of the system can be estimated as follows:
$V_{f}=R_{out}\times R_{side}\times R_{long}$. The excitation function of the
freeze-out volume is shown in Figure 8 as a function of $dN_{ch}/d\eta$. From
the lowest to the highest energies measured, the freeze-out volume increases
linearly with the charged particle multiplicity.
Figure 7: HBT radii as a function of $\sqrt{s_{NN}}$ for central collisions at
$<k_{T}>$=0.3 GeV/c. The red points are the PHENIX measurements. The data from
other experiments can be found elsewhere [14, 15, 16, 17, 18, 19, 20, 21].
Figure 8: The HBT freeze-out volume, $V_{f}$ as a function of $dN_{ch}/d\eta$
for central collisions at $<k_{T}>$=0.3 GeV/c. The red points are the PHENIX
measurements. The data from other experiments can be found elsewhere [14, 15,
16, 17, 18, 19, 20, 21].
## 5 Charged Hadron Flow
Measurements of the anisotropy parameter $v_{2}$ for identified particles
exhibit strong evidence of quark-like degrees of freedom at the top RHIC
energies. A goal of the RHIC beam energy scan is to determine where the
constituent quark scaling of $v_{2}$ no longer holds. PHENIX has measured
$v_{2},v_{3}$, and $v_{4}$ for identified pions, kaons, and protons in 62.4
and 39 GeV Au+Au collisions. Shown in Figure 9 and Figure 10 are $v_{2}$
measurements scaled as $v_{2}/n_{q}^{n/2}$ on the vertical axis and
$KE_{T}/n_{q}$ on the horizontal axis, where $n_{q}$ represents the number of
quarks in the particle species being plotted, and $KE_{T}$ represents the
transverse kinetic energy. At both of these collision energies, the scaling of
$v_{2}$ observed at 200 GeV holds down to 39 GeV.
Figure 9: The scaling of $v_{2}$ for 62.4 GeV Au+Au collisions. Shown are the
measurements for identified pions, kaons, and protons. Figure 10: The scaling
of $v_{2}$ for 39 GeV Au+Au collisions. Shown are the measurements for
identified pions, kaons, and protons.
## 6 Energy Loss
At the top RHIC energies, a very large suppression of hadron production at
high transverse momentum is observed when compared to baseline p+p collisions
[12]. This suppression has been attributed to the dominance of parton energy
loss in the medium. Previous studies of Cu+Cu collisions at $\sqrt{s_{NN}}=$
200 GeV, 62.4 GeV, and 22.4 GeV [13] show that suppression is observed
(suppression factor $R_{AA}<1$) at 200 and 62.4 GeV, but enhancement
($R_{AA}>1$) dominates at all centralities at 22.4 GeV. PHENIX has measured
$R_{AA}$ for neutral pions in 200, 62.4, and 39 GeV Au+Au collisions [22]. The
value of $R_{AA}$ for neutral pions with $p_{T}>6$ GeV/c is shown in Figure 11
for all 3 energies. There is still significant suppression observed in 39 GeV
Au+Au collisions, but the suppression at the lower energy has decreased
compared to the suppression seen in 62.4 and 200 GeV Au+Au collisions. PHENIX
has also measured the suppression of $J/\psi$ particles in 200, 62.4, and 39
GeV Au+Au collisions at forward rapidity [23]. Again, significant suppression
is still observed in 39 GeV collisions, but the amount of suppression is
decreased compared to that in 200 and 62.4 GeV collisions.
Figure 11: The suppression factor, $R_{AA}$, for neutral pions with $p_{T}>6$
GeV/c for 200, 62.4, and 39 GeV Au+Au collisions. Figure 12: The suppression
factor, $R_{AA}$, for $J/\psi$ particles at forward rapidity for 200, 62.4,
and 39 GeV Au+Au collisions.
## 7 Summary
Presented here are some of the PHENIX results from the RHIC beam energy scan
program. From the analyses completed to date, there is no significant
indication of the presence of the QCD critical point. Measurements of the
suppression of neutral pions and $J/\psi$ particles suggest that the point at
which the onset of deconfinement is seen may lie below collision energies of
39 GeV. Many analyses from PHENIX, particularly at $\sqrt{s_{NN}}=$ 27 GeV and
19.6 GeV, will be available soon.
## References
* [1] S.S. Adler et al., Phys. Rev. C 71, 034908 (2005).
* [2] J. D. Bjorken, Phys. Rev. D 27, 140 (1983).
* [3] G. Aad et al., Phys. Lett. B 710, 363 (2012).
* [4] C. Loizides et al., arXiv:1106.6324v1 (2011).
* [5] M. Stephanov et al, Phys. Rev. D 60, 114028 (1999).
* [6] A. Adare et al, Phys. Rev. C 78, 044902 (2008).
* [7] R. V. Gavai and S. Gupta, Phys. Lett. B 696, 459 (2011).
* [8] F. Karsch and K. Redlich, Phys. Lett. B 695, 136 (2011).
* [9] P. Garg et al., arXiv:1305.7327 (2013).
* [10] S. Pratt, Phys. Rev. Lett. 53, 1219 (1984).
* [11] T. Csorgo et al., arXiv:nucl-th/0512060 (2005).
* [12] K. Adcox et al., Phys. Rev. Lett. 88, 022301 (2001).
* [13] A. Adare et al., Phys. Rev. Lett. 101, 162301 (2008).
* [14] M. Lisa et al., Phys. Rev. Lett. 84, 2798 (2000).
* [15] D. Adamova et al., Nucl. Phys. A714, 124–144 (2003).
* [16] C. Alt et al., Phys. Rev. C77, 064908 (2008).
* [17] B. B. Back et al., Phys. Rev. C73, 031901 (2006).
* [18] B. B. Back et al., Phys. Rev. C74, 021901, (2006).
* [19] B. I. Abelev et al., Phys. Rev. C79, 034909, (2009).
* [20] B. I. Abelev et al., Phys. Rev. C80, 024905, (2009).
* [21] K. Aamodt et al., Phys. Lett. B696: 328 (2011).
* [22] A. Adare et al., Phys. Rev. Lett. 109, 152301 (2012).
* [23] A. Adare et al., Phys. Rev. C86, 064901 (2012).
|
arxiv-papers
| 2013-08-09T17:02:04 |
2024-09-04T02:49:49.268835
|
{
"license": "Public Domain",
"authors": "J.T. Mitchell (for the PHENIX Collaboration)",
"submitter": "Jeffery T. Mitchell",
"url": "https://arxiv.org/abs/1308.2185"
}
|
1308.2281
|
# On the determinant of the distance matrix of a bicyclic graph††thanks:
Supported by National Natural Science Foundation of China(11071002,11171373),
and Zhejiang Provincial Natural Science Foundation of China(LY12A01016).
Shi-Cai Gong, Ju-Li Zhang and Guang-Hui Xu
School of Science, Zhejiang A & F University,
Lin’an, 311300, P. R. China Corresponding author. E-mail addresses:
[email protected](S. Gong); [email protected](L. Zhang); ghxu@
zafu.edu.cn(G. Xu).
Abstract: Two cycles are referred as disjoint if they have no common edges. In
this paper, we will investigate the determinant of the distance matrix of a
graph, giving a formula for the determinant of the distance matrix of a
bicyclic graph whose two cycles are disjoint, which extends the formula for
the determinant of the distance matrix of a tree, as well as that of a
unicyclic graph.
Keywords: Distance matrix; bicyclic graph; determinant
AMS Subject Classifications: 05C50; 15A18
## 1 Introduction
In the whole paper all graphs are simple and undirected. Let $G$ be a graph
with vertex set $V=\\{1,2,\cdots,n\\}$ and edge set $E$. The distance between
the vertices $i$ and $j$, denoted by $dis(i,j)$, is the length of a shortest
path between them. The $n$-by-$n$ matrix $D(G)=(d_{i,j})$ with
$d_{i,j}=dis(i,j)$ is referred as the distance matrix of $G$, or the metrics
matrix of $G$.
The determinant of the distance matrices of graphs have been investigated in
the literature. As early, Graham and Pollack [5] showed that if $T$ is a tree
on $n$ vertices with distance matrix $D$, then the determinant of $D$ is
$(-1)^{n-1}(n-1)2^{n-2}$, a formula depending only on $n$. Then Bapat,
Kirkland and Neumann [1] extend this formula to the weighted case and give a
formula for the determinant of the distance matrix of a unicyclic graph,
showing that the determinant of the distance matrix of a unicyclic graph is
related to the length of the cycle contained in it and its order. For more
spectral properties of the distance matrices of a graph, one can see for
example [6, 7, 8, 9, 10] and the references therein.
For a given graph $G$, two cycles of $G$ are referred as disjoint if they have
no common edges. Let $C_{p}$ and $C_{q}$ be two disjoint cycles. Suppose that
$v_{1}\in C_{p},v_{k}\in C_{q}$. Joining $v_{1}$ and $v_{k}$ by a path
$v_{1}v_{2}\cdots v_{k}$ of length $k-1$, where $k\geq 1$ and $k=1$ means
identifying $v_{1}$ with $v_{k}$, the resultant graph, denoted by
$\infty(p,k,q)$, is referred as an $\infty$-graph. A bicyclic graph which
contains an $\infty$-graph as an induced subgraph can be considered a graph
obtained from an $\infty$-graph $\infty(p,k,q)$ by planting some trees to such
an $\infty$-graph.
In this paper, we will investigate the determinant of the distance matrix of a
graph, giving a formula for the determinant of the distance matrix of a
bicyclic graph whose two cycles are disjoint, which extends the formula for
the determinant of the distance matrix of a tree, as well as that of a
unicyclic graph. In addition, as by-product we show that if a graph is
obtained from an induced subgraph by planting some trees on it, then the
determinant of the distance matrix of such a graph is independent to the
structure of those trees.
## 2 Preliminary results
In this section, we will establish some preliminary results, which will be
useful in the following discussion.
Henceforth we use the following notation. For a real matrix $A$, denote by
$A^{T}$ the transpose matrix to $A$. The identity matrix is denoted by $I$ and
the all ones row vector is denoted by 1. The determinant of the matrix $A$ is
denoted by $det(A)$, or $|A|$ for simplify. We refer to D.
Cvetkovi$\acute{c}$, M. Doob and H. Sachs [3] for more terminology and
notation not defined here.
###### Lemma 2.1
Let $C_{k}=\frac{1}{2}B_{k}B_{k}^{T}-2I$, a $k\times k$ matrix, and
$F_{k}=\frac{1}{2}\textbf{1}B^{T}_{k}+\textbf{1}$, a row vector with dimension
$k$, where
$B_{k}=\left(\begin{array}[]{cccccc}-1&0&0&\cdots&0&0\\\ -1&-1&0&\cdots&0&0\\\
0&-1&-1&\cdots&0&0\\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&\cdots&-1&0\\\ 0&0&0&\cdots&-1&-1\end{array}\right)_{k\times k}.$
Then
$detC_{k}=\frac{(-1)^{k}(2k+1)}{2^{k}},$
and
$F_{k}C_{k}^{-1}F_{k}^{T}=-\frac{k}{2(2k+1)}.$
Proof. By a directly calculation, we have
$C_{k}=\frac{1}{2}\left(\begin{array}[]{cccccc}-3&1&0&\cdots&0&0\\\
1&-2&1&\cdots&0&0\\\ 0&1&-2&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\ 0&0&0&\cdots&-2&1\\\
0&0&0&\cdots&1&-2\\\ \end{array}\right)_{k\times k}$
and
$F_{k}=\frac{1}{2}(1~{}0~{}\cdots~{}0~{}0),$
a vector with exactly one nonzero entry. Now let
$\displaystyle H_{k}=\left(\begin{array}[]{cccccc}-2&1&0&\cdots&0&0\\\
1&-2&1&\cdots&0&0\\\ 0&1&-2&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\ 0&0&0&\cdots&-2&1\\\
0&0&0&\cdots&1&-2\\\ \end{array}\right)_{k\times k}.$
As we know that $detH_{k}=(-1)^{k}(k+1),$ then
$\displaystyle 2^{k}detC_{k}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccccc}-3&1&0&\cdots&0&0\\\
1&-2&1&\cdots&0&0\\\ 0&1&-2&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\ 0&0&0&\cdots&-2&1\\\
0&0&0&\cdots&1&-2\\\ \end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccccc}-2&1&0&\cdots&0&0\\\
1&-2&1&\cdots&0&0\\\ 0&1&-2&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\ 0&0&0&\cdots&-2&1\\\
0&0&0&\cdots&1&-2\\\
\end{array}\right|+\left|\begin{array}[]{cccccc}-1&0&0&\cdots&0&0\\\
1&-2&1&\cdots&0&0\\\ 0&1&-2&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\ 0&0&0&\cdots&-2&1\\\
0&0&0&\cdots&1&-2\\\ \end{array}\right|$ $\displaystyle=$ $\displaystyle
detH_{k}-detH_{k-1}$ $\displaystyle=$ $\displaystyle(2k+1)(-1)^{k}.$
Hence,
$detC_{k}=\frac{(2k+1)(-1)^{k}}{2^{k}},$
and
$F_{k}C_{k}^{-1}F_{k}^{T}=\frac{det(C_{1,1}^{*})}{4|C_{k}|}=\frac{det(H_{k-1})}{4|C_{k}|}=-\frac{k}{2(2k+1)},$
where $C_{1,1}^{*}$ denotes the $(1,1)-$th entry of the adjoint matrix of
$C_{k}$. The result thus follows. $\blacksquare$
###### Lemma 2.2
Suppose that the sequence $f(0),f(1),\cdots,f(n)$ satisfies the following
linear recurrence relation
$\left\\{\begin{array}[]{l}f(n)=-4f(n-1)-4f(n-2)\\\ f(0)=f_{0}\\\
f(1)=f_{1}.\end{array}\right.$
Then
$f(n)=[f_{0}-\frac{n}{2}(f_{1}+2f_{0})](-2)^{n}.$
Proof. Since the characteristic equation of this recurrence relation is
$x^{2}+4x+4=0$
and its two roots are $x_{1}=x_{2}=-2$, by Theorem 7 .4.1 in [2] the general
solution is
$f(n)=(c_{1}+nc_{2})(-2)^{n}.$
Combining with the initial values $f(0)=f_{0}$ and $f(1)=f_{1}$, we have
$\left\\{\begin{array}[]{l}c_{1}=f_{0};\\\
c_{2}=-\frac{1}{2}f_{1}-f_{0}.\end{array}\right.$
The result thus follows. $\blacksquare$
###### Lemma 2.3
Let $G$ be the graph obtained from a graph $G_{1}$ by identifying an arbitrary
vertex of $G_{1}$ and one pendent vertex of the path $P_{2}$. Then the
determinant of the distance matrix of the graph $G$ is fixed, regardless the
choice of the vertex of $G_{1}$.
Proof. Let the vertex set of $G$ be $\\{1,2,\cdots,n\\}$. Without loss of
generality, we can take vertex $1$ to be one pendent vertex of $P_{2}$ and
label another pendent vertex, a quasi-pendent vertex of $G$, as $2$. Then
$G_{1}$ can be considered the subgraph of $G$ induced by vertices
$\\{2,\cdots,n\\}$. Let $(0~{}~{}d_{2})$ be the row vector of the distance
matrix of $G_{1}$ corresponding to the vertex $2$ and $D^{*}$ be the distance
matrix of the subgraph of $G$ induced by vertices $\\{3,\cdots,n\\}$. Then
$D(G)$, the distance matrix of $G$, can be partitioned as
$D(G)=\left(\begin{array}[]{ccc}0&1&d_{2}+\textbf{1}\\\ 1&0&d_{2}\\\
d_{2}^{T}+\textbf{1}^{T}&d_{2}^{T}&D^{*}\end{array}\right).$
Hence
$detD(G)=\left|\begin{array}[]{ccc}0&1&d_{2}+\textbf{1}\\\ 1&0&d_{2}\\\
d_{2}^{T}+\textbf{1}^{T}&d_{2}^{T}&D^{*}\end{array}\right|=\left|\begin{array}[]{ccc}-1&1&\textbf{1}\\\
1&0&d_{2}\\\
d_{2}^{T}+\textbf{1}^{T}&d_{2}^{T}&D^{*}\end{array}\right|=\left|\begin{array}[]{ccc}-2&1&\textbf{1}\\\
1&0&d_{2}\\\ \textbf{1}^{T}&d_{2}^{T}&D^{*}\end{array}\right|,$
the last equalition implies that $detD(G)$ is independent to the choice of the
vertex $2$. The result thus follows. $\blacksquare$
###### Lemma 2.4
Let $G_{1}$ and $G_{2}$ be two graphs with vertex sets $\\{1,2,\cdots,k\\}$
and $\\{k+1,k+2\cdots,n\\}$, respectively. Let $G$ be the graph obtained from
$G_{1}$ and $G_{2}$ by adding an edge between vertices $1$ and $n$, and
$\tilde{G}$ the graph obtained from $G_{1}$ and $G_{2}$ by identifying
vertices $1$ and $n$ and then adding a pendent vertex from $1$ (or $n$).
Denote by $D$ and $\tilde{D}$ respectively the distance matrices of $G$ and
$\tilde{G}$. Then
$detD=det\tilde{D}.$
Proof. Without loss of generality, we take the distance matrices of $G_{1}$
and $G_{2}$ as $D(G_{1})=\left(\begin{array}[]{cc}0&d_{1}\\\
d_{1}^{T}&D^{*}\end{array}\right)$ and
$D(G_{2})=\left(\begin{array}[]{cc}D^{**}&d_{n}^{T}\\\
d_{n}&0\end{array}\right)$, where $D^{*}$ and $D^{**}$ denote respectively the
distance matrix of the subgraphs induced by $\\{2,\cdots,k\\}$ and
$\\{k+1,k+2\cdots,n-1\\}$, and $(0~{}~{}d_{1})$ and $(d_{n}~{}~{}0)$ are
respectively the row vectors of $D(G_{1})$ corresponding to the vertex $1$ and
the row vectors of $D(G_{2})$ corresponding to the vertex $n$. Again without
loss of generality, suppose that, in $\tilde{G}$, the vertex $n$ is the
pendent vertex and the vertex $1$ is the quasi-pendent vertex. For
$D=(d_{i,j})$ and $\tilde{D}=(\tilde{d}_{i,j})$, we set the rows and columns
of them correspond to $\\{1,2,\cdots,n\\}$, respectively. Then we have
$d_{i,j}=\left\\{\begin{array}[]{ll}dis_{G_{1}}(i,j),&{\rm if\mbox{ }1\leq
i,j\leq k};\\\ dis_{G_{2}}(i,j),&{\rm if\mbox{ }k+1\leq i,j\leq n};\\\
dis_{G_{1}}(1,i)+dis_{G_{2}}(j,n)+1,&{\rm if\mbox{ }1\leq i\leq k\mbox{
}k+1\leq j\leq n},\end{array}\right.$
and
$\tilde{d}_{i,j}=\left\\{\begin{array}[]{ll}dis_{G_{1}}(i,j),&{\rm if\mbox{
}1\leq i,j\leq k};\\\ dis_{G_{2}}(i,j),&{\rm if\mbox{ }k+1\leq i,j\leq
n-1};\\\ dis_{G_{1}}(1,j)+1,&{\rm if\mbox{ }i=n\mbox{ }and\mbox{ }1\leq j\leq
k};\\\ dis_{G_{2}}(n,j)+1,&{\rm if\mbox{ }i=n\mbox{ }and\mbox{ }k+1\leq j\leq
n-1};\\\ dis_{G_{1}}(1,i)+dis_{G_{2}}(j,n),&{\rm if\mbox{ }1\leq i\leq k\mbox{
}and\mbox{ }k+1\leq j\leq n-1}.\end{array}\right.$
Hence,
$D=\left(\begin{array}[]{cccc}0&d_{1}&d_{n}+\textbf{1}&1\\\
d_{1}^{T}&D^{*}&d_{n}\textbf{1}^{T}+\textbf{1}d_{1}^{T}+\textbf{1}\textbf{1}^{T}&d_{1}^{T}+\textbf{1}^{T}\\\
d_{n}^{T}+\textbf{1}^{T}&\textbf{1}d_{n}^{T}+d_{1}\textbf{1}^{T}+\textbf{1}\textbf{1}^{T}&D^{**}&d_{n}^{T}\\\
1&d_{1}+\textbf{1}&d_{n}&0\end{array}\right),$
$\tilde{D}=\left(\begin{array}[]{cccc}0&d_{1}&d_{n}&1\\\
d_{1}^{T}&D^{*}&d_{n}\textbf{1}^{T}+\textbf{1}d_{1}^{T}&d_{1}^{T}+\textbf{1}^{T}\\\
d_{n}^{T}&\textbf{1}d_{n}^{T}+d_{1}\textbf{1}^{T}&D^{**}&d_{n}^{T}+\textbf{1}^{T}\\\
1&d_{1}+\textbf{1}&d_{n}+\textbf{1}&0\end{array}\right).$ $None$
Consequently, we have
$\displaystyle detD$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}0&d_{1}&d_{n}+\textbf{1}&1\\\
d_{1}^{T}&D^{*}&d_{n}\textbf{1}^{T}+\textbf{1}d_{1}^{T}+\textbf{1}\textbf{1}^{T}&d_{1}^{T}+\textbf{1}^{T}\\\
d_{n}^{T}+\textbf{1}^{T}&\textbf{1}d_{n}^{T}+d_{1}\textbf{1}^{T}+\textbf{1}\textbf{1}^{T}&D^{**}&d_{n}^{T}\\\
1&d_{1}+\textbf{1}&d_{n}&0\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}0&d_{1}&d_{n}&1\\\
d_{1}^{T}&D^{*}-\textbf{1}d_{1}^{T}-d_{1}\textbf{1}^{T}&0&d_{1}^{T}\\\
d_{n}^{T}&0&D^{**}-\textbf{1}d_{n}^{T}-d_{n}\textbf{1}^{T}&d_{n}^{T}\\\
1&d_{1}&d_{n}&0\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}0&d_{1}&d_{n}&1\\\
d_{1}^{T}&D^{*}&d_{n}\textbf{1}^{T}+\textbf{1}d_{1}^{T}&d_{1}^{T}+\textbf{1}^{T}\\\
d_{n}^{T}&\textbf{1}d_{n}^{T}+d_{1}\textbf{1}^{T}&D^{**}&d_{n}^{T}+\textbf{1}^{T}\\\
1&d_{1}+\textbf{1}&d_{n}+\textbf{1}&0\end{array}\right|=det\tilde{D}.$
Then the result follows. $\blacksquare$
## 3 On the determinant of the distance matrix of a bicyclic graph whose two
cycles are disjoint
For a bicyclic graph $G$, if its two cycles are disjoint, then $G$ contains
$\infty(p,k,q)$ as an induced subgraph for some integers $p$, $q$ and $k$.
This subgraph $\infty(p,k,q)$ is sometimes called the center construct of $G$.
In this way, the graph $G$ can be viewed as the graph obtained from
$\infty(p,k,q)$ by planting some trees on it. In the following discussion, the
graph $\infty(p,1,q)$ will play an important role. For convenience, the
vertex, in $\infty(p,1,q)$, with degree $4$ is called the center vertex of
$\infty(p,1,q)$, and denote by $G(p,q;n)$ the graph obtained from
$\infty(p,1,q)$ by planting the path $P_{n}$ on its center vertex. Then
$G(p,q;0)$ denotes $\infty(p,1,q)$ itself and the graph $G(p,q;n)$ has order
$n+p+q-1$.
First of all, combining with Lemmas 2.3 and 2.4, we have the following result,
which tell us that if a graph is obtained from an induced subgraph by planting
some trees on it, then the determinant of the distance matrix of such a graph
is independent to the structure of those trees.
###### Theorem 3.1
Let $G$ be a bicyclic graph of order $n+p+q-1$ which contains $\infty(p,k,q)$
as an induced subgraph for some integers $p$, $q$ and $k$. Suppose that $D$
and $\tilde{D}$ be respectively the distance matrices of $G$ and $G(p,q;n)$.
Then
$detD=det\tilde{D}.$
Proof. First, applying Lemma 2.4 repeatedly, the distance matrices
corresponding respectively to the graphs $\infty(p,k,q)$ and $G(p,q;k-1)$ have
the same determinant.
Then it remain to show that the bicyclic graph, denoted still by $G$, of order
$n+p+q-1$ which contains $\infty(p,1,q)$ as an induced subgraph has the same
determinant as that of the graph $G(p,q;n)$. We label the vertices of $G$ as
$\\{1,2,\cdots,p+q+n-1\\}$ such that the resultant graph obtained from $G$ by
deleting the vertices $\\{n,n-1,\cdots,n-i\\}$ with $i(0\leq i\leq n)$ is
connected. We first consider the vertex $n$, if $n$ is not adjacent to the
center vertex of $G$, then applying Lemma 2.3 to $G$ such that the vertex $n$
adjacent to the center vertex of $G$, the resultant graph is still denoted by
$G$; then applying Lemma 2.3 to $G$ such that the vertex $n-1$ adjacent to the
vertex $n$, the resultant graph is still denoted by $G$; applying Lemma 2.3 to
$G$ such that the vertex $n-2$ adjacent to the vertex $n-1$, and so on. The
graph $G(p,q;n)$ can be obtained. Applying Lemma 2.3 again, each step above,
the origin graph and its resultant graph have the same determinant, the result
thus follows. $\blacksquare$
From Theorem 3.1, to compute the determinant of the distance matrix of a
bicyclic graph of order $n+p+q-1$ which contains $\infty(p,k,q)$ as an induced
subgraph, it is sufficient to compute the determinant of the distance matrix
of the graph $G(p,q;n)$. For convenience, in the following denote by $D_{n}$
the distance matrix of the graph $G(p,q;n)$.
###### Theorem 3.2
Fixed the integers $p$ and $q$. If $n\geq 2$, then
$detD_{n}=-4detD_{n-1}-4detD_{n-2}.$
Proof. Let the vertex set of $G(p,q;n)$ be $\\{1,2,\cdots,p+q+n-1\\}$. Then
$G(p,q;n-1)$ can be considered as the induced subgraph of $G(p,q;n)$ by
deleting the pendent vertex $p+q+n-1$ and $G(p,q;n-2)$ can be considered as
the induced subgraph of $G(p,q;n)$ by deleting the pendent vertex $p+q+n-1$
together with its neighbor $p+q+n-2$. Hence, $D_{n}$ can be partitioned as
$D_{n}=\left(\begin{array}[]{cccc}D_{n-3}&d^{T}&d^{T}+\textbf{1}^{T}&d^{T}+2\textbf{1}^{T}\\\
d&0&1&2\\\ d+\textbf{1}&1&0&1\\\ d+2\textbf{1}&2&1&0\end{array}\right),$
where $(d~{}~{}0)$ denotes the row vector of $D_{n-1}$ corresponding to the
vertex $p+q+n-3$. Hence,
$\displaystyle detD_{n}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}D_{n-3}&d^{T}&d^{T}+\textbf{1}^{T}&d^{T}+2\textbf{1}^{T}\\\
d&0&1&2\\\ d+\textbf{1}&1&0&1\\\ d+2\textbf{1}&2&1&0\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}D_{n-3}&d^{T}&\textbf{1}^{T}&\textbf{1}^{T}\\\
d&0&1&1\\\ d+\textbf{1}&1&-1&1\\\
d+2\textbf{1}&2&-1&-1\end{array}\right|=\left|\begin{array}[]{cccc}D_{n-3}&d^{T}&\textbf{1}^{T}&\textbf{1}^{T}\\\
d&0&1&1\\\ \textbf{1}&1&-2&0\\\ \textbf{1}&1&0&-2\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}D_{n-3}&d^{T}&\textbf{1}^{T}&0\\\
d&0&1&0\\\ \textbf{1}&1&-2&2\\\ 0&0&2&-4\end{array}\right|$ $\displaystyle=$
$\displaystyle-4\left|\begin{array}[]{ccc}D_{n-3}&d^{T}&\textbf{1}^{T}\\\
d&0&1\\\
\textbf{1}&1&-2\end{array}\right|-4\left|\begin{array}[]{cc}D_{n-3}&d^{T}\\\
d&0\end{array}\right|$ $\displaystyle=$
$\displaystyle-4\left|\begin{array}[]{ccc}D_{n-3}&d^{T}&d^{T}+\textbf{1}^{T}\\\
d&0&1\\\
d+\textbf{1}&1&0\end{array}\right|-4\left|\begin{array}[]{cc}D_{n-3}&d^{T}\\\
d&0\end{array}\right|$ $\displaystyle=$
$\displaystyle-4detD_{n-1}-4detD_{n-2}.$
The result follows. $\blacksquare$
###### Theorem 3.3
Fixed the integers $p$ and $q$. Then
$detD_{0}=detD_{1}=0$
if one of the integers $p$ and $q$ is even; and
$\displaystyle detD_{0}$ $\displaystyle=$ $\displaystyle\frac{(pq-1)(p+q)}{4}$
$\displaystyle detD_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(p+q)(pq-1)-pq$
otherwise.
Proof. Without loss of generality, suppose that, in $G(p,q;0)$,
$\\{1,2,\cdots,p\\}$ and $\\{1,p+1,\cdots,p+q-1\\}$ are respectively the
natural sequences of the vertex sets of the cycles $C_{p}$ and $C_{q}$, and in
$G(p,q;1)$ the unique pendent vertex is labeled as $p+q$. Then $D_{1}$ has the
form as (2.1) and $D_{0}$ is the submatrix of (2.1) by deleting the last row
and the last column, where $D^{*}$ and $D^{**}$ are respectively defined as
Lemma 2.4. Hence from the proof of Lemma 2.4, we have
$\displaystyle detD_{1}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}0&d_{1}&d_{p+q-1}&1\\\
d_{1}^{T}&D^{*}&d_{p+q-1}\textbf{1}^{T}+\textbf{1}d_{1}^{T}&d_{1}^{T}+\textbf{1}^{T}\\\
d_{p+q-1}^{T}&\textbf{1}d_{p+q-1}^{T}+d_{1}\textbf{1}^{T}&D^{**}&d_{p+q-1}^{T}+\textbf{1}^{T}\\\
1&d_{1}+\textbf{1}&d_{p+q-1}+\textbf{1}&0\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}0&d_{1}&d_{p+q-1}&1\\\
d_{1}^{T}&D^{*}-\textbf{1}d_{1}^{T}-d_{1}\textbf{1}^{T}&0&d_{1}^{T}\\\
d_{p+q-1}^{T}&0&D^{**}-\textbf{1}d_{p+q-1}^{T}-d_{n}\textbf{1}^{T}&d_{p+q-1}^{T}\\\
1&d_{1}&d_{p+q-1}&0\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}0&d_{1}&d_{p+q-1}&1\\\
d_{1}^{T}&D^{*}-\textbf{1}d_{1}^{T}-d_{1}\textbf{1}^{T}&0&0\\\
d_{p+q-1}^{T}&0&D^{**}-\textbf{1}d_{p+q-1}^{T}-d_{n}\textbf{1}^{T}&0\\\
1&0&0&-2\end{array}\right|,$
and
$\displaystyle detD_{0}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccc}0&d_{1}&d_{p+q-1}\\\
d_{1}^{T}&D^{*}-\textbf{1}d_{1}^{T}-d_{1}\textbf{1}^{T}&0\\\
d_{p+q-1}^{T}&0&D^{**}-\textbf{1}d_{p+q-1}^{T}-d_{p+q-1}\textbf{1}^{T}\end{array}\right|.$
We first discuss the matrix $D^{*}-d_{1}\textbf{1}^{T}-\textbf{1}d_{1}^{T}$
and denote it by $D^{p}=(d_{i,j})$ for simplify. Recall that we set
$\\{1,2,\cdots,p\\}$ is the natural sequences of the vertex sets of $C_{p}$.
Then for $D^{*}=(d^{*}_{i,j})$ we have $d_{ij}^{*}=min\\{p-|i-j|,|i-j|\\}$,
and
$d_{1}=(1,2,\cdots,k-1,k,k-1,\cdots,2,1)$
if $p=2k$;
$d_{1}=(1,2,\cdots,k-1,k,k,k-1,\cdots,2,1)$
if $p=2k+1$. Hence, for $D^{p}=(d^{p}_{i,j})$, we have
$d^{p}_{i,j}=min\\{p-|i-j|,|i-j|\\}-min\\{p-i+1,i-1\\}-min\\{p-j+1,j-1\\}.$
$None$
Thus, if $p=2k$,
$D^{p}=\left(\begin{array}[]{cccccc|c|cccccc}-2&-2&-2&\cdots&-2&-2&-2&0&0&0&\cdots&0&0\\\
-2&-4&-4&\cdots&-4&-4&-4&-2&0&0&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
-2&-4&-6&\cdots&4-2k&4-2k&4-2k&4-2k&6-2k&8-2k&\cdots&0&0\\\
-2&-4&-6&\cdots&4-2k&2-2k&2-2k&2-2k&4-2k&6-2k&\cdots&-2&0\\\
\hline\cr-2&-4&-6&\cdots&4-2k&2-2k&-2k&2-2k&4-2k&6-2k&\cdots&-4&-2\\\
\hline\cr 0&-2&-4&\cdots&4-2k&2-2k&2-2k&2-2k&4-2k&6-2k&\cdots&-4&-2\\\
0&0&-2&\cdots&4-2k&4-2k&4-2k&4-2k&6-2k&8-2k&\cdots&-4&-2\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&0&0&\cdots&0&-2&-4&-4&-4&-4&\cdots&-4&-2\\\
0&0&0&\cdots&0&0&-2&-2&-2&-2&\cdots&-2&-2\end{array}\right),$
and if $p=2k+1$,
$D^{p}=\left(\begin{array}[]{ccccc|ccccc}-2&-2&-2&\cdots&-2&-1&0&0&\cdots&0\\\
-2&-4&-4&\cdots&-4&-3&-1&0&\cdots&0\\\ -2&-4&-6&\cdots&-6&-5&-3&-1&\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
-2&-4&-6&\cdots&-2k&1-2k&3-2k&5-2k&\cdots&-1\\\
\hline\cr-1&\cdots&-3&-5&1-2k&-2k&\cdots&-6&-4&-2\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&\cdots&-1&-3&-5&-6&\cdots&-6&-4&-2\\\ 0&\cdots&0&-1&-3&-4&\cdots&-4&-4&-2\\\
0&\cdots&0&0&-1&-2&\cdots&-2&-2&-2\end{array}\right).$
For $p=2k$, we have
$\displaystyle\left|\begin{array}[]{cc}0&d_{1}\\\
d_{1}^{T}&D^{p}\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{c|cccccc|c|cccccc}0&1&2&3&\cdots&k-2&k-1&k&k-1&k-2&k-3&\cdots&2&1\\\
\hline\cr 1&-2&-2&-2&\cdots&-2&-2&-2&0&0&0&\cdots&0&0\\\
2&-2&-4&-4&\cdots&-4&-4&-4&-2&0&0&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
k-2&-2&-4&-6&\cdots&4-2k&4-2k&4-2k&4-2k&6-2k&8-2k&\cdots&0&0\\\
k-1&-2&-4&-6&\cdots&4-2k&2-2k&2-2k&2-2k&4-2k&6-2k&\cdots&-2&0\\\ \hline\cr
k&-2&-4&-6&\cdots&4-2k&2-2k&-2k&2-2k&4-2k&6-2k&\cdots&-4&-2\\\ \hline\cr
k-1&0&-2&-4&\cdots&4-2k&2-2k&2-2k&2-2k&4-2k&6-2k&\cdots&-4&-2\\\
k-2&0&0&-2&\cdots&4-2k&4-2k&4-2k&4-2k&6-2k&8-2k&\cdots&-4&-2\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
2&0&0&0&\cdots&0&-2&-4&-4&-4&-4&\cdots&-4&-2\\\
1&0&0&0&\cdots&0&0&-2&-2&-2&-2&\cdots&-2&-2\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{c|cccccc|c|cccccc}0&1&2&3&\cdots&k-2&k-1&k&k-1&k-2&k-3&\cdots&2&1\\\
\hline\cr 1&-2&-2&-2&\cdots&-2&-2&-2&0&0&0&\cdots&0&0\\\
1&0&-2&-2&\cdots&-2&-2&-2&-2&0&0&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&0&0&\cdots&-2&-2&-2&-2&-2&-2&\cdots&0&0\\\
1&0&0&0&\cdots&0&-2&-2&-2&-2&-2&\cdots&-2&0\\\ \hline\cr
1&0&0&0&\cdots&0&0&-2&-2&-2&-2&\cdots&-2&-2\\\ \hline\cr
1&0&-2&-2&\cdots&-2&-2&-2&-2&0&0&\cdots&0&0\\\
1&0&0&-2&\cdots&-2&-2&-2&-2&-2&0&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&0&0&\cdots&0&-2&-2&-2&-2&-2&\cdots&-2&0\\\
1&0&0&0&\cdots&0&0&-2&-2&-2&-2&\cdots&-2&-2\end{array}\right|$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{cccccc|c|cccccc|c}0&1&1&1&\cdots&1&1&1&1&1&1&\cdots&1&1\\\
\hline\cr 1&-2&0&0&\cdots&0&0&0&0&0&0&\cdots&0&0\\\
1&0&-2&0&\cdots&0&0&0&-2&0&0&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&0&0&\cdots&-2&0&0&0&0&-2&\cdots&0&0\\\
1&0&0&0&\cdots&0&-2&0&0&0&0&\cdots&-2&0\\\ \hline\cr
1&0&0&0&\cdots&0&0&-2&0&0&0&\cdots&0&-2\\\ \hline\cr
1&0&0&-2&\cdots&0&0&0&0&-2&0&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&0&0&\cdots&-2&0&0&0&0&-2&\cdots&0&0\\\
1&0&0&0&\cdots&0&-2&0&0&0&0&\cdots&-2&0\\\
1&0&0&0&\cdots&0&0&-2&0&0&0&\cdots&0&-2\end{array}\right|.$
Note that all operation above disconcern the first row and the first column,
thus $det\tilde{D}=0$, as well as $detD=0$, if one of the integers $p$ and $q$
is even.
We now consider the case that both of $p$ and $q$ are odd. For $p=2k+1$, we
have
$\displaystyle\left|\begin{array}[]{cc}0&d_{1}\\\
d_{1}^{T}&D^{p}\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{c|ccccc|ccccc}0&1&2&\cdots&k-1&k&k&k-1&\cdots&2&1\\\
\hline\cr 1&-2&-2&-2&\cdots&-2&-1&0&0&\cdots&0\\\
2&-2&-4&-4&\cdots&-4&-3&-1&0&\cdots&0\\\
3&-2&-4&-6&\cdots&-6&-5&-3&-1&\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
k&-2&-4&-6&\cdots&-2k&1-2k&3-2k&5-2k&\cdots&-1\\\ \hline\cr
k&-1&\cdots&-3&-5&1-2k&-2k&\cdots&-6&-4&-2\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
3&0&\cdots&-1&-3&-5&-6&\cdots&-6&-4&-2\\\
2&0&\cdots&0&-1&-3&-4&\cdots&-4&-4&-2\\\
1&0&\cdots&0&0&-1&-2&\cdots&-2&-2&-2\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{c|ccccc|ccccc}0&1&2&\cdots&k-1&k&k&k-1&\cdots&2&1\\\
\hline\cr 1&-2&-2&-2&\cdots&-2&-1&0&0&\cdots&0\\\
1&0&-2&-2&\cdots&-2&-2&-1&0&\cdots&0\\\
1&0&0&-2&\cdots&-2&-2&-2&-1&\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&0&0&\cdots&-2&-2&-2&-2&\cdots&-1\\\ \hline\cr
1&-1&\cdots&-2&-2&-2&-2&\cdots&0&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&\cdots&-1&-2&-2&-2&\cdots&-2&0&0\\\
1&0&\cdots&0&-1&-2&-2&\cdots&-2&-2&0\\\
1&0&\cdots&0&0&-1&-2&\cdots&-2&-2&-2\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{c|cccccc|cccccc}0&1&1&\cdots&1&1&1&1&1&\cdots&1&1&1\\\
\hline\cr 1&-2&0&0&\cdots&0&0&-1&0&0&\cdots&0&0\\\
1&0&-2&0&\cdots&0&0&-1&-1&0&\cdots&0&0\\\
1&0&0&-2&\cdots&0&0&0&-1&-1&\cdots&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&0&0&\cdots&-2&0&0&0&0&\cdots&-1&0\\\
1&0&0&0&\cdots&0&-2&0&0&0&\cdots&-1&-1\\\ \hline\cr
1&-1&-1&\cdots&0&0&0&-2&0&\cdots&0&0&0\\\
1&0&-1&\cdots&0&0&0&0&-2&\cdots&0&0&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
1&0&0&\cdots&-1&-1&0&0&0&\cdots&-2&0&0\\\
1&0&0&\cdots&0&-1&-1&0&0&\cdots&0&-2&0\\\
1&0&0&\cdots&0&0&-1&0&0&\cdots&0&0&-2\end{array}\right|$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}0&\textbf{1}&\textbf{1}\\\
\textbf{1}^{T}&-2I_{k}&B_{k}\\\
\textbf{1}^{T}&B_{k}^{T}&-2I_{k}\end{array}\right|,$
where $B_{k}$ is defined as Lemma 2.1. Similarly, let $q=2h+1$. Then
$\left(\begin{array}[]{cc}0&d_{p+q-1}\\\
d_{p+q-1}^{T}&D^{**}-d_{p+q-1}\textbf{1}^{T}-\textbf{1}d_{p+q-1}^{T}\end{array}\right)=\left(\begin{array}[]{ccc}0&\textbf{1}&\textbf{1}\\\
\textbf{1}^{T}&-2I_{h}&B_{h}\\\
\textbf{1}^{T}&B_{h}^{T}&-2I_{h}\end{array}\right).$
Hence,
$detD_{0}=\left|\begin{array}[]{cccccc}0&\textbf{1}&\textbf{1}&\textbf{1}&\textbf{1}\\\
\textbf{1}^{T}&-2I_{k}&B_{k}&0&0\\\ \textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0\\\
\textbf{1}^{T}&0&0&-2I_{h}&B_{h}\\\
\textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}\end{array}\right|$
and
$detD_{1}=\left|\begin{array}[]{cccccc}0&\textbf{1}&\textbf{1}&\textbf{1}&\textbf{1}&1\\\
\textbf{1}^{T}&-2I_{k}&B_{k}&0&0&0\\\
\textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0&0\\\
\textbf{1}^{T}&0&0&-2I_{h}&B_{h}&0\\\
\textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}&0\\\ 1&0&0&0&0&-2\\\ \end{array}\right|$
Let $C_{k}=\frac{1}{2}B_{k}B_{k}^{T}-2I_{k}$,
$F_{k}=\frac{1}{2}\textbf{1}B_{k}^{T}+\textbf{1}$. Because
$\displaystyle\left(\begin{array}[]{cccccc}1&0&\frac{1}{2}\textbf{1}&0&\frac{1}{2}\textbf{1}&\frac{1}{2}\\\
0&I_{k}&\frac{1}{2}B_{k}&0&0&0\\\ 0&0&I_{k}&0&0&0\\\
0&0&0&I_{h}&\frac{1}{2}B_{h}&0\\\ 0&0&0&0&I_{h}&0\\\
0&0&0&0&0&1\end{array}\right)\left(\begin{array}[]{cccccc}0&\textbf{1}&\textbf{1}&\textbf{1}&\textbf{1}&1\\\
\textbf{1}^{T}&-2I_{k}&B_{k}&0&0&0\\\
\textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0&0\\\
\textbf{1}^{T}&0&0&-2I_{h}&B_{h}&0\\\
\textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}&0\\\ 1&0&0&0&0&-2\end{array}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}\frac{1+k+h}{2}&F_{k}&0&F_{h}&0&0\\\
F_{k}^{T}&C_{k}&0&0&0&0\\\ \textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0&0\\\
F_{h}^{T}&0&0&C_{h}&0&0\\\ \textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}&0\\\
1&0&0&0&0&-2\end{array}\right),$
$\displaystyle\left(\begin{array}[]{ccccc}1&0&\frac{1}{2}\textbf{1}&0&\frac{1}{2}\textbf{1}\\\
0&I_{k}&\frac{1}{2}B_{k}&0&0\\\ 0&0&I_{k}&0&0\\\
0&0&0&I_{h}&\frac{1}{2}B_{h}\\\
0&0&0&0&I_{h}\end{array}\right)\left(\begin{array}[]{ccccc}0&\textbf{1}&\textbf{1}&\textbf{1}&\textbf{1}\\\
\textbf{1}^{T}&-2I_{k}&B_{k}&0&0\\\ \textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0\\\
\textbf{1}^{T}&0&0&-2I_{h}&B_{h}\\\
\textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}\frac{k+h}{2}&F_{k}&0&F_{h}&0\\\
F_{k}^{T}&C_{k}&0&0&0\\\ \textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0\\\
F_{h}^{T}&0&0&C_{h}&0\\\
\textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}\end{array}\right)$
and
$\displaystyle\left(\begin{array}[]{cccccc}1&-F_{k}C_{k}^{-1}&0&-F_{h}C_{h}^{-1}&0&0\\\
0&I_{k}&0&0&0&0\\\ 0&0&I_{k}&0&0&0\\\ 0&0&0&I_{h}&0&0\\\ 0&0&0&0&I_{h}&0\\\
0&0&0&0&0&1\end{array}\right)\left(\begin{array}[]{cccccc}\frac{1+k+h}{2}&F_{k}&0&F_{h}&0&0\\\
F_{k}^{T}&C_{k}&0&0&0&0\\\ \textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0&0\\\
F_{h}^{T}&0&0&C_{h}&0&0\\\ \textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}&0\\\
1&0&0&0&0&-2\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccccc}\frac{1+k+h}{2}-F_{k}C_{k}^{-1}F_{k}^{T}-F_{h}C_{h}^{-1}F_{h}^{T}&0&0&0&0&0\\\
F_{k}^{T}&C_{k}&0&0&0&0\\\ \textbf{1}^{T}&B^{T}_{k}&-2I_{k}&0&0&0\\\
F_{h}^{T}&0&0&C_{h}&0&0\\\ \textbf{1}^{T}&0&0&B^{T}_{h}&-2I_{h}&0\\\
1&0&0&0&0&-2\end{array}\right),$
we have
$detD_{0}=(-2)^{k+h}(\frac{1}{2}k+\frac{1}{2}h-F_{k}C_{k}^{-1}F_{k}^{T}-F_{h}C_{h}^{-1}F_{h}^{T})|C_{k}||C_{h}|$
and
$detD_{1}=-2(-2)^{k+h}(\frac{k+h+1}{2}-F_{k}C_{k}^{-1}F_{k}^{T}-F_{h}C_{h}^{-1}F_{h}^{T})|C_{k}||C_{h}|.$
From Lemma 2.1, we have $detC_{k}=\frac{(2k+1)(-1)^{k}}{2^{k}}$ and
$F_{k}C_{k}^{-1}F_{k}^{T}=-\frac{k}{2(2k+1)}.$ Recall that $p=2k+1$ and
$q=2h+1$, hence
$\displaystyle detD_{0}$ $\displaystyle=$
$\displaystyle\frac{(2k+1)(2h+1)}{2}[k+\frac{k}{2k+1}+h+\frac{h}{2h+1}]$
$\displaystyle=$ $\displaystyle k(k+1)(2h+1)+h(h+1)(2k+1)$ $\displaystyle=$
$\displaystyle\frac{(pq-1)(p+q)}{4}$
and
$\displaystyle detD_{1}$ $\displaystyle=$
$\displaystyle-2\frac{(2k+1)(2h+1)}{2}[k+\frac{k}{2k+1}+h+1+\frac{h}{2h+1}]$
$\displaystyle=$
$\displaystyle-(2k+1)(2h+1)(k+h+1+\frac{k}{2k+1}+\frac{h}{2h+1})$
$\displaystyle=$ $\displaystyle-(2k+1)(2h+1)(k+h+1)-k(2h+1)-h(2k+1)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}(p+q)(pq-1)-pq.$
The result thus follows.
Putting Theorem 3.3 into Lemma 2.2, we have the main result of this paper as
follows.
###### Theorem 3.4
Let $G$ be an arbitrary bicyclic graph of order $p+q-1+n$ which contains
$\infty(p,k,q)$ as an induced subgraph with $n\geq k-1$. Denote by $D$ the
distance matrix of $G$. Then $detD=0$ if one of integers $p$ and $q$ is even,
and
$detD=[\frac{(pq-1)(p+q)}{4}+\frac{n}{2}pq](-2)^{n}$ $None$
otherwise.
Remark. Theorem 3.4 can be considered as a generalization for Graham and
Pollacks’ result on the determinant of trees [5] and Bapat, Kirkland and
Neumanns’ result on the determinant of a unicyclic graph [1]. We view one
vertex as a cycle with length $1$, then each vertex can be viewed as an
$\infty$-graph $\infty(1,1,1)$ and thus each tree contains $\infty(1,1,1)$ as
its induced subgraph. Consequently, the distance matrix of each tree of order
$n$ has the same determinant of that of the graph $G(1,1;n-1)$, then from
(3.2)
$detD=detD(G(1,1;n-1))=\frac{n-1}{2}(-2)^{n-1}=-(n-1)(-2)^{n-2}$
which is coincide with the formula for the determinant of a tree. Let $G$ be a
unicyclic graph of order $n+p$ whose unique cycle has length $p$, then such a
unique cycle can be viewed as an $\infty$-graph $\infty(p,1,1)$. Thus the
distance matrix of such a graph has the same determinant of that of the graph
$G(p,1;n)$ by Theorem 3.4. Hence, $detD=detD(G)=0$ if $p$ is even, and
$detD=detD(G)=[\frac{(p-1)(p+1)}{4}+\frac{n}{2}p](-2)^{n}$
if $p$ is odd from (3.2), which is coincide with Theorems 3.4 and 3.7 of [1].
## References
* [1] R. Bapat, S.J. Kirkland, M. Neumann, On distance matrices and Laplacians, Linear Algebra and its Applications 401(2005) 193-209.
* [2] R. A. Brualdi, Introductory combinatorics, Elsevier Science Publishers, New York.
* [3] D. Cvetkovi$\acute{c}$, M. Doob, H. Sachs, Spectra of Graphs, Academic Press, New York, 1980\.
* [4] S. B. Hu, X. Z. Tan, B. L. Liu, On the nullity of bicyclic graphs, Linear Algebra Appl. 429 (7) (2008) 1387-1391.
* [5] R.L. Graham, H.O. Pollack, On the addressing problem for loop switching, Bell System Tech. J. 50(1971) 2495-2519.
* [6] H. Lin, W. Yang, H. Zhang, J. Shu, Distance spectral radius of digraphs with given connectivity, Discrete Math. 312 (2012) 1849- 1856\.
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|
arxiv-papers
| 2013-08-10T06:42:15 |
2024-09-04T02:49:49.276726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shi-Cai Gong, Ju-Li Zhang and Guang-Hui Xu",
"submitter": "Shicai Gong Mr",
"url": "https://arxiv.org/abs/1308.2281"
}
|
1308.2297
|
# ON THE REGULARITY OF
3D NAVIER-STOKES EQUATION
Qun Lin 11footnotemark: 1 School of Mathematical Sciences, Xiamen University,
P. R. China
(July 12, 2011 )
Abstract. In this paper we will prove that the vorticity belongs to
$L^{\infty}(0,T;L^{2}(\Omega))$ for 3D incompressible Navier-Stokes equation
with periodic initial-boundary value conditions, then the existence of a
global smooth solution is obtained. Our approach is to construct a set of
auxiliary problems to approximate the original one for vorticity equation.
Keywords. Navier-Stokes equation; Regularity; Vorticity.
AMS subject classifications. 35Q30 76N10
1\. Introduction
Let $\Omega=(0,1)^{3}$, and $\mathscr{D}(\Omega)$ be the space of $C^{\infty}$
functions with compact support contained in $\Omega$. Some basic spaces will
be used in this paper:
$\begin{split}&\mathscr{V}=\\{u\in\mathscr{D}(\Omega),\;\,\mbox{div}\,u=0\\}\\\
&V=\mbox{the}\;\mbox{closure}\;\mbox{of}\;\mathscr{V}\;\mbox{in}\;H^{1}(\Omega)\\\
&H=\mbox{the}\;\mbox{closure}\;\mbox{of}\;\mathscr{V}\;\mbox{in}\;L^{2}(\Omega)\\\
\end{split}$
The velocity-pressure form for Navier- Stokes equation is
$\begin{split}&\partial_{t}u_{1}+u_{1}\partial_{x_{1}}u_{1}+u_{2}\partial_{x_{2}}u_{1}+u_{3}\partial_{x_{3}}u_{1}+\partial_{x_{1}}p=\Delta
u_{1}\\\
&\partial_{t}u_{2}+u_{1}\partial_{x_{1}}u_{2}+u_{2}\partial_{x_{2}}u_{2}+u_{3}\partial_{x_{3}}u_{2}+\partial_{x_{2}}p=\Delta
u_{2}\\\
&\partial_{t}u_{3}+u_{1}\partial_{x_{1}}u_{3}+u_{2}\partial_{x_{2}}u_{3}+u_{3}\partial_{x_{3}}u_{3}+\partial_{x_{3}}p=\Delta
u_{3}\\\ \end{split}$ (1)
with periodic boundary conditions and the incompressible condition :
$\partial_{x_{1}}u_{1}+\partial_{x_{2}}u_{2}+\partial_{x_{3}}u_{3}=0$
We will here recall the global $L^{2}$-estimate from [4]. Since
$\begin{split}&\int_{\Omega}{u_{i}(u_{1}\partial_{x_{1}}u_{i}+u_{2}\partial_{x_{2}}u_{i}+u_{3}\partial_{x_{3}}u_{i})}=\frac{1}{2}\int_{\Omega}{(u_{1}\partial_{x_{1}}u_{i}^{2}+u_{2}\partial_{x_{2}}u_{i}^{2}+u_{3}\partial_{x_{3}}u_{i}^{2})}\\\
&\quad=-\frac{1}{2}\int_{\Omega}{u_{i}^{2}(\partial_{x_{1}}u_{1}+\partial_{x_{2}}u_{2}+\partial_{x_{3}}u_{3})}=0\qquad\qquad
i=1,2,3\\\ \end{split}$
$\int_{\Omega}{(u_{1}\partial_{x_{1}}p+u_{2}\partial_{x_{2}}p+u_{3}\partial_{x_{3}}p)}=-\int_{\Omega}{p\,(\partial_{x_{1}}u_{1}+\partial_{x_{2}}u_{2}+\partial_{x_{3}}u_{3})}=0\qquad\quad$
and
$\int_{\Omega}{u_{i}\Delta
u_{i}}=\int_{\Omega}{u_{i}(\partial_{x_{1}}^{2}u_{i}+\partial_{x_{2}}^{2}u_{i}+\partial_{x_{3}}^{2}u_{i})}=-\int_{\Omega}{((\partial_{x_{1}}u_{i})^{2}+(\partial_{x_{2}}u_{i})^{2}+(\partial_{x_{3}}u_{i})^{2})}$
then
$\begin{split}&\int_{\Omega}{u_{1}\partial_{t}\,u_{1}}+\int_{\Omega}{u_{1}(u_{1}\partial_{x_{1}}u_{1}+u_{2}\partial_{x_{2}}u_{1}+u_{3}\partial_{x_{3}}u_{1})}+\int_{\Omega}{u_{1}\partial_{x_{1}}p}=\int_{\Omega}{u_{1}\Delta
u_{1}}\\\
&\int_{\Omega}{u_{2}\partial_{t}\,u_{2}}+\int_{\Omega}{u_{2}(u_{1}\partial_{x_{1}}u_{2}+u_{2}\partial_{x_{2}}u_{2}+u_{3}\partial_{x_{3}}u_{2})}+\int_{\Omega}{u_{2}\partial_{x_{2}}p}=\int_{\Omega}{u_{2}\Delta
u_{2}}\\\
&\int_{\Omega}{u_{3}\partial_{t}\,u_{3}}+\int_{\Omega}{u_{3}(u_{1}\partial_{x_{1}}u_{3}+u_{2}\partial_{x_{2}}u_{3}+u_{3}\partial_{x_{3}}u_{3})}+\int_{\Omega}{u_{3}\partial_{x_{3}}p}=\int_{\Omega}{u_{3}\Delta
u_{3}}\\\ \end{split}$
so that
$\begin{split}&\frac{1}{2}\partial_{t}\;\int_{\Omega}{(u_{1}^{2}+u_{2}^{2}+u_{3}^{2})}+\;\int_{\Omega}{((\partial_{x_{1}}u_{1})^{2}+(\partial_{x_{2}}u_{1})^{2}+(\partial_{x_{3}}u_{1})^{2}+}\\\
&+(\partial_{x_{1}}u_{2})^{2}+(\partial_{x_{2}}u_{2})^{2}+(\partial_{x_{3}}u_{2})^{2}+(\partial_{x_{1}}u_{3})^{2}+(\partial_{x_{2}}u_{3})^{2}+(\partial_{x_{3}}u_{3})^{2})=0\\\
\end{split}$
it follows that
$\int_{\Omega}{(u_{1}^{2}+u_{2}^{2}+u_{3}^{2})}+2\;\int_{0}^{T}{(\,\left\|{\nabla
u_{1}}\right\|_{L^{2}(\Omega)}^{2}+}\left\|{\nabla
u_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{3}}\right\|_{L^{2}(\Omega)}^{2})=\int_{\Omega}{(u_{10}^{2}+u_{20}^{2}+u_{30}^{2})}$
Hence we have
$\mathop{\sup}\limits_{t\in(0,T)}\;\;\int_{\Omega}{(u_{1}^{2}+u_{2}^{2}+u_{3}^{2})}<+\infty$
(2) $\,\int_{0}^{T}{(\,\left\|{\nabla
u_{1}}\right\|_{L^{2}(\Omega)}^{2}+}\left\|{\nabla
u_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{3}}\right\|_{L^{2}(\Omega)}^{2})<+\infty$ (3)
Above $u$ can be interpreted as the Galerkin approximation of the solution,
but (2) and (3) are also true for the solution of problem (1).
2\. Auxiliary problems
For the 3D regularity, we just need to prove that the vorticity belongs to
$L^{\infty}(0,T;L^{2}(\Omega))$. The vorticity-velocity form for Navier-Stokes
equation is
$\begin{split}&\partial_{t}\omega_{1}\,+u_{1}\partial_{x_{1}}\omega_{1}+u_{2}\partial_{x_{2}}\omega_{1}+u_{3}\partial_{x_{3}}\omega_{1}-\omega_{1}\partial_{x_{1}}u_{1}-\omega_{2}\partial_{x_{2}}u_{1}-\omega_{3}\partial_{x_{3}}u_{1}=\Delta\omega_{1}\\\
&\partial_{t}\omega_{2}+u_{1}\partial_{x_{1}}\omega_{2}+u_{2}\partial_{x_{2}}\omega_{2}+u_{3}\partial_{x_{3}}\omega_{2}-\omega_{1}\partial_{x_{1}}u_{2}-\omega_{2}\partial_{x_{2}}u_{2}-\omega_{3}\partial_{x_{3}}u_{2}=\Delta\omega_{2}\\\
&\partial_{t}\omega_{3}+u_{1}\partial_{x_{1}}\omega_{3}+u_{2}\partial_{x_{2}}\omega_{3}+u_{3}\partial_{x_{3}}\omega_{3}-\omega_{1}\partial_{x_{1}}u_{3}-\omega_{2}\partial_{x_{2}}u_{3}-\omega_{3}\partial_{x_{3}}u_{3}=\Delta\omega_{3}\\\
\end{split}$ (4)
with incompressible condition :
$\begin{split}&\partial_{x_{1}}\omega_{1}+\partial_{x_{2}}\omega_{2}+\partial_{x_{3}}\omega_{3}=0\\\
&\partial_{x_{1}}u_{1}+\partial_{x_{2}}u_{2}+\partial_{x_{3}}u_{3}=0\\\
\end{split}$
Given a partition with respect to $t$ as follows:
$0=t_{0}<t_{1}<t_{2}<\cdots<t_{k-1}<t_{k}<\cdots<t_{N}=T$
On each $t\in(t_{k-1},\;t_{k})$, we introduce an auxiliary problem:
$\begin{split}&\partial_{t}\tilde{\omega}_{1}\,+\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{1}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{1}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{1}^{k}-\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{1}^{k}-\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{1}^{k}-\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{1}^{k}+\partial_{x_{1}}q=\Delta\tilde{\omega}_{1}\\\
&\partial_{t}\tilde{\omega}_{2}+\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{2}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{2}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{2}^{k}-\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{2}^{k}-\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{2}^{k}-\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{2}^{k}+\partial_{x_{2}}q=\Delta\tilde{\omega}_{2}\\\
&\partial_{t}\tilde{\omega}_{3}+\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{3}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{3}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{3}^{k}-\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{3}^{k}-\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{3}^{k}-\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{3}^{k}+\partial_{x_{3}}q=\Delta\tilde{\omega}_{3}\\\
\end{split}$ (5)
where the initial value is assumed to be
$\tilde{\omega}_{i}(x,t_{k-1})=\tilde{\omega}_{i}^{k-1}$ and
$\bar{\omega}_{i}^{k}(x)=\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{\tilde{\omega}_{i}(x,t)dt}$
and
$\bar{u}_{i}^{k}(x)=\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}u_{i}(x,t)dt,\qquad\;\;i=1,2,3$
It is easy to check that
$\begin{split}&\partial_{x_{1}}\tilde{\omega}_{1}+\partial_{x_{2}}\tilde{\omega}_{2}+\partial_{x_{3}}\tilde{\omega}_{3}=0\quad\Rightarrow\quad\partial_{x_{1}}\bar{\omega}_{1}^{k}+\partial_{x_{2}}\bar{\omega}_{2}^{k}+\partial_{x_{3}}\bar{\omega}_{3}^{k}=0\\\
&\partial_{x_{1}}u_{1}+\partial_{x_{2}}u_{2}+\partial_{x_{3}}u_{3}=0\quad\,\Rightarrow\quad\partial_{x_{1}}\bar{u}_{1}^{k}+\partial_{x_{2}}\bar{u}_{2}^{k}+\partial_{x_{3}}\bar{u}_{3}^{k}=0\\\
\end{split}$
In the section 3, by means of the Galerkin method and the compactness
imbedding theorem, we can prove the local existences of the weak solutions of
these systems for each $(t_{k-1},\;t_{k})$ being small enough. Below we also
interpret $\tilde{\omega}$ as the Galerkin approximation of the solution of
the problem (5), and first prove that $\tilde{\omega},t\in(0,T)$, belong to
$L^{\infty}(0,T;L^{2}(\Omega))$. In section 4, an approach of approximation is
used to assert that the solution of (4) also belongs to
$L^{\infty}(0,T;L^{2}(\Omega))$ as ${k^{\prime}}\to\infty$, or $\Delta
t_{k}^{\prime}\to 0$. Since
$\begin{split}&\int_{\Omega}{(\tilde{\omega}_{1}(\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{1}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{1}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{1}^{k})}\\\
&\;\,\,+\tilde{\omega}_{2}(\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{2}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{2}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{2}^{k})\\\
&\;\,\,+\tilde{\omega}_{3}(\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{3}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{3}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{3}^{k}))\\\
&=-\int_{\Omega}{(\bar{\omega}_{1}^{k}(\partial_{x_{1}}(\tilde{\omega}_{1}\bar{u}_{1}^{k})+\bar{\omega}_{1}^{k}\partial_{x_{2}}(\tilde{\omega}_{1}\bar{u}_{2}^{k})+\bar{\omega}_{1}^{k}\partial_{x_{3}}(\tilde{\omega}_{1}\bar{u}_{3}^{k})}\\\
&\quad\quad\;\;\,+\bar{\omega}_{2}^{k}\partial_{x_{1}}(\tilde{\omega}_{2}\bar{u}_{1}^{k})+\bar{\omega}_{2}^{k}\partial_{x_{2}}(\tilde{\omega}_{2}\bar{u}_{2}^{k})+\bar{\omega}_{2}^{k}\partial_{x_{3}}(\tilde{\omega}_{2}\bar{u}_{3}^{k})\\\
&\quad\quad\;\;\,+\bar{\omega}_{3}^{k}\partial_{x_{1}}(\tilde{\omega}_{3}\bar{u}_{1}^{k})+\bar{\omega}_{3}^{k}\partial_{x_{2}}(\tilde{\omega}_{3}\bar{u}_{2}^{k})+\bar{\omega}_{3}^{k}\partial_{x_{3}}(\tilde{\omega}_{3}\bar{u}_{3}^{k}))\\\
&=-\int_{\Omega}{(\bar{\omega}_{1}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{1}+\bar{\omega}_{1}^{k}\tilde{\omega}_{1}\partial_{x_{1}}\bar{u}_{1}^{k}+\bar{\omega}_{1}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{1}+\bar{\omega}_{1}^{k}\tilde{\omega}_{1}\partial_{x_{2}}\bar{u}_{2}^{k}+\bar{\omega}_{1}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{1}}+\bar{\omega}_{1}^{k}\tilde{\omega}_{1}\partial_{x_{3}}\bar{u}_{3}^{k}\\\
&\quad\quad\;\;+\bar{\omega}_{2}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\tilde{\omega}_{2}\partial_{x_{1}}\bar{u}_{1}^{k}+\bar{\omega}_{2}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\tilde{\omega}_{2}\partial_{x_{2}}\bar{u}_{2}^{k}+\bar{\omega}_{2}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\tilde{\omega}_{2}\partial_{x_{3}}\bar{u}_{3}^{k}\\\
&\quad\quad\;\;+\bar{\omega}_{3}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\tilde{\omega}_{3}\partial_{x_{1}}\bar{u}_{1}^{k}+\bar{\omega}_{3}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\tilde{\omega}_{3}\partial_{x_{2}}\bar{u}_{2}^{k}+\bar{\omega}_{3}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\tilde{\omega}_{3}\partial_{x_{3}}\bar{u}_{3}^{k})\\\
\end{split}$
$\begin{split}&=-\int_{\Omega}{(\bar{\omega}_{1}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{1}+\bar{\omega}_{1}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{1}+\bar{\omega}_{1}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{1}}\\\
&\quad\quad\;\;\,+\bar{\omega}_{2}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{2}\\\
&\quad\quad\;\;\,+\bar{\omega}_{3}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{3})\\\
\end{split}$
and
$\begin{split}&\int_{\Omega}{(\tilde{\omega}_{1}(\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{1}^{k}+\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{1}^{k}+\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{1}^{k}})\\\
&\;\;+\tilde{\omega}_{2}(\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{2}^{k}+\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{2}^{k}+\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{2}^{k})\\\
&\;\;+\tilde{\omega}_{3}(\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{3}^{k}+\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{3}^{k}+\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{3}^{k}))\\\
\end{split}$
$\begin{split}&=-\int_{\Omega}{\,(\bar{u}_{1}^{k}\partial_{x_{1}}(\tilde{\omega}_{1}\bar{\omega}_{1}^{k})+\bar{u}_{1}^{k}\partial_{x_{2}}(\tilde{\omega}_{1}\bar{\omega}_{2}^{k})+\bar{u}_{1}^{k}\partial_{x_{3}}(\tilde{\omega}_{1}\bar{\omega}_{3}^{k})}\\\
&\quad\quad\;\;\;+\bar{u}_{2}^{k}\partial_{x_{1}}(\tilde{\omega}_{2}\bar{\omega}_{1}^{k})+\bar{u}_{2}^{k}\partial_{x_{2}}(\tilde{\omega}_{2}\bar{\omega}_{2}^{k})+\bar{u}_{2}^{k}\partial_{x_{3}}(\tilde{\omega}_{2}\bar{\omega}_{3}^{k})\\\
&\quad\quad\;\;\;+\bar{u}_{3}^{k}\partial_{x_{1}}(\tilde{\omega}_{3}\bar{\omega}_{1}^{k})+\bar{u}_{3}^{k}\partial_{x_{2}}(\tilde{\omega}_{3}\bar{\omega}_{2}^{k})+\bar{u}_{3}^{k}\partial_{x_{3}}(\tilde{\omega}_{3}\bar{\omega}_{3}^{k}))\\\
&=-\int_{\Omega}{(\bar{\omega}_{1}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{1}+\tilde{\omega}_{1}\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{1}^{k}+\bar{\omega}_{2}^{k}\bar{u}_{1}^{k}\partial_{x_{2}}\tilde{\omega}_{1}+\tilde{\omega}_{1}\bar{u}_{1}^{k}\partial_{x_{2}}\bar{\omega}_{2}^{k}+\bar{\omega}_{3}^{k}\bar{u}_{1}^{k}\partial_{x_{3}}\tilde{\omega}_{1}}+\tilde{\omega}_{1}\bar{u}_{1}^{k}\partial_{x_{3}}\bar{\omega}_{3}^{k}\\\
&\quad\quad\;\;\,+\bar{\omega}_{1}^{k}\bar{u}_{2}^{k}\partial_{x_{1}}\tilde{\omega}_{2}+\tilde{\omega}_{2}\bar{u}_{2}^{k}\partial_{x_{1}}\bar{\omega}_{1}^{k}+\bar{\omega}_{2}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{2}+\tilde{\omega}_{2}\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{2}^{k}+\bar{\omega}_{3}^{k}\bar{u}_{2}^{k}\partial_{x_{3}}\tilde{\omega}_{2}+\tilde{\omega}_{2}\bar{u}_{2}^{k}\partial_{x_{3}}\bar{\omega}_{3}^{k}\\\
&\quad\quad\;\;\,+\bar{\omega}_{1}^{k}\bar{u}_{3}^{k}\partial_{x_{1}}\tilde{\omega}_{3}+\tilde{\omega}_{3}\bar{u}_{3}^{k}\partial_{x_{1}}\bar{\omega}_{1}^{k}+\bar{\omega}_{2}^{k}\bar{u}_{3}^{k}\partial_{x_{2}}\tilde{\omega}_{3}+\tilde{\omega}_{3}\bar{u}_{3}^{k}\partial_{x_{2}}\bar{\omega}_{2}^{k}+\bar{\omega}_{3}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{3}+\tilde{\omega}_{3}\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{3}^{k})\\\
&=-\int_{\Omega}{(\bar{\omega}_{1}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{1}+\bar{\omega}_{2}^{k}\bar{u}_{1}^{k}\partial_{x_{2}}\tilde{\omega}_{1}+\bar{\omega}_{3}^{k}\bar{u}_{1}^{k}\partial_{x_{3}}\tilde{\omega}_{1}}\\\
&\quad\quad\;\;\,+\bar{\omega}_{1}^{k}\bar{u}_{2}^{k}\partial_{x_{1}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{2}+\bar{\omega}_{3}^{k}\bar{u}_{2}^{k}\partial_{x_{3}}\tilde{\omega}_{2}\\\
&\quad\quad\;\;\,+\bar{\omega}_{1}^{k}\bar{u}_{3}^{k}\partial_{x_{1}}\tilde{\omega}_{3}+\bar{\omega}_{2}^{k}\bar{u}_{3}^{k}\partial_{x_{2}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{3})\\\
\end{split}$
$\int_{\Omega}{(\tilde{\omega}_{1}\partial_{x_{1}}q+\tilde{\omega}_{2}\partial_{x_{2}}q+\tilde{\omega}_{3}\partial_{x_{3}}q)}=-\int_{\Omega}{q\,(\partial_{x_{1}}\tilde{\omega}_{1}+\partial_{x_{2}}\tilde{\omega}_{2}+\partial_{x_{3}}\tilde{\omega}_{3})}=0$
furthermore
$\int_{\Omega}{\tilde{\omega}_{i}\Delta\tilde{\omega}_{i}}=\int_{\Omega}{\tilde{\omega}_{i}(\partial_{x_{1}}^{2}\tilde{\omega}_{i}+\partial_{x_{2}}^{2}\tilde{\omega}_{i}+\partial_{x_{3}}^{2}\tilde{\omega}_{i})}=-\int_{\Omega}{((\partial_{x_{1}}\tilde{\omega}_{i})^{2}+(\partial_{x_{2}}\tilde{\omega}_{i})^{2}+(\partial_{x_{3}}\tilde{\omega}_{i})^{2})}$
Then from (5) we have
$\begin{split}&\int_{\Omega}{\tilde{\omega}_{1}\partial_{t}\tilde{\omega}_{1}}\;\,+\int_{\Omega}{\tilde{\omega}_{1}(\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{1}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{1}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{1}^{k})}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad-\int_{\Omega}{\tilde{\omega}_{1}(\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{1}^{k}+\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{1}^{k}+\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{1}^{k})}\,+\int_{\Omega}{\tilde{\omega}_{1}\partial_{x_{1}}q}=\int_{\Omega}{\tilde{\omega}_{1}\Delta\tilde{\omega}_{1}}\\\
&\int_{\Omega}{\tilde{\omega}_{2}\partial_{t}\tilde{\omega}_{2}}+\int_{\Omega}{\tilde{\omega}_{2}(\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{2}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{2}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{2}^{k})}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad-\int_{\Omega}{\tilde{\omega}_{2}(\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{2}^{k}+\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{2}^{k}+\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{2}^{k})}+\int_{\Omega}{\tilde{\omega}_{2}\partial_{x_{2}}q}=\int_{\Omega}{\tilde{\omega}_{2}\Delta\tilde{\omega}_{2}}\\\
&\int_{\Omega}{\tilde{\omega}_{3}\partial_{t}\tilde{\omega}_{3}}+\int_{\Omega}{\tilde{\omega}_{3}(\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{3}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{3}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{3}^{k})}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad-\int_{\Omega}{\tilde{\omega}_{3}(\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{3}^{k}+\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{3}^{k}+\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{3}^{k})}+\int_{\Omega}{\tilde{\omega}_{3}\partial_{x_{3}}q}=\int_{\Omega}{\tilde{\omega}_{3}\Delta\tilde{\omega}_{3}}\\\
\end{split}$
so that
$\begin{split}&\frac{1}{2}\partial_{t}\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})\;}+\,\,\,\int_{\Omega}{((\partial_{x_{1}}\tilde{\omega}_{1})^{2}+(\partial_{x_{2}}\tilde{\omega}_{1})^{2}+(\partial_{x_{3}}\tilde{\omega}_{1})^{2})}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\qquad+(\partial_{x_{1}}\tilde{\omega}_{2})^{2}+(\partial_{x_{2}}\tilde{\omega}_{2})^{2}+(\partial_{x_{3}}\tilde{\omega}_{2})^{2}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\qquad+(\partial_{x_{1}}\tilde{\omega}_{3})^{2}+(\partial_{x_{2}}\tilde{\omega}_{3})^{2}+(\partial_{x_{3}}\tilde{\omega}_{3})^{2})\\\
&\quad\quad-\int_{\Omega}{(\bar{\omega}_{1}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{1}+\bar{\omega}_{1}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{1}+\bar{\omega}_{1}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{1}}\\\
&\quad\quad\quad\;\;+\bar{\omega}_{2}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{2}\\\
&\quad\quad\quad\;\;+\bar{\omega}_{3}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{3}\,\,+\bar{\omega}_{3}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{3})\\\
&\quad\quad+\int_{\Omega}{(\bar{\omega}_{1}^{k}\bar{u}_{1}^{k}\partial_{x_{1}}\tilde{\omega}_{1}+\bar{\omega}_{2}^{k}\bar{u}_{1}^{k}\partial_{x_{2}}\tilde{\omega}_{1}+\bar{\omega}_{3}^{k}\bar{u}_{1}^{k}\partial_{x_{3}}\tilde{\omega}_{1}}\\\
&\quad\quad\quad\;\;\,+\bar{\omega}_{1}^{k}\bar{u}_{2}^{k}\partial_{x_{1}}\tilde{\omega}_{2}+\bar{\omega}_{2}^{k}\bar{u}_{2}^{k}\partial_{x_{2}}\tilde{\omega}_{2}+\bar{\omega}_{3}^{k}\bar{u}_{2}^{k}\partial_{x_{3}}\tilde{\omega}_{2}\\\
&\quad\quad\quad\;\;\,+\bar{\omega}_{1}^{k}\bar{u}_{3}^{k}\partial_{x_{1}}\tilde{\omega}_{3}+\bar{\omega}_{2}^{k}\bar{u}_{3}^{k}\partial_{x_{2}}\tilde{\omega}_{3}+\bar{\omega}_{3}^{k}\bar{u}_{3}^{k}\partial_{x_{3}}\tilde{\omega}_{3})=0\\\
\end{split}$
it follows that
$\begin{split}&\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}\;\,+\,\;2\;\;\int_{t_{k-1}}^{t}{\int_{\Omega}{\;((\partial_{x_{1}}\tilde{\omega}_{1})^{2}+(\partial_{x_{2}}\tilde{\omega}_{1})^{2}+(\partial_{x_{3}}\tilde{\omega}_{1})^{2}}}\\\
&\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad\quad\;\;\;\;\;+(\partial_{x_{1}}\tilde{\omega}_{2})^{2}+(\partial_{x_{2}}\tilde{\omega}_{2})^{2}+(\partial_{x_{3}}\tilde{\omega}_{2})^{2}\\\
&\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad\quad\;\;\;\;\;+(\partial_{x_{1}}\tilde{\omega}_{3})^{2}+(\partial_{x_{2}}\tilde{\omega}_{3})^{2}+(\partial_{x_{3}}\tilde{\omega}_{3})^{2})\\\
&\leq\int_{\Omega}{(\tilde{\omega}_{1}^{k-1^{2}}+\tilde{\omega}_{2}^{k-1^{2}}+\tilde{\omega}_{3}^{k-1^{2}})}\;\,+\;\,2\int_{t_{k-1}}^{t}{\int_{\Omega}{(\bar{\omega}_{1}^{k^{2}}\bar{u}_{1}^{k^{2}}+\bar{\omega}_{1}^{k^{2}}\bar{u}_{2}^{k^{2}}+\bar{\omega}_{1}^{k^{2}}\bar{u}_{3}^{k^{2}}}}\\\
&\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\quad\quad\,\quad\,\quad\quad\quad+\bar{\omega}_{2}^{k^{2}}\bar{u}_{1}^{k^{2}}+\bar{\omega}_{2}^{k^{2}}\bar{u}_{2}^{k^{2}}+\bar{\omega}_{2}^{k^{2}}\bar{u}_{3}^{k^{2}}\\\
&\qquad\qquad\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad\,\,\quad\quad\quad+\bar{\omega}_{3}^{k^{2}}\bar{u}_{1}^{k^{2}}\,+\bar{\omega}_{3}^{k^{2}}\bar{u}_{2}^{k^{2}}\,+\bar{\omega}_{3}^{k^{2}}\bar{u}_{3}^{k^{2}})\\\
&\quad\quad\quad\quad\quad\quad\quad\qquad\qquad\qquad\;\;\;\,+\;\,2\int_{t_{k-1}}^{t}{\int_{\Omega}{(\bar{\omega}_{1}^{k^{2}}\bar{u}_{1}^{k^{2}}+\bar{\omega}_{2}^{k^{2}}\bar{u}_{1}^{k^{2}}+\bar{\omega}_{3}^{k^{2}}\bar{u}_{1}^{k^{2}}}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\,\qquad\qquad\quad\quad\quad+\bar{\omega}_{1}^{k^{2}}\bar{u}_{2}^{k^{2}}+\bar{\omega}_{2}^{k^{2}}\bar{u}_{2}^{k^{2}}+\bar{\omega}_{3}^{k^{2}}\bar{u}_{2}^{k^{2}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\qquad\quad\quad\quad+\bar{\omega}_{1}^{k^{2}}\bar{u}_{3}^{k^{2}}\,+\bar{\omega}_{2}^{k^{2}}\bar{u}_{3}^{k^{2}}\,+\bar{\omega}_{3}^{k^{2}}\bar{u}_{3}^{k^{2}})\\\
&\quad\quad\quad\quad\quad\quad\quad+\int_{t_{k-1}}^{t}{\int_{\Omega}{((\partial_{x_{1}}\tilde{\omega}_{1})^{2}+(\partial_{x_{2}}\tilde{\omega}_{1})^{2}+(\partial_{x_{3}}\tilde{\omega}_{1})^{2}}}\\\
&\qquad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\;\,+(\partial_{x_{1}}\tilde{\omega}_{2})^{2}+(\partial_{x_{2}}\tilde{\omega}_{2})^{2}+(\partial_{x_{3}}\tilde{\omega}_{2})^{2}\\\
&\qquad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\;\,+(\partial_{x_{1}}\tilde{\omega}_{3})^{2}+(\partial_{x_{2}}\tilde{\omega}_{3})^{2}+(\partial_{x_{3}}\tilde{\omega}_{3})^{2})\\\
\end{split}$
by using Young inequality: $uv\leq\frac{1}{4}u^{2}+v^{2}$.
According to Cauchy-Schwarz inequality on
$Q_{T_{k}}=(t_{k-1},t_{k})\times\Omega$ , we have
$\begin{split}&\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}+\int_{t_{k-1}}^{t}{\int_{\Omega}{((\partial_{x_{1}}\tilde{\omega}_{1})^{2}+(\partial_{x_{2}}\tilde{\omega}_{1})^{2}+(\partial_{x_{3}}\tilde{\omega}_{1})^{2}}}\\\
\end{split}$
$\begin{split}&\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\,+(\partial_{x_{1}}\tilde{\omega}_{2})^{2}+(\partial_{x_{2}}\tilde{\omega}_{2})^{2}+(\partial_{x_{3}}\tilde{\omega}_{2})^{2}\\\
&\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\,+(\partial_{x_{1}}\tilde{\omega}_{3})^{2}+(\partial_{x_{2}}\tilde{\omega}_{3})^{2}+(\partial_{x_{3}}\tilde{\omega}_{3})^{2})\\\
\end{split}$
$\begin{split}&\leq\int_{\Omega}{(\tilde{\omega}_{1}^{k-1^{2}}+\tilde{\omega}_{2}^{k-1^{2}}+\tilde{\omega}_{3}^{k-1^{2}})}+\\\
&+4\\{(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{1}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{1}^{k^{4}}}})^{\frac{1}{2}}+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{1}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{2}^{k^{4}}}})^{\frac{1}{2}}+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{1}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{3}^{k^{4}}}})^{\frac{1}{2}}\\\
&\;\;\;+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{2}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{1}^{k^{4}}}})^{\frac{1}{2}}+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{2}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{2}^{k^{4}}}})^{\frac{1}{2}}+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{2}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{3}^{k^{4}}}})^{\frac{1}{2}}\\\
&\;\;\;+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{3}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{1}^{k^{4}}}})^{\frac{1}{2}}+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{3}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{2}^{k^{4}}}})^{\frac{1}{2}}+(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{u}_{3}^{k^{4}}}})^{\frac{1}{2}}(\int_{t_{k-1}}^{t}{\int_{\Omega}{\bar{\omega}_{3}^{k^{4}}}})^{\frac{1}{2}}\\}\\\
\end{split}$
$\begin{split}&\leq\int_{\Omega}{(\tilde{\omega}_{1}^{k-1^{2}}+\tilde{\omega}_{2}^{k-1^{2}}+\tilde{\omega}_{3}^{k-1^{2}})}\;+\\\
&\qquad\qquad\qquad+4\;\\{\,\,\left\|{\bar{u}_{1}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}(\,\left\|{\bar{\omega}_{1}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{2}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{3}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2})\\\
&\qquad\qquad\qquad\quad\;\;+\left\|{\bar{u}_{2}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}(\,\left\|{\bar{\omega}_{1}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{2}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{3}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2})\\\
&\qquad\qquad\qquad\quad\;\;+\left\|{\bar{u}_{3}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}(\,\left\|{\bar{\omega}_{1}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{2}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{3}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2})\,\\}\\\
\end{split}$
$\begin{split}&=\int_{\Omega}{(\tilde{\omega}_{1}^{k-1^{2}}+\tilde{\omega}_{2}^{k-1^{2}}+\tilde{\omega}_{3}^{k-1^{2}})}+\\\
&\quad+4\,(\,\left\|{\bar{u}_{1}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{u}_{2}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{u}_{3}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2})\,(\,\left\|{\bar{\omega}_{1}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{2}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2}+\left\|{\bar{\omega}_{3}^{k}}\right\|_{L^{4}(Q_{T_{k}})}^{2})\\\
\end{split}$
From Sobolev imbedding theorem in [1], there exists a constant $C_{1}>0$
independent of $\omega$ and the size of $Q_{T_{k}}$ such that
$\begin{split}&\left\|\omega\right\|_{L^{4}(Q_{T_{k}})}\leq
C_{1}\;\left\|\omega\right\|_{H^{1}(Q_{T_{k}})}\\\
&(\int_{t_{k-1}}^{t}{\left\|{\bar{\omega}_{i}^{k}}\right\|_{L^{4}(\Omega)}^{4}})^{1/2}\leq
C_{1}\;\int_{t_{k-1}}^{t}{\,\\{\,\left\|{\bar{\omega}_{i}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{\omega}_{i}^{k}}\right\|_{L^{2}(\Omega)}^{2}\\}},\quad\quad
i=1,2,3\\\ \end{split}$
it follows that
$\begin{split}&\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}+\int_{t_{k-1}}^{t}{(\,\left\|{\nabla\tilde{\omega}_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{3}}\right\|_{L^{2}(\Omega)}^{2})}\\\
&\leq\int_{\Omega}{(\tilde{\omega}_{1}^{k-1^{2}}+\tilde{\omega}_{2}^{k-1^{2}}+\tilde{\omega}_{3}^{k-1^{2}})}+\\\
&+4C\,\int_{t_{k-1}}^{t}{(\left\|{\bar{u}_{1}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{u}_{1}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\bar{u}_{2}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{u}_{2}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\bar{u}_{3}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{u}_{3}^{k}}\right\|_{L^{2}(\Omega)}^{2})}\\\
&\times\int_{t_{k-1}}^{t}{(\left\|{\bar{\omega}_{1}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{\omega}_{1}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\bar{\omega}_{2}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{\omega}_{2}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\bar{\omega}_{3}^{k}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{\omega}_{3}^{k}}\right\|_{L^{2}(\Omega)}^{2})}\\\
\end{split}$ (6)
Noting that
$\begin{split}\left\|{\bar{\omega}_{i}^{k}}\right\|_{L^{2}(\Omega)}^{2}=\int_{\Omega}{\left({\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{\tilde{\omega}_{i}(x,t)dt}}\right)}^{2}\leq\frac{1}{\Delta
t_{k}^{2}}\int_{\Omega}{\Delta
t_{k}\int_{t_{k-1}}^{t_{k}}{\tilde{\omega}_{i}^{2}(x,t)dt}}=\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{\left\|{\tilde{\omega}_{i}}\right\|_{L^{2}(\Omega)}^{2}}\\\
\end{split}$
and similarly
$\left\|{\bar{u}_{i}^{k}}\right\|_{L^{2}(\Omega)}^{2}\leq\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{\left\|u_{i}\right\|_{L^{2}(\Omega)}^{2}},\qquad
i=1,2,3\\\ $
from (6) we have
$\begin{split}&\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}+\int_{t_{k-1}}^{t}{(\,\left\|{\nabla\tilde{\omega}_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{3}}\right\|_{L^{2}(\Omega)}^{2})}\\\
&\leq\int_{\Omega}{(\tilde{\omega}_{1}^{k-1^{2}}+\tilde{\omega}_{2}^{k-1^{2}}+\tilde{\omega}_{3}^{k-1^{2}})}\;+\\\
\end{split}$
$\begin{split}&+4C\,\left({(t_{k}-t_{k-1})\mathop{\sup}\limits_{(t_{k-1},\;t)}\left\\{{\left\|{u_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{u_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{u_{3}}\right\|_{L^{2}(\Omega)}^{2}}\right\\}}\right.+\\\
&\quad+\left.{\int_{t_{k-1}}^{t_{k}}{\left\\{{\left\|{\nabla
u_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{3}}\right\|_{L^{2}(\Omega)}^{2}}\right\\}}}\right)\\\
&\;\;\;\times\int_{t_{k-1}}^{t_{k}}{(\,\left\|{\tilde{\omega}_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\tilde{\omega}_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\tilde{\omega}_{3}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{3}}\right\|_{L^{2}(\Omega)}^{2})}\\\
\end{split}$
Set
$K_{0}=\int_{\Omega}{(\omega_{10}^{2}+\omega_{20}^{2}+\omega_{30}^{2})}$
$\begin{split}&K_{k}^{\ast}=\Delta
t_{k}\mathop{\sup}\limits_{(t_{k-1},t_{k})}\left\\{{\left\|{u_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{u_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{u_{3}}\right\|_{L^{2}(\Omega)}^{2}}\right\\}+\\\
&\quad\quad+\int_{t_{k-1}}^{t_{k}}{\left\\{{\left\|{\nabla
u_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{3}}\right\|_{L^{2}(\Omega)}^{2}}\right\\}}\\\ \end{split}$
and
$f_{k}(t)=\mathop{\sup}\limits_{(t_{k-1},\;t)}\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}+\;\varepsilon_{0}\int_{t_{k-1}}^{t}{\left\\{{\left\|{\nabla\tilde{\omega}_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{3}}\right\|_{L^{2}(\Omega)}^{2}}\right\\}}$
where $\varepsilon_{0}>0$ is a constant. By (2),(3) we have
$\begin{split}&K_{k}^{\ast}\leq
T\mathop{\sup}\limits_{t\in(0,T)}\int_{\Omega}{(u_{1}^{2}+u_{2}^{2}+u_{3}^{2})}+\int_{0}^{T}{\left({\left\|{\nabla
u_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla
u_{3}}\right\|_{L^{2}(\Omega)}^{2}}\right)}\\\ &\quad\;<+\infty\\\
\end{split}$
and
$\begin{split}&\mathop{\sup}\limits_{t\in(t_{k-1},t_{k})}\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}+(1-4CK_{k}^{\ast})\int_{t_{k-1}}^{t_{k}}{\left({\left\|{\nabla\tilde{\omega}_{1}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{2}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\tilde{\omega}_{3}}\right\|_{L^{2}(\Omega)}^{2}}\right)}\\\
\end{split}$
$\begin{split}&\;\,\leq\int_{\Omega}{(\bar{\omega}_{1}^{k-1^{2}}+\bar{\omega}_{2}^{k-1^{2}}+\bar{\omega}_{3}^{k-1^{2}})}+4CK_{k}^{\ast}\int_{t_{k-1}}^{t_{k}}{f_{k}(t)}\\\
\end{split}$
On $(0,t_{1})$, $t_{1}$ be small enough, since
$\sum\limits_{k=1}^{N}{K_{k}^{\ast}}<+\infty$, the partition is assumed to be
fine enough such that $1-4CK_{1}^{\ast}\geq\varepsilon_{0}$, that is,
$K_{1}^{\ast}\leq\frac{1-\varepsilon_{0}}{4C}$ is valid because of the
absolute continuity of integration with respect to $t$, thus
$f_{1}(t_{1})\leq K_{0}+4CK_{1}^{\ast}\int_{0}^{t_{1}}{f_{1}(t)}$
By using Gronwall inequality it follows that
$f_{1}(t)\leq K_{0}\,e^{\frac{1-\varepsilon_{0}}{\varepsilon_{0}}t_{1}}$
Therefore we set
$\quad M_{k}=\mathop{\sup}\limits_{t\in
T_{k}}\;\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})},\quad\quad
k=1,\cdots,N\\\ $
then
$M_{1}\leq K_{0}\,e^{\frac{1-\varepsilon_{0}}{\varepsilon_{0}}t_{1}}$ on
$(0,t_{1})$
Similar to above we have
on $(t_{1},t_{2})\quad\Rightarrow\quad M_{2}\leq
M_{1}\;e^{\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\;(t_{2}-t_{1})}$
$\cdots\;\;\cdots\;\;\cdots$
on $(t_{N-1},T)\quad\Rightarrow\quad M_{N}\leq
M_{N-1}\;e^{\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\;(T-t_{N-1})}$
$\qquad\qquad\qquad\qquad\qquad\quad\;\;\;\leq
K_{0}\;e^{\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\;[t_{1}+(t_{2}-t_{1})+\cdots+(T-t_{N-1})]}=K_{0}\;e^{\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\;T}$
Finally we get
$\mathop{\sup}\limits_{t\in(0,T)}\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}\;\leq\mathop{\max}\limits_{k}\\{M_{k}\\}\leq
K_{0}\;e^{\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\;T}$
This conclusion is also true for the weak solution of problem (5), by means of
the result of section 3 and the lower limit of Galerkin sequence according to
the page 196 of [4].
3\. Existence
In this section we have to consider the existence of solutions of the
auxiliary problems. We just need to consider the following systems on
$(0,\delta)$:
$\begin{split}&\partial_{t}\omega_{1}\,+\bar{u}_{1}\partial_{x_{1}}\bar{\omega}_{1}+\bar{u}_{2}\partial_{x_{2}}\bar{\omega}_{1}+\bar{u}_{3}\partial_{x_{3}}\bar{\omega}_{1}-\bar{\omega}_{1}\partial_{x_{1}}\bar{u}_{1}-\bar{\omega}_{2}\partial_{x_{2}}\bar{u}_{1}-\bar{\omega}_{3}\partial_{x_{3}}\bar{u}_{1}+\partial_{x_{1}}q=\Delta\omega_{1}\\\
&\partial_{t}\omega_{2}+\bar{u}_{1}\partial_{x_{1}}\bar{\omega}_{2}+\bar{u}_{2}\partial_{x_{2}}\bar{\omega}_{2}+\bar{u}_{3}\partial_{x_{3}}\bar{\omega}_{2}-\bar{\omega}_{1}\partial_{x_{1}}\bar{u}_{2}-\bar{\omega}_{2}\partial_{x_{2}}\bar{u}_{2}-\bar{\omega}_{3}\partial_{x_{3}}\bar{u}_{2}+\partial_{x_{2}}q=\Delta\omega_{2}\\\
&\partial_{t}\omega_{3}+\bar{u}_{1}\partial_{x_{1}}\bar{\omega}_{3}+\bar{u}_{2}\partial_{x_{2}}\bar{\omega}_{3}+\bar{u}_{3}\partial_{x_{3}}\bar{\omega}_{3}-\bar{\omega}_{1}\partial_{x_{1}}\bar{u}_{3}-\bar{\omega}_{2}\partial_{x_{2}}\bar{u}_{3}-\bar{\omega}_{3}\partial_{x_{3}}\bar{u}_{3}+\partial_{x_{3}}q=\Delta\omega_{3}\\\
\end{split}$ (7)
with initial value $\omega_{i}(x,0)=\omega_{i0}(i=1,2,3)$ and
$\bar{\omega}_{i}(x)=\frac{1}{\delta}\int_{0}^{\delta}{\omega_{i}(x,t)dt}$
and
$\bar{u}_{i}(x)=\frac{1}{\delta}\int_{0}^{\delta}u_{i}(x,t)dt,\quad i=1,2,3$
as well as the incompressible conditions:
$\partial_{x_{1}}\omega_{1}+\partial_{x_{2}}\omega_{2}+\partial_{x_{3}}\omega_{3}=0\quad\Rightarrow\quad\partial_{x_{1}}\bar{\omega}_{1}+\partial_{x_{2}}\bar{\omega}_{2}+\partial_{x_{3}}\bar{\omega}_{3}=0$
$\partial_{x_{1}}u_{1}+\partial_{x_{2}}u_{2}+\partial_{x_{3}}u_{3}=0\quad\Rightarrow\quad\partial_{x_{1}}\bar{u}_{1}+\partial_{x_{2}}\bar{u}_{2}+\partial_{x_{3}}\bar{u}_{3}=0$
(i) The Galerkin procedure is applied. For each $m$ and $i=1,2,3$ we define an
approximate solution $(\omega_{1m},\;\omega_{2m},\;\omega_{3m})$ as follows:
$\omega_{im}=\sum\limits_{j=1}^{m}{g_{ij}(t)w_{ij}}$
where $\\{w_{i1},\;\cdots,\;w_{im},\cdots\\}$ is the basis of $W$, and $W=$
the closure of $\mathscr{V}$ in the Sobolev space $W^{2,4}(\Omega)$, which is
separable and is dense in $V$. Thus
${(\partial_{t}\omega_{im},\;w_{il})}+{(\nabla\omega_{im},\;\nabla
w_{il})}+{((\bar{u}\cdot\nabla)\bar{\omega}_{im},\;w_{il})}-{((\bar{\omega}_{m}\cdot\nabla)\bar{u}_{i},\;w_{il})}=0$
(8) $\begin{array}[]{l}t\in(0,\delta),\quad\quad l=1,\cdots,m\\\
\omega_{im}(0)=\omega_{i0}^{m}\\\ \end{array}$
where $\omega_{i0}^{m}$ is the orthogonal projection in $H$ of $\omega_{i0}$
onto the space spanned by $w_{i1},\;\cdots\;w_{im}$. Therefore,
$\begin{split}&\sum\limits_{j=1}^{m}{{(w_{ij},\;w_{il}){g}^{\prime}_{ij}(t)}}+\sum\limits_{j=1}^{m}{{(\nabla
w_{ij},\;\nabla w_{il})g_{ij}(t)}}+\\\
&\quad+\sum\limits_{j=1}^{m}{{\\{((\bar{u}(t)\cdot\nabla)w_{ij},\;w_{il})-((w_{j}\cdot\nabla)w_{il},\;\bar{u}_{i}(t))\\}}}\;\bar{g}_{ij}(t)\,=0\\\
\end{split}$
where $\bar{g}_{ij}(t)=\frac{1}{\delta}\int_{0}^{\delta}{g_{ij}(t)dt}$ and
${u}_{i}\in L^{\infty}(0,T;H)$ from section 1. Inverting the nonsigular matrix
with elements ${(w_{ij},\;w_{il})},\;\;1\leq j,l\leq m$, we can write above
system in the following form
${g}^{\prime}_{ij}(t)+\sum\limits_{l=1}^{m}{{\alpha_{ijl}g_{il}(t)}}+\sum\limits_{l=1}^{m}{{\beta_{ijl}\;\bar{g}_{il}(t)}}=0$
(9)
where $\alpha_{ijl},\;\,\beta_{ijl}$ are constants.
The initial conditions are equivalent to
$g_{ij}(0)=g^{0}_{ij}=\mbox{the}\;j^{th}\;\mbox{component}\;\mbox{of}\;\omega_{i0}^{m}$
We construct a sequence $\\{g_{ij}^{k}\\}$ by using a successive
approximation:
$\begin{split}&{g_{ij}^{1}}^{\prime}=-\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{0}}-\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{0}}\quad\Rightarrow\quad
g_{ij}^{1}=g_{ij}^{0}-\int_{0}^{t}{\left({\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{0}}+\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{0}}}\right)}\\\
&{g_{ij}^{2}}^{\prime}=-\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{1}}-\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{1}}\quad\Rightarrow\quad
g_{ij}^{2}=g_{ij}^{0}-\int_{0}^{t}{\left({\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{1}}+\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{1}}}\right)}\\\
&\quad\quad\quad\quad\cdots\cdots\cdots\cdots\\\
&{g_{ij}^{k}}^{\prime}=-\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{k-1}}-\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{k-1}}\quad\Rightarrow\quad
g_{ij}^{k}=g_{ij}^{0}-\int_{0}^{t}{\left({\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{k-1}}+\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{k-1}}}\right)}\\\
\end{split}$
$\qquad\left|{g_{ij}^{k}(t)-g_{ij}^{k-1}(t)}\right|\leq\int_{0}^{t}{\left({\sum\limits_{l=1}^{m}{\left|{\alpha_{ijl}}\right|\left|{g_{il}^{k-1}(t)-g_{il}^{k-2}(t)}\right|}+\sum\limits_{l=1}^{m}{\left|{\beta_{ijl}}\right|\left|{\bar{g}_{il}^{k-1}(t)-\bar{g}_{il}^{k-2}(t)}\right|}}\right)}$
Related to the a priori estimates we shall give later on, we have
$\qquad\mathop{\max}\limits_{i,j}\mathop{\sup}\limits_{t}\left|{g_{ij}^{k}(t)-g_{ij}^{k-1}(t)}\right|\leq\mathop{\max}\limits_{i,j}\sum\limits_{l=1}^{m}{\left({\left|{\alpha_{ijl}}\right|+\left|{\beta_{ijl}}\right|}\right)}\;\cdot\;t\;\cdot\;\mathop{\max}\limits_{i,j}\mathop{\sup}\limits_{t}\left|{g_{ij}^{k-1}(t)-g_{ij}^{k-2}(t)}\right|$
Taking
$\delta:\,=\frac{1}{\mathop{\max}\limits_{i,j}\sum\limits_{l=1}^{m}{\left({\left|{\alpha_{ijl}}\right|+2\left|{\beta_{ijl}}\right|}\right)}}$,
as $t\leq\delta$, then choosing $\delta^{\ast}$:
$0<\delta^{\ast}=\frac{\mathop{\max}\limits_{i,j}\sum\limits_{l=1}^{m}{\left({\left|{\alpha_{ijl}}\right|+\left|{\beta_{ijl}}\right|}\right)}}{\mathop{\max}\limits_{i,j}\sum\limits_{l=1}^{m}{\left({\left|{\alpha_{ijl}}\right|+2\left|{\beta_{ijl}}\right|}\right)}}<1$
it follows that
$\qquad\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{k}-g_{ij}^{k-1}}\right\|_{\infty}\leq\delta^{\ast}\cdot\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{k-1}-g_{ij}^{k-2}}\right\|_{\infty}\leq(\delta^{\ast})^{k-1}\cdot\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{1}-g_{ij}^{0}}\right\|_{\infty}$
For any $n,k$ (we can set $n>k$ without loss of generality), we get
$\begin{split}&\qquad\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{n}-g_{ij}^{k}}\right\|_{\infty}\leq\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{n}-g_{ij}^{n-1}}\right\|_{\infty}+\cdots+\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{k+1}-g_{ij}^{k}}\right\|_{\infty}\\\
&\qquad\leq\left({(\delta^{\ast})^{n-1}+\cdots+(\delta^{\ast})^{k}}\right)\cdot\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{1}-g_{ij}^{0}}\right\|_{\infty}=(\delta^{\ast})^{k}\frac{1-(\delta^{\ast})^{n-k}}{1-\delta^{\ast}}\mathop{\max}\limits_{i,j}\left\|{g_{ij}^{1}-g_{ij}^{0}}\right\|_{\infty}\\\
&\qquad\to 0\quad(k\to\infty)\\\ \end{split}$
Thus, for every $i=1,2,3;\;\;j=1,\cdots,m$, $\\{g_{ij}^{k}\\}$ is a Cauchy
sequence in $L^{\infty}(0,\delta)$. Since $L^{\infty}(0,\delta)$ is complete,
then there exists a function $g_{ij}^{\ast}\in L^{\infty}(0,\delta)$ such that
$\left\|{g_{ij}^{k}-g_{ij}^{\ast}}\right\|_{\infty}\to 0$ as $k\to\infty$.
From
$g_{ij}^{k}(t)=g_{ij}^{0}-\int_{0}^{t}{\left({\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{k-1}(t)}+\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{k-1}(t)}}\right)}$
let $k\to\infty$, it follows that
$g_{ij}^{\ast}(t)=g_{ij}^{0}-\int_{0}^{t}{\left({\sum\limits_{l=1}^{m}{\alpha_{ijl}g_{il}^{\ast}(t)}+\sum\limits_{l=1}^{m}{\beta_{ijl}\bar{g}_{il}^{\ast}(t)}}\right)}$
i.e., $g_{ij}^{\ast}$ is a solution of the system (9) on $(0,\delta)$ for
which $g_{ij}^{\ast}(0)=g_{ij}^{0}$, $i=1,2,3;\;\,j=1,\cdots,m$.
(ii)
$\sum\limits_{i=1}^{3}{(\partial_{t}\omega_{im},\;\omega_{im})}+\sum\limits_{i=1}^{3}{(\nabla\omega_{im},\;\nabla\omega_{im})}+\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\bar{\omega}_{im},\;\omega_{im})}-\sum\limits_{i=1}^{3}{((\bar{\omega}_{m}\cdot\nabla)\bar{u}_{i},\;\omega_{im})}=0$
Then we write
$\frac{1}{2}\frac{d}{dt}\left({\sum\limits_{i=1}^{3}{\left\|{\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)+\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}-\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\omega_{im},\;\bar{\omega}_{im})}+\sum\limits_{i=1}^{3}{((\bar{\omega}_{m}\cdot\nabla)\omega_{im},\;\bar{u}_{i})}=0$
Similar to those in the section 2, and $\eta$ is chosen to be small enough, we
have
$\sum\limits_{i=1}^{3}{\left\|{\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}+\varepsilon_{0}\int_{0}^{\eta}{\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)}\leq\left({\sum\limits_{i=1}^{3}{\left\|{\omega_{i0}^{m}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\exp\left({\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\eta}\right)$
Hence
$\mathop{\sup}\limits_{t\in(0,\eta)}\left({\sum\limits_{i=1}^{3}{\left\|{\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\leq\left({\sum\limits_{i=1}^{3}{\left\|{\omega_{i0}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\exp\left({\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\eta}\right)$
(10)
and
$\sum\limits_{i=1}^{3}{\left\|{\omega_{im}(\eta)}\right\|_{L^{2}(\Omega)}^{2}}+\int_{0}^{\eta}{\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\leq\frac{1}{\varepsilon_{0}}\left({\sum\limits_{i=1}^{3}{\left\|{\omega_{i0}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\exp\left({\frac{1-\varepsilon_{0}}{\varepsilon_{0}}\eta}\right)$
(11)
The inequalities (10) and (11) are valid for any fixed $\delta\leq\eta$.
(iii) Let $\tilde{\omega}_{m}$ denote the function from ${\mathbb{R}}$ into
$V$, which is equal to $\omega_{m}$ on $(0,\delta)$ and to 0 on the complement
of this interval. The Fourier transform of $\tilde{\omega}_{m}$ is denoted by
$\hat{\omega}_{m}$. We want to show that
$\int_{-\infty}^{+\infty}{\left|\tau\right|^{2\gamma}\left({\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}}}\right)}\,d\tau<+\infty$
for some $\gamma>0$. Along with (11) this will imply that
$\tilde{\omega}_{m}$ belongs to a bounded set of
$H^{\gamma}({\mathbb{R}},V,H)\\\ $
and will enable us to apply the result of compactness.
We observe that (8) can be written as
$\frac{d}{dt}\left({\sum\limits_{i=1}^{3}{(\tilde{\omega}_{im},\;w_{ij})}}\right)=\sum\limits_{i=1}^{3}{(\tilde{f}_{im},\;w_{ij})}+\sum\limits_{i=1}^{3}{(\omega_{i0}^{m},\;w_{ij})\,}\eta_{0}-\sum\limits_{i=1}^{3}{(\omega_{im}(\delta),\;w_{ij})\,}\eta_{\delta}$
where $\eta_{0},\;\eta_{\delta}$ are Dirac distributions at 0 and $\delta$,
and
$f_{im}=-\Delta\omega_{im}+(\bar{u}\cdot\nabla)\bar{\omega}_{im}-(\bar{\omega}_{m}\cdot\nabla)\bar{u}_{i}\;\;\;$
$\tilde{f}_{im}=f_{im}\;\mbox{on}\;(0,\delta),\quad 0\;\mbox{outside this
interval}$
By the Fourier transform,
$2\mbox{i}\pi\tau\sum\limits_{i=1}^{3}{(\hat{\omega}_{im},\;w_{ij})}=\sum\limits_{i=1}^{3}{(\hat{f}_{im},\;w_{ij})}+\sum\limits_{i=1}^{3}{(\omega_{i0}^{m},\;w_{ij})}-\sum\limits_{i=1}^{3}{(\omega_{im}(\delta),\;w_{ij})\,}\exp(-2\mbox{i}\pi\delta\tau)$
where $\hat{\omega}_{im}$ and $\hat{f}_{im}$ denoting the Fourier transforms
of $\tilde{\omega}_{im}$ and $\tilde{f}_{im}$ respectively.
We multiply above equality by $\hat{g}_{ij}(\tau)=$Fourier transform of
$\tilde{g}_{ij}$ and add the resulting equation for $j=1,\cdots,m$, we get
$\begin{split}&2\mbox{i}\pi\tau\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}}=\sum\limits_{i=1}^{3}{(\hat{f}_{im}(\tau),\;\hat{\omega}_{im}(\tau))}\\\
&\quad\quad+\sum\limits_{i=1}^{3}{(\omega_{i0}^{m},\;\hat{\omega}_{im}(\tau))}-\sum\limits_{i=1}^{3}{(\omega_{im}(\delta),\;\hat{\omega}_{im}(\tau))\,}\exp(-2\mbox{i}\pi\delta\tau)\end{split}$
For some $\varphi_{i}\in V$ and $Q_{\delta}=(0,\delta)\times\Omega$,
$\begin{split}&\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(f_{im},\;\varphi_{i})}}=\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(-\Delta\omega_{im},\;\varphi_{i})}}+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\bar{\omega}_{im},\;\varphi_{i})}}-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{\omega}_{m}\cdot\nabla)\bar{u}_{i},\;\varphi_{i})}}\\\
&=\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\nabla\omega_{im},\;\nabla\varphi_{i})}}-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\varphi_{i},\;\bar{\omega}_{im})}}+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{\omega}_{m}\cdot\nabla)\varphi_{i},\;\bar{u}_{i})}}\\\
&\leq\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}}\right\|_{L^{2}(\Omega)}\left\|{\nabla\varphi_{i}}\right\|_{L^{2}(\Omega)}}}+\\\
&\quad+2\left({\sum\limits_{i=1}^{3}{\left\|{\bar{u}_{i}}\right\|_{L^{4}(Q_{\delta})}^{2}}}\right)^{1/2}\left({\sum\limits_{i=1}^{3}{\left\|{\bar{\omega}_{im}}\right\|_{L^{4}(Q_{\delta})}^{2}}}\right)^{1/2}\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\varphi_{i}}\right\|_{L^{2}(Q_{\delta})}^{2}}}\right)^{1/2}\\\
&\leq\int_{0}^{\delta}{\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}}\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\varphi_{i}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}+\\\
&\quad+2C\sqrt{\delta}\left(\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\\{\left\|{\bar{u}_{i}}\right\|_{L^{2}(\Omega)}^{2}}+{{\left\|{\nabla\bar{u}_{i}}\right\|_{L^{2}(\Omega)}^{2}}\\}}}\right)^{1/2}\\\
&\quad\times\left({\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\\{\,\left\|{\bar{\omega}_{im}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\bar{\omega}_{im}}\right\|_{L^{2}(\Omega)}^{2}\\}}}}\right)^{1/2}\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\varphi_{i}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}\\\
\end{split}$
$\begin{split}&\leq\int_{0}^{\delta}{\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}}\left\|{\nabla\varphi}\right\|_{V}+\\\
&\quad+2C\sqrt{\delta}\left({\delta\mathop{\sup}\limits_{(0,\delta)}\sum\limits_{i=1}^{3}{\left\|{u_{i}}\right\|_{L^{2}(\Omega)}^{2}}+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\left\|{\nabla
u_{i}}\right\|_{L^{2}(\Omega)}^{2}}}}\right)^{1/2}\\\
&\quad\times\left({\delta\;\mathop{\sup}\limits_{(0,\delta)}\sum\limits_{i=1}^{3}{\left\|{\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}}\right)^{1/2}\left\|{\nabla\varphi}\right\|_{V}\\\
\end{split}$
this remains bounded according to (2), (3), and (10), (11). Therefore
$\int_{0}^{\delta}{\left\|{f_{im}(t)}\right\|_{V}dt}=\int_{0}^{\delta}{\;\mathop{\sup}\limits_{\left\|\varphi\right\|_{V}=1}\;\sum\limits_{i=1}^{3}{(f_{im},\;\varphi_{i})}}<+\infty$
it follows that
$\mathop{\sup}\limits_{\tau\in{\mathbb{R}}}\left\|{\hat{f}_{im}(\tau)}\right\|_{V}<+\infty,\quad\;\forall
m$
Due to (10), we have
$\left\|{\omega_{im}(0)}\right\|_{L^{2}(\Omega)}<+\infty,\quad\quad\left\|{\omega_{im}(\delta)}\right\|_{L^{2}(\Omega)}<+\infty$
then by Poincare inequality,
$\begin{split}\left|\tau\right|\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}}&\leq
c_{1}\sum\limits_{i=1}^{3}{\left\|{\hat{f}_{im}(\tau)}\right\|_{V}\;\left\|{\hat{\omega}_{im}(\tau)}\right\|_{V}}+c_{2}\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}}\\\
&\leq
c_{3}\left(\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}}+\sum\limits_{i=1}^{3}{\left\|{\nabla\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}}\right)\\\
\end{split}$ (12)
For $\gamma$ fixed, $\gamma<1/4$, we observe that
$\left|\tau\right|^{2\gamma}\leq
c_{4}(\gamma)\frac{1+\left|\tau\right|}{1+\left|\tau\right|^{1-2\gamma}},\quad\quad\forall\tau\in{\mathbb{R}}$
Thus by (12),
$\begin{split}&\int_{-\infty}^{+\infty}{\left|\tau\right|^{2\gamma}\left({\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}}}\right)}\,d\tau\leq
c_{4}(\gamma)\int_{-\infty}^{+\infty}{\frac{1+\left|\tau\right|}{1+\left|\tau\right|^{1-2\gamma}}\left({\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}}}\right)}\,d\tau\\\
&\leq
c_{5}\int_{-\infty}^{+\infty}{\frac{1}{1+\left|\tau\right|^{1-2\gamma}}\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}}}d\tau\\\
&+\;c_{6}\int_{-\infty}^{+\infty}{\frac{1}{1+\left|\tau\right|^{1-2\gamma}}\sum\limits_{i=1}^{3}{\left\|{\nabla\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}}}d\tau+\;c_{7}\int_{-\infty}^{+\infty}{\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}}}d\tau\\\
\end{split}$
Because of the Parseval equality,
$\begin{split}&\int_{-\infty}^{+\infty}{\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}d\tau}}=\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\left\|{\omega_{im}(t)}\right\|_{L^{2}(\Omega)}^{2}dt}}\\\
&\qquad\qquad\qquad\quad\quad\quad\quad\quad\;\;\,\leq\delta\;\mathop{\sup}\limits_{(0,\delta)}\;\sum\limits_{i=1}^{3}{\left\|{\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}<+\infty\\\
&\int_{-\infty}^{+\infty}{\sum\limits_{i=1}^{3}{\left\|{\nabla\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}^{2}}}\,d\tau=\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}(t)}\right\|_{L^{2}(\Omega)}^{2}}}\,dt<+\infty\\\
\end{split}$
as $m\to\infty$. By Cauchy-Schwarz inequality and the Parseval equality,
$\begin{split}&\int_{-\infty}^{+\infty}{\frac{1}{1+\left|\tau\right|^{1-2\gamma}}\sum\limits_{i=1}^{3}{\left\|{\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}}}d\tau\\\
&\qquad\leq\left({\int_{-\infty}^{+\infty}{\frac{1}{(1+\left|\tau\right|^{1-2\gamma})^{2}}}}\right)^{1/2}\left({\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\left\|{\omega_{im}(t)}\right\|_{L^{2}(\Omega)}^{2}}}dt}\right)^{1/2}<+\infty\\\
&\int_{-\infty}^{+\infty}{\frac{1}{1+\left|\tau\right|^{1-2\gamma}}\sum\limits_{i=1}^{3}{\left\|{\nabla\hat{\omega}_{im}(\tau)}\right\|_{L^{2}(\Omega)}}}d\tau\\\
&\qquad\leq\left({\int_{-\infty}^{+\infty}{\frac{1}{(1+\left|\tau\right|^{1-2\gamma})^{2}}}}\right)^{1/2}\left({\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{\left\|{\nabla\omega_{im}(t)}\right\|_{L^{2}(\Omega)}^{2}}}dt}\right)^{1/2}<+\infty\\\
\end{split}$
as $m\to\infty$ by $\gamma<1/4$ and (11).
(iv) The estimate (10) and (11) enable us to assert the existence of an
element $\omega^{\ast}\in L^{2}(0,\delta;V)\cap L^{\infty}(0,\delta;H)$ and a
subsequence $\omega_{{m}^{\prime}}$ such that
$\omega_{{m}^{\prime}}\to\omega^{\ast}$ in $L^{2}(0,\delta;V)$ weakly, and in
$L^{\infty}(0,\delta;H)$ weak-star, as ${m}^{\prime}\to\infty$
Due to (iii) we also have
$\omega_{{m}^{\prime}}\to\omega^{\ast}$ in $L^{2}(0,\delta;H)$ strongly as
${m}^{\prime}\to\infty\\\ $
This convergence result enable us to pass to the limit.
Let $\psi_{i}$ be a continuously differentiable function on $(0,\delta)$ with
$\psi_{i}(\delta)=0$. We multiply (8) by $\psi_{i}(t)$ then integrate by
parts. This leads to the equation
$\begin{split}&-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\omega_{im}(t),\;\partial_{t}\psi_{i}(t)w_{ij})\,dt}}+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\nabla\omega_{im},\;\psi_{i}(t)\nabla
w_{ij})\,dt}}\\\
&+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\bar{\omega}_{im},\;w_{ij}\psi_{i}(t))}}-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{\omega}_{m}\cdot\nabla)\bar{u}_{i},\;w_{ij}\psi_{i}(t))}}=\sum\limits_{i=1}^{3}{(\omega_{i0}^{m},\;w_{ij})\psi_{i}(0)}\\\
\end{split}$
Since $\omega_{i{m}^{\prime}}$ converges to $\omega_{i}^{\ast}$ in
$L^{2}(0,\delta;H)$ strongly as ${m}^{\prime}\to\infty$, then
$\bar{\omega}_{i{m}^{\prime}}$ also converges strongly to
$\bar{\omega}_{i}^{\ast}$, and
$\begin{split}&\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\omega_{i{m}^{\prime}},\;\partial_{t}\psi_{i}(t)w_{ij})\,dt}}\to\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\omega_{i}^{\ast},\;\partial_{t}\psi_{i}(t)w_{ij})\,dt}}\\\
&\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\nabla\omega_{i{m}^{\prime}},\;\psi_{i}(t)\nabla
w_{ij})\,dt}}=-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\omega_{i{m}^{\prime}},\;\psi_{i}(t)\Delta
w_{ij})\,dt}}\\\
&\quad\;\to-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\omega_{i}^{\ast},\;\psi_{i}(t)\Delta
w_{ij})}}=\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\nabla\omega_{i}^{\ast},\;\psi_{i}(t)\nabla
w_{ij})\,dt}}\\\
&\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\bar{\omega}_{i{m}^{\prime}},\;w_{ij}\psi_{i}(t))}}=-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)w_{ij}\psi_{i}(t),\;\bar{\omega}_{i{m}^{\prime}})}}\\\
&\quad\;\to-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)w_{ij}\psi_{i}(t),\;\bar{\omega}_{i}^{\ast})}}=\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\bar{\omega}_{i}^{\ast},\;w_{ij}\psi_{i}(t))}}\\\
&\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{\omega}_{{m}^{\prime}}\cdot\nabla)\bar{u}_{i},\;w_{ij}\psi_{i}(t))}}\to\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{\omega}^{\ast}\cdot\nabla)\bar{u}_{i},\;w_{ij}\psi_{i}(t))}}\\\
&\sum\limits_{i=1}^{3}{(\omega_{i0}^{{m}^{\prime}},\;w_{ij})\psi_{i}(0)}\to\sum\limits_{i=1}^{3}{(\omega_{i0},\;w_{ij})\psi_{i}(0)}\\\
\end{split}$
Thus, in the limit we find
$\begin{split}&-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\omega_{i}^{\ast},\;\partial_{t}\psi_{i}(t)v_{i})\,dt}}+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\nabla\omega_{i}^{\ast},\;\psi_{i}(t)\nabla
v_{i})\,dt}}\\\
&+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\bar{\omega}_{i}^{\ast},\;v_{i}\psi_{i}(t))}}-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{\omega}^{\ast}\cdot\nabla)\bar{u}_{i},\;v_{i}\psi_{i}(t))}}=\sum\limits_{i=1}^{3}{(\omega_{i0},\;v_{i})\psi_{i}(0)}\\\
\end{split}$ (13)
holds for $v_{i}=w_{i1},\;w_{i2},\cdots$; by this equation holds for
$v_{i}=$any finite linear combination of the $w_{ij}$, and by a continuity
argument above equation is still true for any $v_{i}\in V$. Hence we find that
$\omega_{i}^{\ast}(i=1,2,3)$ is a Leray-Hopf weak solution of the system (7).
Finally it remains to prove that $\omega_{i}^{\ast}$ satisfies the initial
conditions. For this we multiply (7) by $v_{i}\psi_{i}(t)$, after integrating
some terms by parts, we get
$\begin{split}&-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\omega_{i}^{\ast},\;\partial_{t}\psi_{i}(t)v_{i})}}+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{(\nabla\omega_{i}^{\ast},\;\psi_{i}(t)\nabla
v_{i})\,dt}}\\\
&+\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{u}\cdot\nabla)\bar{\omega}_{i}^{\ast},\;v_{i}\psi_{i}(t))}}-\int_{0}^{\delta}{\sum\limits_{i=1}^{3}{((\bar{\omega}^{\ast}\cdot\nabla)\bar{u}_{i},\;v_{i}\psi_{i}(t))}}=\sum\limits_{i=1}^{3}{(\omega_{i}^{\ast}(0),\;v_{i})\psi_{i}(0)}\\\
\end{split}$
By comparison with (13),
$\sum\limits_{i=1}^{3}{(\omega_{i}^{\ast}(0)-\omega_{i0},\;v_{i})\psi_{i}(0)}=0$
Therefore we can choose $\psi_{i}$ particularly such that
$(\omega_{i}^{\ast}(0)-\omega_{i0},\;v_{i})=0,\quad\quad\forall v_{i}\in V$
4\. Convergence
Now the partition is refined infinitely, we will prove that there exists some
subsequence of the solutions of auxiliary problems which converges to a weak
solution of (4).
Since
$\mathop{\sup}\limits_{t\in(0,T)}\int_{\Omega}{(\tilde{\omega}_{1}^{2}+\tilde{\omega}_{2}^{2}+\tilde{\omega}_{3}^{2})}\;<+\infty$
the family $(\tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{\omega}_{3})$ is
uniformly bounded in $L^{2}(0,T;H)\cap L^{\infty}(0,T;H)$, then we can choose
${k}^{\prime}\to\infty$, or $\Delta t_{k}^{\prime}\to 0$, such that there
exists a subsequence
$({\tilde{\omega}}^{\prime}_{1},{\tilde{\omega}}^{\prime}_{2},{\tilde{\omega}}^{\prime}_{3})$
converging weakly in $L^{2}(0,T;H)$ and weak-star in $L^{\infty}(0,T;H)$ to
some element $(\omega_{1}^{\ast},\omega_{2}^{\ast},\omega_{3}^{\ast})$. On the
other hand, because $\tilde{\omega}_{i}(i=1,2,3)$ belong to $L^{2}(0,T;H)$, we
can verify that
$\bar{\omega}_{i}(x,t)=\left\\{{\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{\tilde{\omega}_{i}(x,t)dt},\;\;t\in(t_{k-1},t_{k})\subset(0,T)}\right\\}$
also belongs to $L^{2}(0,T;H)$. In fact,
$\begin{split}&\int_{0}^{T}\int_{\Omega}{\bar{\omega}_{i}^{2}(x,t)}=\sum\limits_{k}{\int_{t_{k-1}}^{t_{k}}\int_{\Omega}{\left({\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{\tilde{\omega}_{i}(x,t)}}\right)}}^{2}=\sum\limits_{k}{\frac{1}{\Delta
t_{k}^{2}}\cdot\Delta
t_{k}\cdot\int_{\Omega}\left({\int_{t_{k-1}}^{t_{k}}{\tilde{\omega}_{i}(x,t)}}\right)}^{2}\\\
&\quad\leq\sum\limits_{k}{\frac{1}{\Delta
t_{k}}\cdot\int_{\Omega}\left(\int_{t_{k-1}}^{t_{k}}1\cdot\int_{t_{k-1}}^{t_{k}}{\tilde{\omega}_{i}^{2}(x,t)}\right)}=\sum\limits_{k}{\int_{t_{k-1}}^{t_{k}}\int_{\Omega}{\tilde{\omega}_{i}^{2}(x,t)}}=\int_{0}^{T}\int_{\Omega}{\tilde{\omega}_{i}^{2}(x,t)}<+\infty\\\
\end{split}$
In the same way, we know from (2) that the function
$\bar{u}_{i}(x,t)=\left\\{{\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{u_{i}(x,t)dt},\;\;t\in(t_{k-1},t_{k})\subset(0,T)}\right\\}$
belongs to $L^{2}(0,T;H)$.
Finally we will prove that
$(\omega_{1}^{\ast},\omega_{2}^{\ast},\omega_{3}^{\ast})$ is a solution of the
vorticity-velocity form of Navier-Stokes equation (4).
Taking $\varphi_{i}\in C^{\infty}((0,T)\times{\mathbb{R}}^{3}),\;\;(i=1,2,3)$
with a period on $\Omega$, and
$\partial_{x_{1}}\varphi_{1}+\partial_{x_{2}}\varphi_{2}+\partial_{x_{3}}\varphi_{3}=0$
we have
$\begin{split}&\sum\limits_{k=1}^{N}{\int_{t_{k-1}}^{t_{k}}{\int_{\Omega}{\varphi_{1}(\partial_{t}\tilde{\omega}_{1}\,+\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{1}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{1}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{1}^{k}-}}}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad-\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{1}^{k}-\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{1}^{k}-\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{1}^{k}+\partial_{x_{1}}q-\Delta\tilde{\omega}_{1})=0\\\
\end{split}$
$\begin{split}&\sum\limits_{k=1}^{N}{\int_{t_{k-1}}^{t_{k}}{\int_{\Omega}{\varphi_{2}(\partial_{t}\tilde{\omega}_{2}+\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{2}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{2}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{2}^{k}}}}-\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad-\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{2}^{k}-\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{2}^{k}-\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{2}^{k}+\partial_{x_{2}}q-\Delta\tilde{\omega}_{2})=0\\\
&\sum\limits_{k=1}^{N}{\int_{t_{k-1}}^{t_{k}}{\int_{\Omega}{\varphi_{3}(\partial_{t}\tilde{\omega}_{3}+\bar{u}_{1}^{k}\partial_{x_{1}}\bar{\omega}_{3}^{k}+\bar{u}_{2}^{k}\partial_{x_{2}}\bar{\omega}_{3}^{k}+\bar{u}_{3}^{k}\partial_{x_{3}}\bar{\omega}_{3}^{k}}}}-\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad-\bar{\omega}_{1}^{k}\partial_{x_{1}}\bar{u}_{3}^{k}-\bar{\omega}_{2}^{k}\partial_{x_{2}}\bar{u}_{3}^{k}-\bar{\omega}_{3}^{k}\partial_{x_{3}}\bar{u}_{3}^{k}+\partial_{x_{3}}q-\Delta\tilde{\omega}_{3})=0\\\
\end{split}$
Here $\tilde{\omega}_{i}\;(i=1,2,3)$ denote the collection of those solutions
of problem (5) defined on every $(t_{k-1},t_{k})$. Integrating by parts we get
$\begin{split}&\sum\limits_{k=1}^{N}{\int_{t_{k-1}}^{t_{k}}{\int_{\Omega}{(\tilde{\omega}_{1}\partial_{t}\varphi_{1}\,+\bar{\omega}_{1}^{k}((\bar{u}_{1}^{k}\partial_{x_{1}}\varphi_{1}+\varphi_{1}\,\partial_{x_{1}}\bar{u}_{1}^{k})+(\bar{u}_{2}^{k}\partial_{x_{2}}\varphi_{1}+\varphi_{1}\,\partial_{x_{2}}\bar{u}_{2}^{k})+(\bar{u}_{3}^{k}\partial_{x_{3}}\varphi_{1}+\varphi_{1}\,\partial_{x_{3}}\bar{u}_{3}^{k}))-}}}\\\
&\quad-\bar{u}_{1}^{k}((\bar{\omega}_{1}^{k}\partial_{x_{1}}\varphi_{1}+\varphi_{1}\,\partial_{x_{1}}\bar{\omega}_{1}^{k})+(\bar{\omega}_{2}^{k}\partial_{x_{2}}\varphi_{1}+\varphi_{1}\,\partial_{x_{2}}\bar{\omega}_{2}^{k})+(\bar{\omega}_{3}^{k}\partial_{x_{3}}\varphi_{1}+\varphi_{1}\,\partial_{x_{3}}\bar{\omega}_{3}^{k}))+\\\
&\quad+q\partial_{x_{1}}\varphi_{1}+\tilde{\omega}_{1}\Delta\varphi_{1})=\sum\limits_{k=1}^{N}{\int_{\Omega}{(\varphi_{1}(x,t_{k})\tilde{\omega}_{1}(x,t_{k})-\varphi_{1}(x,t_{k-1})\tilde{\omega}_{1}(x,t_{k-1}))}}\\\
\end{split}$
$\begin{split}&\sum\limits_{k=1}^{N}{\int_{t_{k-1}}^{t_{k}}{\int_{\Omega}{(\tilde{\omega}_{2}\partial_{t}\varphi_{2}\,+\bar{\omega}_{2}^{k}((\bar{u}_{1}^{k}\partial_{x_{1}}\varphi_{2}+\varphi_{2}\,\partial_{x_{1}}\bar{u}_{1}^{k})+(\bar{u}_{2}^{k}\partial_{x_{2}}\varphi_{2}+\varphi_{2}\,\partial_{x_{2}}\bar{u}_{2}^{k})+(\bar{u}_{3}^{k}\partial_{x_{3}}\varphi_{2}+\varphi_{2}\,\partial_{x_{3}}\bar{u}_{3}^{k}))-}}}\\\
&\quad-\bar{u}_{2}^{k}((\bar{\omega}_{1}^{k}\partial_{x_{1}}\varphi_{2}+\varphi_{2}\,\partial_{x_{1}}\bar{\omega}_{1}^{k})+(\bar{\omega}_{2}^{k}\partial_{x_{2}}\varphi_{2}+\varphi_{2}\,\partial_{x_{2}}\bar{\omega}_{2}^{k})+(\bar{\omega}_{3}^{k}\partial_{x_{3}}\varphi_{2}+\varphi_{2}\,\partial_{x_{3}}\bar{\omega}_{3}^{k}))+\\\
&\quad+q\partial_{x_{2}}\varphi_{2}+\tilde{\omega}_{2}\Delta\varphi_{2})=\sum\limits_{k=1}^{N}{\int_{\Omega}{(\varphi_{2}(x,t_{k})\tilde{\omega}_{2}(x,t_{k})-\varphi_{2}(x,t_{k-1})\tilde{\omega}_{2}(x,t_{k-1}))}}\\\
&\sum\limits_{k=1}^{N}{\int_{t_{k-1}}^{t_{k}}{\int_{\Omega}{(\tilde{\omega}_{3}\partial_{t}\varphi_{3}\,+\bar{\omega}_{3}^{k}((\bar{u}_{1}^{k}\partial_{x_{1}}\varphi_{3}+\varphi_{3}\,\partial_{x_{1}}\bar{u}_{1}^{k})+(\bar{u}_{2}^{k}\partial_{x_{2}}\varphi_{3}+\varphi_{3}\,\partial_{x_{2}}\bar{u}_{2}^{k})+(\bar{u}_{3}^{k}\partial_{x_{3}}\varphi_{3}+\varphi_{3}\,\partial_{x_{3}}\bar{u}_{3}^{k}))-}}}\\\
&\quad-\bar{u}_{3}^{k}((\bar{\omega}_{1}^{k}\partial_{x_{1}}\varphi_{3}+\varphi_{3}\,\partial_{x_{1}}\bar{\omega}_{1}^{k})+(\bar{\omega}_{2}^{k}\partial_{x_{2}}\varphi_{3}+\varphi_{3}\,\partial_{x_{2}}\bar{\omega}_{2}^{k})+(\bar{\omega}_{3}^{k}\partial_{x_{3}}\varphi_{3}+\varphi_{3}\,\partial_{x_{3}}\bar{\omega}_{3}^{k}))+\\\
&\quad+q\partial_{x_{3}}\varphi_{3}+\tilde{\omega}_{3}\Delta\varphi_{3})=\sum\limits_{k=1}^{N}{\int_{\Omega}{(\varphi_{3}(x,t_{k})\tilde{\omega}_{3}(x,t_{k})-\varphi_{3}(x,t_{k-1})\tilde{\omega}_{3}(x,t_{k-1}))}}\\\
\end{split}$
From section 2 we have the following conclusions:
$\tilde{\omega}_{i}\to\omega_{i}^{\ast}$ in $L^{2}(0,T;H)$ weakly, and in
$L^{\infty}(0,T;H)$ weak-star
$\bar{\omega}_{i}\to\omega_{i}^{\ast}$ in $L^{2}(0,T;H)$ weakly
as ${k^{\prime}}\to\infty$, or $\Delta t_{k}^{\prime}\to 0$.
In addition, for a certain solution $u$ of (1), we can prove due to (2) and
(3) that
$\bar{u}_{i}\to u_{i}$ in $L^{2}(0,T;H)$ strongly
as ${k}\to\infty$, or $\Delta t_{k}\to 0$.
In fact, set $Q=(0,T)\times\bar{\Omega}$, $\Delta
t=\mathop{\max}\limits_{k}\\{\Delta t_{k}\\}$. $\forall\varepsilon>0$, and
$u_{i}\in L^{2}(0,T;L^{2}(\Omega))$, there exists a $v_{i}\in
C^{\infty}(0,T;L^{2}(\Omega))$ such that
$\left\|{u_{i}-v_{i}}\right\|_{L^{2}(Q)}<\varepsilon$
By means of the same partition as that for $\bar{u}_{i}$ to construct
$\bar{v}_{i}$, since there exists a constant $C>0$ such that
$\left\|{\partial_{t}v_{i}}\right\|_{L^{2}(\Omega)}\leq C$, and
$\mathop{\max}\limits_{t}\,\,\left\|{\bar{v}_{i}-v_{i}}\right\|_{L^{2}(\Omega)}\leq
C\;\Delta t$, it follows that
$\left\|{\bar{v}_{i}-v_{i}}\right\|_{L^{2}(Q)}=\left({\int_{0}^{T}{\left\|{\bar{v}_{i}-v_{i}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}\leq
CT^{1/2}\;\Delta t$
Thus
$\bar{v}_{i}\to
v_{i}\;\;\left({L^{\infty}(0,T;L^{2}(\Omega))}\right),\;\;\;\;\mbox{as}\;\;\Delta
t\to 0$
Take $\Delta t$ such that
$\left\|{\bar{v}_{i}-v_{i}}\right\|_{L^{2}(Q)}<\varepsilon$. Moreover,
$\begin{split}&\int_{0}^{T}{\left\|{\bar{u}_{i}-\bar{v}_{i}}\right\|_{L^{2}(\Omega)}^{2}}=\sum\limits_{k=1}^{N}{\left\|{\frac{1}{\Delta
t_{k}}\int_{t_{k-1}}^{t_{k}}{(u_{i}-v_{i})}}\right\|}_{L^{2}(\Omega)}^{2}\Delta
t_{k}\\\
&\leq\sum\limits_{k=1}^{N}{\left\|{\left({\int_{t_{k-1}}^{t_{k}}{(u_{i}-v_{i})^{2}}}\right)^{1/2}}\right\|}_{L^{2}(\Omega)}^{2}\leq\int_{0}^{T}{\left\|{u_{i}-v_{i}}\right\|_{L^{2}(\Omega)}^{2}}\\\
\end{split}$
so that
$\left\|{\bar{u}_{i}-\bar{v}_{i}}\right\|_{L^{2}(Q)}\leq\left\|{u_{i}-v_{i}}\right\|_{L^{2}(Q)}<\varepsilon$.
Therefore,
$\left\|{\bar{u}_{i}-u_{i}}\right\|_{L^{2}(Q)}\leq\left\|{u_{i}-v_{i}}\right\|_{L^{2}(Q)}+\left\|{v_{i}-\bar{v}_{i}}\right\|_{L^{2}(Q)}+\left\|{\bar{v}_{i}-\bar{u}_{i}}\right\|_{L^{2}(Q)}<3\varepsilon$
Hence as $\Delta t\to 0$, we have
$\left\|{\bar{u}_{i}-u_{i}}\right\|_{L^{2}(Q)}\to 0$.
These convergence results enable us to pass the limit. That is,
$\begin{split}&\sum\limits_{k^{\prime}}{\int_{t_{k^{\prime}-1}}^{t_{k}^{\prime}}{\int_{\Omega}{(\tilde{\omega}_{1}\partial_{t}\varphi_{1}\,+\bar{\omega}_{1}^{k^{\prime}}(\bar{u}_{1}^{k^{\prime}}\partial_{x_{1}}\varphi_{1}+\bar{u}_{2}^{k^{\prime}}\partial_{x_{2}}\varphi_{1}+\bar{u}_{3}^{k^{\prime}}\partial_{x_{3}}\varphi_{1})-}}}\\\
&\quad\quad\quad\quad\quad\quad-\bar{u}_{1}^{k^{\prime}}(\bar{\omega}_{1}^{k^{\prime}}\partial_{x_{1}}\varphi_{1}+\bar{\omega}_{2}^{k^{\prime}}\partial_{x_{2}}\varphi_{1}+\bar{\omega}_{3}^{k^{\prime}}\partial_{x_{3}}\varphi_{1})+q\partial_{x_{1}}\varphi_{1}+\tilde{\omega}_{1}\Delta\varphi_{1})\\\
&\quad\quad\quad\quad\quad\quad=\int_{\Omega}{(\varphi_{1}(x,T)\tilde{\omega}_{1}(x,T)-\varphi_{1}(x,0)\tilde{\omega}_{1}(x,0))}\\\
\end{split}$
$\begin{split}&\sum\limits_{k^{\prime}}{\int_{t_{k^{\prime}-1}}^{t_{k}^{\prime}}{\int_{\Omega}{(\tilde{\omega}_{2}\partial_{t}\varphi_{2}\,+\bar{\omega}_{2}^{k^{\prime}}(\bar{u}_{1}^{k^{\prime}}\partial_{x_{1}}\varphi_{2}+\bar{u}_{2}^{k^{\prime}}\partial_{x_{2}}\varphi_{2}+\bar{u}_{3}^{k^{\prime}}\partial_{x_{3}}\varphi_{2})-}}}\\\
&\quad\quad\quad\quad\quad\quad-\bar{u}_{2}^{k^{\prime}}(\bar{\omega}_{1}^{k^{\prime}}\partial_{x_{1}}\varphi_{2}+\bar{\omega}_{2}^{k^{\prime}}\partial_{x_{2}}\varphi_{2}+\bar{\omega}_{3}^{k^{\prime}}\partial_{x_{3}}\varphi_{2})+q\partial_{x_{2}}\varphi_{2}+\tilde{\omega}_{2}\Delta\varphi_{2})\\\
&\quad\quad\quad\quad\quad\quad=\int_{\Omega}{(\varphi_{2}(x,T)\tilde{\omega}_{2}(x,T)-\varphi_{2}(x,0)\tilde{\omega}_{2}(x,0))}\\\
&\sum\limits_{k^{\prime}}{\int_{t_{k^{\prime}-1}}^{t_{k}^{\prime}}{\int_{\Omega}{(\tilde{\omega}_{3}\partial_{t}\varphi_{3}\,+\bar{\omega}_{3}^{k^{\prime}}(\bar{u}_{1}^{k^{\prime}}\partial_{x_{1}}\varphi_{3}+\bar{u}_{2}^{k^{\prime}}\partial_{x_{2}}\varphi_{3}+\bar{u}_{3}^{k^{\prime}}\partial_{x_{3}}\varphi_{3})-}}}\\\
&\quad\quad\quad\quad\quad\quad-\bar{u}_{3}^{k^{\prime}}(\bar{\omega}_{1}^{k^{\prime}}\partial_{x_{1}}\varphi_{3}+\bar{\omega}_{2}^{k^{\prime}}\partial_{x_{2}}\varphi_{3}+\bar{\omega}_{3}^{k^{\prime}}\partial_{x_{3}}\varphi_{3})+q\partial_{x_{3}}\varphi_{3}+\tilde{\omega}_{3}\Delta\varphi_{3})\\\
&\quad\quad\quad\quad\quad\quad=\int_{\Omega}{(\varphi_{3}(x,T)\tilde{\omega}_{3}(x,T)-\varphi_{3}(x,0)\tilde{\omega}_{3}(x,0))}\\\
\end{split}$
This is equivalent to
$\begin{split}&\int_{0}^{T}{\int_{\Omega}{\,\\{(\omega_{1}^{\ast}\partial_{t}\varphi_{1}\,+\omega_{2}^{\ast}\partial_{t}\varphi_{2}\,+\omega_{3}^{\ast}\partial_{t}\varphi_{3})+}}\\\
&\quad\quad+(\omega_{1}^{\ast}\Delta\varphi_{1}+\omega_{2}^{\ast}\Delta\varphi_{2}+\omega_{3}^{\ast}\Delta\varphi_{3})+\\\
&\quad\quad+\omega_{1}^{\ast}(u_{1}\partial_{x_{1}}\varphi_{1}+u_{2}\partial_{x_{2}}\varphi_{1}+u_{3}\partial_{x_{3}}\varphi_{1})+\omega_{2}^{\ast}(u_{1}\partial_{x_{1}}\varphi_{2}+u_{2}\partial_{x_{2}}\varphi_{2}+u_{3}\partial_{x_{3}}\varphi_{2})+\\\
&\quad\quad+\omega_{3}^{\ast}(u_{1}\partial_{x_{1}}\varphi_{3}+u_{2}\partial_{x_{2}}\varphi_{3}+u_{3}\partial_{x_{3}}\varphi_{3})\\\
&\quad\quad-
u_{1}(\omega_{1}^{\ast}\partial_{x_{1}}\varphi_{1}+\omega_{2}^{\ast}\partial_{x_{2}}\varphi_{1}+\omega_{3}^{\ast}\partial_{x_{3}}\varphi_{1})-u_{2}(\omega_{1}^{\ast}\partial_{x_{1}}\varphi_{2}+\omega_{2}^{\ast}\partial_{x_{2}}\varphi_{2}+\omega_{3}^{\ast}\partial_{x_{3}}\varphi_{2})-\\\
&\quad\quad-
u_{3}(\omega_{1}^{\ast}\partial_{x_{1}}\varphi_{3}+\omega_{2}^{\ast}\partial_{x_{2}}\varphi_{3}+\omega_{3}^{\ast}\partial_{x_{3}}\varphi_{3})\\}\\\
&=\int_{\Omega}{\\{(\varphi_{1}(x,T)\omega_{1}^{\ast}(x,T)+\varphi_{2}(x,T)\omega_{2}^{\ast}(x,T)+\varphi_{3}(x,T)\omega_{3}^{\ast}(x,T))-}\\\
&\quad\quad\;\;-(\varphi_{10}(x)\omega_{10}(x)+\varphi_{20}(x)\omega_{20}(x)+\varphi_{30}(x)\omega_{30}(x))\\}\\\
\end{split}$
Here we also have
$\omega_{i}^{\ast}(x,0)=\omega_{i0}(x),\quad\varphi_{i}(x,0)=\varphi_{i0}(x),\quad
i=1,2,3$
Hence we know that there exists some $\omega_{i}^{\ast}$ which belongs to
$L^{\infty}(0,T;L^{2}(\Omega))$ and is a Leray-Hopf weak solution of (4).
5\. Regularity
We can still use Galerkin procedure as in section 3. Since $V$ is separable
there exists a sequence of linearly independent elements
$w_{i1},\cdots,w_{im},\cdots$ which is total in $V$. For each $m$ we define an
approximate solution $u_{im}$ of (1) as follows:
$u_{im}=\sum\limits_{j=1}^{m}{g_{ij}(t)w_{ij}}$
and
$\begin{split}&\int_{\Omega}{\partial_{t}u_{1m}w_{1j}}+\int_{\Omega}{(u_{1m}\partial_{x_{1}}u_{1m}+u_{2m}\partial_{x_{2}}u_{1m}+u_{3m}\partial_{x_{3}}u_{1m})}\,w_{1j}+\int_{\Omega}{\partial_{x_{1}}p\,w_{1j}}=\int_{\Omega}{\Delta
u_{1m}\,w_{1j}}\\\ \end{split}$
$\begin{split}&\int_{\Omega}{\partial_{t}u_{2m}w_{2j}}+\int_{\Omega}{(u_{1m}\partial_{x_{1}}u_{2m}+u_{2m}\partial_{x_{2}}u_{2m}+u_{3m}\partial_{x_{3}}u_{2m})}\,w_{2j}+\int_{\Omega}{\partial_{x_{2}}p\,w_{2j}}=\int_{\Omega}{\Delta
u_{2m}\,w_{2j}}\\\
&\int_{\Omega}{\partial_{t}u_{3m}w_{3j}}+\int_{\Omega}{(u_{1m}\partial_{x_{1}}u_{3m}+u_{2m}\partial_{x_{2}}u_{3m}+u_{3m}\partial_{x_{3}}u_{3m})}\,w_{3j}+\int_{\Omega}{\partial_{x_{3}}p\,w_{3j}}=\int_{\Omega}{\Delta
u_{3m}\,w_{3j}}\\\ &\quad\quad u_{im}(0)=u_{i0}^{m},\quad\quad j=1,\cdots,m\\\
\end{split}$ (14)
where $u_{i0}^{m}$ is the orthogonal projection in $H$ of $u_{i0}$ on the
space spanned by $w_{i1},\cdots,w_{im}$.
We now are allowed to differentiate (14) in the $t$, we get
$\begin{split}&\int_{\Omega}{\partial_{t}^{2}u_{1m}w_{1j}}+\int_{\Omega}{(\partial_{t}u_{1m}\partial_{x_{1}}u_{1m}+\partial_{t}u_{2m}\partial_{x_{2}}u_{1m}+\partial_{t}u_{3m}\partial_{x_{3}}u_{1m})}\,w_{1j}+\\\
&\quad+\int_{\Omega}{(u_{1m}\partial_{x_{1}}\partial_{t}u_{1m}+u_{2m}\partial_{x_{2}}\partial_{t}u_{1m}+u_{3m}\partial_{x_{3}}\partial_{t}u_{1m})}\,w_{1j}+\int_{\Omega}{\partial_{x_{1}}\partial_{t}p\,w_{1j}}=\int_{\Omega}{\Delta\partial_{t}u_{1m}\,w_{1j}}\\\
&\int_{\Omega}{\partial_{t}^{2}u_{2m}w_{2j}}+\int_{\Omega}{(\partial_{t}u_{1m}\partial_{x_{1}}u_{2m}+\partial_{t}u_{2m}\partial_{x_{2}}u_{2m}+\partial_{t}u_{3m}\partial_{x_{3}}u_{2m})}\,w_{2j}+\\\
&\quad+\int_{\Omega}{(u_{1m}\partial_{x_{1}}\partial_{t}u_{2m}+u_{2m}\partial_{x_{2}}\partial_{t}u_{2m}+u_{3m}\partial_{x_{3}}\partial_{t}u_{2m})}\,w_{2j}+\int_{\Omega}{\partial_{x_{2}}\partial_{t}p\,w_{2j}}=\int_{\Omega}{\Delta\partial_{t}u_{2m}\,w_{2j}}\\\
&\int_{\Omega}{\partial_{t}^{2}u_{3m}w_{3j}}+\int_{\Omega}{(\partial_{t}u_{1m}\partial_{x_{1}}u_{3m}+\partial_{t}u_{2m}\partial_{x_{2}}u_{3m}+\partial_{t}u_{3m}\partial_{x_{3}}u_{3m})}\,w_{3j}+\\\
&\quad+\int_{\Omega}{(u_{1m}\partial_{x_{1}}\partial_{t}u_{3m}+u_{2m}\partial_{x_{2}}\partial_{t}u_{3m}+u_{3m}\partial_{x_{3}}\partial_{t}u_{3m})}\,w_{3j}+\int_{\Omega}{\partial_{x_{3}}\partial_{t}p\,w_{3j}}=\int_{\Omega}{\Delta\partial_{t}u_{3m}\,w_{3j}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad j=1,\cdots,m\\\ \end{split}$
(15)
We multiply (15) by ${g}^{\prime}_{ij}(t)$ and add the resulting equations for
$j=1,\cdots,m$, we find
$\begin{split}&\frac{1}{2}\partial_{t}\int_{\Omega}{(\partial_{t}u_{1m})^{2}}+\int_{\Omega}{\partial_{t}u_{1m}(\partial_{t}u_{1m}\partial_{x_{1}}u_{1m}+\partial_{t}u_{2m}\partial_{x_{2}}u_{1m}+\partial_{t}u_{3m}\partial_{x_{3}}u_{1m})}+\\\
&\quad\quad+\int_{\Omega}{\partial_{t}u_{1m}(u_{1m}\partial_{x_{1}}\partial_{t}u_{1m}+u_{2m}\partial_{x_{2}}\partial_{t}u_{1m}+u_{3m}\partial_{x_{3}}\partial_{t}u_{1m})}+\int_{\Omega}{\partial_{t}u_{1m}\partial_{x_{1}}\partial_{t}p}=\int_{\Omega}{\partial_{t}u_{1m}\,\Delta\partial_{t}u_{1m}}\\\
&\frac{1}{2}\partial_{t}\int_{\Omega}{(\partial_{t}u_{2m})^{2}}+\int_{\Omega}{\partial_{t}u_{2m}(\partial_{t}u_{1m}\partial_{x_{1}}u_{2m}+\partial_{t}u_{2m}\partial_{x_{2}}u_{2m}+\partial_{t}u_{3m}\partial_{x_{3}}u_{2m})}+\\\
&\quad\quad+\int_{\Omega}{\partial_{t}u_{2m}(u_{1m}\partial_{x_{1}}\partial_{t}u_{2m}+u_{2m}\partial_{x_{2}}\partial_{t}u_{2m}+u_{3m}\partial_{x_{3}}\partial_{t}u_{2m})}+\int_{\Omega}{\partial_{t}u_{2m}\partial_{x_{2}}\partial_{t}p}=\int_{\Omega}{\partial_{t}u_{2m}\,\Delta\partial_{t}u_{2m}}\\\
&\frac{1}{2}\partial_{t}\int_{\Omega}{(\partial_{t}u_{3m})^{2}}+\int_{\Omega}{\partial_{t}u_{3m}(\partial_{t}u_{1m}\partial_{x_{1}}u_{3m}+\partial_{t}u_{2m}\partial_{x_{2}}u_{3m}+\partial_{t}u_{3m}\partial_{x_{3}}u_{3m})}+\\\
&\quad\quad+\int_{\Omega}{\partial_{t}u_{3m}(u_{1m}\partial_{x_{1}}\partial_{t}u_{3m}+u_{2m}\partial_{x_{2}}\partial_{t}u_{3m}+u_{3m}\partial_{x_{3}}\partial_{t}u_{3m})}+\int_{\Omega}{\partial_{t}u_{3m}\partial_{x_{3}}\partial_{t}p}=\int_{\Omega}{\partial_{t}u_{3m}\,\Delta\partial_{t}u_{3m}}\\\
\end{split}$
and
$\int_{\Omega}{(\partial_{t}u_{1m}\partial_{x_{1}}\partial_{t}p+\partial_{t}u_{2m}\partial_{x_{2}}\partial_{t}p+\partial_{t}u_{3m}\partial_{x_{3}}\partial_{t}p)}=-\int_{\Omega}{\partial_{t}p\,\partial_{t}(\partial_{x_{1}}u_{1m}+\partial_{x_{2}}u_{2m}+\partial_{x_{3}}u_{3m})}=0$
Since
$\begin{split}&\int_{\Omega}{\partial_{t}u_{im}(u_{1m}\partial_{x_{1}}\partial_{t}u_{im}+u_{2m}\partial_{x_{2}}\partial_{t}u_{im}+u_{3m}\partial_{x_{3}}\partial_{t}u_{im})}=\\\
&\quad=\frac{1}{2}\int_{\Omega}{(u_{1m}\partial_{x_{1}}(\partial_{t}u_{im})^{2}+u_{2m}\partial_{x_{2}}(\partial_{t}u_{im})^{2}+u_{3m}\partial_{x_{3}}(\partial_{t}u_{im})^{2})}\\\
&\quad=-\frac{1}{2}\int_{\Omega}{(\partial_{t}u_{im})^{2}(\partial_{x_{1}}u_{1m}+\partial_{x_{2}}u_{2m}+\partial_{x_{3}}u_{3m})}=0\\\
\end{split}$
and
$\begin{split}&\int_{\Omega}{\partial_{t}u_{im}\,\Delta\partial_{t}u_{im}}=\int_{\Omega}{\partial_{t}u_{im}(\partial_{x_{1}}^{2}\partial_{t}u_{im}+\partial_{x_{2}}^{2}\partial_{t}u_{im}+\partial_{x_{3}}^{2}\partial_{t}u_{im})}=\\\
&=-\int_{\Omega}{((\partial_{x_{1}}\partial_{t}u_{im})^{2}+(\partial_{x_{2}}\partial_{t}u_{im})^{2}+(\partial_{x_{3}}\partial_{t}u_{im})^{2})},\quad\quad
i=1,2,3\\\ \end{split}$
then
$\begin{split}&\frac{1}{2}\partial_{t}\int_{\Omega}{((\partial_{t}u_{1m})^{2}+(\partial_{t}u_{2m})^{2}+(\partial_{t}u_{3m})^{2})}+\\\
&\qquad\quad+\left\|{\nabla\partial_{t}u_{1m}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\partial_{t}u_{2m}}\right\|_{L^{2}(\Omega)}^{2}+\left\|{\nabla\partial_{t}u_{3m}}\right\|_{L^{2}(\Omega)}^{2}\\\
&\\\
&\leq\left\|{\partial_{t}u_{1m}}\right\|_{L^{4}(\Omega)}\left({\left\|{\partial_{t}u_{1m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{1}}u_{1m}}\right\|_{L^{2}(\Omega)}+\left\|{\partial_{t}u_{2m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{2}}u_{1m}}\right\|_{L^{2}(\Omega)}\;+}\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\left.{\left\|{\partial_{t}u_{3m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{3}}u_{1m}}\right\|_{L^{2}(\Omega)}}\right)\\\
&+\left\|{\partial_{t}u_{2m}}\right\|_{L^{4}(\Omega)}\left({\left\|{\partial_{t}u_{1m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{1}}u_{2m}}\right\|_{L^{2}(\Omega)}+\left\|{\partial_{t}u_{2m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{2}}u_{2m}}\right\|_{L^{2}(\Omega)}\;+}\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\left.{\left\|{\partial_{t}u_{3m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{3}}u_{2m}}\right\|_{L^{2}(\Omega)}}\right)\\\
&+\left\|{\partial_{t}u_{3m}}\right\|_{L^{4}(\Omega)}\left({\left\|{\partial_{t}u_{1m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{1}}u_{3m}}\right\|_{L^{2}(\Omega)}+\left\|{\partial_{t}u_{2m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{2}}u_{3m}}\right\|_{L^{2}(\Omega)}\;+}\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\left.{\left\|{\partial_{t}u_{3m}}\right\|_{L^{4}(\Omega)}\left\|{\partial_{x_{3}}u_{3m}}\right\|_{L^{2}(\Omega)}}\right)\\\
&\leq\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{4}(\Omega)}^{2}}}\right)^{1/2}\left({\sum\limits_{j=1}^{3}{\left\|{\partial_{t}u_{jm}}\right\|_{L^{4}(\Omega)}^{2}}}\right)^{1/2}\left({\sum\limits_{i,j=1}^{3}{\left\|{\partial_{x_{i}}u_{jm}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}\\\
\end{split}$
where
$\begin{split}&\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{4}(\Omega)}^{2}}\leq
2\sum\limits_{i=1}^{3}{\left({\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{1/2}\left\|{\nabla\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{3/2}}\right)}\\\
&\quad\leq
2\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/4}\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{3/4}\\\
\end{split}$
so that
$\begin{split}&\partial_{t}\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)+2\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\\\
&\quad\leq
2^{2}\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/4}\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{3/4}\left({\sum\limits_{i=1}^{3}{\left\|{\nabla
u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}\\\ &\quad\leq
3^{3}\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\left({\sum\limits_{i=1}^{3}{\left\|{\nabla
u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{2}+\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\\\
\end{split}$
it follows that
$\partial_{t}\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)+\left({\sum\limits_{i=1}^{3}{\left\|{\nabla\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\leq\phi_{m}(t)\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)$
where $\phi_{m}(t)=3^{3}\left({\sum\limits_{i=1}^{3}{\left\|{\nabla
u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{2}$.
Introducing a stream function: $\psi=(\psi_{2},\psi_{2},\psi_{3})$,
$\mbox{curl}\psi=(\partial_{x_{2}}\psi_{3}-\partial_{x_{3}}\psi_{2},\;\,\;\partial_{x_{3}}\psi_{1}-\partial_{x_{1}}\psi_{3},\;\,\;\partial_{x_{1}}\psi_{2}-\partial_{x_{2}}\psi_{1})$
According to $\omega=\mbox{curl}u$, $u=\mbox{curl}\psi$ and
$\mbox{div}\psi=0$, we have
$\mbox{curlcurl}\psi=-\Delta\psi=\omega,\quad-\Delta\mbox{curl}\psi=\mbox{curl}\omega$
That is, $-\Delta u=\mbox{curl}\omega$. Then $(-\Delta
u,\;\,u)=(\mbox{curl}\omega,\;\,u)$, where
$\begin{split}&(-\Delta u,\;\,u)=\sum\limits_{i=1}^{3}{(-\Delta
u_{i},\;\,u_{i})}=\sum\limits_{i=1}^{3}{(\nabla u_{i},\;\,\nabla
u_{i})}=\sum\limits_{i=1}^{3}{\left\|{\nabla
u_{i}}\right\|_{L^{2}(\Omega)}^{2}}\\\ \\\
&(\mbox{curl}\omega,\;\,u)=(\partial_{x_{2}}\omega_{3}-\partial_{x_{3}}\omega_{2},\;\;u_{1})+(\partial_{x_{3}}\omega_{1}-\partial_{x_{1}}\omega_{3},\;\;u_{2})+(\partial_{x_{1}}\omega_{2}-\partial_{x_{2}}\omega_{1},\;\;u_{3})\\\
&\quad\quad\quad=-(\omega_{3},\;\partial_{x_{2}}u_{1})+(\omega_{2},\;\partial_{x_{3}}u_{1})-(\omega_{1},\;\partial_{x_{3}}u_{2})+(\omega_{3},\;\partial_{x_{1}}u_{2})-(\omega_{2},\;\partial_{x_{1}}u_{3})+(\omega_{1},\;\partial_{x_{2}}u_{3})\\\
&\quad\quad\quad=(\omega_{1},\;\;\partial_{x_{2}}u_{3}-\partial_{x_{3}}u_{2})+(\omega_{2},\;\;\partial_{x_{3}}u_{1}-\partial_{x_{1}}u_{3})+(\omega_{3},\;\;\partial_{x_{1}}u_{2}-\partial_{x_{2}}u_{1})\\\
&\quad\quad\quad=(\omega,\;\,\mbox{curl}u)=(\omega,\omega)=\sum\limits_{i=1}^{3}{\left\|{\omega_{i}}\right\|_{L^{2}(\Omega)}^{2}}\\\
\end{split}$
Hence,
$\left({\sum\limits_{i=1}^{3}{\left\|{\nabla
u_{i}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}=\left({\sum\limits_{i=1}^{3}{\left\|{\omega_{i}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{1/2}$
it follows that
$\phi_{m}(t)=3^{3}\left({\sum\limits_{i=1}^{3}{\left\|{\omega_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)^{2}<+\infty$
By the Gronwall inequality,
$\frac{d}{dt}\left\\{{\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}}\right\|_{L^{2}(\Omega)}^{2}}}\right)\;\exp\left({-\int_{0}^{t}{\phi_{m}(s)ds}}\right)}\right\\}\leq
0$
whence
$\mathop{\sup}\limits_{t\in(0,T)}\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}(t)}\right\|_{L^{2}(\Omega)}^{2}}}\right)\leq\left({\sum\limits_{i=1}^{3}{\left\|{\partial_{t}u_{im}(0)}\right\|_{L^{2}(\Omega)}^{2}}}\right)\;\exp\left({\int_{0}^{T}{\phi_{m}(s)ds}}\right)$
Therefore
$\partial_{t}u_{im}\in L^{\infty}(0,T;\;H)\cap L^{2}(0,T;\;V),\qquad\qquad
i=1,2,3$
Similar to the Theorem 3.8 in Chapter 3 of [4], we obtain
$u_{i}\in L^{\infty}(0,T;\;H^{2}(\Omega)),\qquad\qquad i=1,2,3$
Remark 1. Noting that $(-\Delta
u,\;v)=(-\partial_{t}u-(u\cdot\nabla)u,\;\,v)$. If $\partial_{t}u$ and
$(u\cdot\nabla)u$ are of some degree of continuity, then $u$ can reach a
higher degree of continuity, based on the smoothing effect of inverse elliptic
operator $\Delta^{-1}$. By repeated application of this process one can prove
that the solution $u$ is in $C^{\infty}(\Omega\times(0,T))$.
Remark 2. For handling the initial value problem of 3D Navier-Stokes
equation, a weighted function is introduced and some conditions for the
initial value $u_{0}$ are needed. Based on problems separated and potential
theory of fluid flow, we may keep the same result for the general initial-
boundary value problems under the assumptions of regularity on the boundary
and data.
## References
* [1] R. A. Adams, and J. J. F. Fournier, Sobolev Spaces, Second ed., Pure and Applied Mathematics, Elsevier, Oxford, (2003);
* [2] O.A.Ladyženskaya, V.A.Solonnikov, and N.N.Ural’ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, (1988);
* [3] Qun Lin, and Lung-an Ying, Interval Vorticity Methods, (2009);
* [4] R. Temam, Navier-Stokes equations Theory and numerical analysis, Reprint of the 1984, AMS Chelsea Publishing, Providence, R.I., (2001).
|
arxiv-papers
| 2013-08-10T08:45:05 |
2024-09-04T02:49:49.285818
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Qun Lin",
"submitter": "Qun Lin",
"url": "https://arxiv.org/abs/1308.2297"
}
|
1308.2376
|
# Saari’s Conjecture for Elliptical Type $N$-Body Problem and An
Application††thanks: Supported partially by NSF of China
Xiang Yu111Email:[email protected] and Shiqing
Zhang222Email:[email protected]
Department of Mathematics, Sichuan University, Chengdu 610064, People’s
Republic of China
Abstract: By using an arithmetic fact, we will firstly prove Saari’s
conjecture in a particular case, which is called the Elliptical Type N-Body
Problem, and then we apply it to prove that the variational minimal solution
of the planar Newtonian N-body problem is precisely a relative equilibrium
solution whose configuration minimizes the function $IU^{2}$, it’s worth
noticing that we don’t need the hypothesis of Finiteness of Central
Configurations. In the Planetary Restricted Problem (which ignore all the
mutual gravitational interactions between the planets), the corresponding
Saari’s conjecture is stated and proved.
Key Words: N-body problems, Central configurations, Saari’s conjecture,
Variational minimization, the Planetary Restricted Problem, Homographic
solutions.
2000AMS Subject Classification 11J17, 11J71, 34C25, 42A16, 70F10, 70F15,
70G75.
## 1 Introduction
In 1970, Donald Saari [31] proposed the following conjecture : In the
Newtonian $N$-body problem, if the moment of inertia,
$I=\Sigma^{n}_{k=1}m_{k}|q_{k}|^{2}$, is constant, where
$q_{1},q_{2},\cdots,q_{n}$ represent the position vectors of the bodies of
masses $m_{1},\cdots,m_{n}$, then the corresponding solution is a relative
equilibrium. In other words: Newtonian particle systems of constant moment of
inertia rotate like rigid bodies.
A lot of energies have been spent to understand Saari’s conjecture, but most
of those works ( such as [27, 28]) failed to achieve crucial results. However
there have been a few successes in the struggle to understand Saari’s
conjecture. McCord [23] proved that the conjecture is true for three bodies of
equal masses. Llibre and Pina [21] gave an alternative proof of this case, but
they never published it.In particular, Moeckel [25, 26] obtained a computer-
assisted proof for the Newtonian three-body problem with positive masses when
physical space is $\mathbb{R}^{d}$ for all positive integer $d\geq 2$. Diacu,
P$\acute{\rm e}$rez-Chavela, and Santoprete [15] showed that the conjectre is
true for any $n$ in the collinear case for potentials that depend only on the
mutual distances between point masses. Roberts and Melanson [30] showed that
the conjecture is true for the restricted three-body problem using a computer-
assisted proof. There have been results, such as [29, 32, 33], which studied
the conjecture in other contexts than the Newtonian $N$-body problem.
Recently the interest in this conjecture has grown considerably due to the
discovery of the figure eight solution [10], which, as numerical arguments
show, has an approximately constant moment of inertia but is not a relative
equilibrium. In recent years, for a natural extension of the original Saari’s
conjecture, namely Saari’s homographic conjecture, some mathematicians have
made some progress [14, 17, 18].
The variational minimal solutions of the N-body problem are attractive, since
they are nature from the viewpoint of the principle of least action.
Unfortunately, there were very few works about the variational minimal
solutions before 2000. It’s worth noticing that a lot of results have been got
by the action minimization methods in recent years, please see [3, 4, 5, 6, 7,
8, 9, 10, 11, 16, 22, 36, 37, 38, 39] and the references there.
Let $\mathcal{X}_{d}$ denote the space of configurations of $N\geq 2$ point
particles with masses $m_{1},\ldots,m_{N}$ in Euclidean space $\mathbb{R}^{d}$
of dimension $d$, whose center of masses is at the origin, that is,
$\mathcal{X}_{d}=\\{q=(q_{1},\cdots,q_{N})\in(\mathbb{R}^{d})^{N}:\sum_{i=1}^{N}{m_{i}q_{i}}=0\\}$.
Let $\mathbb{T}=\mathbb{R}/T\mathbb{Z}$ denote the circle of length
$T=|\mathbb{T}|$, embedded as $\mathbb{T}\subset\mathbb{R}^{2}$.By the loop
space $\Lambda$, we mean the Sobolev space
$\Lambda=H^{1}(\mathbb{T},\mathcal{X}_{d})$. We consider the opposite of the
potential energy (force function) defined by
$U(q)=\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}.$ (1.1)
The kinetic energy is defined (on the tangent bundle of $\mathcal{X}_{d}$) by
$K=\sum_{i=1}^{N}{\frac{1}{2}{m_{i}|\dot{q}_{i}|^{2}}}$, the total energy is
$E=K-U$ and the Lagrangian is
$L(q,\dot{q})=L=K+U=\sum_{i}\frac{1}{2}m_{i}|\dot{q}|^{2}+\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}$.
Given the Lagrangian L, the positive definite functional
$\mathcal{{A}}:\Lambda\rightarrow\mathbb{R}\cup\\{+\infty\\}$ defined by
$\mathcal{{A}}(q)=\int_{\mathbb{T}}{L(q(t),\dot{q}(t))dt}.$ (1.2)
is termed as action functional (or the Lagrangian action).
The action functional $\mathcal{{A}}$ is of class $C^{1}$ on the subspace
$\hat{\Lambda}\subset\Lambda$, which is collision-free space. Hence critical
point of $\mathcal{{A}}$ in $\hat{\Lambda}$ are T-periodic classical solutions
(of class $C^{2}$) of Newton’s equations
$m_{i}\ddot{q}_{i}=\frac{\partial U}{\partial q_{i}}.$ (1.3)
Definition [35]. A configuration
$q=(q_{1},\cdots,q_{N})\in{\mathcal{X}}_{d}\setminus\Delta_{d}$ is called a
central configuration if there exists a constant $\lambda\in{\mathbb{R}}$ such
that
$\sum_{j=1,j\neq
k}^{N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leq k\leq N$ (1.4)
The value of $\lambda$ in (1.1) is uniquely determined by
$\lambda=\frac{U(q)}{I(q)}$ (1.5)
Where
$\Delta_{d}=\left\\{q=(q_{1},\cdots,q_{N})\in(\mathbb{R}^{d})^{N}:q_{j}=q_{k}~{}\mbox{for~{}some}~{}j\neq
k\right\\}$ (1.6) $I(q)=\sum_{1\leq j\leq
N}m_{j}|q_{j}|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(1.7)
It’s well known that the central configurations are the critical points of the
function $IU^{2}$, and $IU^{2}$ attains its infimum on
${\mathcal{X}}_{d}\setminus\Delta_{d}$. Furthermore, we know [24] that
$inf_{{\mathcal{X}}_{2}\setminus\Delta_{2}}{IU^{2}}<inf_{{\mathcal{X}}_{1}\setminus\Delta_{1}}{{IU^{2}}}$and
$inf_{{\mathcal{X}}_{3}\setminus\Delta_{3}}{{IU^{2}}}<inf_{{\mathcal{X}}_{2}\setminus\Delta_{2}}{IU^{2}}$
when$N\geq 4$. When $N\geq 4$ and ${\mathbb{R}^{d}}={\mathbb{R}^{3}}$, it is
well known that the homographic solutions derived by the central
configurations minimizing the function $IU^{2}$ are homothetic, furthermore, a
homographic motion in ${\mathbb{R}^{3}}$ which is not homothetic takes place
in a fixed plane[1, 2, 8, 35].This is an important reason for us only to
consider $d=2$. In fact, A. Chenciner [8] and Zhang-Zhou [38] had proved that
the minimizer of Lagrangian action among (anti)symmetric loops for the spatial
$N$-body($N\geq 4$) problem is a collision-free non-planar solution. From the
results of A. Albouy and A. Chenciner [1], our idea can be applied to the case
that $d$ is any positive even number, however, for the sake of simplicity, we
only consider the case $d=2$.
The paper is structured as follows. Section 2 introduces the Planetary
Restricted Problem and gives a precise statement of Saari’s Conjecture for the
Planetary Restricted Problem. Section 3 gives our main results. Section 4
gives the statements and proofs of some lemmas which are useful and
interesting for themselves. Finally, Section 5 gives the proofs of the main
results in Section 3 by using the lemmas in Section 4.
## 2 Saari’s Conjecture for the Planetary Restricted Problem
The evolution of $(1+N)$-body systems (one can see [12]) interacting only
through gravitational attraction is governed by Newton’s equations (1.3).
Equations (1.3) are equivalent to the standard Hamilton’s equations
corresponding to the Hamiltonian function
$H(p,q)=K-U=\sum_{0\leq i\leq N}\frac{1}{2m_{i}}|p_{i}|^{2}-\sum_{0\leq
i<j\leq N}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}$ (2.8)
where $(p,q)=(p_{0},\cdots,p_{N};q_{0},\cdots,q_{N})$ are standard symplectic
variables. The symplectic form is the standard one.
Introducing the symplectic coordinate change $(p,q)=\phi_{hel}(P,Q)$:
$\phi_{hel}:\begin{array}[]{c}q_{0}=Q_{0},q_{i}=Q_{0}+Q_{i}(i=1,\cdots,N)\\\
p_{0}=P_{0}-\sum_{1\leq i\leq N}P_{i},p_{i}=P_{i}(i=1,\cdots,N)\end{array}$
(2.9)
one sees that the new Hamiltonian $H_{hel}=H\circ\phi_{hel}$ does not depend
upon $Q_{0}$. This means that $P_{0}$ (total linear momentum) is a global
integral of motion. Without loss of generality, one can suppose that $P_{0}=0$
since the invariance of the equation (1.3) under the changes of inertial
reference frames.
In the “planetary” case, one assumes that one of the bodies, say $i=0$ (the
Sun), has mass much larger than that of the other bodies (this accounts for
the index ”hel”, which stands for “heliocentric”).To make the problem
transparent, one may introduce the following rescalings. Let
$m_{i}=\epsilon\widetilde{m}_{i},y_{i}=\frac{P_{i}}{\epsilon
m^{{5}/{3}}_{0}},x_{i}=\frac{Q_{i}}{m^{{2}/{3}}_{0}},(i=1,\cdots,N)$, we
rescale time by a factor $\epsilon m^{{7}/{3}}_{0}$ (which amounts to dividing
the new Hamiltonian by such a factor); then, the flow of the Hamiltonian
function $H_{hel}$ is equivalent to the flow of the following Hamiltonian
function:
$H_{new}(y,x)=\sum_{1\leq i\leq
N}(\frac{|y_{i}|^{2}}{2\mu_{i}}-\frac{\mu_{i}M_{i}}{|x_{i}|})+\epsilon\sum_{1\leq
i<j\leq N}(y_{i}\cdot
y_{j}-\frac{\widetilde{m}_{i}\widetilde{m}_{j}/{m^{2}_{0}}}{|x_{i}-x_{j}|}),$
(2.10)
where the mass parameters are defined as
$M_{i}\triangleq
1+\epsilon\frac{\widetilde{m}_{i}}{m_{0}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mu_{i}\triangleq\frac{\widetilde{m}_{i}}{m_{0}+\epsilon\widetilde{m}_{i}}=\frac{\widetilde{m}_{i}}{m_{0}}\frac{1}{M_{i}}$
(2.11)
By using these elements, the moment of inertia
$I=\Sigma^{N}_{i=0}m_{i}|q_{i}|^{2}$ and force function
$U(q)=\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}$ can be expressed as
$I=\Sigma^{N}_{i=0}m_{i}|q_{i}|^{2}=\epsilon m^{{4}/{3}}_{0}[\sum_{1\leq i\leq
N}\widetilde{m}_{i}|x_{i}|^{2}-\frac{\epsilon(\sum_{1\leq i\leq
N}\widetilde{m}_{i}x_{i})^{2}}{\epsilon\sum_{1\leq i\leq
N}\widetilde{m}_{i}+m_{0}}]$ (2.12) $U=\epsilon m^{{4}/{3}}_{0}[\sum_{1\leq
i\leq N}\frac{\mu_{i}M_{i}}{|x_{i}|}+\epsilon\sum_{1\leq i<j\leq
N}{\frac{\widetilde{m}_{i}\widetilde{m}_{j}/m^{2}_{0}}{|x_{i}-x_{j}|}}]$
(2.13)
By using rescalings, we can think that
$I=\sum_{1\leq i\leq N}\widetilde{m}_{i}|x_{i}|^{2}-\frac{\epsilon(\sum_{1\leq
i\leq N}\widetilde{m}_{i}x_{i})^{2}}{\epsilon\sum_{1\leq i\leq
N}\widetilde{m}_{i}+m_{0}}$ (2.14) $U=\sum_{1\leq i\leq
N}\frac{\mu_{i}M_{i}}{|x_{i}|}+\epsilon\sum_{1\leq i<j\leq
N}{\frac{\widetilde{m}_{i}\widetilde{m}_{j}/m^{2}_{0}}{|x_{i}-x_{j}|}}$ (2.15)
For the Planetary Restricted Problem, that is the Planetary Problem when
$\epsilon=0$, the Hamiltonian becomes
$H_{0}(y,x)=\sum_{1\leq i\leq
N}(\frac{|y_{i}|^{2}}{2\varrho_{i}}-\frac{\varrho_{i}}{|x_{i}|}),$ (2.16)
where $\varrho_{i}=\frac{\widetilde{m}_{i}}{m_{0}}$. The systems with
Hamiltonian $H_{0}$ are integrable and represent the sum of N two-body systems
formed by the Sun and the $i$-th planet (disregarding the interaction with the
other planets). In the same time, the moment of inertia $I$ and force function
$U$ become
$I_{0}=\sum_{1\leq i\leq N}\widetilde{m}_{i}|x_{i}|^{2}$ (2.17)
$U_{0}=\sum_{1\leq i\leq N}\frac{\varrho_{i}}{|x_{i}|}$ (2.18)
For Two-body Problem (one can see [19]), Newton’s equation is
$\ddot{\mathbf{r}}=-\frac{\kappa\mathbf{r}}{|\mathbf{r}|^{3}},$ (2.19)
suppose the solution $\mathbf{r}(t)$ is ellipse, $a$ denotes semi-major axis,
$e$ denotes eccentricity, $T$ denotes period, $\tilde{n}=2\pi/T$ denotes mean
motion, $E$ denotes eccentric anomaly, $\tau=\tilde{n}(t-\iota)$ denotes mean
anomaly, where $\iota$ denotes time of perihelion passage. There are Kepler’s
Third Law: $\tilde{n}^{2}a^{3}=\kappa$ and Kepler equation: $E-e\sin E=\tau$.
Let $r=|\mathbf{r}|$, then $r(t)=a[1-e\cos E]$, furthermore, $E(mod2\pi)$ is
periodic with period $T$. For the Two-body Problem corresponds to the
Planetary Restricted Problem
$\ddot{x_{i}}=-\frac{x_{i}}{|x_{i}|^{3}},$ (2.20)
suppose the solution $x_{i}(t)$ is ellipse, then ${|x_{i}|}=a_{i}(1-e_{i}\cos
E_{i})$, where $E_{i}(mod2\pi)$ is periodic with period $T_{i}$.
It is obvious that, in the Planetary Restricted Problem, if every point
particle moves uniformly in circular orbit, then the moment of inertia,
$I_{0}=\sum_{1\leq i\leq N}\widetilde{m}_{i}|x_{i}|^{2}$, is constant. In the
Planetary Restricted Problem, the Saari’s Conjecture says this is the only
case: if the moment of inertia, $I_{0}=\sum_{1\leq i\leq
N}\widetilde{m}_{i}|x_{i}|^{2}$, is constant, then every point particle moves
uniformly in circular orbit, that is, every eccentricity $e_{i}(i=1,\cdots,N)$
must be zero.
## 3 Main Results
The main results in this paper are the following theorems:
###### Theorem 3.1
Saari’s Conjecture is true if $i$-th point particle has mode of motion
$q_{i}(t)=a_{i}\cos(\theta(t))+b_{i}\sin(\theta(t)),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (3.21)
and $a_{i},b_{i}\in\mathbb{R}^{d}$ for all $i=1,\ldots,N$,
$[\varphi,\varphi+\pi]\subseteq\\{\theta(t):t\in\mathbb{T}\\}$ for some
$\varphi\in\mathbb{R}$. In particular, Saari’s Conjecture is true when
$\theta(t)=\frac{2\pi}{T}t$.
###### Corollary 3.2
Saari’s Conjecture is true if in a barycentric reference frame the
configurations formed by the bodies remain the central configurations all the
time.
Remark. If the Conjecture on the Finiteness of Central Configurations is true
[20, 34, 35], then the Corollary 3.2 is obvious, but we don’t need this
hypothesis here, so the Corollary 3.2 is not trivial.
###### Theorem 3.3
In the Planetary Restricted Problem, the Saari’s Conjecture is true.
###### Theorem 3.4
For Newtonian N-body problem, the regular solutions minimizing the functional
${\mathcal{A}}$ in $\mathcal{S}={\\{q\in
H^{1}(\mathbb{T},(\mathbb{R}^{2})^{N}):\int_{\mathbb{T}}{q(t)dt}=0}\\}$ are
precisely the relative equilibrium solutions whose configurations minimize the
function $IU^{2}$ in ${\mathbb{R}^{2}}$.
Remark. Compared with the result of A.Chenciner [8] and Checiner-Desolneux
[9]: For the planar $N$-body problem, a relative equilibrium solution whose
configuration minimizes $I^{\frac{1}{2}}U$ is always a minimizer of the action
on $\mathcal{S}$; moreover, all minimizers are of this form provided there
exists only a finite number of similitude classes of $N$-body central
configurations. For the second part, he could only prove rigorously for 3-body
and 4-body problems, since we know that the Conjecture on the Finiteness of
Central Configurations have only been proved for 3-body and 4-body problems
until now [20].
## 4 Some Lemmas
Let $[t]$ denote the unique integer such that $t-1<[x]\leq t$ for any real
$t$. The difference $t-[t]$ is written as $\\{t\\}$ and satisfies
$0\leq\\{t\\}<1$.
First of all, we need a famous arithmetic fact which belongs to Kronecker:
###### Lemma 4.1
If 1,$\theta_{1}$, …, $\theta_{n}$ are linearly independent over the rational
field, then the set {($\\{k\theta_{1}\\}$, …, $\\{k\theta_{n}\\}$):
$k\in\mathbb{N}\\}$ are dense in the $n$-dim unite cube
$\\{(\varphi_{1},\ldots,\varphi_{n}):0\leq\varphi_{i}\leq 1,i=1,\ldots,n\\}$.
In the following, we will prove three lemmas which are needed to prove our
main results, and these lemmas are also interesting for themselves.
###### Lemma 4.2
Given $\theta_{1}$, …, $\theta_{n}$ and any $\epsilon>0$, there are infinitely
many integers $k\in\mathbb{N}$ such that $\\{k\theta_{i}\\}<\epsilon$ or
$\\{k\theta_{i}\\}>1-\epsilon$ for every $i=1,\ldots,n$.
Proof of Lemma 4.2:
If all of $\theta_{1}$, …, $\theta_{n}$ are rational, the proposition is
obviously right. Hence, without loss of generality, we will suppose that
1,$\theta_{1}$, …, $\theta_{l}$($1\leq l\leq n$) are linearly independent over
the rational field and $\theta_{l+1}$, …, $\theta_{n}$ can be spanned by
rational linear combination, that is, we have
$\theta_{i}=x_{i}^{0}+\sum_{1\leq j\leq l}x_{i}^{j}\theta_{j}$, where $l<i\leq
n$ and $x_{i}^{j}$ are rational numbers for $0\leq j\leq l$. Let integer $p$
satisfy that all of $px_{i}^{0}$ are integers for $l<i$. It is easy to know
that 1,$p\theta_{1}$, …, $p\theta_{l}$ are still linearly independent over the
rational field. Then for any $\delta>0$, there are infinitely many integers
$k\in\mathbb{N}$ such that $\\{kp\theta_{i}\\}<\delta$ or
$\\{kp\theta_{i}\\}>1-\delta$ for every $i=1,\ldots,l$ by the
${\mathbf{Lemma~{}\ref{Kronecker}}}$ , and it is easy to know that
$\\{kp\theta_{i}\\}<C\delta$ or $\\{kp\theta_{i}\\}>1-C\delta$ for some
constant $C$ which only depends on $x_{i}^{j}$. So for any $\epsilon>0$, there
are infinitely many integers $k\in\mathbb{N}$ such that
$\\{k\theta_{i}\\}<\epsilon$ or $\\{k\theta_{i}\\}>1-\epsilon$ for every
$i=1,\ldots,n$.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
###### Lemma 4.3
If $U(q)\equiv const$, where $q=(q_{1},\cdots,q_{N})$,
$q_{i}(t)=a_{i}\cos(\theta(t))+b_{i}\sin(\theta(t)),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (4.22)
and $a_{i},b_{i}\in\mathbb{R}^{d}$ for all $i=1,\ldots,N$,
$[\varphi,\varphi+\pi]\subseteq\\{\theta(t):t\in\mathbb{T}\\}$ for some
$\varphi\in\mathbb{R}$. Then $q_{i}(t)(i=1,\ldots,N)$ is is a rigid motion.
Proof of Lemma 4.3:
Firstly, we expand $U(q(t))$ as Fourier series:
$\displaystyle U$ $\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|}$ $\displaystyle=$ $\displaystyle\sum_{1\leq
j<k\leq
N}\frac{m_{j}m_{k}}{[|a_{j}-a_{k}|^{2}\cos^{2}\theta(t)+|b_{j}-b_{k}|^{2}\sin^{2}\theta(t)+2(a_{j}-a_{k})\cdot(b_{j}-b_{k})\sin\theta(t)\cos\theta(t)]^{\frac{1}{2}}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{[\frac{|a_{j}-a_{k}|^{2}+|b_{j}-b_{k}|^{2}}{2}+(\frac{|a_{j}-a_{k}|^{2}-|b_{j}-b_{k}|^{2}}{2})\cos(2\theta(t))+(a_{j}-a_{k})\cdot(b_{j}-b_{k})\sin(2\theta(t))]^{\frac{1}{2}}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{[\frac{|a_{j}-a_{k}|^{2}+|b_{j}-b_{k}|^{2}}{2}+(\frac{|a_{j}-a_{k}|^{2}-|b_{j}-b_{k}|^{2}}{2})\cos(2\theta(t))+(a_{j}-a_{k})\cdot(b_{j}-b_{k})\sin(2\theta(t))]^{\frac{1}{2}}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{[A_{jk}+B_{jk}\cos(2\theta(t)+\theta_{jk})]^{\frac{1}{2}}}$
where
$A_{jk}=\frac{|a_{j}-a_{k}|^{2}+|b_{j}-b_{k}|^{2}}{2}$ (4.23)
$B_{jk}=[(\frac{|a_{j}-a_{k}|^{2}-|b_{j}-b_{k}|^{2}}{2})^{2}+((a_{j}-a_{k})\cdot(b_{j}-b_{k}))^{2}]^{\frac{1}{2}}$
(4.24)
and $\theta_{jk}$ can be determined when $B_{jk}>0$. In the following, we will
prove $B_{jk}=0$ for any $j,k\in\\{{1,\ldots,N}\\}$. It is easy to know that
$A_{jk}\geq B_{jk}$, let $C_{jk}=\frac{B_{jk}}{A_{jk}}$, then we have
$\displaystyle U$ $\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[1+(-\frac{1}{2})C_{jk}\cos(2\theta(t)+\theta_{jk})+\ldots+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}(C_{jk})^{n}\cos^{n}(2\theta(t)+\theta_{jk})+\ldots]$
$\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}\\{1+(-\frac{1}{2})C_{jk}\frac{\exp\sqrt{-1}(2\theta(t)+\theta_{jk})+\exp-\sqrt{-1}(2\theta(t)+\theta_{jk})}{2}+\ldots+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}(C_{jk})^{n}[\frac{\exp\sqrt{-1}(2\theta(t)+\theta_{jk})+\exp-\sqrt{-1}(2\theta(t)+\theta_{jk})}{2}]^{n}$
$\displaystyle+\ldots\\}$ $\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[1+(-\frac{1}{2})C_{jk}\frac{\exp\sqrt{-1}(2\theta(t)+\theta_{jk})+\exp-\sqrt{-1}(2\theta(t)+\theta_{jk})}{2}+\ldots+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}(C_{jk})^{n}\frac{\sum_{0\leq
l\leq n}\left(\begin{array}[]{c}n\\\ l\\\
\end{array}\right)\exp\sqrt{-1}((2\theta(t)+\theta_{jk})(2l-n))}{2^{n}}+$
$\displaystyle\ldots]$ $\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}\\{1+\sum_{1\leq
l}\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-2l+1)}{(2l)!}(C_{jk})^{2l}\frac{\left(\begin{array}[]{c}2l\\\
l\\\ \end{array}\right)}{2^{2l}}+$ $\displaystyle\sum_{1\leq
n}\exp\sqrt{-1}(2n\theta(t))[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp\sqrt{-1}(n\theta_{jk})}{2^{n}}+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp\sqrt{-1}(n\theta_{jk})}{2^{n+2}}+\ldots]+$
$\displaystyle\sum_{1\leq
n}\exp\sqrt{-1}(-2n\theta(t))[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp\sqrt{-1}(-n\theta_{jk})}{2^{n}}+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp\sqrt{-1}(-n\theta_{jk})}{2^{n+2}}+\ldots]\\}$
Since $U\equiv const$, then by the uniqueness of Fourier series we have
$\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp\sqrt{-1}(n\theta_{jk})}{2^{n}}+$
(4.25)
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp\sqrt{-1}(n\theta_{jk})}{2^{n+2}}+\ldots]=0$
$\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp-\sqrt{-1}(n\theta_{jk})}{2^{n}}+$
(4.26)
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp-\sqrt{-1}(n\theta_{jk})}{2^{n+2}}+\ldots]=0$
for any $n\geq 1$. Hence we have
$\sum_{1\leq j<k\leq N}D^{(n)}_{jk}\exp
2\pi\sqrt{-1}(n\frac{\theta_{jk}}{2\pi})=0$ (4.27)
for any $n\geq 1$, where
$D^{(n)}_{jk}=\frac{m_{j}m_{k}C_{jk}^{n}}{A_{jk}^{\frac{1}{2}}}[1+\frac{(\frac{1}{2}+n)(\frac{1}{2}+n+1)}{(n+1)(n+2)}\frac{(C_{jk})^{2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)}{2^{2}}+\ldots]$ (4.28)
We claim that the right side of the equation (4.28) is convergent. In fact,
let
$\displaystyle f_{jk}$
$\displaystyle=1+\frac{(\frac{1}{2}+n)(\frac{1}{2}+n+1)}{(n+1)(n+2)}\frac{(C_{jk})^{2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)}{2^{2}}+\ldots$
$\displaystyle=1+c_{1}(C_{jk})^{2}+c_{2}(C_{jk})^{4}+\ldots+c_{l}(C_{jk})^{2l}+\ldots$
where
$c_{l}=\frac{(\frac{1}{2}+n)(\frac{1}{2}+n+1)\ldots(2l-1-\frac{1}{2}+n)(2l-\frac{1}{2}+n)}{(n+1)(n+2)\ldots(n+2l-1)(n+2l)}\frac{\left(\begin{array}[]{c}n+2l\\\
n+l\\\ \end{array}\right)}{2^{2l}}$ (4.29)
Then we have
$\frac{c_{l+1}}{c_{l}}=\frac{(2l+\frac{1}{2}+n)(2l+1+\frac{1}{2}+n)}{4(l+1)(l+1+n)}$
(4.30) $\lim_{l\rightarrow\infty}\frac{c_{l+1}}{c_{l}}=1$ (4.31)
Hence the series of the equation (4.28) is convergent when $(C_{jk})^{2}<1$.
Furthermore, we can prove the convergence of the series for the equation
(4.28) by using Gauss’ text when $(C_{jk})^{2}=1$. In fact, we have
$\frac{c_{l}}{c_{l+1}}=1+\frac{\frac{n+2}{2}}{l}+\beta_{l}$ (4.32)
where
$\beta_{l}=-\frac{2n^{2}+2n+\frac{3}{4}+\frac{(n+\frac{1}{2})(n+\frac{3}{2})(n+2)}{2l}}{4l^{2}+2l(n+2)+(n+\frac{1}{2})(n+\frac{3}{2})}$
(4.33)
Since $\frac{n+2}{2}>1$ and $|\beta_{l}|\sim\frac{c}{l^{2}}$, where $c$ is a
constant, then it is easy to know that the series of the equation (4.28) is
convergent when $C_{jk}^{2}=1$.
From ${\mathbf{Lemma~{}\ref{shulun}}}$, we know there exists some $n$ such
that $n\frac{\theta_{jk}}{2\pi}=k_{n}+\varphi_{jk}$, where $k_{n}$ is an
integer and $-\frac{1}{4}<\varphi_{jk}<\frac{1}{4}$. Since $D^{(n)}_{jk}\geq
0$, there must be $D^{(n)}_{jk}=0$ for any $j,k$ by the equation (4.27). So we
have $C_{jk}=0$, $|q_{j}-q_{k}|\equiv\sqrt{A_{jk}}$.
Hence $q_{i}(t)(i=1,\ldots,N)$ is a rigid motion.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Remark. It is easy to know that the same result is still true when the
potential function is defined by
$U(q)=\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|^{\alpha}}}$ for any
$\alpha>0$ and if $U(q(t))$ is a trigonometric polynomial when $i$-th point
particle has the following mode of motion
$q_{i}(t)=a_{i}\cos{\theta(t)}+b_{i}\sin{\theta(t)},~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (4.34)
and $a_{i},b_{i}\in\mathbb{R}^{d}$, for all $i=1,\ldots,N$.
Two numbers $t_{1}$ and $t_{2}$ are called to be linearly dependent over the
rational field, if there exist two rational numbers $s_{1}$ and $s_{2}$ (at
least one of them is nonvanishing) such that $t_{1}s_{1}+t_{2}s_{2}=0$. It is
easy to know that linear dependence for two numbers over the rational field is
a equivalence relation on the set $\mathbb{R}\backslash\\{0\\}$. Hence we can
get a partition of any subset of $\mathbb{R}\backslash\\{0\\}$.
###### Lemma 4.4
Given some continuous periodic functions $u_{i}(t)(i\in\Lambda$,
$t\in\mathbb{R})$, for the set of all the periods of $u_{i}(t)(i\in\Lambda)$,
suppose there are only finite equivalence relations according to linear
dependence over the rational field, that is, there are index subsets
$\Lambda_{i}(i=1,\cdots,n)$ such that $\bigcup^{n}_{j=1}\Lambda_{j}=\Lambda$
and $\Lambda_{i}\bigcap\Lambda_{j}=\emptyset(1\leq i\neq j\leq n)$, moreover,
the functions $u_{i}(t)(i\in\Lambda_{1})$ have a common period $T_{1}$,
$\cdots$, the functions $u_{i}(t)$ ($i\in\Lambda_{n}$) have a common period
$T_{n}$, and $T_{i},T_{j}$ are linearly independent over the rational field
for any $1\leq i,j\leq n$. If $\sum_{i\in\Lambda}u_{i}(t)\equiv const$, then
$\sum_{i\in\Lambda_{j}}u_{i}(t)\equiv const$ for every $j\in\\{1,\cdots,n\\}$.
Proof of Lemma 4.4:
For a function $u(t)$, we define
$\triangle_{i}u\triangleq u(t-T_{i})-u(t)$,
$\triangle_{j}\triangle_{i}u\triangleq\triangle_{i}u(t-T_{j})-\triangle_{i}u(t)$,
$\triangle^{k}u\triangleq\triangle_{k}\cdots\triangle_{1}u$ for any
$k\in\\{1,\cdots,n\\}$,
and
$\widetilde{\triangle}_{i}u\triangleq u(t+T_{i})-u(t)$,
$\widetilde{\triangle}_{j}\widetilde{\triangle}_{i}u\triangleq\widetilde{\triangle}_{i}u(t+T_{j})-\widetilde{\triangle}_{i}u(t)$,
$\widetilde{\triangle}^{k}u\triangleq\widetilde{\triangle}_{n-k+1}\cdots\widetilde{\triangle}_{n}u$
for any $k\in\\{1,\cdots,n\\}$.
From
$\sum_{i\in\Lambda}u_{i}(t)=\sum_{1\leq j\leq
n}\sum_{i\in\Lambda_{j}}u_{i}(t)\equiv const,$ (4.35)
we can get
$\triangle_{1}\sum_{1\leq j\leq
n}\sum_{i\in\Lambda_{j}}u_{i}(t)=\triangle_{1}\sum_{2\leq j\leq
n}\sum_{i\in\Lambda_{j}}u_{i}(t)=0,$ (4.36)
$\triangle_{2}\triangle_{1}\sum_{2\leq j\leq
n}\sum_{i\in\Lambda_{j}}u_{i}(t)=\triangle_{2}\triangle_{1}\sum_{3\leq j\leq
n}\sum_{i\in\Lambda_{j}}u_{i}(t)=\triangle^{2}\sum_{3\leq j\leq
n}\sum_{i\in\Lambda_{j}}u_{i}(t)=0,$ (4.37)
$\cdots$
$\triangle^{n-1}\sum_{i\in\Lambda_{n}}u_{i}(t)=0,$ (4.38)
Then
$\int^{T_{n}}_{0}\triangle^{n-1}\sum_{i\in\Lambda_{n}}u_{i}(t)\exp\sqrt{-1}(k\frac{2\pi}{T_{n}}t)dt=0,$
(4.39)
for any $k\in\mathbb{Z}\backslash\\{0\\}$.
The above equations can be changed as
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int^{T_{n}}_{0}[\triangle^{n-2}\sum_{i\in\Lambda_{n}}u_{i}(t-T_{n-1})-\triangle^{n-2}\sum_{i\in\Lambda_{n}}u_{i}(t)]\exp\sqrt{-1}(k\frac{2\pi}{T_{n}}t)dt$
$\displaystyle=$
$\displaystyle\int^{T_{n}}_{0}\triangle^{n-2}\sum_{i\in\Lambda_{n}}u_{i}(t)\widetilde{\triangle}_{n-1}\exp\sqrt{-1}(k\frac{2\pi}{T_{n}}t)dt$
$\displaystyle=$ $\displaystyle(\exp\sqrt{-1}(k\frac{2\pi
T_{n-1}}{T_{n}})-1)\int^{T_{n}}_{0}\triangle^{n-2}\sum_{i\in\Lambda_{n}}u_{i}(t)\exp\sqrt{-1}(k\frac{2\pi}{T_{n}}t)dt$
$\displaystyle\cdots$ $\displaystyle=$
$\displaystyle(\exp\sqrt{-1}(k\frac{2\pi
T_{1}}{T_{n}})-1)\cdots(\exp\sqrt{-1}(k\frac{2\pi T_{n-1}}{T_{n}})-1)$
$\displaystyle\int^{T_{n}}_{0}\sum_{i\in\Lambda_{n}}u_{i}(t)\exp\sqrt{-1}(k\frac{2\pi}{T_{n}}t)dt$
for any $k\in\mathbb{Z}\backslash\\{0\\}$.
Since $T_{n},T_{j}$ are linearly independent over the rational field for any
$1\leq j\leq n-1$, we can get
$\int^{T_{n}}_{0}\sum_{i\in\Lambda_{n}}u_{i}(t)\exp\sqrt{-1}(k\frac{2\pi}{T_{n}}t)dt=0,$
(4.41)
for any $k\in\mathbb{Z}\backslash\\{0\\}$.
Hence $\sum_{i\in\Lambda_{n}}u_{i}(t)\equiv const$ holds.
Similarly, we can also get $\sum_{i\in\Lambda_{j}}u_{i}(t)\equiv const$ for
every $j\in\\{1,\cdots,n-1\\}$.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
## 5 The Proofs of Main Results
Proof of Theorem 3.1:
From the Jacobi’s identity, we known that $U$ is constant on the solution for
Newtonian particle systems of constant moment of inertia, so we can get
Theorem 3.1 by Lemma 4.3.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Proof of Corollary 3.2:
From the conditions of $\mathbf{Corollary~{}\ref{centralconfigurations}}$, we
have
$m_{i}\ddot{q}_{i}=-\lambda m_{i}q_{i}.$ (5.42)
where $\lambda=\frac{U(q)}{I(q)}$ is a constant. It is easy to know that
$q_{i}(t)=a_{i}\cos(\sqrt{\lambda}t)+b_{i}\sin(\sqrt{\lambda}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (5.43)
for some $a_{i},b_{i}\in\mathbb{R}^{d}$, $i=1,\ldots,N$.
Then by Theorem 3.1, we know that the Saari’s Conjecture is true.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Proof of Theorem 3.3:
If the solution $(x_{1}(t),\cdots,x_{N}(t))$ of the Planetary Restricted
Problem satisfies $I_{0}=\sum_{1\leq i\leq
N}\widetilde{m}_{i}|x_{i}|^{2}\equiv const$, it is easy to know that
$U_{0}=\sum_{1\leq i\leq N}\frac{\varrho_{i}}{|x_{i}|}\equiv const$ is true.
Then we know that every point particle does not collide with the sun,
otherwise, $U_{0}$ can not be constant since $U_{0}$ will tend to $\infty$ for
the collision orbit; every point particle moves in elliptic orbit, otherwise,
the moment of inertia $I_{0}$ can not be constant since $T_{0}$ will tend to
$\infty$ for the parabolic or hyperbolic orbit. So we have
$I_{0}=\sum_{1\leq i\leq N}{\widetilde{m}_{i}}{a^{2}_{i}}(1-e_{i}\cos
E_{i})^{2}$ (5.44) $U_{0}=\sum_{1\leq i\leq
N}\frac{\varrho_{i}}{a_{i}(1-e_{i}\cos E_{i})}$ (5.45)
Our aim is to prove that every eccentricity $e_{i},(i=1,\cdots,N)$ must be
zero. We will mainly use the equation (5.45), it will be convenient to divide
the proof into several steps.
Step 1.
If N point particles have the same period $T$, then N point particles have the
same semi-major axis $a$ by Kepler’s Third Law, their mean anomaly are
respectively $\tau_{i}=\tilde{n}t-\tilde{n}\iota_{i}$. We will prove
$e_{i},(i=1,\cdots,N)$ must be zero in this case.
From Kepler equation, one can get (one can see [2]):
$\frac{1}{1-e_{i}\cos E_{i}}=1+2\sum_{n\geq 1}J_{n}(ne_{i})\cos({n\tau_{i}})$
(5.46)
where
$J_{n}(z)=\frac{1}{2\pi}\int^{2\pi}_{0}\cos(n\theta-z\sin\theta)d\theta=\sum_{k\geq
0}\frac{(-1)^{k}(z/2)^{n+2k}}{k!(n+k)!}$ (5.47)
is the Bessel function of order $n$.
Then we have
$\displaystyle U_{0}$ $\displaystyle=$ $\displaystyle\sum_{1\leq i\leq
N}\frac{\varrho_{i}}{a(1-e_{i}\cos E_{i})}$ (5.48) $\displaystyle=$
$\displaystyle\sum_{1\leq i\leq N}\frac{\varrho_{i}}{a}[1+2\sum_{n\geq
1}J_{n}(ne_{i})\cos({n\tau_{i}})]$ $\displaystyle=$ $\displaystyle\sum_{1\leq
i\leq N}\frac{\varrho_{i}}{a}+\sum_{n\geq 1}[\sum_{1\leq i\leq
N}\frac{2\varrho_{i}}{a}J_{n}(ne_{i})\cos({n\tilde{n}\iota_{i}})\cos(n\tilde{n}t)$
$\displaystyle+$ $\displaystyle\sum_{1\leq i\leq
N}\frac{2\varrho_{i}}{a}J_{n}(ne_{i})\sin(n\tilde{n}\iota_{i})\sin(n\tilde{n}t)]$
Since $U_{0}\equiv const$, we get
$\sum_{1\leq i\leq N}{\varrho_{i}}J_{n}(ne_{i})\cos({n\tilde{n}\iota_{i}})=0$
(5.49) $\sum_{1\leq i\leq
N}{\varrho_{i}}J_{n}(ne_{i})\sin(n\tilde{n}\iota_{i})=0$ (5.50)
If $e_{i}>0$, then we can find the asymptotic formula for $J_{n}(ne_{i})$ (one
can see [13]):
$J_{n}(ne_{i})=\frac{2}{\sqrt{2\pi n\tanh\gamma_{i}}}\exp
n(\tanh\gamma_{i}-\gamma_{i})(1+{\it O}(n^{-1/5})),$ (5.51)
where $e_{i}=\frac{1}{\cosh\gamma_{i}}$ and $\gamma_{i}>0$, hence
$J_{n}(ne_{i})>0$ holds for sufficiently large $n$. By Lemma 4.2, we know
there exists some sufficiently large $n$ such that
$n\tilde{n}\iota_{i}=2\pi(k_{ni}+\varphi_{ni})$, where $k_{ni}$ is an integer
and $-\frac{1}{4}<\varphi_{ni}<\frac{1}{4}$. Since
${\varrho_{i}}J_{n}(ne_{i})>0$, we will get
$\sum_{1\leq i\leq N}{\varrho_{i}}J_{n}(ne_{i})\cos({n\tilde{n}\iota_{i}})>0$
(5.52)
this is a contradiction with the equation (5.49). So there must be $e_{i}=0$
for any $i\in\\{1,\cdots,N\\}$ .
Step 2.
If N point particles have different periods but they have a common period $T$.
Then one can suppose that $1$-th body, $\cdots$, $N$-th body have respectively
the period $T_{1}$, $\cdots$, $T_{N}$, and $T=k_{i}T_{i}$, where $k_{i}$ is
positive integer, $i\in\\{1,\cdots,N\\}$.
Since
$\displaystyle U_{0}$ $\displaystyle=$ $\displaystyle\sum_{1\leq i\leq
N}\frac{\varrho_{i}}{a_{i}(1-e_{i}\cos E_{i})}$ $\displaystyle=$
$\displaystyle\sum_{1\leq i\leq N}\frac{\varrho_{i}}{a_{i}}[1+2\sum_{n\geq
1}J_{n}(ne_{i})\cos({nk_{i}\frac{2\pi}{T}(t-\iota_{i})})]$ $\displaystyle=$
$\displaystyle\sum_{1\leq i\leq N}\frac{\varrho_{i}}{a_{i}}+\sum_{n\geq
1}[\sum_{1\leq i\leq
N}\frac{2\varrho_{i}}{a_{i}}J_{n}(ne_{i})\cos({nk_{i}\frac{2\pi}{T}\iota_{i}})\cos(nk_{i}\frac{2\pi}{T}t)$
$\displaystyle+$ $\displaystyle\sum_{1\leq i\leq
N}\frac{2\varrho_{i}}{a_{i}}J_{n}(ne_{i})\sin(nk_{i}\frac{2\pi}{T}\iota_{i})\sin(nk_{i}\frac{2\pi}{T}t)]$
$\displaystyle=$ $\displaystyle\sum_{1\leq i\leq
N}\frac{\varrho_{i}}{a_{i}}+\sum_{n\geq
1}[\sum_{i\in\Sigma_{n}}\frac{2\varrho_{i}}{a_{i}}J_{n/k_{i}}(\frac{n}{k_{i}}e_{i})\cos({n\frac{2\pi}{T}\iota_{i}})\cos(n\frac{2\pi}{T}t)$
$\displaystyle+$
$\displaystyle\sum_{i\in\Sigma_{n}}\frac{2\varrho_{i}}{a_{i}}J_{n/k_{i}}(\frac{n}{k_{i}}e_{i})\sin(n\frac{2\pi}{T}\iota_{i})\sin(n\frac{2\pi}{T}t)]$
where $\Sigma_{n}$ is the subset of $\\{1,\cdots,N\\}$, whose element $i$ is a
divisor of $n$.
We have
$\sum_{i\in\Sigma_{n}}\frac{2\varrho_{i}}{a_{i}}J_{n/k_{i}}(\frac{n}{k_{i}}e_{i})\cos({n\frac{2\pi}{T}\iota_{i}})=0$
(5.54)
$\sum_{i\in\Sigma_{n}}\frac{2\varrho_{i}}{a_{i}}J_{n/k_{i}}(\frac{n}{k_{i}}e_{i})\sin(n\frac{2\pi}{T}\iota_{i})=0$
(5.55)
Then it is similar to Step 1, if some $e_{i}>0$, then we can find some
sufficiently large $n$ such that
$\sum_{i\in\Sigma_{n}}\frac{2\varrho_{i}}{a_{i}}J_{n/k_{i}}(\frac{n}{k_{i}}e_{i})\cos({n\frac{2\pi}{T}\iota_{i}})>0.$
(5.56)
However this result contradicts with the equation (5.54). So there must be
$e_{i}=0$ for any $i\in\\{1,\cdots,N\\}$.
Step 3.
If N point particles have different periods and they don’t have a common
period. We firstly divide these periods according to the equivalence relations
of linear dependence over the rational field. One can suppose that the family
of sets $\Omega_{1}$, $\cdots$, $\Omega_{n}$ ($1\leq n\leq N$) is the
partition of these periods, and the corresponding point particles constitute
respectively the sets $\Sigma_{1}$, $\cdots$, $\Sigma_{n}$ ($1\leq n\leq N$).
By Lemma 4.4, we have
$\sum_{i\in\Sigma_{1}}\frac{\varrho_{i}}{a_{i}(1-e_{i}\cos E_{i})}\equiv
const$ (5.57)
$\cdots$
$\sum_{i\in\Sigma_{n}}\frac{\varrho_{i}}{a_{i}(1-e_{i}\cos E_{i})}\equiv
const$ (5.58)
Then by Step 2, we know that the Saari’s Conjecture is true in the Planetary
Restricted Problem.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Proof of Theorem 3.4:
We have
$\displaystyle{\mathcal{A}}(q)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}{[\sum_{i}\frac{1}{2}m_{i}|\dot{q_{i}}|^{2}+\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}]dt}$
$\displaystyle\geq$
$\displaystyle\int_{\mathbb{T}}{[(\frac{2\pi}{T})^{2}\sum_{i}\frac{1}{2}m_{i}|{q_{i}}|^{2}+\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}]dt}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}{[\frac{1}{2}(\frac{2\pi}{T})^{2}I(q)+\frac{1}{2}U(q)+\frac{1}{2}U(q)]dt}$
$\displaystyle\geq$ $\displaystyle
3\int_{\mathbb{T}}{[(\frac{1}{2})^{3}(\frac{2\pi}{T})^{2}I(q)U^{2}(q)]^{\frac{1}{3}}dt}$
$\displaystyle\geq$ $\displaystyle
3[\frac{(inf_{\mathcal{X}_{2}\setminus\Delta_{2}}{IU^{2}})\pi^{2}}{2}]^{\frac{1}{3}}T^{\frac{1}{3}}$
then,
${\mathcal{A}}(q)=3[\frac{(inf_{\mathcal{X}_{2}\setminus\Delta_{2}}{IU^{2}})\pi^{2}}{2}]^{\frac{1}{3}}T^{\frac{1}{3}}$
if and only if:
${(\textit{i})}.$ there exist $a_{i},b_{i}\in\mathbb{R}^{2}$, for all
$i=1,\ldots,N$, such that
$q_{i}(t)=a_{i}\cos(\frac{2\pi}{T}t)+b_{i}\sin(\frac{2\pi}{T}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (5.59)
${(\textit{ii})}.$ $(\frac{2\pi}{T})^{2}I(q)=U(q).$
${(\textit{iii})}.$ $q$ minimizes the function $IU^{2}$.
By ${(\textit{ii})}$ and ${(\textit{iii})}$ we know $I(q)\equiv
const,U(q)\equiv const$, and $q(t)$ is always a central configuration. Then
$q$ is a relative equilibrium solution whose configuration minimizes the
function $IU^{2}$ by ${(\textit{i})}$ and Theorem 3.1.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Remark. We notice that as in A.Chenciner [7] [8] and Checiner-Desolneux [9],
if the Conjecture on the Finiteness of Central Configurations is true,
${(\textit{ii})}$ and ${(\textit{iii})}$ are sufficient to prove Theorem 3.4;
in fact, as [7] pointed that if a weaker conjecture: “the minimum points of
the function $IU^{2}$ are finite” could be proved, ${(\textit{ii})}$ and
${(\textit{iii})}$ are also sufficient to prove Theorem 3.4. However, we don’t
know any rigorous proofs for the above conjectures, hence we exploit the
condition ${(\textit{i})}$ as far as possible, after we prove Saari’s
conjecture in the elliptical type N-Body Problem, we can get over the
obstacle.
## Acknowledgements
The authors sincerely thank Professor F.Diacu who told us the new progress of
the Saari’s conjecture.
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|
arxiv-papers
| 2013-08-11T08:49:26 |
2024-09-04T02:49:49.305459
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiang Yu and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1308.2376"
}
|
1308.2454
|
# Understanding the Benefits of Open Access in Femtocell Networks: Stochastic
Geometric Analysis in the Uplink
Wei Bao and Ben Liang
Department of Electrical and Computer Engineering University of Toronto
{wbao,liang}@comm.utoronto.ca
###### Abstract
We introduce a comprehensive analytical framework to compare between open
access and closed access in two-tier femtocell networks, with regard to uplink
interference and outage. Interference at both the macrocell and femtocell
levels is considered. A stochastic geometric approach is employed as the basis
for our analysis. We further derive sufficient conditions for open access and
closed access to outperform each other in terms of the outage probability,
leading to closed-form expressions to upper and lower bound the difference in
the targeted received power between the two access modes. Simulations are
conducted to validate the accuracy of the analytical model and the correctness
of the bounds.
###### category:
C.2.1 Network Architecture and Design Wireless communication
###### keywords:
Femtocell, uplink interference, stochastic geometry, open access
††terms: Theory
## 1 Introduction
In deploying wireless celluar networks, some of the most important objectives
are to provide higher capacity, better service quality, lower power usage, and
ubiquitous coverage. To achieve these goals, one efficient approach is to
install a second tier of smaller cells, which are referred to as femtocells,
overlapping the original macrocell network [16]. Each femtocell is equipped
with a short-range and low-cost base station (BS).
In the presence of femtocells, whenever a User Equipment (UE) is near a
femtocell BS, two different access mechanisms may be applied: closed access
and open access. Under closed access, a femtocell BS only provides service to
its local users, without further admitting nearby macrocell users. In
contrast, under open access, all nearby macrocell users are allowed to access
the femtocell BS. The open access mode increases the interference level from
within a femtocell, but it also allows macrocell UEs that might otherwise
transmit at a high power toward their faraway macrocell BS to potentially
switch to lower-power transmission toward the femtocell BS, therefore reducing
the overall interference in the system. However, the relative merits between
open access and closed access remain unresolved within the research community,
as they may concern diverse factors in communication efficiency, control
overhead, system security, and regulatory policies.
In this work, we contribute to the current debate by providing new technical
insights on how the two access modes may affect both macrocell users and local
femtocell users, in terms of the uplink interference and outage probabilities.
We seek to quantify the conditions to guarantee that one access mode improves
the performance of macrocell or femtocell users. It is a challenging task, as
we need to account for the diverse spatial patterns of different network
components. Macrocell BSs are usually deployed regularly by the network
operator, while femtocell BSs are spread irregularly, sometimes in an anywhere
plug-and-play manner, leading to a high level of spatial randomness.
Furthermore, macrocell users are randomly distributed throughout the system,
while femtocell users show strong spatial locality and correlation, since they
aggregate around femtocell BSs. Whenever open access is applied, we also need
to consider the effects of handoffs made by open access users, which brings
even more complication to the analytical model.
We develop stochastic geometric analysis schemes to derive numerical
expressions for the uplink interference and outage probabilities of open
access and closed access by modeling macrocell BSs as a regular grid,
macrocell UEs as a Poission point process (PPP), and femtocell UEs as a two-
level clustered Poisson point process, which captures the spatial patterns of
different network components. However, uplink interference analysis is
notoriously complex even for traditional single-tier cellular networks. For
the two-tier network under consideration, our analysis yields non-closed forms
requiring numerical integrations. This motivates us to further develop closed-
form sufficient conditions for open access and closed access to outperform
each other, at both the macrocell and femtocell levels.
Based on the above analysis, we are able to extract a key factor that
influences the performance difference between open access and closed access:
the power enhancement factor $\rho$, which is the ratio of the targeted
received power of an open access user to its original targeted received power
in the macrocell. We investigate the threshold value $\rho^{*}$ (resp.
$\rho^{**}$) such that macrocell (resp. femtocell) users may benefit through
open access if $\rho<\rho^{*}$ (resp. $\rho<\rho^{**}$) as we apply open
access to replace closed access. Tight upper and lower bounds of $\rho^{*}$
are derived in closed forms, and the bounds of $\rho^{**}$ can be found by
numerically searching through a closed-form equation, providing system design
guidelines with low computational complexity. To the best of our knowledge,
this is the first paper to theoretically analyze the uplink performance
difference between open access and closed access of femtocell networks that
considers the impact of random spatial patterns of BSs and UEs.
The rest of the paper is organized as follows: In Section 2, we discuss the
relation between our work and prior works. In Section 3, we present the system
model. In Section 4 and 5, we analyze the performance at the macrocell and
femtocll levels, respectively. In Section 6, we validate our analysis with
simulation results. Finally, concluding remarks are given in Section 7.
## 2 related works
The downlink interference and outage performance in cellular networks have
been extensively studied using the stochastic geometric approach. [8, 9]
analyzed the downlink performance of heterogeneous networks with multiple
tiers by assuming the signal-to-interference plus noise ratio (SINR) threshold
is greater than $1$. [13] studied the maximum tier-1 user and tier-2 cell
densities under downlink outage constraints. [10] studied the downlink
interference considering load balance. [18] studied the downlink user
achievable rate in a heterogeneous network considering both SINR and spatial
user distributions. [12] studied open access versus closed access in femtocell
networks in terms of downlink performance.
The analysis of uplink interference in multi-tier networks is more challenging
compared with the downlink case. For uplink analysis, the interference
generators are the set of UEs, which are more complicatedly distributed
compared with the interference generators (i.e., BSs) in downlink analysis.
Under closed access, without considering random spatial patterns, [14] studied
the uplink performance of a single tier-1 cell and a single femtocell, while
[15] extended it to the case of multiple tier-1 cells and multiple femtocells.
[1] studied the co-channel uplink interference in LTE-based multi-tier
cellular networks, considering a constant number of femtocells in a macrocell.
However, none of [14, 15, 1] considered the random spatial patterns of users
or femtocells.
By considering random spatial patterns, [17] analyzed uplink performance of
cellular networks, but it was limited to the one-tier case. [6] evaluated the
uplink performance of two-tier networks considering random spatial patterns.
However, several interference components were analyzed based on
approximations, such as (1) BSs see a femtocell as a point interference source
and (2) Femtocell UEs transmit at the maximum power at the edge of cells. [7]
studied both uplink and downlink interference of femtocell networks based on a
Neyman-Scott Process. However it assumed that each UE transmits at the same
power and femtocell users are uniformly distributed in an infinitesimally thin
ring around the femtocell BS. With a more general system model, [4] derived
the uplink interference in a two-tier network with multiple types of users and
small cell BSs, but no closed-form result was obtained. Moreover, both [4, 6,
7] considered only the closed access case.
The analysis of open access in femtocell networks is even more complicated.
This is because the model for open access needs to capture the impact of the
users disconnecting from the original macrocell BS and connecting to a
femtocll BS. In order to satisfy mathematical tractability, the previous
analysis of open access was based on simplified assumptions. [22] compared the
performance of open access and closed access based on a model with one
macorcell, one femtocell, and a given number of macrocell users, while [20]
was based on a model with one macorcell, a constant number of macrocell users,
and randomly distributed femtocells. Although [22] and [20] provide useful
insights into the performance comparison between open access and closed
access, due to their limited system models, they have not addressed the
challenging issues brought by the diverse spatial patterns of BSs and UEs.
Finally, several other works studied the performance of femtocells based on
experiments [2, 23], which provided important practical knowledge in designing
a real system. Compared with these works, our theoretical approach is an
essential alternative that allows more rigorous reasoning to understand the
performance benefits of open access compared with closed access, by
considering more general system models and behaviors instead of specific
experimental scenarios.
## 3 System Model
Figure 1: Two-tier network with macrocells and femtocells.
### 3.1 Two-Tier Network
We consider a two-tier network with macrocells and femtocells as shown in Fig.
1. Following the convention in literature, we assume that the macrocells form
an infinite hexagonal grid in the two-dimensional Euclidean space
$\mathbb{R}^{2}$. Macrocell BSs are located at the centers of the hexagons
$\mathbb{B}=\\{(\frac{3}{2}aR_{c},\frac{\sqrt{3}}{2}aR_{c}+\sqrt{3}bR_{c})|a,b\in\mathbb{Z}\\}$,
where $R_{c}$ is the radius of the hexagon. Macrocell UEs are randomly
distributed in the system, which are modeled as a homogeneous Poisson point
process (PPP) $\Phi$ with intensity $\lambda$.
Because femtocell BSs are operated in a plug-and-play fashion, inducing a high
level of spatial randomness, we assume femtocell BSs form a homogeneous PPP
$\Theta$ with intensity $\mu$. Each femtocell BS is connected to the core
network by high-capacity wired links that has no influence on our wireless
performance analysis.
Each femtocell BS communicates with local femtocell UEs surrounding it,
constituting a femtocell. We assume $R$ as the communication radius of each
femtocell BS. Given the location of a femtocell BS at $\mathbf{x}_{0}$, we
assume that its femtocell UEs, denoted by $\Psi(\mathbf{x}_{0})$, are
distributed as a non-homogenous PPP in the disk centered at $\mathbf{x}_{0}$
with radius $R$. Its intensity at $\mathbf{x}$ is described by
$\nu(\mathbf{x}-\mathbf{x}_{0})$, a non-negative function of the vector
$\mathbf{x}-\mathbf{x}_{0}$. Note that the user intensity
$\nu(\mathbf{x}-\mathbf{x}_{0})=0$ if $|\mathbf{x}-\mathbf{x}_{0}|>R$. The
femtocell UEs in one femtocell are independent with femtocell UEs in other
femtocells, as well as the macrocell UEs. We assume the scale of femtocells is
much small than the scale of macrocells [16], $R\ll R_{c}$.
To better understand the spatial distribution of femtocell BSs and femtocell
UEs, the femtocell BSs $\Theta$ can be regarded as a parent point process in
$\mathbb{R}^{2}$, while femtocell UEs $\Psi$ is a daughter process associated
with a point in the parent point process, forming a two-level random pattern.
Note that the aggregating of femtocell UEs around a femtocell BS implicitly
defines the location correlation among femtocell UEs.
Let $\mathcal{H}(\mathbf{x})$ denote the hexagon region centered at
$\mathbf{x}$ with radius $R_{c}$; let $\mathcal{B}(\mathbf{x},R)$ denote the
disk region centered at $\mathbf{x}$ with radius $R$; let
$\mathcal{BS}(\mathbf{x})$ denote the hexagon center nearest to $\mathbf{x}$
(i.e., $\mathcal{BS}(\mathbf{x})=\mathbf{x}_{0}$ $\Leftrightarrow$
$\mathbf{x}\in\mathcal{H}(\mathbf{x}_{0})$).
### 3.2 Open Access versus Closed Access
If a macrocell UE is covered by a femtocell BS (i.e., within a distance of $R$
from a femtocell BS), under closed access, the UE still connects to the
macrocell BS. Under open access, the UE is handed-off to connect to the
femtocell BS and _disconnects_ from the original macrocell BS; the UE is then
referred to as an _open access UE_.
Given a femtocell BS located at $\mathbf{x}_{0}$, let $\Omega(\mathbf{x}_{0})$
denote the point process corresponding to the open access UEs connecting to
it. Note that because the radius of a femtocell is much smaller than that of
macrocells, the probability of two femtocells overlapping is small. Thus,
$\Omega(\mathbf{x}_{0})$ corresponds to points of $\Phi$ inside the range of
the femtocell BS at $\mathbf{x}_{0}$, which is a PPP with intensity $\lambda$
inside $\mathcal{B}(\mathbf{x}_{0},R)$.
### 3.3 Pathloss and Power Control
Let $P_{t}(\mathbf{x})$ denote the transmission power at $\mathbf{x}$ and
$P_{r}(\mathbf{y})$ denote the received power at $\mathbf{y}$. We assume that
$P_{r}(\mathbf{y})=\frac{P_{t}(\mathbf{x})h_{\mathbf{x},\mathbf{y}}}{A|\mathbf{x}-\mathbf{y}|^{\gamma}}$,
where $A|\mathbf{x}-\mathbf{y}|^{\gamma}$ is the propagation loss function
with predetermined constants $A$ and $\gamma$ (where $\gamma>2$ in practice),
and $h_{\mathbf{x},\mathbf{y}}$ is the fast fading term. Corresponding to
common Rayleigh fading with power normalization, $h_{\mathbf{x},\mathbf{y}}$
is independently exponentially distributed with unit mean. Let $H(\cdot)$ be
the cumulative distribution function of $h_{\mathbf{x},\mathbf{y}}$.
We follow the conventional assumption that uplink power control adjusts for
propagation losses [5, 6, 11, 21]. The targeted received power level of
macrocell UEs, femtocell UEs and open access UEs are $P$, $Q$, and
$P^{\prime}$, respectively111We assume a single fixed level of targeted
received power at the macrocell or femtocell level for mathematical
tractability. We show that our model is still valid when the targeted received
power is randomly distributed through simulations in Section 6.. Given the
targeted received power $P_{T}$ ($P_{T}=P$, $P_{T}=Q$, or $P_{T}=P^{\prime}$)
at $\mathbf{y}$ and transmitter at $\mathbf{x}$, the transmission power is
$P_{T}A|\mathbf{x}-\mathbf{y}|^{\gamma}$. Then, the resultant interference at
$\mathbf{y}^{\prime}$ is
$\frac{P_{T}|\mathbf{x}-\mathbf{y}|^{\gamma}h_{\mathbf{x},\mathbf{y}^{\prime}}}{|\mathbf{x}-\mathbf{y}^{\prime}|^{\gamma}}$.
Let $\rho=P^{\prime}/P$, which is the targeted received power enhancement if a
macrocell UE becomes an open access UE. In this paper, we study the
performance variation when open access is applied to replace closed access.
Therefore, as a parameter corresponding to open access UEs, $\rho$ is regarded
as an important designed parameter. Other parameters, such as $P,Q$, and
$\gamma$ are considered as predetermined system-level constants.
### 3.4 Outage Performance
In this paper, the performance of macrocell UEs and femtocell UEs (under open
access or closed access) is examined through the outage probability, which is
defined as the probability that the signal to interference ratio (SIR) is
smaller than a given threshold value $T$. Because we focus on the interference
analysis, the thermal noise is assumed to be negligible in this paper.
### 3.5 Scope of This Work
The above model assumes a single shared channel for all UEs. However, the
model is applicable for the orthogonal multiplexing case (e.g., OFDMA) [9]. In
that case, the spectrum is partitioned into $n$ orthogonal resource blocks,
and thus the density of UEs is equivalently reduced by a factor of $n$ when we
assume random access of each resource block.
In this case,
$\overline{\nu}=\int_{\mathcal{B}(\mathbf{0},R)}\nu(\mathbf{x})d\mathbf{x}$ is
the average number of local femtocell UEs inside a femtocell sharing the same
resource block, and $\overline{\lambda}=\pi R^{2}\lambda$ is the average
number of open access UEs inside a femtocell sharing the same resource block
(in the open access case only).
## 4 Open Access vs. Closed Access at the Macrocell Level
In this section, we analyze the uplink interference and outage performance of
macrocell UEs. Consider a reference macrocell UE, termed the typical UE,
communicating with its macrocell BS, termed the typical BS. We aim to
investigate the performance of the typical UE.
Due to stationarity of point processes corresponding to macrocell UEs,
femtocell BSs, and femtocell UEs, throughout this section we will re-define
the coordinates so that the typical BS is located at $\mathbf{0}$ [3].
Correspondingly, the typical UE is located at some $\mathbf{x}_{U}$ that is
uniformly distributed in $\mathcal{H}(\mathbf{0})$, since macrocell BSs form a
deterministic hexagonal grid [3].
Let $\Phi^{\prime}$ be the point process of all other macrocell UEs
conditioned on the typical UE, which is called the reduced Palm point process
[3] with respect to (w.r.t.) $\Phi$. Because the reduced Palm point process of
a PPP has the same distribution as its original PPP, $\Phi^{\prime}$ is still
a PPP with intensity $\lambda$ [3]. Therefore, for presentation convenience,
we still use $\Phi$ to denote this reduced Palm point process.
### 4.1 Open Access Case
#### 4.1.1 Interference Components
The overall interference in the uplink has three parts: from macrocell UEs not
inside any femtocell (denoted by $I_{1}$), from open access UEs (denoted by
$I_{2}$), and from femtocell UEs (denoted by $I_{3}$).
$I_{1}$ can be computed as the sum of interference from each macrocell UE:
$\displaystyle
I_{1}=\sum_{\mathbf{x}\in\Phi^{0}}\frac{P|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}h_{\mathbf{x},\mathbf{0}}}{|\mathbf{x}|^{\gamma}},$
(1)
where $\Phi^{0}$ denotes the points of $\Phi$ not inside any femtocell.
$I_{2}$ can be computed as the sum of interference from all open access UEs of
all femtocells:
$\displaystyle
I_{2}=\sum_{\mathbf{x}_{0}\in\Theta}\sum_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}\frac{P^{\prime}|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}h_{\mathbf{x},\mathbf{0}}}{|\mathbf{x}|^{\gamma}}.$
(2)
$I_{3}$ can be computed as the sum of interference from all femtocell UEs of
all femtocells:
$\displaystyle
I_{3}=\sum_{\mathbf{x}_{0}\in\Theta}\sum_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}\frac{Q|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}h_{\mathbf{x},\mathbf{0}}}{|\mathbf{x}|^{\gamma}}.$
(3)
The overall interference of open access is $I=I_{1}+I_{2}+I_{3}$.
#### 4.1.2 Laplace Transform of $I$
In this subsection, we study the Laplace transform of $I$, denoted by
$\mathcal{L}_{I}$, which leads to the following theorem222For presentation
convenience, we omit the variable $s$ in all Laplace transform expressions.:
###### Theorem 1.
$\displaystyle\mathcal{L}_{I}=$
$\displaystyle\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi}u(\mathbf{x})\bigg{)}\cdot\mathbf{E}\Bigg{[}\prod_{\mathbf{x}_{0}\in\Theta}\frac{\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\Big{)}}{\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}u(\mathbf{x})\Big{)}}$
$\displaystyle\qquad\qquad\qquad\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\Big{)}\bigg{)}\Bigg{]},$
(4)
where
$u(\mathbf{x})=\exp\left(-\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}h_{\mathbf{x},0}}{|\mathbf{x}|^{\gamma}}\right)$,
$v(\mathbf{x},\mathbf{x}_{0})=\\\ \exp\left(-\frac{s\rho
P|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}h_{\mathbf{x},\mathbf{0}}}{|\mathbf{x}|^{\gamma}}\right)$,
and
$w(\mathbf{x},\mathbf{x}_{0})=\exp\left(-\frac{sQ|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}h_{\mathbf{x},\mathbf{0}}}{|\mathbf{x}|^{\gamma}}\right)$.
Proof: See Appendix for the proof.
#### 4.1.3 Numeric Computation of $\mathcal{L}_{I}$
In this subsection, we present a numeric approach to compute $\mathcal{L}_{I}$
derived in (1), which will facilitate later comparison between open access and
closed access. Let
$\mathcal{L}_{0}=\mathbf{E}\left(\prod_{x\in\Phi}u(\mathbf{x})\right)$, which
is a generating functional corresponding to $\Phi$ [3, 19]. It can be re-
written in a standard integral form as follows:
$\displaystyle\mathcal{L}_{0}=\exp\Bigg{(}-\lambda\int\limits_{\mathbb{R}^{2}}\bigg{(}1-\int\limits_{\mathbb{R}+}e^{-\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}h}{|\mathbf{x}|^{\gamma}}}H(dh)\bigg{)}d\mathbf{x}\Bigg{)}$
$\displaystyle=\exp\Bigg{(}-\lambda\int_{\mathbb{R}^{2}}\frac{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}\Bigg{)}d\mathbf{x}.$
(5)
Given the location of a femtocell BS at $\mathbf{x}_{0}$, let
$\mathcal{W}(\mathbf{x}_{0})=\mathbf{E}\left(\prod\limits_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\right)$,
which is a generating functional corresponding to $\Psi(\mathbf{x}_{0})$. It
can be expressed in a standard form through the Laplace functional of PPP
$\Psi(\mathbf{x}_{0})$,
$\displaystyle\mathcal{W}(\mathbf{x}_{0})=\exp\Bigg{(}-\int\limits_{\mathcal{B}(\mathbf{0},R)}\frac{\frac{sQ|\mathbf{x}|^{\gamma}}{|\mathbf{x}+\mathbf{x}_{0}|^{\gamma}}}{\frac{sQ|\mathbf{x}|^{\gamma}}{|\mathbf{x}+\mathbf{x}_{0}|^{\gamma}}+1}\nu(\mathbf{x})d\mathbf{x}\Bigg{)}.$
(6)
Similarly, let
$\mathcal{V}(\mathbf{x}_{0})=\mathbf{E}\left(\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\right)$,
and
$\mathcal{U}(\mathbf{x}_{0})=\mathbf{E}\left(\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}u(\mathbf{x})\right)$,
we have
$\displaystyle\mathcal{V}(\mathbf{x}_{0})=\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{0},R)}\frac{\frac{s\rho
P|\mathbf{x}|^{\gamma}}{|\mathbf{x}+\mathbf{x}_{0}|^{\gamma}}}{\frac{s\rho
P|\mathbf{x}|^{\gamma}}{|\mathbf{x}+\mathbf{x}_{0}|^{\gamma}}+1}d\mathbf{x}\Bigg{)},$
(7)
$\displaystyle\mathcal{U}(\mathbf{x}_{0})=\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}d\mathbf{x}\Bigg{)}.$
(8)
Let
$\mathcal{J}(\mathbf{x}_{0})=\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}\mathcal{W}(\mathbf{x}_{0})$,
which is numerically computable through (6)-(8). Finally, we note that
$\displaystyle\mathbf{E}\Bigg{[}\prod_{\mathbf{x}_{0}\in\Theta}\frac{\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\Big{)}}{\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}u(\mathbf{x})\Big{)}}\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\Big{)}\bigg{)}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbf{E}\Bigg{[}\prod_{\mathbf{x}_{0}\in\Theta}\bigg{(}\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}\mathcal{W}(\mathbf{x}_{0})\bigg{)}\Bigg{]}=\mathbf{E}\left(\prod_{\mathbf{x}_{0}\in\Theta}\mathcal{J}(\mathbf{x}_{0})\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\mu\int_{\mathbb{R}^{2}}\left(1-\mathcal{J}(\mathbf{x}_{0})\right)d\mathbf{x}_{0}\right),$
(9)
where (9) is derived from the generating functional with respect to PPP
$\Theta$. Substituting (5) and (9) into (1), we can numerically compute
$\mathcal{L}_{I}$:
$\displaystyle\mathcal{L}_{I}=\mathcal{L}_{0}\exp\left(-\mu\int_{\mathbb{R}^{2}}\left(1-\mathcal{J}(\mathbf{x}_{0})\right)d\mathbf{x}_{0}\right).$
(10)
The overall logic to the above is as follows: First, in terms of the Laplace
transform, additive interference is in the _product_ form, and interference
decrease is in the _division_ form. Suppose that there are no femtocells at
the beginning, and $\mathcal{L}_{0}$ corresponds to the interference from
macrocell UEs. Then, we add femtocells to the system. Given a femtocell BS at
$\mathbf{x}_{0}$, $\mathcal{W}(\mathbf{x}_{0})$ corresponds to the
interference from local femtocell UEs inside the femtocell,
$\mathcal{V}(\mathbf{x}_{0})$ corresponds to interference from open access UEs
inside the femtocell, and $\mathcal{U}(\mathbf{x}_{0})$ corresponds to
interference _decrease_ of open access UEs as they disconnect from their
original macrocell BS. Thus,
$\mathcal{J}(\mathbf{x}_{0})=\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}\mathcal{W}(\mathbf{x}_{0})$
represents the overall interference variation when a femtocell centered at
$\mathbf{x}_{0}$ is added. Finally,
$\exp\left(-\mu\int_{\mathbb{R}^{2}}(1-\mathcal{J}(\mathbf{x}_{0}))d\mathbf{x}_{0}\right)$
is the overall interference variation after adding all femtocells. As a
consequence, the overall interference can be computed in formula (10).
#### 4.1.4 Outage Probability
Given the SIR threshold $T$, the outage probability of the typical UE can be
computed as the probability that the signal strength
$Ph_{\mathbf{x}_{U},\mathbf{0}}$ over the interference $I$ is less than $T$:
$\displaystyle
P^{o}_{out}=\mathbf{P}(Ph_{\mathbf{x}_{U},\mathbf{0}}<TI)=1-\mathcal{L}_{I}|_{s=\frac{T}{P}}.$
(11)
The last equality above is due to $h_{\mathbf{x}_{U},\mathbf{0}}$ being
exponentially distributed with unit mean. As a result, $P^{o}_{out}$ can be
derived directly from $\mathcal{L}_{I}$.
### 4.2 Closed Access Case
Different from the open access case, the overall interference has only two
parts: from macrocell UEs (denoted by $\widehat{I}_{1}$) and from femtocell
UEs (denoted by $\widehat{I}_{3}$).
$\widehat{I}_{1}$ can be computed as the sum of interference from each
macrocell UE:
$\displaystyle\widehat{I}_{1}=\sum_{\mathbf{x}\in\Phi}\frac{P|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}h_{\mathbf{x},\mathbf{0}}}{|\mathbf{x}|^{\gamma}}.$
(12)
$\widehat{I}_{3}$ is exactly the same as $I_{3}$ in (3).
Then, the total interference can be computed as
$\widehat{I}=\widehat{I}_{1}+\widehat{I}_{3}$. Similar to Section 4.1.3, the
Laplace transform of $\widehat{I}$ is
$\displaystyle\mathcal{L}_{\widehat{I}}=\mathbf{E}\Bigg{[}\prod_{\mathbf{x}\in\Phi}u(\mathbf{x})\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathcal{L}_{0}\mathbf{E}\Bigg{[}\prod_{\mathbf{x}_{0}\in\Theta}\bigg{(}\mathcal{W}(\mathbf{x}_{0})\bigg{)}\Bigg{]}=\mathcal{L}_{0}\exp\left(-\mu\int_{\mathbb{R}^{2}}(1-\mathcal{W}(\mathbf{x}_{0}))d\mathbf{x}_{0}\right),$
(13)
where $\mathcal{L}_{0}$ is the same as (5), and $\mathcal{W}(\mathbf{x}_{0})$
is the same as (6).
The overall logic to the above is as follows: First, $\mathcal{L}_{0}$
corresponds to the interference of all macrocell UEs. Given a femtocell BS at
$\mathbf{x}_{0}$, $\mathcal{W}(\mathbf{x}_{0})$ corresponds to interference
from local femtocell UEs inside the femtocell. Then,
$\exp\left(-\mu\int_{\mathbb{R}^{2}}(1-\mathcal{W}(\mathbf{x}_{0}))d\mathbf{x}_{0}\right)$
is the overall interference from all femtocells. As a consequence, the overall
interference can be computed as formula (13).
Finally, the outage probability of the typical UE can be computed as
$\displaystyle
P^{c}_{out}=\mathbf{P}(Ph_{\mathbf{x}_{U},\mathbf{0}}<T\widehat{I})=1-\mathcal{L}_{\widehat{I}}|_{s=\frac{T}{P}}.$
(14)
### 4.3 Parameter Normalization
From the above performance analysis of both open access and closed access, we
see that one can can normalize the radius of macrocells $R_{c}$ to $1$, so
that $R$ is equivalent to the ratio of the radius of femtocells to that of
macrocells ($R\ll 1$). Also, we can normalize the target received power of
macrocell UEs $P$ to $1$, so that $Q$ is equivalent to the ratio of the target
received power of femtocell UEs to that of macrocell UEs, and
$P^{\prime}=\rho$. Therefore, in the rest of this section, without loss of
generality, we set $R_{c}=1$ and $P=1$.
### 4.4 Open Access vs. Closed Access
We compare the outage performance of open access and closed access at the
macrocell level. Due to the integral form of the Laplace transform, the
expressions of outage probabilities for both open and closed access cases are
in non-closed forms, requiring multiple levels of integration. As a
consequence, we are motivated to derive closed-form bounds to compare open
access and closed access.
Let
$\mathbf{V}_{\max}=4\pi^{2}R^{4}(T\rho)^{\frac{2}{\gamma}}\Big{(}\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\Big{)}$,
$\mathbf{V}_{\min}=2\pi^{2}R^{4}(T\rho)^{\frac{2}{\gamma}}\Big{(}\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\Big{)}$,
and $C_{u}$ be a system-level constant predetermined by $T$ and $\gamma$,
shown in (45) of the proof to Theorem 2. The closed-form bounds are presented
in the following theorem:
###### Theorem 2.
$-\mathbf{V}_{\max}+\pi R^{2}C_{u}e^{-\overline{\nu}}>0$ is a sufficient
condition for $P^{o}_{out}<P^{c}_{out}$, and $-\pi
R^{2}C_{u}e^{\overline{\lambda}}+\mathbf{V}_{\min}e^{-\overline{\lambda}-\overline{\nu}}>0$
is a sufficient condition for $P^{c}_{out}<P^{o}_{out}$.
Proof: See Appendix for the proof.
Through Theorem 2, the closed-form expressions can be used to compare the
outage probabilities between open access and closed access without the
computational complexity introduced by numeric integrations in (10) and (13).
In the following, we focus on the performance variation if open access is
applied to replace closed access. The parameter corresponding to open access
UEs, $\rho$, is regarded as a designed parameter. If we fix all the other
network parameters, increasing $\rho$ implies better performance for open
access UEs, but it will also increase the interference from open access UEs to
macrocell BSs. As a consequence, we aim to derive $\rho^{*}$, such that
$P^{o}_{out}=P^{c}_{out}$. At the macrocell level, macrocell UEs experience
less outage iff $\rho<\rho^{*}$. Thus, $\rho^{*}$ is referred to as the
_maximum power enhancement tolerated at the macrocell level_. Thus, in the
deployment of open access femtocells, the network operator is motivated to
limit $\rho$ below $\rho^{*}$ to guarantee that the performance of macrocell
UEs under open access is no worse than that under closed access. One way to
derive $\rho^{*}$ is through numerical computation of (10) and (13) and
numerical search, which introduces high computational complexity due to the
multiple levels of integrations. A more efficient alternative is to find the
bounds of $\rho^{*}$ through Theorem 2. Simple algebra manipulation leads to
$\displaystyle\rho^{*}_{\min}=$
$\displaystyle\frac{1}{T}\left(\frac{C_{u}e^{-\overline{\nu}}}{4\pi
R^{2}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right)}\right)^{\frac{\gamma}{2}},$
(15) $\displaystyle\rho^{*}_{\max}=$
$\displaystyle\frac{1}{T}\left(\frac{C_{u}e^{\overline{\nu}+2\overline{\lambda}}}{2\pi
R^{2}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right)}\right)^{\frac{\gamma}{2}},$
(16)
where $\rho^{*}_{\min}$ and $\rho^{*}_{\max}$ are the lower bound and upper
bound of $\rho^{*}$, respectively. If the network operator limits
$\rho<\rho^{*}_{\min}$, the performance of macrocell UEs under open access can
be guaranteed no worse than their performance under closed access.
Through (15) and (16), we observe that
$\rho^{*}_{\min}=\mathcal{O}(\frac{1}{R^{\gamma}})$ and
$\rho^{*}_{\max}=\mathcal{O}(\frac{1}{R^{\gamma}})$, leading to the following
corollary:
###### Corollary 1.
$\rho^{*}=\mathcal{O}(\frac{1}{R^{\gamma}})$.
Intuitively, as a rough estimation, open access UEs have their distance to the
BS reduced approximately by a factor of $R$, leading to the capability to
increase their received power by the corresponding gain in the propagation
loss function, as their average interference level is maintained. However,
Corollary 1 cannot be trivially obtained from the above intuition. This is
because the outage probability does not only depend on the average
interference, but also depends on the distribution of the interference (i.e.,
the Laplace transform of the interference). By comparing (10) with (13), if we
switch from closed access to open access, the distribution of the interference
will change drastically. Corollary 1 can be derived only after rigorously
comparing and bounding the Laplace transforms of interference under open
access and closed access.
Finally, because $\rho^{*}_{\min}$ and $\rho^{*}_{\max}$ have the same scaling
behavior, Corollary 1 also demonstrates the tightness of the bounds in (15)
and (16).
## 5 Open Access vs. Closed Access at the Femtocell Level
In this section, we analyze the uplink interference and outage performance of
femtocell UEs. Given a reference femtocell UE, termed as the typical femtocell
UE, connecting with its femtocell BS, termed as the typical femtocell BS, we
aim to study the interference at the typical femtocell BS. We also define the
femtocell corresponding to the typical femtocell BS as the typical femtocell,
and the macrocell BS nearest to the typical femtocell BS as the typical
macrocell BS.
Similar to Section 4, we re-define the coordinate of the typical macrocell BS
as $\mathbf{0}$. Correspondingly, the typical femtocell BS is locating at some
$\mathbf{x}_{B}$ that is uniformly distributed in $\mathcal{H}(\mathbf{0})$
[3]. Given the typical femtocell centered at $\mathbf{x}_{B}$, let
$\Theta^{\prime}$ denote the point process of other femtocell BSs conditioned
on the typical femtocell BS, i.e., the reduced Palm point process w.r.t.
$\Theta$. Then, $\Theta^{\prime}$ is still a PPP with intensity $\mu$ [3]. For
presentation convenience, we still use $\Theta$ to denote this reduced Palm
point process. Let $\widetilde{\Psi}(\mathbf{x}_{B})$ denote the other
femtocell UEs inside the typical femtocell conditioned on the typical
femtocell UE. Similarly, $\widetilde{\Psi}(\mathbf{x}_{B})$ has the same
distribution as $\Psi(\mathbf{x}_{B})$. Let
$\widetilde{\Omega}(\mathbf{x}_{B})$ denote open access UEs connecting to the
typical femtocell BS.
### 5.1 Open Access Case
The overall interference in the uplink of the typical femtocell UE has five
parts: from macrocell UEs not inside any femtocell
($I_{1}^{\prime}(\mathbf{x}_{B})$), from open access UEs outside the typical
femtocell ($I_{2}^{\prime}(\mathbf{x}_{B})$), from femtocell UEs outside the
typical femtocell ($I_{3}^{\prime}(\mathbf{x}_{B})$), from local femtocell UEs
inside the typical femtocell ($I_{4}^{\prime}(\mathbf{x}_{B})$), and from open
access UEs inside the typical femtocell ($I_{5}^{\prime}(\mathbf{x}_{B})$). We
have
$\displaystyle I_{1}^{\prime}(\mathbf{x}_{B})=$
$\displaystyle\sum_{\mathbf{x}\in\Phi^{0}}\frac{P|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}h_{\mathbf{x},\mathbf{x}_{B}}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}},$
(17) $\displaystyle I_{2}^{\prime}(\mathbf{x}_{B})=$
$\displaystyle\sum_{\mathbf{x}_{0}\in\Theta}\sum_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}\frac{\rho
P|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}h_{\mathbf{x},\mathbf{x}_{B}}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}},$
(18) $\displaystyle I_{3}^{\prime}(\mathbf{x}_{B})=$
$\displaystyle\sum_{\mathbf{x}_{0}\in\Theta}\sum_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}\frac{Q|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}h_{\mathbf{x},\mathbf{x}_{B}}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}},$
(19) $\displaystyle I_{4}^{\prime}(\mathbf{x}_{B})=$
$\displaystyle\sum_{\mathbf{x}\in\widetilde{\Psi}(\mathbf{x}_{B})}Qh_{\mathbf{x},\mathbf{x}_{B}},$
(20) $\displaystyle I_{5}^{\prime}(\mathbf{x}_{B})=$
$\displaystyle\sum_{\mathbf{x}\in\widetilde{\Omega}(\mathbf{x}_{B})}\rho
Ph_{\mathbf{x},\mathbf{x}_{B}}.$ (21)
The overall interference is
$I^{\prime}(\mathbf{x}_{B})=\sum_{i=1}^{5}I_{i}^{\prime}(\mathbf{x}_{B})$.
Similar to the derivations in Sections 4.1.2 and 4.1.3, the Laplace transform
of $I^{\prime}(\mathbf{x}_{B})$, denoted by
$\mathcal{L}_{I^{\prime}}(\mathbf{x}_{B})$, is derived as
$\displaystyle\mathcal{L}_{I^{\prime}}(\mathbf{x}_{B})=\mathcal{L}^{\prime}_{0}(\mathbf{x}_{B})\exp\Bigg{(}-\mu\int_{\mathbb{R}^{2}}1-$
$\displaystyle\quad\frac{\mathcal{V}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}{\mathcal{U}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}d\mathbf{x}_{0}\Bigg{)}\frac{\mathcal{W}^{\prime\prime}(\mathbf{x}_{B})\mathcal{V}^{\prime\prime}(\mathbf{x}_{B})}{\mathcal{U}^{\prime\prime}(\mathbf{x}_{B})},$
(22)
where
$\displaystyle\mathcal{L}_{0}^{\prime}(\mathbf{x}_{B})=$
$\displaystyle\exp\Bigg{(}-\lambda\int_{\mathbb{R}^{2}}\frac{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}\Bigg{)},$
(23) $\displaystyle\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=$
$\displaystyle\exp\Bigg{(}-\int\limits_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{sQ|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{sQ|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}\nu(\mathbf{x}-\mathbf{x}_{0})d\mathbf{x}\Bigg{)},$
(24) $\displaystyle\mathcal{V}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=$
$\displaystyle\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{s\rho
P|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{s\rho
P|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}\Bigg{)},$
(25) $\displaystyle\mathcal{U}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=$
$\displaystyle\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}\Bigg{)},$
(26) $\displaystyle\mathcal{W}^{\prime\prime}(\mathbf{x}_{B})=$ $\displaystyle
e^{-\frac{sQ\overline{\nu}}{sQ+1}},\mathcal{V}^{\prime\prime}(\mathbf{x}_{B})=e^{-\frac{s\rho
P\overline{\lambda}}{s\rho P+1}},$ (27)
$\displaystyle\mathcal{U}^{\prime\prime}(\mathbf{x}_{B})=$
$\displaystyle\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{B},R)}\frac{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{sP|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}\Bigg{)}.$
(28)
Similar to (11), the outage probability (given $\mathbf{x}_{B}$) is
$\displaystyle\widehat{P}_{out}^{o}(\mathbf{x}_{B})=\mathbf{P}(Qh_{\mathbf{x}_{U},\mathbf{x}_{B}}<TI^{\prime}(\mathbf{x}_{B}))=1-\mathcal{L}_{I^{\prime}}(\mathbf{x}_{B})|_{s=T^{\prime}},$
(29)
where $\mathbf{x}_{U}$ is the coordinate of the typical femtocell UE
(irrelevant to the result), $T^{\prime}=\frac{T}{Q}$, and $T$ is the SIR
threshold. Because $\mathbf{x}_{B}$ is uniformly distributed in
$\mathcal{H}(\mathbf{0})$, the average outage probability can be computed as
$\int_{\mathcal{H}(\mathbf{0})}\widehat{P}_{out}^{o}(\mathbf{x}_{B})d\mathbf{x}_{B}/|\mathcal{H}(\mathbf{0})|$,
where $|\mathcal{H}(\mathbf{0})|=\frac{3\sqrt{3}R_{c}^{2}}{2}$ is the area of
a macrocell.
### 5.2 Closed Access Case
The overall interference has three parts: from macrocell UEs
($\widehat{I}^{\prime}_{1}(\mathbf{x}_{B})$), from femtocell UEs outside the
typical femtocell ($\widehat{I}_{3}^{\prime}(\mathbf{x}_{B})$), and from
femtocell UEs inside the typical femtocell
($\widehat{I}_{4}^{\prime}(\mathbf{x}_{B})$).
$\widehat{I}^{\prime}_{1}(\mathbf{x}_{B})$ can be computed as
$\displaystyle\widehat{I}_{1}^{\prime}(\mathbf{x}_{B})=\sum_{\mathbf{x}\in\Phi}\frac{P|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}h_{\mathbf{x},\mathbf{x}_{B}}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}},$
(30)
and $\widehat{I}_{3}^{\prime}(\mathbf{x}_{B})$ and
$\widehat{I}_{4}^{\prime}(\mathbf{x}_{B})$ are exactly the same as
$I^{\prime}_{3}(\mathbf{x}_{B})$ in (19) and $I^{\prime}_{4}(\mathbf{x}_{B})$
in (20), respectively.
Thus, the overall interference is
$\widehat{I}^{\prime}(\mathbf{x}_{B})=\widehat{I}_{1}^{\prime}(\mathbf{x}_{B})+\widehat{I}_{3}^{\prime}(\mathbf{x}_{B})+\widehat{I}_{4}^{\prime}(\mathbf{x}_{B})$.
Then, the Laplace transform of $\widehat{I}^{\prime}(\mathbf{x}_{B})$ is
$\displaystyle\mathcal{L}_{\widehat{I}^{\prime}}(\mathbf{x}_{B})=$ (31)
$\displaystyle\qquad\mathcal{L}^{\prime}_{0}(\mathbf{x}_{B})\cdot\exp\left(-\mu\int_{\mathbb{R}^{2}}1-\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})d\mathbf{x}_{0}\right)\cdot\mathcal{W}^{\prime\prime}(\mathbf{x}_{B}).$
The outage probability (given $\mathbf{x}_{B}$) is
$\displaystyle\widehat{P}_{out}^{c}(\mathbf{x}_{B})=1-\mathcal{L}_{\widehat{I}^{\prime}}(\mathbf{x}_{B})|_{s=T^{\prime}}.$
(32)
The average outage probability is
$\int_{\mathcal{H}(\mathbf{0})}\widehat{P}_{out}^{c}(\mathbf{x}_{B})d\mathbf{x}_{B}/|\mathcal{H}(\mathbf{0})|$.
Similar to the discussion in Section 4.3, we still can normalize $R_{c}$ and
$P$. Hence, in the rest of this section, without loss of generality, we set
$R_{c}=1$ and $P=1$.
Figure 2: Numerical results.
### 5.3 Open Access vs. Closed Access
In this subsection, we compare the outage performance of open access and
closed access at the femtocell level.
Let
$\mathbf{V}_{\max}^{\prime}=4\pi^{2}R^{4}(T^{\prime}\rho)^{\frac{2}{\gamma}}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right)$,
$\mathbf{V}_{\min}^{\prime}\\\
=2\pi^{2}R^{4}(T^{\prime}\rho)^{\frac{2}{\gamma}}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right)$;
$C_{u}^{\prime}$ be a system-level parameter predetermined by $T^{\prime}$ and
$\gamma$ similar to $C_{u}$ in Theorem 2; $\mathcal{R}_{\min}(\mathbf{x}_{B})$
and $\mathcal{R}_{\max}(\mathbf{x}_{B})$ be as shown in (60) and (61) in the
proof of Theorem 3, which are in the closed forms if $\gamma$ is a rational
number333It is acceptable to assume $\gamma$ as a rational number in reality,
because each real number can be approximated by a rational number with
arbitrary precision.. Then we have the following theorem:
###### Theorem 3.
Given $\mathbf{x}_{B}$, $K_{1}\triangleq-\mu\mathbf{V}_{\max}^{\prime}+\mu\pi
R^{2}C_{u}^{\prime}e^{-\overline{\nu}}-\frac{\pi
R^{2}T^{\prime}\rho}{T^{\prime}\rho+1}+\mathcal{R}_{\min}(\mathbf{x}_{B})>0$
is a sufficient condition for
$\widehat{P}^{o}_{out}(\mathbf{x}_{B})<\widehat{P}^{c}_{out}(\mathbf{x}_{B})$,
and $K_{2}\triangleq-\mu\pi
R^{2}C_{u}^{\prime}e^{\overline{\lambda}}+\mu\mathbf{V}_{\min}^{\prime}e^{-\overline{\nu}-\overline{\lambda}}+\frac{\pi
R^{2}T^{\prime}\rho}{T^{\prime}\rho+1}-\mathcal{R}_{\max}(\mathbf{x}_{B})>0$
is a sufficient condition for
$\widehat{P}^{c}_{out}(\mathbf{x}_{B})<\widehat{P}^{o}_{out}(\mathbf{x}_{B})$.
Proof: See Appendix for the proof.
Through Theorem 3, the closed-form expressions can be used to compare the
outage probabilities between open access and closed access without the
computational complexity introduced by numeric integrations in (29) and (32).
Similar to the discussion in Section 4.4, let $\rho^{**}$ denote the value of
$\rho$, such that
$\widehat{P}^{o}_{out}(\mathbf{x}_{B})=\widehat{P}^{c}_{out}(\mathbf{x}_{B})$.
At the femtocell level, given that a femtocell BS is located at
$\mathbf{x}_{B}$ (the relative coordinate w.r.t. the nearest macrocell), its
local femtocell UEs experience less outage iff $\rho<\rho^{**}$. Thus,
$\rho^{**}$ is referred to as the _maximum power enhancement tolerated by the
femtocell_.
Instead of deriving $\rho^{**}$ through (29) and (32), which introduces high
computational complexity due to multiple levels of integrations, we can find
the lower bound $\rho^{**}_{\min}$ and upper bound $\rho^{**}_{\max}$ of
$\rho^{**}$ through Theorem 3. Accordingly, $\rho^{**}_{\min}$ is the value
satisfying $K_{1}=0$ and $\rho^{**}_{\max}$ is the value satisfying $K_{2}=0$.
Thus, $\rho^{**}_{\min}$ and $\rho^{**}_{\max}$ can be found by a numerical
search approach w.r.t. the closed-form expressions.
## 6 Numerical Study
We present simulation and numerical studies on the outage performance in the
two-tier network with femtocells. First, we study the performance of open
access and closed access under different user and femtocell densities. Second,
we present the numerical results of $\rho^{*}$ and $\rho^{**}$. Unless
otherwise stated, $R_{c}=500$ m, $R=50$ m, $\gamma=3$; and fast fading is
Rayleigh with unit mean. Each simulation data point is averaged over $50000$
trials. The SIR threshold $T$ is set to $0.1$.
First, we study the performance under different user and femtocell
densities444As discuss in Section 3.5, these intensities may already account
for the multiplicative factor introduced by orthogonal multiplexing.. The
network parameters are as follows: $R_{c}=500$ m; $\nu(\mathbf{x})=80$
units/km2 if $|\mathbf{x}|<R$, and $\nu(\mathbf{x})=0$ otherwise; $P=-60$ dBm,
and $Q=P^{\prime}=-54$ dBm ($\rho=6$ dB).
Fig. 2 (a) and (b) show the uplink outage probability of macrocell UEs under
different $\lambda$ and $\mu$ respectively. Fig. 2 (c) and (d) show the uplink
outage probability of femtocell UEs under different $\lambda$ and $\mu$
respectively. The analytical results are derived from the exact expressions in
Sections 4.1, 4.2, 5.1, and 5.2, without applying any bounds. The error bars
show the $95\%$ confidence intervals for simulation results. The plot points
are slightly shifted to avoid overlapping error bars for easier inspection.
The figures illustrate the accuracy of our analytical results. In addition,
the figures show that the macrocell UE density strongly influences the outage
probability of both macrocell and femtocell UEs, while the femtocell density
only has a slight influence. At the macrocell level, increasing the density of
femtocell leads to more proportion of macrocell UEs becoming open access UEs,
which gives higher performance gap between open access and closed access. At
the femtocell level, the interference is observed at femtocell BSs, and the
average number of macrocell UEs in a femtocell becomes a more important factor
influencing the performance gap.
Next, we present the numerical results of $\rho^{*}$ and $\rho^{**}$. The
network parameters are as follows: $\lambda=4$ units/km2, $\mu=4$ units/km2;
$\nu(\mathbf{x})=20$ units/km2 if $|\mathbf{x}|<R$, and $\nu(\mathbf{x})=0$
otherwise; $P=-60$ dBm, and $Q=-54$ dBm.
Fig. 2 (e) presents the value of $\rho^{*}$ at the macrocell level. We compute
the actual value of $\rho^{*}$ by numerically searching for the value such
that (11) is equal to (14). Through the closed-form expression in Theorem 2,
we are able to derive the upper and lower bounds of $\rho^{*}$. Through
simulation, we can also search for the value of $\rho^{*}$ such that the
simulated outage probability of open access is equal to that of closed access.
Furthermore, we also simulate a more realistic scenario, in which each
macrocell UE randomly selects a targeted received power level among $0.5P$,
$P$, $1.5P$, and $2P$ with equal probability. If a macrocell UE is handed-off
to a femtocell, then its targeted received power is multiplied by $\rho$ no
matter which power level it has selected. The figure shows that $\rho^{*}$ is
indeed within the upper bound and the lower bound, and the simulated
$\rho^{*}$ agrees with the analytical $\rho^{*}$, validating the correctness
of our analysis. Furthermore, this remains the case when the targeted received
power is random, indicating the usefulness of our analysis in more practical
scenarios.
Figs. 2 (f) and (g) present the value of $\rho^{**}$ at the femtocell level.
Fig. 2 (f) shows $\rho^{**}$ under different $R$ as we fixed
$\mathbf{x}_{B}=(0,100\textrm{m})$. Fig. 2 (g) shows $\rho^{**}$ under
different $\mathbf{x}_{B}$ ($\mathbf{x}_{B}=(x_{B},0)$) as we fixed $R=50$ m.
The results show that $\rho^{**}$ is indeed within the upper and lower bounds,
and the simulated values of $\rho^{**}$ agree with their analytical values,
validating the correctness of our analysis. Furthermore, $\rho^{**}$ decreases
in $R$ at a rate slightly faster than that of $\rho^{*}$, while it increases
in $x_{B}$, until saturating when the femtocell BS is near the macrocell edge.
This quantifies when femtocells are more beneficial as they decrease in size
and increase in distance away from the macrocell BS.
## 7 Conclusions
In this work, we provide a theoretical framework to analyze the performance
difference between open access and closed access in a two-tier femtocell
network. Through establishing a stochastic geometric model, we capture the
spatial patterns of different network components. Then, we derive the
analytical outage performance of open access and closed access at the
macrocell and femtocell levels. As in most uplink interference analysis, the
outage probability expressions are in non-closed forms. Hence, we derive
closed-form bounds to compare open access and closed access. Simulations and
numerical studies are conducted, validating the correctness of the analytical
model as well as the usefulness of the bounds.
## APPENDIX
##### Proof of Theorem 1
$\displaystyle\mathcal{L}_{I}(s)=\mathbf{E}\left(\exp(-sI)\right)=\mathbf{E}\Bigg{[}\prod_{\mathbf{x}\in\Phi^{0}}u(\mathbf{x})\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\Bigg{]}$
(33) $\displaystyle=$
$\displaystyle\mathbf{E}\Bigg{[}\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi^{0}}u(\mathbf{x})\bigg{|}\Theta\bigg{)}\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\bigg{|}\Theta\bigg{)}\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\bigg{)}\bigg{|}\Theta\bigg{)}\Bigg{]}$
(34) $\displaystyle=$
$\displaystyle\mathbf{E}\Bigg{[}\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi^{0}}u(\mathbf{x})\bigg{|}\Theta\bigg{)}\frac{\mathbf{E}\bigg{(}\prod\limits_{\mathbf{x}\in\Phi^{1}}u(\mathbf{x})\bigg{|}\Theta\bigg{)}}{\mathbf{E}\bigg{(}\prod\limits_{\mathbf{x}\in\Phi^{1}}u(\mathbf{x})\bigg{|}\Theta\bigg{)}}\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\bigg{|}\Theta\bigg{)}\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\bigg{|}\Theta\bigg{)}\Bigg{]}$
(35) $\displaystyle=$
$\displaystyle\mathbf{E}\Bigg{[}\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi}u(\mathbf{x})\bigg{|}\Theta\bigg{)}\frac{\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\bigg{|}\Theta\bigg{)}}{\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}u(\mathbf{x})\bigg{|}\Theta\bigg{)}}\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\bigg{|}\Theta\bigg{)}\Bigg{]}$
(36) $\displaystyle=$
$\displaystyle\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi}u(\mathbf{x})\bigg{)}\mathbf{E}\Bigg{[}\prod_{\mathbf{x}_{0}\in\Theta}\bigg{(}\frac{\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}v(\mathbf{x},\mathbf{x}_{0})\Big{)}}{\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}u(\mathbf{x})\Big{)}}\mathbf{E}\Big{(}\prod_{\mathbf{x}\in\Psi(\mathbf{x}_{0})}w(\mathbf{x},\mathbf{x}_{0})\Big{)}\bigg{)}\Bigg{]}.$
(37)
###### Proof.
The steps to derive Theorem 1 is shown in (33)-(37), where $\Phi^{0}$ is the
point process corresponding to macrocell UEs not inside any femtocell,
$\Phi^{1}$ is the point process corresponding to macrocell UEs inside some
femtocell, and $\Phi$ is the aggregation of $\Phi^{0}$ and $\Phi^{1}$.
By the law of total expectation, we derive (34) from (33). $\Phi^{1}$ can be
rewritten as the union of all the open access UEs in each femtocell, thus
$\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi^{1}}u(\mathbf{x})\bigg{|}\Theta\bigg{)}$
is equal to
$\mathbf{E}\bigg{(}\prod_{\mathbf{x}_{0}\in\Theta}\prod_{\mathbf{x}\in\Omega(\mathbf{x}_{0})}u(\mathbf{x})\bigg{|}\Theta\bigg{)}$.
In addition, because $\Phi$ is the aggregation of $\Phi^{0}$ and $\Phi^{1}$,
$\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi^{0}}u(\mathbf{x})\bigg{|}\Theta\bigg{)}$
$\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi^{1}}u(\mathbf{x})\bigg{|}\Theta\bigg{)}$
is equal to
$\mathbf{E}\bigg{(}\prod_{\mathbf{x}\in\Phi}u(\mathbf{x})\bigg{|}\Theta\bigg{)}$.
By considering the two equalities, we derive (36) from (35). Finally, we
obtain (37) from the conditional expectation theorem. ∎
##### Proof of Theorem 2
###### Proof.
In this proof, we use the fact that $P$ and $R_{c}$ can be normalized and set
$P=R_{c}=1$. Furthermore, we substitute $s=T$ into the integrals in (7) and
(8) to define
$\mathcal{V}(\mathbf{x}_{0})=\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{0},R)}\frac{\frac{T\rho|\mathbf{x}|^{\gamma}}{|\mathbf{x}+\mathbf{x}_{0}|^{\gamma}}}{\frac{T\rho|\mathbf{x}|^{\gamma}}{|\mathbf{x}+\mathbf{x}_{0}|^{\gamma}}+1}d\mathbf{x}\Bigg{)}$
and
$\mathcal{U}(\mathbf{x}_{0})=\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}d\mathbf{x}\Bigg{)}.$
(a) A sufficient condition for $P^{o}_{out}<P^{c}_{out}$
According to (10), (11), (13), and (14), $P^{o}_{out}<P^{c}_{out}$ iff
$\displaystyle\frac{\exp\bigg{(}-\mu\int_{\mathbb{R}^{2}}\Big{(}1-\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}\mathcal{W}(\mathbf{x}_{0})\Big{)}d\mathbf{x}_{0}\bigg{)}}{\exp\bigg{(}-\mu\int_{\mathbb{R}^{2}}\Big{(}1-\mathcal{W}(\mathbf{x}_{0})\Big{)}d\mathbf{x}_{0}\bigg{)}}>1,$
(38)
which is equivalent to
$\displaystyle\int_{\mathbb{R}^{2}}\left(\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}-1\right)\mathcal{W}(\mathbf{x}_{0})d\mathbf{x}_{0}>0.$
(39)
Let
$V(\mathbf{x}_{0})=\int_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}d\mathbf{x}$,
and $U(\mathbf{x}_{0})=\\\
\int_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}d\mathbf{x}$.
Substitute $V(\mathbf{x}_{0})$ and $U(\mathbf{x}_{0})$ into (39), we have
$\displaystyle\int_{\mathbb{R}^{2}}\left(\frac{\exp(-\lambda
V(\mathbf{x}_{0}))}{\exp(-\lambda
U(\mathbf{x}_{0}))}-1\right)\mathcal{W}(\mathbf{x}_{0})d\mathbf{x}_{0}>0.$
(40)
It is easy to see that the following inequality is a sufficient condition for
(40):
$\displaystyle\int_{\mathbb{R}^{2}}\left(-\lambda V(\mathbf{x}_{0})+\lambda
U(\mathbf{x}_{0})\right)\mathcal{W}(\mathbf{x}_{0})d\mathbf{x}_{0}>0.$ (41)
Let $W_{\min}$ and $W_{\max}$ be the lower bound and upper bound of
$\mathcal{W}(\mathbf{x}_{0})$, respectively. According to (6), $W_{\max}=1$
and $W_{\min}=e^{-\overline{\nu}}$. Thus, the following is a sufficient
condition for (41):
$\displaystyle-
W_{\max}\int_{\mathbb{R}^{2}}V(\mathbf{x}_{0})d\mathbf{x}_{0}+W_{\min}\int_{\mathbb{R}^{2}}U(\mathbf{x}_{0})d\mathbf{x}_{0}>0.$
(42)
Let $\mathbf{V}=\int_{\mathbb{R}^{2}}V(\mathbf{x}_{0})d\mathbf{x}_{0}$, we
have the following lemma corresponding to the upper and lower bounds of
$\mathbf{V}$. Hence, the following is a sufficient condition for (42):
$\displaystyle-
W_{\max}\mathbf{V}_{\max}+W_{\min}\int_{\mathbb{R}^{2}}U(\mathbf{x}_{0})d\mathbf{x}_{0}>0.$
(43)
###### Lemma 1.
$\mathbf{V}_{\max}=4\pi^{2}R^{4}(T\rho)^{\frac{2}{\gamma}}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right)$,
$\mathbf{V}_{\min}=2\pi^{2}R^{4}(T\rho)^{\frac{2}{\gamma}}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right)$,
then $\mathbf{V}_{\min}\leq\mathbf{V}\leq\mathbf{V}_{\max}$.
Proof: See the next subsection.
In addition, we have
$\displaystyle\int_{\mathbb{R}^{2}}U(\mathbf{x}_{0})d\mathbf{x}_{0}=\int_{\mathbb{R}^{2}}\int_{\mathcal{B}(\mathbf{x}_{0},R)}\left(\frac{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}\right)d\mathbf{x}d\mathbf{x}_{0}$
$\displaystyle=$ $\displaystyle\pi
R^{2}\int_{\mathbb{R}^{2}}\left(\frac{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}\right)d\mathbf{x}=\pi
R^{2}C_{u},$ (44)
where
$\displaystyle
C_{u}=\int_{\mathbb{R}^{2}}\left(\frac{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}\right)d\mathbf{x}$
(45)
is only related to predetermined system-level constants $T$ and $\gamma$.
As a consequence (43) becomes
$\displaystyle-W_{\max}\mathbf{V}_{\max}+W_{\min}\pi R^{2}C_{u}>0.$ (46)
(b) A sufficient condition for $P^{o}_{out}>P^{c}_{out}$
According to (10), (11), (13), and (14), $P^{o}_{out}>P^{c}_{out}$ iff
$\displaystyle\frac{\exp\bigg{(}-\mu\int_{\mathbb{R}^{2}}\Big{(}1-\mathcal{W}(\mathbf{x}_{0})\Big{)}d\mathbf{x}_{0}\bigg{)}}{\exp\bigg{(}-\mu\int_{\mathbb{R}^{2}}\Big{(}1-\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}\mathcal{W}(\mathbf{x}_{0})\Big{)}d\mathbf{x}_{0}\bigg{)}}>1,$
(47)
Then the following is a sufficient condition for (47):
$\displaystyle\int_{\mathbb{R}^{2}}\left(-\lambda U(\mathbf{x}_{0})+\lambda
V(\mathbf{x}_{0})\right)\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}\mathcal{W}(\mathbf{x}_{0})d\mathbf{x}_{0}>0.$
(48)
Let $W_{\min}^{\prime}$ and $W_{\max}^{\prime}$ be the lower bound and upper
bound of
$\frac{\mathcal{V}(\mathbf{x}_{0})}{\mathcal{U}(\mathbf{x}_{0})}\mathcal{W}(\mathbf{x}_{0})$,
respectively. According to (6), (7), and (8),
$W_{\max}^{\prime}=\exp\left(\overline{\lambda}\right)$ and
$W_{\min}^{\prime}=\exp\left(-\overline{\lambda}-\overline{\nu}\right)$.
Finally, (49) is a sufficient condition for (48)
$\displaystyle-W_{\max}^{\prime}\pi
R^{2}C_{u}+W_{\min}^{\prime}\mathbf{V}_{\min}>0.$ (49)
∎
##### Proof of Lemma 1
###### Proof.
Upper Bound of $\mathbf{V}$
$\displaystyle\mathbf{V}=$
$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}d\mathbf{x}d\mathbf{x}_{0}$
(50) $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{2}}\int_{\mathcal{B}(\mathbf{x},R)}\frac{\frac{T\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}d\mathbf{x}_{0}d\mathbf{x}$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}2\pi
r_{1}\int_{0}^{R}\frac{\frac{T\rho
r_{2}^{\gamma}}{r_{1}^{\gamma}}}{\frac{T\rho
r_{2}^{\gamma}}{r_{1}^{\gamma}}+1}2\pi r_{2}dr_{2}dr_{1}$ (51)
$\displaystyle\leq$ $\displaystyle\int_{0}^{\infty}2\pi
r_{1}\int_{0}^{R}\mathbf{1}(T\rho\frac{r_{2}^{\gamma}}{r_{1}^{\gamma}}\geq
1)2\pi r_{2}dr_{2}dr_{1}+$ $\displaystyle\int_{0}^{\infty}2\pi
r_{1}\int_{0}^{R}\mathbf{1}(T\rho\frac{r_{2}^{\gamma}}{r_{1}^{\gamma}}<1)\frac{T\rho
r_{2}^{\gamma}}{r_{1}^{\gamma}}2\pi r_{2}dr_{2}dr_{1}$ (52) $\displaystyle=$
$\displaystyle
4\pi^{2}R^{4}(T\rho)^{\frac{2}{\gamma}}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right).$
(53)
In (51), the integrated item is in the form of $\frac{X}{X+1}$, where
$X=\frac{T\rho r_{2}^{\gamma}}{r_{1}^{\gamma}}\geq 0$. The bound of the
integrated item can be found as follows: if $X\geq 1$,
$\frac{1}{2}\leq\frac{X}{X+1}\leq 1$; otherwise, if $X<1$,
$\frac{X}{2}\leq\frac{X}{X+1}\leq X$. Accordingly, we can separate the
integration region into $\frac{T\rho r_{2}^{\gamma}}{r_{1}^{\gamma}}\geq 1$
region and $\frac{T\rho r_{2}^{\gamma}}{r_{1}^{\gamma}}<1$ region. As a
consequence, the upper bound of (51) can be derived as (52).
Lower Bound of $\mathbf{V}$
Following a similar approach as above, we have
$\displaystyle\mathbf{V}=$ $\displaystyle\int_{0}^{\infty}2\pi
r_{1}\int_{0}^{R}\frac{\frac{T\rho
r_{2}^{\gamma}}{r_{1}^{\gamma}}}{\frac{T\rho
r_{2}^{\gamma}}{r_{1}^{\gamma}}+1}2\pi r_{2}dr_{2}dr_{1}$ (54)
$\displaystyle\geq$ $\displaystyle\int_{0}^{\infty}2\pi
r_{1}\int_{0}^{R}\mathbf{1}(T\rho\frac{r_{2}^{\gamma}}{r_{1}^{\gamma}}\geq
1)\pi r_{2}dr_{2}dr_{1}+$ $\displaystyle\int_{0}^{\infty}2\pi
r_{1}\int_{0}^{R}\mathbf{1}(T\rho\frac{r_{2}^{\gamma}}{r_{1}^{\gamma}}<1)\frac{T\rho
r_{2}^{\gamma}}{r_{1}^{\gamma}}\pi r_{2}dr_{2}dr_{1}$ (55) $\displaystyle=$
$\displaystyle
2\pi^{2}R^{4}(T\rho)^{\frac{2}{\gamma}}\left(\frac{1}{8}+\frac{1}{4(\gamma+2)}+\frac{1}{(\gamma+2)(\gamma-2)}\right).$
(56)
∎
##### Proof of Theorem 3
###### Proof.
In this proof, we use the fact that $P$ and $R_{c}$ can be normalized and set
$P=R_{c}=1$. Furthermore, we substitute $s=T^{\prime}$ into the integrals in
(25)-(28) to define
$\mathcal{V}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T^{\prime}\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{T^{\prime}\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}\Bigg{)}$,
$\mathcal{U}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}\Bigg{)}$,
$\mathcal{V}^{\prime\prime}(\mathbf{x}_{B})=e^{-\frac{T^{\prime}\rho\overline{\lambda}}{T^{\prime}\rho+1}}$,
and
$\mathcal{U}^{\prime\prime}(\mathbf{x}_{B})=\exp\Bigg{(}-\lambda\int\limits_{\mathcal{B}(\mathbf{x}_{B},R)}\frac{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}\Bigg{)}.$
(a) A sufficient condition for
$\widehat{P}^{o}_{out}(\mathbf{x}_{B})<\widehat{P}^{c}_{out}(\mathbf{x}_{B})$
According to (22), (29), (31), and (32),
$\widehat{P}^{o}_{out}(\mathbf{x}_{B})<\widehat{P}^{c}_{out}(\mathbf{x}_{B})$
iff
$\displaystyle\frac{\exp\bigg{(}-\mu\int_{\mathbb{R}^{2}}\Big{(}1-\frac{\mathcal{V}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}{\mathcal{U}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})\Big{)}d\mathbf{x}_{0}\bigg{)}}{\exp\bigg{(}-\mu\int_{\mathbb{R}^{2}}\Big{(}1-\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})\Big{)}d\mathbf{x}_{0}\bigg{)}}\frac{\mathcal{V}^{\prime\prime}(\mathbf{x}_{B})}{\mathcal{U}^{\prime\prime}(\mathbf{x}_{B})}>1.$
(57)
Let
$V^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=\int_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T^{\prime}\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{T^{\prime}\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}$,
$U^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=\\\
\int_{\mathcal{B}(\mathbf{x}_{0},R)}\left(\frac{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}\right)d\mathbf{x}$,
and $\mathcal{R}(\mathbf{x}_{B})=\int_{\mathcal{B}(\mathbf{x}_{B},R)}\\\
\frac{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}d\mathbf{x}$.
Substituting $V^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})$,
$U^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})$ and $\mathcal{R}(\mathbf{x}_{B})$
into (57), similar to (41), the following is a sufficient condition for (57):
$\displaystyle\mu\int_{\mathbb{R}^{2}}\left(-\lambda
V^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})+\lambda
U(\mathbf{x}_{0},\mathbf{x}_{B})\right)\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})d\mathbf{x}_{0}$
$\displaystyle-\frac{\lambda\pi
R^{2}T^{\prime}\rho}{T^{\prime}\rho+1}+\lambda\mathcal{R}(\mathbf{x}_{B})>0.$
(58)
Let $W_{\min}^{\prime\prime}$ and $W_{\max}^{\prime\prime}$ be the lower bound
and upper bound of $\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})$,
respectively. According to (24), $W_{\max}^{\prime\prime}=1$ and
$W_{\min}^{\prime\prime}=\exp\left(-\overline{\nu}\right)$. Thus, the
following is a sufficient condition for (58):
$\displaystyle\mu\int_{\mathbb{R}^{2}}-\left(V^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})W^{\prime\prime}_{\max}+U(\mathbf{x}_{0},\mathbf{x}_{B})W^{\prime\prime}_{\min}\right)d\mathbf{x}_{0}$
$\displaystyle-\frac{\pi
R^{2}T^{\prime}\rho}{T^{\prime}\rho+1}+\mathcal{R}(\mathbf{x}_{B})>0,$ (59)
where
$\int_{\mathbb{R}^{2}}V^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})d\mathbf{x}_{0}=\int_{\mathbb{R}^{2}}\int_{\mathcal{B}(\mathbf{x}_{0},R)}\frac{\frac{T^{\prime}\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}}{\frac{T^{\prime}\rho|\mathbf{x}-\mathbf{x}_{0}|^{\gamma}}{|\mathbf{x}|^{\gamma}}+1}d\mathbf{x}d\mathbf{x}_{0}$
is in the same form as (50). Thus, by applying Lemma 1, we can derive its
upper bound and lower bound as $\mathbf{V}_{\max}^{\prime}$ and
$\mathbf{V}_{\min}^{\prime}$ from (53) and (56), respectively. Similar to the
derivation of (44),
$\int_{\mathbb{R}^{2}}U^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})=\pi
R^{2}C_{u}^{\prime}$ where
$C_{u}^{\prime}=\int_{\mathbb{R}^{2}}\left(\frac{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}}{\frac{T^{\prime}|\mathbf{x}-\mathcal{BS}(\mathbf{x})|^{\gamma}}{|\mathbf{x}-\mathbf{x}_{B}|^{\gamma}}+1}\right)d\mathbf{x}$
is a constant predetermined by $T^{\prime}$ and $\gamma$.
In addition, the lower bound $\mathcal{R}_{\min}(\mathbf{x}_{B})$ and the
upper bound $\mathcal{R}_{\max}(\mathbf{x}_{B})$ of
$\mathcal{R}(\mathbf{x}_{B})$ can be derived as follows:
$\displaystyle\mathcal{R}_{\min}(\mathbf{x}_{B})=$ (60)
$\displaystyle\begin{cases}\pi\int_{0}^{R}\frac{\frac{T^{\prime}(|\mathbf{x}_{B}|)^{\gamma}}{r^{\gamma}}r}{\frac{T^{\prime}(|\mathbf{x}_{B}|)^{\gamma}}{r^{\gamma}}+1}dr&\mbox{if
}|\mathbf{x}_{B}|\leq R,\\\
\pi\int_{0}^{R}\frac{\frac{T^{\prime}(|\mathbf{x}_{B}|-R)^{\gamma}}{r^{\gamma}}r}{\frac{T^{\prime}(|\mathbf{x}_{B}|-R)^{\gamma}}{r^{\gamma}}+1}dr+\pi\int_{0}^{R}\frac{\frac{T^{\prime}(|\mathbf{x}_{B}|)^{\gamma}}{r^{\gamma}}r}{\frac{T^{\prime}(|\mathbf{x}_{B}|)^{\gamma}}{r^{\gamma}}+1}dr&\mbox{if
}|\mathbf{x}_{B}|>R,\end{cases}$
and
$\displaystyle\mathcal{R}_{\max}(\mathbf{x}_{B})=$
$\displaystyle\pi\int_{0}^{R}\frac{\frac{T^{\prime}(|\mathbf{x}_{B}|+R)^{\gamma}}{r^{\gamma}}r}{\frac{T^{\prime}(|\mathbf{x}_{B}|+R)^{\gamma}}{r^{\gamma}}+1}dr+$
$\displaystyle\qquad\qquad\pi\int_{0}^{R}\frac{\frac{T^{\prime}(\sqrt{|\mathbf{x}_{B}|^{2}+R^{2}})^{\gamma}}{r^{\gamma}}r}{\frac{T^{\prime}(\sqrt{|\mathbf{x}_{B}|^{2}+R^{2}})^{\gamma}}{r^{\gamma}}+1}dr.$
(61)
Note that $\int\frac{Br}{r^{\gamma}+B}dr$ is in closed form when $\gamma$ is a
rational number. Therefore, both $\mathcal{R}_{\min}(\mathbf{x}_{B})$ and
$\mathcal{R}_{\max}(\mathbf{x}_{B})$ are expressed in closed forms.
Finally, the following is a sufficient condition for (59):
$\displaystyle-\mu\mathbf{V}_{\max}^{\prime}+\mu\pi
R^{2}C_{u}^{\prime}W_{\min}^{\prime\prime}-\frac{\pi
R^{2}T^{\prime}\rho}{T^{\prime}\rho+1}+\mathcal{R}_{\min}(\mathbf{x}_{B})>0.$
(62)
(b) A sufficient condition for
$\widehat{P}^{o}_{out}(\mathbf{x}_{B})>\widehat{P}^{c}_{out}(\mathbf{x}_{B})$
$\widehat{P}^{o}_{out}(\mathbf{x}_{B})>\widehat{P}^{c}_{out}(\mathbf{x}_{B})$
iff
$\displaystyle\mu\int_{\mathbb{R}^{2}}\left(-\lambda
U^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})+\lambda
V^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})\right)\frac{\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})\mathcal{V}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}{\mathcal{U}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}d\mathbf{x}_{0}$
$\displaystyle+\frac{\lambda\pi
R^{2}T^{\prime}\rho}{T^{\prime}\rho+1}-\lambda\mathcal{R}(\mathbf{x}_{0})>0.$
(63)
Let $W_{\min}^{\prime\prime\prime}$ and $W_{\max}^{\prime\prime\prime}$ be the
lower bound and upper bound value of
$\frac{\mathcal{W}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})\mathcal{V}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}{\mathcal{U}^{\prime}(\mathbf{x}_{0},\mathbf{x}_{B})}$,
respectively. According to (24)-(26),
$W_{\max}^{\prime\prime\prime}=\exp\left(\overline{\lambda}\right)$ and
$W_{\min}^{\prime\prime\prime}=\exp\left(-\overline{\lambda}-\overline{\nu}\right)$.
Then similarly to the derivation of (62), we see that the following is a
sufficient condition for (63):
$\displaystyle-\mu\pi
R^{2}C_{u}^{\prime}W_{\max}^{\prime\prime\prime}+\mu\mathbf{V}_{\min}^{\prime}W_{\min}^{\prime\prime\prime}+\frac{\pi
R^{2}T^{\prime}\rho}{T^{\prime}\rho+1}-\mathcal{R}_{\max}(\mathbf{x}_{B})>0.$
(64)
∎
## References
* [1] X. An and F. Pianese. Understanding co-channel interference in LTE-based multi-tier cellular networks. In Proc. of ACM PE-WASUN, Paphos, Cyprus, Oct. 2012.
* [2] M. Y. Arslan, J. Yoon, K. Sundaresan, S. V. Krishnamurthy, and S. Banerjee. FERMI: a femtocell resource management system forinterference mitigation in OFDMA networks. In Proc. of ACM MobiCom, Las Vegas, NV, Sept. 2011.
* [3] F. Baccelli and B. Blaszczyszyn. Stochastic geometry and wireless networks, volume 1: Theory. Foundations and Trends in Networking, 3(3-4):249 – 449, 2009.
* [4] W. Bao and B. Liang. Uplink interference analysis for two-tier cellular networks with diverse users under random spatial patterns. In Proc. of IEEE/CIC International Conference on Communications in China (ICCC), Xi’an, China, Aug. 2013.
* [5] C. C. Chan and S. Hanly. Calculating the outage probability in a CDMA network with spatial Poisson traffic. IEEE Trans. on Vehicular Technology, 50(1):183 – 204, Jan. 2001\.
* [6] V. Chandrasekhar and J. Andrews. Uplink capacity and interference avoidance for two-tier femtocell networks. IEEE Trans. on Wireless Communications, 8(7):3498–3509, Jul. 2009\.
* [7] W. C. Cheung, T. Q. S. Quek, and M. Kountouris. Stochastic analysis of two-tier networks: Effect of spectrum allocation. In Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic, May 2011.
* [8] H. Dhillon, R. Ganti, F. Baccelli, and J. Andrews. A tractable framework for coverage and outage in heterogeneous cellular networks. In Information Theory and Applications Workshop, San Diego, CA, Feb. 2011.
* [9] H. Dhillon, R. Ganti, F. Baccelli, and J. Andrews. Modeling and analysis of K-tier downlink heterogeneous cellular networks. IEEE Journal on Selected Areas in Communications, 30(3):550–560, Apr. 2012.
* [10] H. S. Dhillon, R. K. Ganti, and J. G. Andrews. Load-aware heterogeneous cellular networks: Modeling and SIR distribution. In Proc. of IEEE GLOBECOM, Anaheim, CA, Dec. 2012.
* [11] K. Gilhousen, I. Jacobs, R. Padovani, A. Viterbi, L. Weaver, and C. Wheatley. On the capacity of a cellular CDMA system. IEEE Trans. on Vehicular Technology, 40(2):303–312, May 1991.
* [12] H.-S. Jo, P. Xia, and J. G. Andrews. Open, closed, and shared access femtocells in the downlink. arXiv:1009.3522 [cs.NI], 2010.
* [13] Y. Kim, S. Lee, and D. Hong. Performance analysis of two-tier femtocell networks with outage constraints. IEEE Trans. on Wireless Communications, 9(9):2695– 2700, Sept. 2010\.
* [14] S. Kishore, L. Greenstein, H. Poor, and S. Schwartz. Uplink user capacity in a CDMA macrocell with a hotspot microcell: exact and approximate analyses. IEEE Trans. on Wireless Communications, 2(2):364–374, Mar. 2003\.
* [15] S. Kishore, L. Greenstein, H. Poor, and S. Schwartz. Uplink user capacity in a multicell CDMA system with hotspot microcells. IEEE Trans. on Wireless Communications, 5(6):1333–1342, June 2006\.
* [16] D. Knisely, T. Yoshizawa, and F. Favichia. Standardization of femtocells in 3GPP. IEEE Communications Magazine, 47(9):68–75, Sept. 2009.
* [17] T. D. Novlan, H. S. Dhillon, and J. G. Andrews. Analytical modeling of uplink cellular networks. arXiv:1203.1304 [cs.IT], 2012.
* [18] S. Singh, H. S. Dhillon, and J. G. Andrews. Offloading in heterogeneous networks: Modeling, analysis and design insights. arXiv:1208.1977 [cs.IT], 2012.
* [19] D. Stoyan, W. Kendall, and J. Mecke. Stochastic Geometry and Its Applications. Wiley, second edition, 1995.
* [20] P. Tarasak, T. Q. S. Quek, and F. P. S. Chin. Uplink timing misalignment in open and closed access OFDMA femtocell networks. IEEE Communications Letters, 15(9):926–928, Sept. 2011.
* [21] A. J. Viterbi, A. M. Viterbi, K. Gilhousen, and E. Zehavi. Soft handoff extends CDMA cell coverage and increases reverse link capacity. IEEE Journal on Selected Areas in Communications, 12(8):1281–1288, Oct. 1994.
* [22] P. Xia, V. Chandrasekhar, and J. Andrews. Open vs. closed access femtocells in the uplink. IEEE Trans. on Wireless Communications, 9(12):3798–3809, Dec. 2010\.
* [23] J. Yoon, M. Y. Arslan, K. Sundaresan, S. V. Krishnamurthy, and S. Banerjee. A distributed resource management framework for interference mitigation in OFDMA femtocell networks. In Proc. of ACM MobiHoc, Hilton Head Island, SC, June 2012 2012\.
|
arxiv-papers
| 2013-08-12T03:42:31 |
2024-09-04T02:49:49.318287
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Bao and Ben Liang",
"submitter": "Wei Bao",
"url": "https://arxiv.org/abs/1308.2454"
}
|
1308.2461
|
# Space-time fractional diffusion equations and asymptotic behaviors of a
coupled continuous time random walk model
Long Shi1,2, Zuguo Yu1, Zhi Mao1, Aiguo Xiao1 and Hailan Huang1
1School of Mathematics and Computational Science, Xiangtan University, Hunan
411105, China.
2Institute of Mathematics and Physics, Central South University of Forest and
Technology, Changsha, Hunan 410004, China. Corresponding author, email:
[email protected]
###### Abstract
In this paper, we consider a type of continuous time random walk model where
the jump length is correlated with the waiting time. The asymptotic behaviors
of the coupled jump probability density function in the Fourier-Laplace domain
are discussed. The corresponding fractional diffusion equations are derived
from the given asymptotic behaviors. Corresponding to the asymptotic behaviors
of the joint probability density function in the Fourier-Laplace space, the
asymptotic behaviors of the waiting time probability density and the
conditional probability density for jump length are also discussed.
Keywords: Space-time fractional diffusion equation, Caputo fractional
derivative, Riesz fractional derivative, coupled continuous time random walk,
asymptotic behavior
## 1 Introduction
The continuous time random walk (CTRW) theory, which was introduced in the
1960s by Montroll and Weiss to describe a walker hopping randomly on a
periodic lattice with the steps occurring at random time intervals [1], has
been applied successfully in many fields (e.g. the reviews [2-4] and
references therein).
In a continuum one-dimensional space, the CTRW scheme is characterised by a
jump probability density function (PDF) $\psi(x,t)$, which is the probability
density that the walker makes a jump of length $x$ after some waiting time
$t$. Let $P(x,t)$ be the PDF of finding the walker at a given place $x$ and at
time $t$ with the initial condition $P(x,0)=\delta(x)$. A CTRW process can be
described by the following integral equation [3]:
$P(x,t)=\int_{-\infty}^{+\infty}dx^{\prime}\int_{0}^{t}\psi(x-x^{\prime},t-t^{\prime})P(x^{\prime},t^{\prime})dt^{\prime}+\delta(x)\Phi(t),$
(1)
where $\Phi(t)=1-\int_{0}^{t}\varphi(\tau)d\tau$ is the probability of not
having made a jump until time $t$ and
$\varphi(t)=\int_{-\infty}^{+\infty}\psi(x,t)dx$ is the waiting time PDF.
Fractional diffusion equations (FDEs) arise quite naturally as the limiting
dynamic equations of the CTRW models with temporal and/or space memories [5].
The asymptotic relation between the CTRW models and fractional diffusion
processes was studied firstly by Balakrishnan in 1985, dealing with the
anomalous diffusion in one dimension [6]. Later, many authors discussed the
relation between CTRW and FDEs [3-5,7-19]. However, the usual assumption in
most of these works is that the CTRW is decoupled, which means that the jump
lengths and the waiting times are independent. Recently the coupled CTRW
models have attracted more attention [20-25]. Here we focus on the coupled
CTRW models with the jump length correlated with the waiting time [25], i.e.
$\psi(x,t)=\varphi(t)\lambda(x|t)$, and derive the corresponding FDEs from the
asymptotic behaviors of the waiting time PDF $\varphi(t)$ and the jump PDF
$\psi(x,t)$ in the Fourier-Laplace space.
This paper is organized as follows. In section 2, we introduce a space-time
fractional diffusion equation which can be obtained from the standard
diffusion equation by replacing the first-order time derivative and/or the
second-order space derivative by a Caputo derivative of order $\alpha\in(0,2]$
and/or a Riesz derivative of order $\beta\in(0,2]$, respectively. In section
3, the asymptotic behaviors of the jump PDF $\psi(x,t)$ in the Fourier-Laplace
domain are given and the corresponding FDEs are derived. In section 4,
corresponding to the asymptotic behaviors of the jump PDF $\psi(x,t)$ in the
Fourier-Laplace domain, the asymptotic behaviors of the waiting time PDF
$\varphi(t)$ and the conditional PDF of jump length $\lambda(x|t)$ are
discussed. In section 5, some conclusions are presented.
## 2 The space-time fractional diffusion equation
We consider a space-time FDE [10]
$_{0}^{C}D_{t}^{\alpha}u(x,t)=K\frac{\partial^{\beta}u(x,t)}{\partial|x|^{\beta}},\hskip
14.22636ptx\in R,t>0,$ (2)
where $u(x,t)$ is the field variable, $K$ is the generalized diffusion
constant and the real paraments $\alpha,\beta$ are restricted to the range
$0<\alpha\leq 2,\ 0<\beta\leq 2$.
In Eq. (2), the time derivative is the Caputo fractional derivative of order
$\alpha$, defined as [26]
$_{0}^{C}D_{t}^{\alpha}g(t)=\left\\{\begin{array}[]{cl}\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}\frac{g^{(n)}(\tau)d\tau}{(t-\tau)^{\alpha+1-n}},&n-1<\alpha<n,\\\
\\\ g^{(n)}(t),&\alpha=n\in N,\end{array}\right.$ (3)
and the space derivative is the Riesz fractional derivative of order $\beta$,
defined as [27]
$\frac{d^{\beta}}{d|x|^{\beta}}f(x)=\left\\{\begin{array}[]{cl}\Gamma(1+\beta)\frac{\sin(\beta\pi/2)}{\pi}\int_{0}^{+\infty}\frac{f(x+\xi)-2f(x)+f(x-\xi)}{\xi^{1+\beta}}d\xi,&0<\beta<2,\\\
\\\ \frac{d^{2}f(x)}{dx^{2}},&\beta=2.\end{array}\right.$ (4)
Let
$\widehat{f}(k)={\cal F}\\{f(x)\\}=\int_{-\infty}^{+\infty}f(x)e^{ikx}dx$ (5)
be the Fourier transform of $f(x)$ and
$\widetilde{g}(s)={\cal L}\\{g(t)\\}=\int_{0}^{+\infty}g(t)e^{-st}dt$ (6)
be the Laplace transform of $g(t)$.
Now, let us recall the following fundamental formulas about the Laplace
transform of the Caputo fractional derivative of order $\alpha$ and the
Fourier transform of the Riesz fractional derivative of order $\beta$:
${\cal
L}\\{_{0}^{C}D_{t}^{\alpha}g(t)\\}=s^{\alpha}\widetilde{g}(s)-\sum\limits_{m=0}^{n-1}s^{\alpha-1-m}g^{(m)}(0),\hskip
14.22636ptn-1<\alpha\leq n,$ (7)
${\cal F}\\{\frac{d^{\beta}}{d|x|^{\beta}}f(x)\\}=-|k|^{\beta}\widehat{f}(k).$
(8)
After applying the formula (7), in the Laplace space, the space-time FDE (2)
appears in the form
$s^{\alpha}\widetilde{u}(x,s)-s^{\alpha-1}u(x,0)=K\frac{\partial^{\beta}\widetilde{u}(x,s)}{\partial|x|^{\beta}}$
(9)
for $0<\alpha\leq 1$ and in the form
$s^{\alpha}\widetilde{u}(x,s)-s^{\alpha-1}u(x,0)-s^{\alpha-2}u_{t}(x,0)=K\frac{\partial^{\beta}\widetilde{u}(x,s)}{\partial|x|^{\beta}}$
(10)
for $1<\alpha\leq 2$.
Taking the Fourier transform of Eq.(9) with initial condition
$u(x,0)=\delta(x)$, or of Eq.(10) with initial conditions $u(x,0)=\delta(x),\
u_{t}(x,0)=0$, we get
$s^{\alpha}\widehat{\widetilde{u}}(k,s)-s^{\alpha-1}=-K|k|^{\beta}\widehat{\widetilde{u}}(k,s),$
(11)
and obtain immediately
$\widehat{\widetilde{u}}(k,s)=\frac{s^{\alpha-1}}{s^{\alpha}+K|k|^{\beta}},\hskip
14.22636pt0<\alpha\leq 2,0<\beta\leq 2.$ (12)
Remark 1: The fundamental solutions and the asymptotic solutions of Eq.(2)
(containing its special cases) have been considered in many previous works
[10,16,28-34]. In the following section we will consider two types of
asymptotic behaviors of the jump PDF $\psi(x,t)$ in the Fourier-Laplace space
and derive the corresponding space-time FDEs.
## 3 From the coupled CTRW models to FDEs
After taking the Laplace transform in the variable $t$ and the Fourier
transform in the variable $x$ of Eq.(1), we get the following well-known
relation [3]
$\widehat{\widetilde{P}}(k,s)=\frac{1-\widetilde{\varphi}(s)}{s}\cdot\frac{1}{1-\widehat{\widetilde{\psi}}(k,s)},$
(13)
which is called the Montroll-Weiss equation.
Different types of CTRW processes can be categorised by the existence or non-
existence of the characteristic waiting time [3]
$T=\int_{0}^{+\infty}dt\int_{-\infty}^{+\infty}t\psi(x,t)dx,$ (14)
and the second moment of the jump length
$\sigma^{2}=\int_{-\infty}^{+\infty}dx\int_{0}^{+\infty}x^{2}\psi(x,t)dt.$
(15)
For finite $T$ and $\sigma^{2}$, the Laplace transform of the waiting time PDF
$\varphi(t)$ and the Fourier transform of the jump length PDF $\lambda(x)$ are
of the forms
$\widetilde{\varphi}(s)=1-sT+o(s),\hskip 14.22636pts\rightarrow 0,$ (16)
$\widehat{\lambda}(k)=1-\sigma^{2}k^{2}+o(k^{2}),\hskip 14.22636ptk\rightarrow
0.$ (17)
In many applications, one needs to consider long waiting time and/or long jump
length, meaning that the characteristic waiting time and/or the second moment
of the jump length are infinite. It is natural to generalize Eq. (16) and Eq.
(17) to the following forms [3]:
$\widetilde{\varphi}(s)=1-A_{\alpha}s^{\alpha}+o(s^{\alpha}),\hskip
14.22636pts\rightarrow 0,0<\alpha\leq 1,$ (18)
and/or
$\widehat{\lambda}(k)=1-A_{\beta}|k|^{\beta}+o(|k|^{\beta}),\hskip
14.22636ptk\rightarrow 0,0<\beta\leq 2,$ (19)
where $A_{\alpha}$ and $A_{\beta}$ are two positive normal constants.
Therefore, for the decoupled case, in the limit $(k,s)\rightarrow(0,0)$, one
has
$\begin{array}[]{lll}\widehat{\widetilde{\psi}}(k,s)&=&(1-A_{\alpha}s^{\alpha}+o(s^{\alpha}))(1-A_{\beta}|k|^{\beta}+o(|k|^{\beta}))\\\
\\\
&=&1-A_{\alpha}s^{\alpha}-A_{\beta}|k|^{\beta}+O(s^{\alpha}|k|^{\beta}).\end{array}$
(20)
In Eq. (20), the term $1-A_{\alpha}s^{\alpha}-A_{\beta}|k|^{\beta}$ has main
influence on $\widehat{\widetilde{\psi}}(k,s)$ in the limit
$(k,s)\rightarrow(0,0)$. So we can weaken the independent condition
$\psi(x,t)=\lambda(x)\varphi(t)$ and assume $\psi(x,t)$ has the following form
in the Fourier-Laplace domain:
$\widehat{\widetilde{\psi}}(k,s)=1-A_{\alpha}s^{\alpha}-A_{\beta}|k|^{\beta}+o(s^{\alpha},|k|^{\beta}),$
(21)
which implies that $\psi(x,t)$ is coupled. If
$o(s^{\alpha},|k|^{\beta})=O(s^{\alpha}|k|^{\beta})$, Eq. (21) reduces to the
decoupled case.
Inserting Eq. (18) and Eq. (21) into Eq. (13), in the limit
$(k,s)\rightarrow(0,0)$, we obtain
$\widehat{\widetilde{P}}(k,s)=\frac{s^{\alpha-1}}{s^{\alpha}+K|k|^{\beta}},\hskip
14.22636pt0<\alpha\leq 1,0<\beta\leq 2,$ (22)
where $K=\frac{A_{\beta}}{A_{\alpha}}$.
By comparing Eq. (22) with Eq. (12), with the initial condition
$P(x,0)=\delta(x)$, the following space-time fractional equation is derived
immediately:
$_{0}^{c}D_{t}^{\alpha}P(x,t)=K\frac{\partial^{\beta}P(x,t)}{\partial|x|^{\beta}},\hskip
14.22636pt0<\alpha\leq 1,0<\beta\leq 2.$ (23)
Remark 2: We derived Eq. (23) by using the coupled CTRW model with the
asymptotic relations Eqs. (18) and (21). The same space-time FDE has been also
derived using the decoupled CTRW models in Refs. [14,18,35], where the
distribution of the waiting times and that of the jump lengths are required to
be independent of each other. In Refs. [18,35], the authors showed how the
integral equation for the CTRW reduces to the space-time fractional diffusion
equation by a properly scaled passage to the limit of compressed waiting times
and jump lengths. Here we extend their consideration to the coupled case. In
Ref. [14], the authors noted that the same result can be derived by weakening
the independent hypothesis and replacing it with
$\widehat{\widetilde{\psi}}(k,s)\sim 1-s^{\gamma}-|k|^{\beta}$. But they did
not discuss under what conditions one has the above limiting behavior for the
joint distribution $\psi(x,t)$. In the following section, we will explore the
problem and consider a specific case.
Next, let us extend the asymptotic relation (21) further and suppose
$\psi(x,t)$ has the following form of in the Fourier-Laplace space:
$\widehat{\widetilde{\psi}}(k,s)\sim
1-A_{\alpha}s^{\alpha}-A_{\beta}\frac{|k|^{\beta}}{s^{\gamma-\alpha}},\hskip
14.22636pt0<\alpha\leq 1,\alpha<\gamma\leq 2,0<\beta\leq 2,$ (24)
which implies that $\psi(x,t)$ cannot be decoupled in any event. When
$\beta=2$ the asymptotic behavior of $\widehat{\widetilde{\psi}}(k,s)$ in (24)
has been discussed in Ref. [36].
Inserting Eq. (18) and Eq. (24) into Eq. (13), in the limit
$(k,s)\rightarrow(0,0)$, we obtain
$\begin{array}[]{lll}\widehat{\widetilde{P}}(k,s)&=&\frac{A_{\alpha}s^{\alpha-1}}{A_{\alpha}s^{\alpha}+A_{\beta}|k|^{\beta}s^{\alpha-\gamma}}\\\
\\\ &=&\frac{s^{\gamma-1}}{s^{\gamma}+K|k|^{\beta}},\hskip
14.22636pt0<\alpha\leq 1,\alpha<\gamma\leq 2,\end{array}$ (25)
where $K=\frac{A_{\beta}}{A_{\alpha}}$.
By comparing Eq.(25) with Eq.(12), we obtain the following space-time FDE:
$_{0}^{c}D_{t}^{\gamma}P(x,t)=K\frac{\partial^{\beta}P(x,t)}{\partial|x|^{\beta}},\hskip
14.22636pt0<\alpha\leq 1,\alpha<\gamma\leq 2,0<\beta\leq 2,$ (26)
with the initial condition $P(x,t=0)=\delta(x)$ for $\alpha<\gamma\leq 1$ or
the initial conditions $P(x,t=0)=\delta(x),P_{t}(x,t=0)=0$ for $1<\gamma\leq
2$.
## 4 The derivation of the asymptotic behaviors of the jump PDF $\psi(x,t)$
In this work, we focus on the coupled CTRW model where the jump PDF
$\psi(x,t)$ has the form $\psi(x,t)=\varphi(t)\lambda(x|t)$, meaning that the
jump length is correlated with the waiting time [25]. In the following, in the
Fourier-Laplace domain, we derive the asymptotic behaviors of
$\widehat{\widetilde{\psi}}(k,s)$ in the limit $(k,s)\rightarrow(0,0)$ which
are introduced in the previous section.
For the waiting time PDF $\varphi(t)$, we assume in the Laplace space
$\widetilde{\varphi}(s)\sim 1-A_{\alpha}s^{\alpha},\hskip
14.22636pts\rightarrow 0,0<\alpha\leq 1.$ (27)
For the conditional PDF $\lambda(x|t)$, we assume
$\lambda(x|t)=\left\\{\begin{array}[]{lll}\frac{1}{\sqrt{4\pi
g(t)}}\exp(-\frac{x^{2}}{4g(t)}),&if&\beta=2,\\\ \\\
\frac{1}{(g(t))^{1/\beta}}L_{\beta}(\frac{x}{(g(t))^{1/\beta}}),&if&0<\beta<2,\end{array}\right.$
(28)
where $L_{\beta}(x)$ is two-sided L$\acute{e}$vy stable probability density,
defined in Ref. [37]
$L_{\beta}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\exp(-|k|^{\beta})e^{-ikx}dk$
(29)
and $g(t)>0$ is an auxiliary function. In the Fourier space, we obtain
$\widehat{\lambda}(k|t)=\exp(-g(t)|k|^{\beta}),\hskip 14.22636pt0<\beta\leq
2.$ (30)
Then, in the limit $k\rightarrow 0$, we have the asymptotic relation
$\widehat{\lambda}(k|t)\sim 1-g(t)|k|^{\beta},\hskip 14.22636ptk\rightarrow
0,0<\beta\leq 2.$ (31)
In the Fourier-Laplace space, in the limit $(k,s)\rightarrow(0,0)$ we have
$\begin{array}[]{lll}\widehat{\widetilde{\psi}}(k,s)-\widetilde{\varphi}(s)&=&\int_{0}^{+\infty}dt\int_{-\infty}^{+\infty}\psi(x,t)\exp(ikx-
st)dx-\int_{0}^{+\infty}\varphi(t)\exp(-st)dt\\\ \\\
&=&\int_{0}^{+\infty}[\widehat{\lambda}(k|t)-1]\varphi(t)\exp(-st)dt\\\ \\\
&\sim&-|k|^{\beta}\int_{0}^{+\infty}g(t)\varphi(t)\exp(-st)dt\\\ \\\
&=&-|k|^{\beta}{\cal L}\\{g(t)\varphi(t)\\}.\end{array}$ (32)
So
$\widehat{\widetilde{\psi}}(k,s)\sim 1-A_{\alpha}s^{\alpha}-|k|^{\beta}{\cal
L}\\{g(t)\varphi(t)\\},\hskip 14.22636pt0<\alpha\leq 1,0<\beta\leq 2.$ (33)
If
${\cal L}\\{g(t)\varphi(t)\\}\sim 1-s^{\mu},\hskip
14.22636pt\mu>0,s\rightarrow 0,$ (34)
we can obtain the asymptotic relation
$\widehat{\widetilde{\psi}}(k,s)\sim 1-A_{\alpha}s^{\alpha}-|k|^{\beta},\hskip
14.22636pt0<\alpha\leq 1,0<\beta\leq 2,$ (35)
which is the same as Eq. (21).
If
${\cal
L}\\{g(t)\varphi(t)\\}=\frac{\Gamma(\gamma-\alpha)}{s^{\gamma-\alpha}},\hskip
14.22636pt0<\alpha<\gamma,s\rightarrow 0,$ (36)
we have
$\widehat{\widetilde{\psi}}(k,s)\sim
1-A_{\alpha}s^{\alpha}-A_{\beta}\frac{|k|^{\beta}}{s^{\gamma-\alpha}},\hskip
14.22636pt0<\alpha\leq 1,\alpha<\gamma,0<\beta\leq 2.$ (37)
which is the same as Eq. (24).
Now we consider the specific case
$\varphi(t)\sim t^{-1-\alpha},\hskip 14.22636pt0<\alpha<1,$ (38)
and
$g(t)=t^{\gamma},\hskip 14.22636pt0<\gamma\leq 2.$ (39)
Then
$g(t)\varphi(t)\sim t^{-1-\alpha+\gamma},\hskip
14.22636pt0<\alpha<1,0<\gamma\leq 2.$ (40)
If $\gamma<\alpha$, according to the Tauberian theorem [38], we have
${\cal L}\\{g(t)\varphi(t)\\}\sim 1-s^{\alpha-\gamma},\hskip
14.22636pt0<\gamma<\alpha<1.$ (41)
After taking $\alpha-\gamma=\mu$, Eq. (41) satisfies the condition Eq. (34).
We then obtain the asymptotic relation Eq. (35), and the corresponding space-
time FDE is
$_{0}^{c}D_{t}^{\alpha}P(x,t)=K\frac{\partial^{\beta}P(x,t)}{\partial|x|^{\beta}},\hskip
14.22636pt0<\gamma<\alpha<1,0<\beta\leq 2,$ (42)
which implies that the order of time fractional derivative in Eq. (42) is
determined by the parameter $\alpha$ of the waiting time PDF $\varphi(t)$.
If $\gamma>\alpha$, then $-1-\alpha+\gamma>-1$. Using the Laplace transform
formula of power function, we have
${\cal
L}\\{g(t)\varphi(t)\\}=\frac{\Gamma(\gamma-\alpha)}{s^{\gamma-\alpha}},\hskip
14.22636pt0<\alpha<1,\alpha<\gamma\leq 2.$ (43)
It satisfies the condition Eq. (36). So we obtain the asymptotic relation
(37), and the corresponding space-time FDE is
$_{0}^{c}D_{t}^{\gamma}P(x,t)=K\frac{\partial^{\beta}P(x,t)}{\partial|x|^{\beta}},\hskip
14.22636pt0<\alpha<1,\alpha<\gamma\leq 2,0<\beta\leq 2,$ (44)
which implies that the order of time fractional derivative in Eq. (44) is
determined by the parameter $\gamma$ of the auxiliary function $g(t)$.
According to above discussions, we find that for long waiting time, i.e.
$0<\alpha<1$, there exists a competition between the waiting time PDF
$\varphi(t)$ and the auxiliary function $g(t)$ to decide the order of the time
fractional derivative in the space-time FDEs.
## 5 Conclusions
In this work, we discuss the asymptotic behaviors of the jump PDF $\psi(x,t)$
in the Fourier-Laplace space in the coupled CTRW model with
$\psi(x,t)=\varphi(t)\lambda(x|t)$. The corresponding space-time FDEs are
derived from the asymptotic behaviors of the jump PDF $\psi(x,t)$ in the
Fourier-Laplace space and the waiting time PDF $\varphi(t)$ in the Laplace
space. We also discuss the asymptotic behaviors of the conditional PDF of jump
length $\lambda(x|t)$ and show that there exists a competition between the
waiting time PDF $\varphi(t)$ and an auxiliary function $g(t)$ of the
conditional PDF of jump length $\lambda(x|t)$ to determine the order of the
time derivative in the space-time FDE. We also conclude that when $\beta=2$,
the derived FDE Eq.(42) from the given coupled CTRW model yields subdiffusion.
Moreover, FDE (44) yields subdiffusion for the case of $0<\alpha<\gamma<1$,
normal diffusion for the case of $0<\alpha<\gamma=1$, superdiffusion for the
case of $0<\alpha<1<\gamma\leq 2$.
## Acknowledgements
This project was supported by the Natural Science Foundation of China (Grant
no. 11071282 and 10971175), the Chinese Program for Changjiang Scholars and
Innovative Research Team in University (PCSIRT) (Grant No. IRT1179), the
Research Foundation of Education Commission of Hunan Province of China (grant
no. 11A122), the Lotus Scholars Program of Hunan province of China, the Aid
program for Science and Technology Innovative Research Team in Higher
Educational Institutions of Hunan Province of China. The authors would like to
thank Prof. Vo Anh in Queensland University of Technology for his useful
comments and suggestions to improve this paper.
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|
arxiv-papers
| 2013-08-12T04:34:57 |
2024-09-04T02:49:49.329176
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Long Shi, Zuguo Yu, Zhi Mao, Aiguo Xiao and Hailan Huang",
"submitter": "Zu-Guo Yu",
"url": "https://arxiv.org/abs/1308.2461"
}
|
1308.2487
|
[labelstyle=]
# Fifty Shades of Black
Alexandre Borovik School of Mathematics, University of Manchester, UK;
[email protected] and Şükrü Yalçınkaya Nesin Mathematics Village,
Izmir, Turkey; [email protected]
###### Abstract.
The paper proposes a new and systematic approach to the so-called black box
group methods in computational group theory. As the starting point of our
programme, we construct Frobenius maps on black box groups of untwisted Lie
type in odd characteristic and then apply them to black box groups $X$
encrypting groups $({\rm{P}}){\rm{SL}}_{2}(q)$ in small odd characteristics.
We propose an algorithm constructing a black box field $\mathbb{K}$ isomorphic
to $\mathbb{F}_{q}$, and an isomorphism from
$({\rm{P}}){\rm{SL}}_{2}(\mathbb{K})$ to $X$. The algorithm runs in time
quadratic in the characteristic of the underlying field and polynomial in
$\log q$.
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
###### 1991 Mathematics Subject Classification:
Primary 20P05, Secondary 03C65
###### Contents
1. 1 Introduction
2. 2 Black box groups and their automorphisms
1. 2.1 Axioms BB1 – BB3
2. 2.2 Global exponent and Axiom BB4
3. 2.3 Relations with other black box groups projects
4. 2.4 Morphisms
5. 2.5 Shades of black
6. 2.6 Automorphisms as lighter shades of black
7. 2.7 Construction of Frobenius maps
3. 3 Oracles and revelations
1. 3.1 Monte-Carlo and Las Vegas
2. 3.2 Constructive recognition
3. 3.3 On oracles and revelations: an example from even characteristic
4. 3.4 Structure recovery
5. 3.5 Black box fields
4. 4 Application of Frobenius maps: structure recovery of $({\rm{P}}){\rm{SL}}_{2}(q)$, $q\equiv 1\bmod 4$
1. 4.1 Proof of Theorem 4.1, general case
2. 4.2 A more straightforward treatment of ${\rm{SL}}_{2}(p)$
5. 5 A Revelation and Its Reverberations: Proof of Theorem 3.1
1. 5.1 Proof of Theorem 3.1
2. 5.2 Other groups of characteristic $2$
## 1\. Introduction
Black box groups were introduced by Babai and Szemeredi [7] as an idealized
setting for randomized algorithms for solving permutation and matrix group
problems in computational group theory. This paper belongs to a series of
works aimed at development of systematic structural analysis of black box
groups [11, 12, 13, 15, 16, 49, 50].
The principal results of this paper are concerned with construction of
Frobenius maps on black box Chevalley groups of untwisted type and odd
characteristic, they are stated and proven in Section 2.7.
In Section 4.2, these constructions are applied to prove Theorem 4.1 concerned
with recognition of black box groups $({\rm{P}}){\rm{SL}}_{2}(q)$ for $q\equiv
1\bmod 4$ and $q=p^{k}$ for some $k\geqslant 1$.
Our approach requires a detailed discussion of some methodological issues of
black box group theory; this discussion is spread all over the paper and is
supported by some “toy” mathematical results, such as Theorem 3.1 that
provides recognition of black box groups $SL_{2}(2^{n})$ under a (rather
hypothetical) assumption that we are given an involution in the group.
## 2\. Black box groups and their automorphisms
### 2.1. Axioms BB1 – BB3
A black box group $X$ is a black box (or an oracle, or a device, or an
algorithm) operating with $0$–$1$ strings of bounded length which encrypt (not
necessarily in a unique way) elements of some finite group $G$ (in various
classes of black box problems the isomorphism type of $G$ could be known in
advance or unknown). The functionality of a black box is specified by the
following axioms, where every operation is carried out in time polynomial in
terms of $\log|G|$.
* BB1
$X$ produces strings of fixed length $l(X)$ encrypting random (almost)
uniformly distributed elements from $G$; the string length $l(X)$ is
polynomially bounded in terms of $\log|G|$.
* BB2
$X$ computes, in time polynomial in $l(X)$, a string encrypting the product of
two group elements given by strings or a string encrypting the inverse of an
element given by a string.
* BB3
$X$ compares, in time polynomial in $l(X)$, whether two strings encrypt the
same element in $G$—therefore identification of strings is a canonical
projection
We shall say in this situation that $X$ is a _black box over $G$_ or that a
black box $X$ _encrypts_ the group $G$. Notice that we are not making any
assumptions on computability of the projection $\pi$.
A typical example of a black box group is provided by a group $G$ generated in
a big matrix group ${\rm{GL}}_{n}(r^{k})$ by several matrices
$g_{1},\dots,g_{l}$. The product replacement algorithm [26] produces a sample
of (almost) independent elements from a distribution on $G$ which is close to
the uniform distribution (see a discussion and further development in [5, 6,
17, 30, 37, 39, 41, 40, 42]). We can, of course, multiply, invert, compare
matrices. Therefore the computer routines for these operations together with
the sampling of the product replacement algorithm run on the tuple of
generators $(g_{1},\dots,g_{l})$ can be viewed as a black box $X$ encrypting
the group $G$. The group $G$ could be unknown—in which case we are interested
in its isomorphism type—or it could be known, as it happens in a variety of
other black box problems.
### 2.2. Global exponent and Axiom BB4
Notice that even in routine examples the number of elements of a matrix group
$G$ could be astronomical, thus making many natural questions about the black
box $X$ over $G$—for example, finding the isomorphism type or the order of
$G$—inaccessible for all known deterministic methods. Even when $G$ is cyclic
and thus is characterized by its order, existing approaches to finding
multiplicative orders of matrices over finite fields are conditional and
involve oracles either for the discrete logarithm problem in finite fields or
for prime factorization of integers.
Nevertheless black box problems for matrix groups have a feature which makes
them more accessible:
* BB4
We are given a _global exponent_ of $X$, that is, a natural number $E$ such
that it is expected that $\pi(x)^{E}=1$ for all strings $x\in X$ while
computation of $x^{E}$ is computationally feasible (say, $\log E$ is
polynomially bounded in terms of $\log|G|$).
Usually, for a black box group $X$ arising from a subgroup in the ambient
group ${\rm{GL}}_{n}(r^{k})$, the exponent of ${\rm{GL}}_{n}(r^{k})$ can be
taken for a global exponent of $X$.
One of the reasons why the axioms BB1–BB4, and, in particular, the concept of
global exponent, appear to be natural, is provided by some surprising model-
theoretic analogies. For example, D’Aquino and Macintyre [29] studied non-
standard finite fields defined in a certain fragment of bounded Peano
arithmetic; it is called $I\Delta_{0}+\Omega_{1}$ and imitates proofs and
computations of polynomial time complexity in modular arithmetic. It appears
that such basic and fundamental fact as the Fermat Little Theorem has no proof
which can be encoded in $I\Delta_{0}+\Omega_{1}$; the best that had so far
been proven in $I\Delta_{0}+\Omega_{1}$ is that the multiplicative group
$\mathbb{F}_{p}^{*}$ of the prime field $\mathbb{F}_{p}$ has a global exponent
$E<2p$ [29].
We shall discuss model theory and logic connections of black box group theory
in some details elsewhere.
### 2.3. Relations with other black box groups projects
> _In this paper, we assume that all our black box groups satisfy assumptions
> BB1–BB4._
We emphasize that we do not assume that black box groups under consideration
in this paper are given as subgroups of ambient matrix groups; thus our
approach is wider than the setup of the computational matrix group project
[34]. Notice that we are not using the Discrete Logarithm Oracles for finite
fields $\mathbb{F}_{q}$: in our original setup, we do not have fields.
Nevertheless we are frequently concerned with black box groups encrypting
classical linear groups; even so, some of our results (such as Theorems 3.2
and 3.3) do not even involve the assumption that we know the underlying field
of the group but instead assume the knowledge of the characteristic of the
field without imposing bounds on the size of the field. Finally, in the case
of groups over fields of small characteristics we can prove much sharper
results, see, for example, Theorem 4.1. Here, it is natural to call
characteristic $p$ “small”, if it is known and if a linear or quadratic
dependency of the running time of algorithm on $p$ does not cause trouble and
algorithms are computationally feasible.
So we attach to statements of our results one of the two labels:
* •
Known characteristic,
* •
Small characteristic.
Our next paper [15] is dominated by “known characteristic” results. In this
one, we concentrate on black box groups of known or small characteristics.
### 2.4. Morphisms
Given two black boxes $X$ and $Y$ encrypting finite groups $G$ and $H$,
correspondingly, we say that a map $\alpha$ which assigns strings from $Y$ to
strings from $X$ is a _morphism_ of black box groups, if
* •
the map $\alpha$ is computable in probabilistic time polynomial in $l(X)$ and
$l(Y)$, and
* •
there is an abstract homomorphism $\beta:G\to H$ such that the following
diagram is commutative: where $\pi_{X}$ and $\pi_{Y}$ are the canonical
projections of $X$ and $Y$ onto $G$ and $H$, correspondingly.
We shall say in this situation that a morphism $\alpha$ _encrypts_ the
homomorphism $\beta$. For example, morphisms arise naturally when we replace a
generating set for black box group $X$ by a more convenient one and start
sampling the product replacement algorithm for the new generating set; in
fact, we replace a black box for $X$ and deal with a morphism
$Y\longrightarrow X$ from the new black box into $X$. Also, a black box
subgroup $Z$ of $X$ is a morphism $Z\hookrightarrow X$.
Slightly abusing terminology, we say that a morphism $\alpha$ is an embedding,
or an epimorphism, etc., if $\beta$ has these properties. In accordance with
standard conventions, hooked arrows stand for embeddings and doubleheaded
arrows for epimorphisms; dotted arrows are reserved for abstract
homomorphisms, including natural projections the latter are not necessarily
morphisms, since, by the very nature of black box problems, we do not have
efficient procedures for constructing the projection of a black box onto the
(abstract) group it encrypts.
We further discuss morphisms in Sections 2.6 and 3.4.
### 2.5. Shades of black
Polynomial time complexity is an asymptotic concept, to work with it we need
an infinite class of objects. Therefore our theory refers to some infinite
family $\mathcal{X}$ of black box groups ($\mathcal{X}$ of course varies from
one black box problem to another). For $X\in\mathcal{X}$, we denote by $l(X)$
the length of $0$–$1$ strings representing elements in $X$. We assume that,
for every $X\in\mathcal{X}$, basic operations of generating, multiplying,
comparing strings in $X$ can be done in probabilistic polynomial time in
$l(X)$. We assume that encryption of group elements in $X$ is sufficiently
economical and $l(X)$ is bounded by a polynomial in $\log|\pi(X)|$.
We also assume that the lengths $\log E(X)$ of global exponents $E(X)$ for
$X\in\mathcal{X}$ are bounded by a polynomial in $l(X)$.
Morphism $X\longrightarrow Y$ in $\mathcal{X}$ are understood as defined in
Section 2.4 and their running times are bounded by a polynomial in $l(X)$ and
$l(Y)$.
At the expense of slightly increasing $\mathcal{X}$ and its bounds for
complexity, we can include in $\mathcal{X}$ a collection of explicitly given
“known” finite groups. Indeed, using standard computer implementations of
finite field arithmetic, we can represent every group
$Y={\rm{GL}}_{n}(\mathbb{F}_{p^{k}})$ as an algorithm or computer routine
operating on $0$–$1$ strings of length $l(Y)=n^{2}k\log p$. Using standard
matrix representations for simple algebraic groups, we can represent every
group of points $Y={\rm G}(\mathbb{F}_{p^{k}})$ of a reductive algebraic group
$\rm G$ defined over $\mathbb{F}_{p^{k}}$ as a black box $Y$ generating and
processing strings of length $l(Y)$ polynomial in $\log|\mathbb{F}_{p^{k}}|$
and the Lie rank of $Y$. Therefore an “explicitly defined” group can be seen a
black box group, perhaps of a lighter shade of black.
We shall use direct products of black boxes: if $X$ encrypts $G$ and $Y$
encrypts $H$ then the black box $X\times Y$ generates pairs of strings $(x,y)$
by sampling $X$ and $Y$ independently, with operations carried out
componentwise in $X$ and $Y$; of course, $X\times Y$ encrypts $G\times H$.
Figure 1. M.C. Escher, _Day and Night_ , 1938
We feel that the best way to understand a black box group is a step-by-step
construction of a chain of morphisms at each step changing the shade of black
and increasing amount of information provided by black boxes $X_{i}$.
Even in relatively simple black box problems we may end up dealing with a
sophisticated category of black boxes and their morphisms. Step-by-step
transformation of black boxes into “white boxes” and their complex
entanglement is captured well by Escher’s famous woodcut, Figure 1.
### 2.6. Automorphisms as lighter shades of black
The first application of the “shadows of black” philosophy is the following
self-evident theorem which explains how an automorphism of a group can be
added to a black box encrypting this group.
###### Theorem 2.1.
Let $X$ be a black box group encrypting a finite group $G$ and assume that
each of $k$ tuples of strings
$\tilde{x}^{(i)}=(x^{(i)}_{1},\dots,x^{(i)}_{m}),\quad i=1,\dots,k,$
generate $X$ in the sense that the projections
$\pi\left(x^{(i)}_{1}\right),\dots,\pi\left(x^{(i)}_{m})\right)$ generate $G$.
Assume that the map
$\pi:x^{(i)}_{j}\mapsto\pi(x^{(i+1\bmod k)}_{j}),\quad i=0,\dots,k-1,\quad
j=1,\dots,m,$
can be extended to an automorphism $a\in{\rm Aut}\,G$ of order $k$. The black
box group $Y$ generated in $X^{k}$ by the strings
$\bar{x}_{j}=\left(x^{(0)}_{j},x^{(1)}_{j},\dots,x^{(k-1)}_{j}\right),\quad
j=1,\dots,m,$
encrypts $G$ via the canonical projection on the first component
$(y_{0},\dots,y_{k-1})\mapsto\pi(y_{o}),$
and possess an additional unary operation, cyclic shift
$\displaystyle\alpha:Y$ $\displaystyle\longrightarrow$ $\displaystyle Y$
$\displaystyle(y_{0},y_{2},\dots,y_{k-1},y_{k-1})$ $\displaystyle\mapsto$
$\displaystyle(y_{1},y_{2},\dots,y_{k-1},y_{0})$
which encrypts the automorphism $a$ of $G$ in the sense that the following
diagram commutes:
A somewhat more precise formulation of Theorem 2.1 is that we can construct,
in polynomial in $k$ and $m$ time, a commutative diagram
(1) $\begin{diagram}$
where $d$ is the twisted diagonal embedding
$\displaystyle d:G$ $\displaystyle\longrightarrow$ $\displaystyle G^{k}$
$\displaystyle x$ $\displaystyle\mapsto$
$\displaystyle(x,x^{a},x^{a^{2}},x^{a^{k-1}}),$
and $p_{i}$ is the projection
$\displaystyle p_{i}:G^{k}$ $\displaystyle\longrightarrow$ $\displaystyle G$
$\displaystyle(g_{0},\dots,g_{i},\dots,g_{k-1})$ $\displaystyle\mapsto$
$\displaystyle g_{i}.$
Of course, this construction leads to memory requirements increasing by factor
of $k$, but, as our subsequent papers [15, 16] show, this is price worth
paying. After all, in most practical problems the value of $k$ is not that
big, in most interesting cases $k=2$.
A useful special case of Theorem 2.1 is the following result about
amalgamation of black box automorphisms, stated here in an informal wording
rather than expressed by a formal commutative diagram.
###### Theorem 2.2.
Let $X$ be a black box group encrypting a group $G$. Assume that $G$ contains
subgroups $G_{1},\dots,G_{l}$ invariant under an automorphism
$\alpha\in\mathop{{\rm Aut}}G$ and that these subgroups are encrypted in $X$
as black boxes $X_{i}$, $i=1,\dots,l$, supplied with morphisms
$\phi_{i}:X_{i}\longrightarrow X_{i}$ which encrypt restrictions
$\alpha\\!\mid_{G_{i}}$ of $\alpha$ on $G_{i}$.
Finally, assume $\langle G_{i},i=1,\dots,l\rangle=G$.
Then we can construct, in polynomial in $l(X)$ time, a morphism
$\phi:X\longrightarrow X$ which encrypts $\alpha$.
### 2.7. Construction of Frobenius maps
We now use Theorem 2.1 to construct a Frobenius map on a black box group $X$
encrypting $({\rm{P}}){\rm{SL}}_{2}(q)$ with $q\equiv 1\bmod 4$ and $q=p^{k}$
for some $k\geqslant 1$. We make sure that the Frobenius map constructed
leaves invariant the specified Borel subgroup, thus giving us access to
subtler structural properties of the group.
We shall use the following result from [12].
###### Theorem 2.3 (Small characteristic).
[12, Theorem 1.2] Let $X$ be a black box group encrypting
$({\rm{P}}){\rm{SL}}_{2}(q)$, where $q\equiv 1\bmod 4$ and $q=p^{k}$ for some
$k\geqslant 1$. If $p\neq 5,7$, then there is a Monte-Carlo algorithm which
constructs in $X$ strings $u$, $h$, $n$ such that there exists an _(abstract)_
isomorphism
$\Phi:X\longrightarrow({\rm{P}}){\rm{SL}}_{2}(q)$
with
$\Phi(u)=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix},\Phi(h)=\begin{bmatrix}t&0\\\
0&t^{-1}\end{bmatrix},\Phi(n)=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix},$
where $t$ is some primitive element of the field $\mathbb{F}_{q}$. The running
time of the algorithm is quadratic in $p$ and polynomial in $\log q$.
If $p=5$ or $7$, and $k$ has a small divisor $\ell$, the same result holds
where the running time is polynomial in $\log q$ and quadratic in $p^{\ell}$.
In notation of Theorem 2.3, Theorem 2.1 immediately yields the following
remarkably useful result, see its extensions and applications in our
subsequent papers [15, 16].
###### Theorem 2.4 (Small characteristic).
(Informal formulation) Let $X$ be as in Theorem 2.3. Then there is a Monte-
Carlo algorithm which constructs a map that corresponds to the Frobenius
automorphism $a\mapsto a^{p}$ of the field $\mathbb{F}_{q}$ and leaves
invariant subgroups $U$ and $T$ and the elements $u$ and $w$ of $X$.
The running time of the algorithm is quadratic in $p$ and polynomial in $\log
q$.
###### Proof.
It suffices to observe that the action of the canonical Frobenius map
$F:\begin{bmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{bmatrix}\mapsto\begin{bmatrix}a_{11}^{p}&a_{12}^{p}\\\
a_{21}^{p}&a_{22}^{p}\end{bmatrix}$
on the preimages of $\bar{u},\bar{w},\bar{h}$ in ${\rm{PSL}}_{2}(q)$ and their
images under the powers of the Frobenius map looks like that:
$\displaystyle\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}^{F^{i}}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}1&1\\\ 0&1\end{bmatrix},$
$\displaystyle\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}^{F^{i}}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}0&1\\\ -1&0\end{bmatrix},$
$\displaystyle\begin{bmatrix}t^{p^{l}}&0\\\ 0&t^{p^{-l}}\end{bmatrix}^{F^{i}}$
$\displaystyle=$ $\displaystyle\begin{bmatrix}t^{p^{l+i\pmod{k}}}&0\\\
0&t^{p^{-l-i\pmod{k}}}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}t^{p^{l}}&0\\\
0&t^{p^{-1}}\end{bmatrix}^{p^{i}}.$
Therefore the black box group $Y$ is generated in the direct product $X^{k}$
by elements
$\displaystyle\bar{u}$ $\displaystyle=$ $\displaystyle(u,u,\dots,u)$
$\displaystyle\bar{w}$ $\displaystyle=$ $\displaystyle(w,w,\dots,w)$
$\displaystyle\bar{h}$ $\displaystyle=$
$\displaystyle(h,h^{p},\dots,h^{p^{k-1}})$
fits precisely in the construction described in Theorem 2.1.
It remains to notice that, by nature of its construction, the map $\alpha$ in
Theorem 2.1 leaves invariant elements $\bar{u}$, $\bar{w}$ and the torus
$\bar{T}$ generated by $\bar{h}$ and hence leaves invariant the unipotent
group $\bar{U}=\langle\bar{u}^{\bar{T}}\rangle$ and the Borel subgroup
$\bar{U}\bar{T}$ of $Y$. ∎
Actually we have a more general construction of Frobenius maps on all
untwisted Chevalley groups over finite field of odd characteristic; unlike
Theorem 2.4, it does not use unipotent elements.
###### Theorem 2.5 (Known characteristic).
Let $X$ be a black box group encrypting a simple Lie type group $G=G(q)$ of
untwisted type over a field of order $q=p^{k}$ for $p$ odd (and known) and
$k>1$. Then we can construct, in time polynomial in $\log|G|$,
* •
a black box $Y$ encrypting $G$,
* •
a morphism $X\longleftarrow Y$, and
* •
a morphism $\phi:Y\longrightarrow Y$ which encrypts a Frobenius automorphism
of $G$ induced by the map $x\mapsto x^{p}$ on the field $\mathbb{F}_{q}$.
###### Proof.
The proof is based on two applications of Theorem 2.2. First we consider the
case when $X$ encrypts ${\rm{PSL}}_{2}(\mathbb{F}_{q})$. Using the standard
technique for dealing with involution centralizers, we can find in $X$ a
$4$-subgroup $V$; let $E$ be the subgroup in $G={\rm{PSL}}_{2}(q)$ encrypted
by $V$. Since all $4$-subgroups in ${\rm{PSL}}_{2}(\mathbb{F}_{q})$ are
conjugate to a subgroup in ${\rm{PSL}}_{2}(\mathbb{F}_{p})$, we can assume
without loss of generality that $E$ belongs to a subfield subgroup
$H={\rm{PSL}}_{2}(\mathbb{F}_{p})$ of $G$ and therefore $E$ is fixed by a
Frobenius map $F$ on $G$. Now let $e_{1}$ and $e_{2}$ be two involutions in
$E$, and $C_{1}$ and $C_{2}$ maximal cyclic subgroups in their centralizers in
$G$; notice that $C_{1}$ and $C_{2}$ are conjugate by an element from $H$ and
are $F$-invariant.
It follows from the basic Galois cohomology considerations that $F$ acts on
$C_{1}$ and $C_{2}$ as power maps $\alpha_{i}:c\mapsto c^{\epsilon p}$ for
$p\equiv\epsilon\bmod 4$. If now we take images $X_{i}$ of groups $C_{i}$, we
see that the morphisms $\phi_{i}:x\mapsto x^{\epsilon p}$ of $X_{i}$ encrypt
restrictions of $F$ to $C_{i}$. Obviously, $X_{1}$ and $X_{2}$ generate a
black box $Y\longrightarrow X$, and we can use Theorem 2.2 to amalgamate
$\phi_{1}$ and $\phi_{2}$ into a morphism $\phi$ which encrypts $F$.
As usual, for groups ${\rm{SL}}_{2}(q)$ the same result can be achieved by
essentially the same arguments as for ${\rm{PSL}}_{2}(q)$. Moving to other
untwisted Chevalley groups, we apply amalgamation to (encryptions of)
restrictions of a Frobenius map on $G$ to (encryptions in $X$) of a family of
root $({\rm{P}}){\rm{SL}}_{2}$-subgroups $K_{i}$ in $G$ forming a Curtis-Tits
system in $G$ (and therefore generating $G$). Black boxes for Curtis-Tits
system in classical groups of odd characteristic are constructed in [11], in
exceptional groups in [14]. This completes the proof. ∎
## 3\. Oracles and revelations
In this section, we revise the classification of black box group problems and
briefly discuss the role of “oracles”.
### 3.1. Monte-Carlo and Las Vegas
This is a brief reminder of two canonical concepts for the benefit of those
readers who came from the pure group theory rather than computational group
theory background.
A Monte-Carlo algorithm is a randomized algorithm which gives a correct output
to a decision problem with probability strictly bigger than $1/2$. The
probability of having incorrect output can be made arbitrarily small by
running the algorithm sufficiently many times. A Monte-Carlo algorithm with
outputs “yes” and “no” is called one-sided if the output “yes” is always
correct. A special subclass of Monte-Carlo algorithm is a Las Vegas algorithm
which either outputs a correct answer or reports failure (the latter with
probability less than $1/2$). The probability of having a report of failure is
prescribed by the user. A detailed comparison of Monte-Carlo and Las Vegas
algorithms, both from practical and theoretical point, can be found in Babai’s
paper [4].
### 3.2. Constructive recognition
We shall outline an hierarchy of typical black box group problems.
Verification Problem:
Is the unknown group encrypted by a black box group $X$ isomorphic to the
given group $G$ (“target group”)?
Recognition Problem:
Determine the isomorphism class of the group encrypted by $X$.
The Verification Problem arises as a sub-problem within more complicated
Recognition Problems. The two problems have dramatically different complexity.
For example, the celebrated Miller-Rabin algorithm [43] for testing primality
of the given odd natural number $n$ in nothing else but a black box algorithm
for solving the verification problem for the multiplicative group
$\mathbb{Z}/n\mathbb{Z}^{*}$ of residues modulo $n$ (given by a simple black
box: take your favorite random numbers generator and generate random integers
between $1$ and $n$) and the cyclic group $\mathbb{Z}/(n-1)\mathbb{Z}$ of
order $n-1$ as the target group. On the other hand, if $n=pq$ is the product
of primes $p$ and $q$, the recognition problem for the same black box group
means finding the direct product decomposition
$\mathbb{Z}/n\mathbb{Z}^{*}\cong\mathbb{Z}/(p-1)\mathbb{Z}\oplus\mathbb{Z}/(q-1)\mathbb{Z}$
which is equivalent to factorization of $n$ into product of primes.
The next step after finding the isomorphism type of the black box group $X$ is
Constructive Recognition:
Suppose that a black box group $X$ encrypts a concrete and explicitly given
group $G$. Rewording a definition given in [21],
> _The goal of a constructive recognition algorithm is to construct an
> effective isomorphism $\Psi:G\longrightarrow X$. That is, given $g\in G$,
> there is an efficient procedure to construct a string $\Psi(g)$ encrypting
> $g$ in $X$ and given a string $x$ produced by $X$, there is an efficient
> procedure to construct the element $\Psi^{-1}(x)\in G$ encrypted by $X$._
However, there are still no really efficient constructive recognition
algorithms for black box groups $X$ of (known) Lie type over a finite field of
large order $q=p^{k}$. The first computational obstacles for known algorithms
[19, 20, 21, 22, 23, 25, 28, 35] are the need to construct unipotent elements
in black box groups, [19, 20, 21, 23, 22, 25] or to solve discrete logarithm
problem for matrix groups [27, 28, 35].
Unfortunately, the proportion of the unipotent elements in $X$ is $O(1/q)$
[31]. Moreover the probability that the order of a random element is divisible
by $p$ is also $O(1/q)$, so one has to make $O(q)$ (that is, _exponentially
many_ , in terms of the input length $O(\log q)$ of the black boxes and the
algorithms) random selections of elements in a given group to construct a
unipotent element. However, this brute force approach is still working for
small values of $q$, and Kantor and Seress [33] used it to develop an
algorithm for recognition of black box classical groups. Later the algorithms
of [33] were upgraded to polynomial time constructive recognition algorithms
[20, 21, 22, 23] by assuming the availability of additional _oracles_ :
* •
the _discrete logarithm oracle_ in $\mathbb{F}_{q}^{*}$, and
* •
the _${\rm{SL}}_{2}(q)$ -oracle_.
Here, the _${\rm{SL}}_{2}(q)$ -oracle_ is a procedure for constructive
recognition of ${\rm{SL}}_{2}(q)$; see discussion in [21, Section 3].
> _We emphasize that in this and subsequent papers we are using neither the
> discrete logarithm oracle in $\mathbb{F}_{q}^{*}$ nor the
> ${\rm{SL}}_{2}(q)$-oracle._
### 3.3. On oracles and revelations: an example from even characteristic
We have to admit that the concept of constructive recognition modulo the use
of unrealistically powerful oracles makes us uncomfortable. We feel that the
use of excessively powerful and blunt tools leads to loss of essential (and
frequently very beautiful) theoretical details. Instead, we propose to use all
“ _fifty shades of black_ ” and exploit all available gradations of black
(that is, a subtler hierarchy of complexity of black box problems) in
development of practically useful algorithms. Our papers [12, 15, 16] provide
a number of concrete examples where this alternative approach has happened to
be fruitful.
In the present paper, we wish to dispel some mystic of the
${\rm{SL}}_{2}(q)$-oracle by analyzing the structure of the black box group
$X$ encrypting ${\rm{SL}}_{2}(2^{n})$ using formally a more modest assumption:
that we are given an involution $r\in X$. We shall say that $r$ is obtained
_by revelation_ , to acknowledge that this assumption is quite unnatural in
practical applications.
Still, we feel that there is a difference between a revelation or epiphany
(which, by their nature, are non-reproducible, unique events) and an appeal to
an oracle; indeed, there is an implicit assumption that the oracle can be
approached for advice again and again.
###### Theorem 3.1 (Small characteristic).
Let $X$ be a black box group encrypting ${\rm{SL}}_{2}(2^{n})$ for some
(perhaps unknown) $n$. We assume that we are given an involution $u\in X$.
Then there exists a Monte-Carlo algorithm which constructs, in polynomial in
$l(X)$ time,
* •
a black box field $\mathbb{U}$ encrypting $\mathbb{F}_{2^{n}}$, and
* •
a polynomial in $l(X)$ time isomorphism
$\Phi:{\rm{SL}}_{2}(\mathbb{U})\longrightarrow X.$
### 3.4. Structure recovery
Theorem 3.1 is an example of a class of results which we call _structure
recovery theorems_.111We extend our definition from [12] where it refers to a
special case of the present one.
Suppose that a black box group $X$ encrypts a concrete and explicitly given
group $G=G(\mathbb{F}_{q})$ of Chevalley type $G$ over a explicitly given
finite field $\mathbb{F}_{q}$. To achieve _structure recovery_ in $X$ means to
construct, in probabilistic polynomial time in $\log|G|$,
* •
a black box field $\mathbb{K}$ encrypting $\mathbb{F}_{q}$, and
* •
a probabilistic polynomial time morphism
$\Psi:G(\mathbb{K})\longrightarrow X.$
This new concept requires a detailed discussion.
Recall that simple algebraic groups (in particular, Chevalley groups over
finite fields) are understood in the theory of algebraic groups as functors
from the category of unital commutative rings into the category of groups;
most structural properties of a Chevalley group are encoded in the functor;
the field mostly provides the flesh on the bones. Remarkably, this separation
of flesh from the bones is very prominent in the black box group theory. Here,
we wish to mention a few from many constructions from our subsequent paper
[15] which illustrate this point.
###### Theorem 3.2 (Known characteristic).
[15] Let $X$ be a black box group encrypting the group ${\rm{SL}}_{n}(q^{2})$
for $q$ odd, $q=p^{k}$ for some $k$ (perhaps unknown) and a known prime number
$p$. Then we can construct, in time polynomial in $\log q$ and $n$, a black
box group $Y$ encrypting the group ${\rm{SU}}_{n}(q)$ and a morphism
$Y\hookrightarrow X$. If in addition $n$ is even and $n=2m$, we can do the
same with a black box group $Z$ encrypting ${\rm{Sp}}_{2m}(q)$ and a morphism
$Z\hookrightarrow Y$.
An important feature of the proofs of this and other similar results in [15]
is that they never refer to the ground fields of groups and do not involved
any computations with unipotent elements. In fact, we interpret morphisms
between functors
${\rm{Sp}}_{2m}(\cdot)\hookrightarrow{\rm{SU}}_{n}(\cdot)\hookrightarrow{\rm{SL}}_{n}(\cdot^{2}).$
within our black boxes.
This example shows that a modicum of categorical language is useful for the
theory as well as for its implementation in the code since it suggests a
natural structural approach to development of the computer code.
Another example of a “category-theoretical” approach is provided by a very
elementary, but also very important observation that the graph of a group
homomorphism $G\longrightarrow H$ is a subgroup of $G\times H$. Therefore it
is natural to identify a morphism $\mu:X\longrightarrow Y$ of black box groups
with its graph $M<X\times Y$. In its turn, the black box $M$ is a morphism
$M\longrightarrow X\times Y$. In practice this could mean (although in some
cases a more sophisticated construction is used) that we take some strings
$x_{1},\dots,x_{k}$ generating $X$ and their images
$y_{1}=\mu(x_{1}),\dots,y_{k}=\mu(x_{k})$ in $Y$ and use the product
replacement algorithm to run a black box for the subgroup
$M=\langle(x_{1},y_{1}),\dots,(x_{k},y_{k})\rangle\leqslant X\times Y$
which is of course exactly the graph $\\{\,(x,\mu(x))\,\\}$ of the
homomorphism $\mu$. Random sampling of the black box $M$ returns strings $x\in
X$ with their images $\mu(x)\in Y$ already attached. This doubles the
computational cost of the black box for $X$, but allows us to do constructions
like the following one.
###### Theorem 3.3 (Known characteristic).
[15] Let $X$ be a black box group encrypting the group ${\rm{SL}}_{8}(F)$ for
a field $F$ of (unknown) odd order $q=p^{k}$ but known $p={\rm char}\,F$. Then
we can construct, in time polynomial in $\log|F|$, a chain of black box groups
and morphisms
$U\hookrightarrow V\hookrightarrow W\hookrightarrow X$
that encrypts the chain of canonical embeddings
$\mathop{G_{2}}(F)\hookrightarrow{\rm{SO}}_{7}(F)\hookrightarrow{\rm{SO}}_{8}^{+}(F)\hookrightarrow{\rm{SL}}_{8}(F).$
Again, these our constructions (and even the embedding ${}^{3}{\rm
D}_{4}(q)\hookrightarrow{\rm{SO}}_{8}^{+}(q^{3})$, also done in [15]) are
“field-free” and, moreover, “characteristic-free”.
Another aspect of the concept of “structure recovery” is that it follows an
important technique from the model-theoretic algebra: interpretability of one
algebraic structure in another, see, for example, [10]. Construction of a
black box field in a black box group in Theorems 3.1 and 4.1 closely follows
this model-theoretic paradigm.
### 3.5. Black box fields
We define black box fields by analogy with black box groups, the reader may
wish to compare the exposition in this section with [8].
A _black box_ (finite) _field_ $\mathbb{K}$ is an oracle or an algorithm
operating on $0$-$1$ strings of uniform length (input length) which encrypts a
field of known characteristic $p$. The oracle can compute $x+y$, $xy$ and
compares whether $x=y$ for any strings $x,y\in\mathbb{K}$. We refer the reader
to [8, 38] for more details of black box fields and their applications to
cryptography.
In this paper, we shall be using some results about the isomorphism problem of
black box fields [38], that is, the problem of constructing an isomorphism and
its inverse between $\mathbb{K}$ and an explicitly given finite field
$\mathbb{F}_{p^{n}}$. The explicit data for a finite field of cardinality
$p^{n}$ is defined to be a system of _structure constants_ over the prime
field, that is $n^{3}$ elements $(c_{ijk})_{i,j,k=1}^{n}$ of the prime field
$\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ (represented as integers in $[0,p-1]$)
so that $\mathbb{F}_{p^{n}}$ becomes a field with ordinary addition and
multiplication by elements of $\mathbb{F}_{p}$ and multiplication is
determined by
$s_{i}s_{j}=\sum_{k=1}^{n}c_{ijk}s_{k},$
where $s_{1},s_{2},\dots,s_{n}$ denotes a basis of $\mathbb{F}_{p^{n}}$ over
$\mathbb{F}_{p}$. The concept of explicitly given field of order $p^{n}$ is
robust; indeed, Lenstra Jr. has shown in [36, Theorem 1.2] that for any two
fields $A$ and $B$ of order $p^{n}$ given by two sets of structure constants
$(a_{ijk})_{i,j,k=1}^{n}$ and $(b_{ijk})_{i,j,k=1}^{n}$ an isomorphism
$A\longrightarrow B$ can be constructed in polynomial in $n\log p$ time.
Maurer and Raub [38] proved that the isomorphism problem for a black box field
$\mathbb{K}$ and an explicitly given field $\mathbb{F}_{p^{n}}$ is reducible
in polynomial time to the same problem for the prime subfield in $\mathbb{K}$
and $\mathbb{F}_{p}$. Hence, for small primes $p$, one can construct an
isomorphism between $\mathbb{K}$ and $\mathbb{F}_{p^{n}}$ in time polynomial
in $n\log p$ and linear in $p$.
In our construction of a black box field, we use the so called primitive prime
divisor elements in the field of size $p^{n}$. A prime number $r$ is said to
be a primitive prime divisor of $p^{n}-1$ if $r$ divides $p^{n}-1$ but not
$p^{i}-1$ for $1\leqslant i<n$. By [51], there exists a primitive prime
divisor of $p^{n}-1$ except when $(p,n)=(2,6)$, or $n=2$ and $p$ is a Mersenne
prime. Here, we shall note that the Mersenne primes which are less than 1000
are 3, 7, 31, 127. We call a group element a $ppd(n,p)$-element if its order
is odd and divisible by a primitive prime divisor of $p^{n}-1$.
## 4\. Application of Frobenius maps: structure recovery of
$({\rm{P}}){\rm{SL}}_{2}(q)$, $q\equiv 1\bmod 4$
We remind that in all theorems and conjectures stated in this paper, we assume
that black boxes for groups satisfy Axioms BB1–BB4; in particular, they come
with known and computationally feasible global exponent (Axiom BB4).
For the structure recovery of $({\rm{P}}){\rm{SL}}_{2}(q)$, we need to recall
the Steinberg generators of $({\rm{P}}){\rm{SL}}_{2}(q)$ as introduced by
Steinberg [44, Theorem 8]. We use notation from [12].
Let $G={\rm{SL}}_{2}(q)$. Then set the Steinberg generators of $G$ as
$\displaystyle\mathbf{u}(t)=\left[\begin{array}[]{cc}1&t\\\ 0&1\\\
\end{array}\right],\,\mathbf{v}(t)=\left[\begin{array}[]{cc}1&0\\\ t&1\\\
\end{array}\right],\,\mathbf{h}(t)=\left[\begin{array}[]{cc}t&0\\\ 0&t^{-1}\\\
\end{array}\right],\,\mathbf{n}(t)=\left[\begin{array}[]{cc}0&t\\\
-t^{-1}&0\\\ \end{array}\right]$
where $t\in\mathbb{F}_{q}$ and in addition $t\neq 0$ in $\mathbf{h}(t)$ and
$\mathbf{n}(t)$.
The group ${\rm{PSL}}_{2}(q)$ is obtained from ${\rm{SL}}_{2}(q)$ by
factorizing over the relation
$\mathbf{h}(t)=\mathbf{h}(-t).$
Abusing notation, we are using for elements in ${\rm{PSL}}_{2}(q)$ the same
matrix notation as for their pre-images in ${\rm{SL}}_{2}(q)$.
It is straightforward to check that
(3)
$\displaystyle\mathbf{u}(t)^{\mathbf{n}(s)}=\mathbf{v}(-s^{-2}t),\,\mathbf{u}(1)^{\mathbf{h}(t)}=\mathbf{u}(t^{-2})\mbox{
and }\mathbf{n}(1)^{\mathbf{h}(t)}=\mathbf{n}(t^{-2}).$
Moreover,
(4)
$\displaystyle\mathbf{n}(t)=\mathbf{u}(t)\mathbf{v}(-t^{-1})\mathbf{u}(t)\mbox{
and }\mathbf{h}(t)=\mathbf{n}(t)\mathbf{n}(-1).$
It is well-known that
$G=\langle\mathbf{u}(t),\mathbf{v}(t)\mid t\in\mathbb{F}_{q}\rangle,$
see, for example, [24, Lemma 6.1.1]. Therefore, by (3) and (4),
$G=\langle\mathbf{u}(1),\mathbf{h}(t),\mathbf{n}(1)\mid
t\in\mathbb{F}_{q}^{*}\rangle;$
notice that actually $G$ is generated by three elements
$G=\langle\mathbf{u}(1),\mathbf{h}(t),\mathbf{n}(1)\rangle$
where we can take $t$ as an arbitrary $ppd(k,p)$-element of the field
$\mathbb{F}_{p^{k}}$.
In this section, we prove the following theorem.
###### Theorem 4.1 (Small characteristic).
Let $X$ be a black box group encrypting the group
$G\cong({\rm{P}}){\rm{SL}}_{2}(q)$, where $q\equiv 1\bmod 4$ and $q=p^{k}$ for
some $k\geqslant 1$ (perhaps unknown). If $p\neq 5,7$, then there is a Monte-
Carlo algorithm which constructs, in time quadratic in $p$ and polynomial in
$\log q$,
* •
a black box field $\mathbb{K}$ encrypting $\mathbb{F}_{q}$, and
* •
a quadratic in $p$ and polynomial in $\log q$ time isomorphism
$\Phi:(P)SL_{2}(\mathbb{K})\longrightarrow X.$
If $p=5$ or $7$, and $k$ has a small divisor $\ell$, the same result holds
where the running time is polynomial in $\log q$ and quadratic in $p^{\ell}$.
Theorem 4.1 is used in our paper [13] as the basis of recursion in the proof
of the following structure recovery theorem for classical groups in small
characteristics.
###### Theorem 4.2 (Small characteristic).
[13] Let $X$ be a black box group encrypting one of the classical groups ${\rm
G}(q)\simeq({\rm{P}}){\rm{SL}}_{n+1}(q)$, $({\rm{P}}){\rm{Sp}}_{2n}(q)$,
${\rm{\Omega}}_{2n+1}(q)$ or ${\rm{(P)\Omega}}_{2n}^{+}(q)$, where $q\equiv
1\bmod 4$ and $q=p^{k}$ for some $k\geqslant 1$ ($k$ and the type of the group
are perhaps unknown).
If $p\neq 5,7$, then there is a Monte-Carlo algorithm which constructs, in
time quadratic in $p$ and polynomial in $\log q$,
* •
a black box field $\mathbb{K}$ encrypting $\mathbb{F}_{q}$, and
* •
a quadratic in $p$ and polynomial in $\log q$ time isomorphism
$\Phi:{\rm G}(\mathbb{K})\longrightarrow X.$
If $p=5$ or $7$, and $k$ has a small divisor $\ell$, the same result holds
where the running time is polynomial in $\log q$ and quadratic in $p^{\ell}$.
### 4.1. Proof of Theorem 4.1, general case
Our aim is to present an algorithm which produces a black box field
$\mathbb{K}$ and an isomorphism
$\varphi:{\rm{SL}}_{2}(\mathbb{K})\rightarrow X.$
1. (1)
We use Theorem 2.3 as applied to our black box group $X$, so $u$, $h$, $n$ are
string in $X$ such that
$\Phi(u)=\begin{bmatrix}1&1\\\ 0&1\end{bmatrix},\Phi(h)=\begin{bmatrix}t&0\\\
0&t^{-1}\end{bmatrix},\Phi(n)=\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}$
for some abstract isomorphism
$\Phi:X\longrightarrow({\rm{P}}){\rm{SL}}_{2}(q);$
here, $t$ is some primitive element in $\mathbb{F}_{q}$, $q=p^{k}$. We shall
note that we only know the existence of the map $\Phi$. Let $\tilde{h}$ be a
$ppd(k,p)$-element produced by taking some power of $h$ and
$\Phi(\tilde{h})=\begin{bmatrix}\tilde{t}&0\\\ 0&\tilde{t}^{-1}\end{bmatrix}.$
2. (2)
We consider the cyclic subgroup $T=\langle\tilde{h}\rangle$ and the unipotent
subgroup $U=\langle u^{T}\rangle$ in $X$. Observe that $U$ is the full
unipotent subgroup of $X$ since the order of $\tilde{h}$ is a
$ppd(k,p)$-element in $\mathbb{F}_{p^{k}}$.
3. (3)
Now we start introducing on $U$ a structure of field $\mathbb{K}$ isomorphic
to $\mathbb{F}_{q}$. First, for any $u_{1},u_{2}\in U$, we define an addition
on $\mathbb{K}$ by setting
$u_{1}\oplus u_{2}=u_{1}u_{2}.$
For the multiplication on $\mathbb{K}$, we set the element $u$ as the unity of
$\mathbb{K}$. Since $\tilde{h}$ is a $ppd(k,p)$-element, it has odd order $m$
and the element $\sqrt{\tilde{h}}:=\tilde{h}^{(m+1)/2}$ has the property that
$\sqrt{\tilde{h}}^{2}=\tilde{h}$. We also set
$s:=u^{\sqrt{\tilde{h}}}.$
Notice that
$\left[\begin{array}[]{cc}1&1\\\ 0&1\\\
\end{array}\right]^{\,\left[\begin{array}[]{cc}\sqrt{\tilde{t}}&0\\\
0&\sqrt{\tilde{t}^{-1}}\\\
\end{array}\right]}=\left[\begin{array}[]{cc}1&\tilde{t}^{-1}\\\ 0&1\\\
\end{array}\right].$
Hence $s$ can be seen as an element in $\mathbb{K}$ corresponding to
$\tilde{t}^{-1}$, and after setting $s^{i}=u^{(\sqrt{\tilde{h}})^{i}}$, the
elements
$s,s^{2},\dots,s^{{k-1}},s^{k}$
form a polynomial basis of $\mathbb{K}$ over the prime field
$\mathbb{L}\simeq\mathbb{F}_{p}$. The additive groups of $\mathbb{L}$ is
cyclic of order $p$. We have already fixed the identity element $1$ of
$\mathbb{K}$ and hence of $\mathbb{L}$, which uniquely defines the
multiplicative structure on $\mathbb{L}$.
For $w\in U$, we define the product
$w\otimes s^{l}=w^{h^{l}}$
and expanded by linearity to product of any two elements in $\mathbb{K}$. We
still do not know, however, why this operation can be carried out in feasible
time—but we should be reassured that at least the product $w\otimes s^{l}$ can
be computed in time polynomial in $\log q$. So at this stage we treat
$\mathbb{K}$ a _partially_ polynomial time black box field: random generation,
comparison, and addition of elements in $\mathbb{K}$ can be carried out in
polynomial in $\log q$, as well as multiplication of an arbitrary element in
$\mathbb{K}$ by some specific elements.
4. (4)
In view of Theorem 2.4, we have the Frobenius map $\phi$ on our black box
group $X\simeq({\rm{P}}){\rm{SL}}_{2}(q)$ which leaves $U$ and $T$ invariant
and induces the Frobenius map $F$ on $U$. This allows us to introduce on $U$
the Frobenius trace ${\rm Tr}:U\to\mathbb{F}_{p}$
${\rm Tr}(x)=x\oplus x^{F}\oplus x^{F^{2}}\oplus\cdots\oplus x^{F^{k-1}}$
and the trace form, that is, the non-degenerate symmetric
$\mathbb{F}_{p}$-bilinear form given by
$\langle x,y\rangle={\rm Tr}(x\otimes y).$
It is interesting that the Frobenius map and the trace form of our future
black box field are introduced _before_ the field multiplication!
We do not know yet whether the evaluation of the trace form on $\mathbb{K}$ is
computationally feasible, but we can compute in polynomial in $\log q$ time
the values $w\otimes s^{l}=w^{h^{l}}$ and of $\langle w,s^{l}\rangle$ for
arbitrary $w\in\mathbb{K}$ and powers of $s$. In particular, this allows us to
compute the matrix of the trace form
$A=(a_{ij})_{k\times k},\quad a_{i,j}=\langle s^{{i}},s^{{j}}\rangle,\quad
i,j,=1,2,\dots,k.$
5. (5)
We are now in position to introduce in $\mathbb{K}$ an explicit structure of a
$\mathbb{L}$ vector space by computing the decomposition of an arbitrary
element $w\in\mathbb{K}$ with respect to the basis $s,s^{2},\dots,s^{{k}}$.
Indeed, for an arbitrary element $w\in\mathbb{K}$, set
$w=\alpha_{1}s\oplus\alpha_{2}s^{2}\oplus\cdots\oplus\alpha_{k}s^{{k}}$
and
$\beta_{j}=\langle w,s^{j}\rangle,\quad j=1,2,\dots,k.$
The coefficients $\beta_{j}$ are computable in time polynomial in $\log p$ and
$k$:
$\beta_{j}=\langle w,s^{j}\rangle=\sum_{i=1}^{k}\alpha_{i}a_{ij},$
which in matrix notation becomes
$(\beta_{1},\dots,\beta_{k})=(\alpha_{1},\dots,\alpha_{k})\cdot A,$
and therefore
$(\alpha_{0},\dots,\alpha_{k-1})=(\beta_{0},\dots,\beta_{k-1})\cdot A^{-1}.$
6. (6)
We can now decompose products $s^{i}\otimes s^{j}$ with respect to the basis
$s,s^{2},\dots,s^{{k}}$ and thus find the structure constants $c_{ijl}$ for
this basis:
$s^{i}\otimes s^{j}=\sum_{l=1}^{k}c_{ijl}s^{l}.$
Of course now we are in position to multiply any two elements in $\mathbb{K}$,
and, as we can easily see, in time polynomial in $\log q$. Now, we shall use
the algorithms in [1, 36, 38] to construct the isomorphism between
$\mathbb{F}_{p^{k}}$ and $\mathbb{K}$, see discussion in Section 3.5.
7. (7)
Now, we construct $({\rm{P}}){\rm{SL}}_{2}(\mathbb{K})$ by using the Steinberg
generators, see Section 4. Recall that the element $s\in\mathbb{K}$
corresponds to the element $\tilde{t}^{-1}\in\mathbb{F}_{q}$ where $\tilde{t}$
is a $ppd(k,p)$-element in $\mathbb{F}_{q}$, so $s$ is a $ppd(k,p)$-element in
$\mathbb{K}$. We construct, in $({\rm{P}}){\rm{SL}}_{2}(\mathbb{K})$, the
elements encrypting the strings
$\mathbf{u}(1),\mathbf{h}(s^{-1}),\mathbf{n}(1)$
by using the isomorphism between the fields $\mathbb{F}_{p^{k}}$ and
$\mathbb{K}$ constructed in Step 6.
8. (8)
Our first assignments are $\mathbf{u}(1)\mapsto u$ and
$\mathbf{h}(s^{-1})\mapsto\tilde{h}$. Now we need to construct the element in
$X$ encrypting the string $\mathbf{n}(1)$. Note that the element $n\in X$,
which was constructed in Step 1, need not necessarily be the element
corresponding to $\mathbf{n}(1)$. Therefore we shall replace the original
element $n$ by the one that corresponds to $\mathbf{n}(1)$. Recall that the
elements $u,n\in X$ are indeed computed inside a subgroup isomorphic to
$({\rm{P}}){\rm{SL}}_{2}(p)$ or ${\rm{PSL}}_{2}(p^{2})$ depending on $p\equiv
1\bmod 4$ or $p\equiv-1\bmod 4$, respectively [12]. For simplicity, we may
assume that this subgroup encrypts $({\rm{P}}){\rm{SL}}_{2}(p)$ and the
following computations are carried out in this black box subgroup. Note that
raising the element $h$ to the power so that the resulting element $h_{0}$ has
order $(p-1)/2$ and belongs to this subgroup isomorphic to
$({\rm{P}}){\rm{SL}}_{2}(p)$.
We compute all $v:=(u^{-1})^{h_{0}^{k}n}$ for $k=1,\ldots p-1$, and check
which of the elements of the form
$uv^{-1}u$
has order $4$ (Recall that, by (3) and (4), we have
$\mathbf{u}(1)^{\mathbf{n}(s)}=\mathbf{v}(-s^{-2})$ and
$\mathbf{n}(t)=\mathbf{u}(t)\mathbf{v}(-t^{-1})\mathbf{u}(t)$). Observe that
there are only two elements of the form $uv^{-1}u$ of order 4 and they
correspond to the elements $\mathbf{n}(1)$ and $\mathbf{n}(-1)$. Now we need
to distinguish $\mathbf{n}(1)$ from $\mathbf{n}(-1)$. Recall also that, by (3)
and (4), we have
$\mathbf{n}(1)^{\mathbf{h}(t)}=\mathbf{n}(t^{-2}),\mathbf{u}(1)^{\mathbf{h}(t)}=\mathbf{u}(t^{-2}),\mathbf{v}(1)^{\mathbf{h}(t)}=\mathbf{v}(t^{2})$
and
(5)
$\mathbf{n}(t^{-2})=\mathbf{u}(t^{-2})\mathbf{v}(-t^{2})\mathbf{u}(t^{-2}).$
Now it is easy to see that if one of the elements of the form $uv^{-1}u$ of
order 4 corresponds to the Weyl group element $\mathbf{n}(-1)$, then Equation
(5) is not satisfied. Hence the Weyl group element $h_{0}^{k}n$ which produces
the element $uv^{-1}u$ satisfying Equation (5) is the desired Weyl group
element, say $\tilde{n}$.
9. (9)
Observe that the following map
$\displaystyle({\rm{P}}){\rm{SL}}_{2}(\mathbb{K})$
$\displaystyle\longrightarrow$ $\displaystyle Y$ $\displaystyle\mathbf{u}(1)$
$\displaystyle\mapsto$ $\displaystyle u$ $\displaystyle\mathbf{h}(s^{-1})$
$\displaystyle\mapsto$ $\displaystyle\tilde{h}$ $\displaystyle\mathbf{n}(1)$
$\displaystyle\mapsto$ $\displaystyle\tilde{n}$
is an isomorphism.
Notice that the algorithm described above provides a proof of Theorem 4.1.
### 4.2. A more straightforward treatment of ${\rm{SL}}_{2}(p)$
Because of its importance, we give a streamlined construction of an
isomorphism between ${\rm{SL}}_{2}(p)$, $p\equiv 1\bmod 4$, and a black box
group $X$ encrypting ${\rm{SL}}_{2}(p)$. Notice, in this case, that we may
assume that the field structure of $\mathbb{F}_{p}$ is available. Hence, we
shall construct the elements in $X$ encrypting the images of
$\mathbf{u}(1),\mathbf{h}(t)$ and $\mathbf{n}(1)$ where $0,1\neq
t\in\mathbb{F}_{p}$ in $X$.
Step 1: Using Theorem 2.3, we select in $X$ a unipotent element $u$, a toral
element $h$ normalizing the root subgroup containing $u$, and $n$ a Weyl group
element for the torus containing $h$. Our fist assignment is
$\mathbf{u}(1)\mapsto u$.
Step 2: Recall that for a given $\mathbf{h}(t)$ we have
$\mathbf{u}(1)^{\mathbf{h}(t)}=\mathbf{u}(t^{-2})$. Assume that
$\mathbf{u}(t^{-2})=\mathbf{u}(k)=\mathbf{u}(1)^{k}$ for some
$k\in\\{1,2,\ldots,p-1\\}$.
Now we check whether $u^{h}=u^{k}$ in $X$. If not, then some power $\ell$ of
$h$ has this property, that is, $u^{h^{\ell}}=u^{k}$. Observe that $\ell$ is
necessarily relatively prime to $p-1$ so that the resulting element $h^{\ell}$
generates the torus. We replace $h$ with $h^{\ell}$ and assign
$\mathbf{h}(t)\mapsto h$.
Step 3: Now we compute $\mathbf{n}(1)$ by using the same arguments in Step 9
of the algorithm in Section 4.1. Thus we have an isomorphism
$\displaystyle{\rm{SL}}_{2}(p)$ $\displaystyle\longrightarrow$ $\displaystyle
X$ $\displaystyle\mathbf{u}(1)$ $\displaystyle\mapsto$ $\displaystyle u$
$\displaystyle\mathbf{h}(t)$ $\displaystyle\mapsto$ $\displaystyle h$
$\displaystyle\mathbf{n}(1)$ $\displaystyle\mapsto$ $\displaystyle n.$
## 5\. A Revelation and Its Reverberations: Proof of Theorem 3.1
### 5.1. Proof of Theorem 3.1
We describe an algorithm which produces a black box field $\mathbb{U}$ and an
isomorphism
$\Phi:{\rm{SL}}_{2}(\mathbb{U})\longrightarrow X.$
1. (1)
We take our revelation involution $r$ and consider strongly real elements of
the form $r^{x}\cdot r$ for random $x\in X$, and raising them to appropriate
powers, find an element $\theta$ of order $3$ inverted by $r$.
2. (2)
Set $v=\theta r$ and $w=\theta^{2}r$. Observe that $v$ and $w$ are involutions
and $L=\langle\theta\rangle\langle r\rangle$ is the dihedral group of order
$6$.
3. (3)
Observe that all dihedral subgroups of order $6$ in $X$ are conjugate in $X$
and therefore we can assume without loss of generality that
$L\cong{\rm{SL}}_{2}(2)$ encrypts a subfield subgroup of
${\rm{SL}}_{2}(2^{n})$. In particular, there exist a system of Steinberg
generators of ${\rm{SL}}_{2}(2^{n})$,
$\displaystyle\mathbf{u}(t)=\left[\begin{array}[]{cc}1&t\\\ 0&1\\\
\end{array}\right],\,\mathbf{v}(t)=\left[\begin{array}[]{cc}1&0\\\ t&1\\\
\end{array}\right],\,\mathbf{h}(t)=\left[\begin{array}[]{cc}t&0\\\ 0&t^{-1}\\\
\end{array}\right],\,\mathbf{n}(t)=\left[\begin{array}[]{cc}0&t\\\ t^{-1}&0\\\
\end{array}\right]$
for $t\in\mathbb{F}_{2^{n}}$ and $t\neq 0$ for $\mathbf{h}(t)$ and
$\mathbf{n}(t)$, and such that $r$, $v$ and $n$ encrypt $\mathbf{u}(1)$,
$\mathbf{v}(1)$, and $\mathbf{n}(1)$, correspondingly.
4. (4)
The standard procedure for construction of centralizers of involutions [9, 18]
produces unipotent subgroups $U=C_{X}(r)$ and $V=C_{X}(v)$. If we set
$H=\langle h(t)\mid t\in\mathbb{F}_{2^{n}}\rangle$
(warning: this subgroup is not constructed yet) then $B^{+}=UH=N_{X}(U)$ and
$B^{-}=VH=N_{X}(V)$ are Borel subgroups in $X$.
5. (5)
Observe that if $x\in X$ is such that $u^{x}\in U$ for some $1\neq u\in U$
then $x\in B$.
6. (6)
We can identify action of $H$ on $U$ by conjugation with the action of $B/U$
on $U$. Observe that for any two involutions $s,t\in U$ there is a unique
$\bar{b}\in B/U$ such that $s^{\bar{b}}=t$.
7. (7)
Using the double conjugation trick, we can find, for any given involutions $s$
and $t$ in $U$ an element $x$ in $X$ (and hence in $B$) such that $s^{x}=t$.
This is done in the following way: notice that the exponent of
${\rm{SL}}_{2}(2^{n})$ is $2\cdot(2^{n}-1)(2^{n}+1)$ and therefore if $y\in X$
is an element of odd order than $y^{2^{2n}-1}=1$. By conjugating $s$ by a
random element $z\in X$, find an involution $r=s^{z}$ such that elements
$y_{1}=sr$ and $y_{2}=rt$ have odd order. Then it can be checked directly that
$s^{\left((sr)^{2^{2n-1}}\right)}=r\quad\mbox{ and }\quad
r^{\left((rt)^{2^{2n-1}}\right)}=t$
and
$x=(sr)^{2^{2n-1}}\cdot(rt)^{2^{2n-1}}$
has the desired property $s^{x}=t$. By the previous point, the coset $xU$ in
$B/U$ is uniquely determined.
(The same idea of “local conjugation” of involutions is used by Ballantyne and
Rowley for construction of centralizers of involutions in black box groups
with expensive generation of random elements [32].)
8. (8)
Treating the subgroup $B$ as a black box, we have
$U=\\{\,x\in B\mid x^{2}=1\\}.$
Therefore after introducing on $B$ a new equality relation
$x\equiv y\mbox{ if and only if }(xy^{-1})^{2}=1$
we get a black box $\mathbb{T}$ for the factor group $T=B/U$. Notice that
there is a natural action of $\mathbb{T}$ on $U$ by conjugation and that
notation $u^{t}$ for $u\in U$ and $t\in\mathbb{T}$ is not ambiguous.
9. (9)
Now we construct a black box field $\mathbb{U}$. We start with the
multiplicative group $\mathbb{U}^{*}$ of $\mathbb{U}$ which we define as the
graph of the orbit action map of $\mathbb{T}$ onto the orbit $r^{\mathbb{T}}$.
Namely, $\mathbb{U}^{*}$ is the set of all pairs $(t,s)$ with $t\in\mathbb{T}$
and $s\in U\smallsetminus\\{1\\}$ such that $r^{t}=s$. We define in
$\mathbb{U}$ multiplication $\otimes$ by the rule
$(t_{1},u_{1})\otimes(t_{2},u_{2})=(t_{1}t_{s},r^{t_{1}t_{2}}).$
In particular, the element $\mathbf{1}=(1,r)$ plays the role of the identity
element in $\mathbb{U}^{*}$.
Then we define the zero element of $\mathbb{U}$ as
$\mathbf{0}=(1,1),$
set
$\mathbb{U}=\mathbb{U}^{*}\cup\\{0\\}$
(and use lower case boldfaced letter to denote elements
$\mathbf{u}\in\mathbb{U}$), and define
$\mathbf{0}\otimes\mathbf{u}=\mathbf{u}\otimes\mathbf{0}\mbox{ for all
}\mathbf{u}\in\mathbb{U}^{*}.$
Finally, we define on $\mathbb{U}$ addition $\oplus$ by setting
$\displaystyle\mathbf{0}\oplus\mathbf{u}=\mathbf{u}\oplus\mathbf{0}$
$\displaystyle=$ $\displaystyle\mathbf{u}$
$\displaystyle\mathbf{u}\oplus\mathbf{u}$ $\displaystyle=$
$\displaystyle\mathbf{0}$ $\displaystyle(t_{1},u_{1})\oplus(t_{2},u_{2})$
$\displaystyle=$ $\displaystyle(t,u_{1}u_{2})$
where in the last line $u_{1}\neq u_{2}$ (and thus $u_{1}u_{2}\neq 1$) and
$t\in\mathbb{T}$ is chosen to send $r$ to $u_{1}u_{2}$, that is,
$r^{t}=u_{1}u_{2}$. It follows that the inverse $\mathbf{u}^{-1}$ of
$\mathbf{u}=(t,u)\neq\mathbf{0}$ with respect to multiplication $\otimes$ is
equal to $(t^{-1},r^{t^{-1}})$.
10. (10)
So we have a black box field $\mathbb{U}$ interpreted in the Borel subgroup
$B=N_{X}(C_{X}(r)))$ of the black box group $X$ and such that $X$ encrypts
${\rm{SL}}_{2}(\mathbb{U})$.
It will be convenient to use traditional notation and denote
$1=u(\mathbf{0})$, and write, for elements $\mathbf{t}\in\mathbb{U}^{*}$,
$u=u(\mathbf{t})$ if $\mathbf{t}=(t,u)$. In particular, $r=u(\mathbf{1})$.
This gives us a parametrization of $U$ by elements of the black box field
$\mathbb{U}$.
11. (11)
Now we transfer the black box field parametrization from $U$ to $V$ by setting
$v(\mathbf{0})=1$ and for setting for non-identity elements $v\in V$
$v=v(\mathbf{t})\mbox{ if }v^{w}=u(\mathbf{t}).$
We set further
$n(\mathbf{t})=u(\mathbf{t})v(\mathbf{t}^{-1})u(\mathbf{t}),$
so that this agrees with computation in $L\cong{\rm{SL}}_{2}(2)$, yielding
$n(\mathbf{1})=w,$
and finally set
$h(\mathbf{t})=n(\mathbf{t})n(\mathbf{1}).$
Notice that
$\left\\{h(\mathbf{t})\mid\mathbf{t}\in\mathbb{U}^{*}\right\\}=N_{X}(V)\cap
N_{X}(U)$
is the uniquely determined maximal torus in $X$ normalizing the both $V$ and
$U$. We denote it by $H$.
12. (12)
We can now construct an isomorphism
$\Psi:{\rm{SL}}_{2}(\mathbb{U})\longrightarrow X.$
First of all, recall that matrices from ${\rm{SL}}_{2}(\mathbb{U})$ are
quadruples
$\begin{bmatrix}a_{11}&a_{12}\\\ a_{21}&a_{22}\end{bmatrix}$
of strings $a_{ij}$ generated by black box $\mathbb{U}$, with matrix addition
and multiplication defined with respect to operations $\oplus$ and $\otimes$.
1. (a)
Notice easy-to-check identities over any field of characteristic $2$:
1. (i)
given $a$, $b$, and $d$ such that $bc=1$, we have
$\begin{bmatrix}0&b\\\ c&d\end{bmatrix}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\begin{bmatrix}c&0\\\ 0&b\end{bmatrix}\begin{bmatrix}1&bd\\\
0&1\end{bmatrix};$
2. (ii)
for $a\neq 0$ and $ad-bc=1$,
$\begin{bmatrix}a&b\\\ c&d\end{bmatrix}=\begin{bmatrix}a&0\\\
0&a^{-1}\end{bmatrix}\begin{bmatrix}1&0\\\
ac&1\end{bmatrix}\begin{bmatrix}1&a^{-1}b\\\ 0&1\end{bmatrix}.$
2. (b)
Therefore we can map
$\displaystyle\Psi:\begin{bmatrix}\mathbf{0}&\mathbf{b}\\\
\mathbf{c}&\mathbf{d}\end{bmatrix}$ $\displaystyle\mapsto$ $\displaystyle
n(\mathbf{1})h(\mathbf{c})u(\mathbf{b}\otimes\mathbf{d})$
$\displaystyle\Psi:\begin{bmatrix}\mathbf{a}&\mathbf{b}\\\
\mathbf{c}&\mathbf{d}\end{bmatrix}$ $\displaystyle\mapsto$ $\displaystyle
h(\mathbf{a})v(\mathbf{a}\otimes\mathbf{c})u(\mathbf{a}^{-1}\otimes\mathbf{b}).$
This is an isomorphism.
This completes the proof of Theorem 3.1. $\Box$
### 5.2. Other groups of characteristic $2$
We expect that Theorem 4.2 is mirrored by the following conjecture.
###### Conjecture 5.1.
Let $X$ be a black box group encrypting one of the untwisted Chevalley groups
${\rm G}(2^{n})$. We assume that we are given an involution $u\in X$.
Then there is a Monte-Carlo algorithm which constructs a polynomial time (in
$l(X)$) isomorphism
$\Phi:{\rm G}(2^{n})\longrightarrow X.$
The running time of the algorithm is polynomial in $n$ and the Lie rank of
${\rm G}(2^{n})$.
As a comment to Conjecture 5.1, we formulate here the following easy result.
###### Theorem 5.2.
Let $X$ be a black box group encrypting an untwisted Chevalley group ${\rm
G}(2^{n})$ (with $n$ known) and $U<X$ an unipotent long root subgroup given as
a black box subgroup of $X$. Then there is a polynomial time, in $n$ and Lie
rank of ${\rm G}(2^{n})$, Monte-Carlo algorithm which constructs a black box
for $N_{X}(U)$ and a black box $\mathbb{U}$ for the field $\mathbb{F}_{2^{n}}$
interpreted in the action of $N_{X}(U)$ on $U$, with $U$ becoming the additive
group of the field $\mathbb{U}$.
The proof of this theorem is an immediate and obvious generalization of Step 8
in the proof of Theorem 3.1 in Section 5. Indeed, it suffices to observe that
$U$ is a TI-subgroup of $X$ (that is, $U\cap U^{g}=1$ or $U$ for all $g\in G$)
and that all involutions in $U$ are conjugate in $N_{X}(U)$.
Theorem 5.2 suggests that structure recovery of black box Chevalley groups
${\rm G}(2^{n})$ is likely to share some of the conceptual framework of Franz
Timmesfeld’s classification of groups generated by root type subgroups [45,
46, 47, 48]. If so, then this will be strikingly similar to the use of
Aschbacher’s classical involutions [2, 3] and root ${\rm{SL}}_{2}$-subgroups
in our structural theory of classical black box groups in odd characteristic
[11, 13, 12, 15, 49, 50].
## Acknowledgements
This paper would have never been written if the authors did not enjoy the warm
hospitality offered to them at the Nesin Mathematics Village (in Şirince,
Izmir Province, Turkey) in August 2011, August 2012, and July 2013; our thanks
go to Ali Nesin and to all volunteers and staff who have made the Village a
mathematical paradise.
We thank Adrien Deloro for many fruitful discussions, in Şirince and
elsewhere, and Bill Kantor and Rob Wilson for their helpful comments.
Special thanks go to our logician colleagues: Paola D’Aquino, Gregory Cherlin,
Jan Krajíček, Angus Macintyre, Jeff Paris, Jonathan Pila, and Alex Wilkie for
pointing to fascinating connections with logic and complexity theory.
We gratefully acknowledge the use of Paul Taylor’s _Commutative Diagrams_
package, http://www.paultaylor.eu/diagrams/.
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|
arxiv-papers
| 2013-08-12T08:08:09 |
2024-09-04T02:49:49.339696
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandre Borovik and \\c{S}\\\"ukr\\\"u Yal\\c{c}{\\i}nkaya",
"submitter": "Alexandre Borovik",
"url": "https://arxiv.org/abs/1308.2487"
}
|
1308.2566
|
Graphene has exhibited a wealth of fascinating properties, but is also known
not to be a superconductor. Remarkably, we show that graphene can be made a
conventional Bardeen-Cooper-Schrieffer superconductor by the combined effect
of charge doping and tensile strain. While the effect of doping is obvious to
enlarge Fermi surface, the effect of strain is profound to greatly increase
the electron-phonon coupling. At the experimental accessible doping ($\sim
4\times 10^{14}$ cm-2) and strain ($\sim 16$%) levels, the superconducting
critical temperature $T_{c}$ reaches as high as $\sim 30$ K, the highest for a
single-element material above the liquid hydrogen temperature. This
significantly makes graphene a commercially viable superconductor.
# Superconducting Graphene: the conspiracy of doping and strain
Chen Si1,2, Zheng Liu111This author contributed equally as the first author.
,2 Wenhui Duan,1 and Feng [email protected] 1Department of Physics and
State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University,
Beijing 100084, People’s Republic of China
2Department of Materials Science and Engineering, University of Utah, Salt
Lake City, Utah 84112, USA
###### pacs:
73.22.Pr, 74.10.+v, 74.20.Fg
Since its first makingnovoselov2004electric , graphene has fascinated the
scientific community by a seemingly endless discovery of extraordinary
properties, such as electronically the highest carrier mobility with massless
Dirac Fermionsnovoselov2005two ; morozov2008giant , optically the largest
adsorption per atomic layer in the visible rangebonaccorso2010graphene , and
mechanically the strongest 2D material in naturelee2008measurement . However,
graphene is considered not to be a good superconductor. In particular, two
fundamental conditions of intrinsic graphene define an overall very weak
electron-phonon coupling (EPC), rendering itself not to be a Bardeen-Cooper-
Schrieffer (BCS) superconductorbardeen1957microscopic ; bardeen1957theory .
First, graphene has a point-like Fermi surface (Dirac point) with vanishing
density of states (DOS); second, it has a weak electron-phonon (e-ph) pairing
potential. While the first condition can be obviously modified by doping; the
second condition is not known amenable to change.
Several ideas related to doping have been proposed to induce superconductivity
in graphene. For example, adsorption of alkali metal atoms on graphene has
been found to introduce large DOS around the Fermi level as well as increase
the e-ph pairing potential, and hence to enhance the e-ph coupling (EPC) for
BCS superconductivityprofeta2012phonon . However, the EPC enhancement is
largely due to the DOS and phonon modes of metal atoms, rather than the
intrinsic properties of graphene. A recent theoretic work suggests that in a
highly-doped graphene up to the point of Van Hove singularity the greatly
increased e-e interaction can induce a pairing potential in the d-wave
channel, possibly giving rise to chiral superconductivitynandkishore2012chiral
, but the critical superconducting transition temperature ($T_{c}$) is unknown
and likely to be low.
In this Letter, we demonstrate, using first-principles calculations, that in
combination with doping of either electrons or holes, biaxial tensile strain
can greatly enhance the EPC of graphene in a nonlinear fashion so as to
convert it into a BCS superconductor. Most remarkably, within the experimental
accessible dopingefetov2010controlling and strainlee2008measurement levels,
$T_{c}$ can reach $\sim 30$ K, the highest known for a single-element
superconductor.
According to BCS theorybardeen1957microscopic ; bardeen1957theory , when the
phonon-mediated attraction is strong enough to overcome the Coulomb repulsion,
electrons form ”Cooper pairs”, leading to the emergence of superconductivity
below $T_{c}$. The former is characterized by a dimensionless parameter
$\lambda=N_{F}V_{ep}$, where $N_{F}$ is the electron DOS and $V_{ep}$ is the
mean e-ph pairing potential at the Fermi level; the latter by a dimensionless
parameter $\mu=N_{F}V_{ee}$, where $V_{ee}$ is the mean e-e repulsive
potential at the Fermi level. Superconductivity occurs for $\lambda\gg\mu$,
and $T_{c}$ increases with the increasing Fermi surface (more Cooper pairs be
formed) and e-ph paring potential (easier Cooper pairs be formed). Because
$\mu$ is rather material insensitive (see discussion below), generally,
materials are classified into three regimes of EPC: weak $\lambda\ll 1$,
intermediate $\lambda\sim 1$, and strong $\lambda>1$; a good BCS
superconductor requires $\lambda\geq 1$.
As a promising material for the next-generation electronic devices, the EPC in
graphene has been extensively studied both theoretically and
experimentallyferrari2007raman . Because of a diminishing Fermi surface (a
point for intrinsic graphene) and a very weak e-ph pairing potential (because
of its high Fermi temperature of the massless carriers and high energy of the
optical phonons), the EPC in graphene is found to be very weak. This feature
is actually responsible for some of its other extraordinary properties like
extremely high electricalchen2008intrinsic and thermal
conductivitybalandin2008superior . But on the other hand, it prevents graphene
from being a BCS superconductor. To increase EPC of graphene ($\lambda$), one
must increase the DOS ($N_{F}$) and/or the e-ph pairing potential ($V_{ep}$)
at the Fermi level. Obviously, $N_{F}$ can be increased by doping of either
electrons or holes. Figure 1a shows our calculated $N_{F}$ and $V_{ep}$ as a
function of hole concentration ($n$) for a p-type graphene SI . Both $V_{ep}$
and $\lambda$ increase with the increase of doping, and $\lambda\approx 0.19$
at a doping level of $6.2\times 10^{14}$ cm-2 (corresponding to doping of
$\sim$1/3 of a hole per unit cell).
Figure 1: (Color online) (a) $N_{F}$ and $\lambda$ of p-type graphene under
different doping level ($n$) calculated from first-principles. (b)
$\lambda(\varepsilon)$ for the $4.65\times 10^{14}$ cm-2 hole-doped graphene.
The solid lines are fits to the data.
Because we are interested in doping levels beyond the linear-Dirac-band
regime, we expand $E(k)$ around the Dirac point to the second
orderneto2009electronic ; wang2010manipulation , $E=\alpha k+\beta k^{2}$,
then we have $N_{F}=\sqrt{n}(a+b\sqrt{n})^{-1}$, which gives a very good fit
to the calculated $N_{F}$ with $a=11.93$ and $b=-2.19$, as shown in Fig. 1(a).
Since $\lambda=N_{F}V_{ep}$, we found that $V_{ep}$ remains a constant,
$\sim$0.5 eV, independent of doping, as shown by the very good fitting of
$\lambda=0.5N_{F}$ shown in Fig. 1(a). Thus, we obtain a relation of
$\lambda(n)$ for the hole-doped graphene as
$\lambda=0.5\sqrt{n}(11.93-2.19\sqrt{n})^{-1}$ (1)
We noticed that the value of $V_{ep}\sim 0.5$ eV is much smaller than the
typical values found in BCS superconductors, such as 1.4 eVsavini2010first ;
an2001superconductivity for MgB2 and 3 eVsavini2010first ;
giustino2007electron for B-doped diamond. This means that doping is a
necessary but insufficient condition to make graphene a BCS superconductor.
The above results indicate that in order to make graphene a superconductor,
one must find a way to increase $V_{ep}$ in addition to doping. It has been
shown recently that applying biaxial tensile strain can significantly soften
the in-plane optical modes of graphenemarianetti2010failure , hinting that it
may also enhance $V_{ep}$ and hence the EPC. To explore this possibility, we
have calculated $\lambda$ as a function of biaxial tensile strain
($\varepsilon$) for a hole-doped graphene at a $4.65\times 10^{14}$ cm-2
doping level, as shown in Fig. 1(b). Clearly we see $\lambda$ increases
dramatically with the strain. In particular, at the 16.5% of strain, $\lambda$
reaches as high as 1.45, entering the strong coupling regime, with a
corresponding value of $V_{ep}\sim 3.25$ eV, even larger than that in the
B-doped diamondsavini2010first ; giustino2007electron .
To understand such remarkable strain induced enhancement of EPC, we first
recall that McMillan has shown thatmcmillan1968transition
$\lambda\propto(\langle\omega^{2}\rangle)^{-1}$, where $\omega$ is the
frequency of all the phonon modes contributing to the EPC. Interestingly, we
found that for graphene there exists a characteristic phonon mode
($\omega_{0}$) that dominates the EPC. In general, the frequency of this
characteristic mode must change with strain as
$\omega_{0}(\varepsilon)=\omega_{0}(\varepsilon=0)+p\varepsilon+q\varepsilon^{2}$,
where $p$ and $q$ are the first- and second-order phonon deformation
potential, respectively; the second-order nonlinear term is needed here
because of the large strain involved. Then, we can fit the
$\lambda(\varepsilon)$ curve using an empirical formula
$\lambda(\varepsilon)=0.5(1+t_{1}\varepsilon+t_{2}\varepsilon^{2})^{-2}$ (2)
and a very good fit is obtained with $t_{1}=-0.007$, $t_{2}=-0.002$, as shown
in Fig. 1(b).
Figure 2: (Color online) The Eliashberg spectral functions and $\omega_{0}$
as a function of strain in the hole-doped graphene. (a) $n=1.55\times 10^{14}$
cm-2 (c) $n=3.10\times 10^{14}$ cm-2 (e) $n=4.65\times 10^{14}$ cm-2 under 6%
(black line), 14% (blue line) and 16.5% (red line) strain. (b), (d) and (f),
$\omega_{0}$ versus $\varepsilon$ (black dots) and ${\omega_{0}}^{-2}$ versus
$\varepsilon$ (red triangle) for $n=1.55\times 10^{14}$ cm-2, $3.10\times
10^{14}$ cm-2 and $4.65\times 10^{14}$ cm-2. The solid lines are empirical fit
to the data.
Next we perform a more vigorous analysis of the strain dependence of EPC in
graphene. Figure 2 shows the Eliashberg spectral function
$\alpha^{2}F(\omega)$, which describes the averaged coupling strength between
the electrons of Fermi energy ($E_{F}$) and the phonons of energy $\omega$:
$\alpha^{2}F(\omega)=\frac{1}{N_{F}N_{k}N_{q}}\sum_{mn}\sum_{\textbf{q}\nu}\delta(\omega-\omega_{\textbf{q}\nu})\sum_{\textbf{k}}|g_{\textbf{k}+\textbf{q},\textbf{k}}^{\textbf{q}\nu,mn}|^{2}\delta(E_{\textbf{k}+\textbf{q},m}-E_{F})\delta(E_{\textbf{k},n}-E_{F})$
(3)
and the frequency-dependent EPC functiongrimvall1981electron
$\lambda(\omega)=2\int_{0}^{\omega}\frac{\alpha^{2}F(\omega^{{}^{\prime}})}{\omega^{{}^{\prime}}}d\omega^{{}^{\prime}}$
(4)
where the phonon frequency $\omega$ is indexed with wavevector (q) and mode
number ($\nu$), and the electron eigenvalue $E$ is indexed with wavevector (k)
and the band index ($m$ and $n$), and
$g_{\textbf{k}+\textbf{q},\textbf{k}}^{\textbf{q}\nu,mn}$ represents the
electron-phonon matrix element. For $1.55\times 10^{14}$ cm-2 (Fig. 2(a)),
$3.10\times 10^{14}$ cm-2 (Fig. 2(c)) and 4.65 $\times 10^{14}$ cm-2 (Fig.
3(e)) hole doped graphene, the Eliashberg function is found sharply peaked at
certain energy with a $\delta$-like shape. For clarity, we shaded this peak
that dominates the EPC. It corresponds to the SH∗ (shear horizontal optical)
in-plane C-C stretching mode. As the tensile strain increases, on the one
hand, the shaded peak moves towards lower energy, reflecting the softening of
this particular optical modemarianetti2010failure ; on the other hand, the
shaded peak value is intensified. From Eq. (4), we see that both the red shift
(decreasing $\omega$) and the increase of peak intensity (increasing
$\alpha^{2}F(\omega)$) will increase $\lambda$.
The spectral features in Fig. 2(a), (c) and (e) suggest that the strain
induced phonon softening plays a key role in the strain enhanced EPC. To
further quantify the $\lambda-\varepsilon$ relation, we define a
characteristic phonon mode ($\omega_{0}$) by averaging over all phonon modes
weighted by the Eliashberg spectral function $\alpha^{2}F(\omega)$,
$\omega_{0}=\langle\omega^{2}\rangle^{1/2}=\sqrt{\int
d\omega\,\omega\alpha^{2}F(\omega)/\int\frac{d\omega\,\alpha^{2}F(\omega)}{\omega}}$
(5)
i.e., each phonon mode is weighted by its EPC strength, so that the calculated
$\omega_{0}$ represents the average phonon mode contribution to $\lambda$. The
calculated results (data points) of $\omega_{0}$ as a function of strain are
shown in Fig. 2(b), (d) and (f) for the $1.55\times 10^{14}$ cm-2, $3.10\times
10^{14}$ cm-2, and $4.65\times 10^{14}$ cm-2 hole-doped graphene,
respectively. Clearly, the $\omega_{0}$ decreases with the increasing
$\varepsilon$ and can be fit nicely by
$\omega_{0}(\varepsilon)=\omega_{0}(\varepsilon=0)+p\varepsilon+q\varepsilon^{2}$,
as mentioned above.
Also plotted in Fig. 2(b), (d) and (f) are $\lambda$ as a function of
${\omega_{0}}^{-2}$, illustrating the scaling relation of
$\lambda\sim{\omega_{0}}^{-2}$. Thus, we arrived at the empirical formula of
Eq. (2) used to fit the data in Fig. 1(b). The exponent $-2$ in the
$\lambda\sim\omega_{0}$ relation is more exotic, and serves as the key to
understand the nonlinear enhancement of EPC by strain in graphene. In view of
the $\delta$-like shaped Eliashberg function that dominates the EPC (Fig. 2),
we can assume that the characteristic phonons can be approximated by the
Einstein model, all having the same energy $\omega_{0}$. As a nonpolar
centrosymmetric crystal, the only EPC type in graphene is the deformation
potential interactiongrimvall1981electron
$|g_{\textbf{k}+\textbf{q},\textbf{k}}^{\textbf{q}\nu}|^{2}={|\langle\textbf{k}+\textbf{q}|D_{\textbf{q}\nu}\delta
u|\textbf{k}\rangle|}^{2}=\frac{{|\langle\textbf{k}+\textbf{q}|D_{\textbf{q}\nu}|\textbf{k}\rangle|}^{2}}{2M\omega_{0}}$
(6)
where $D_{q\nu}$ is the deformation potential operator associated with the
phonon mode (q, $\nu$) and $\delta u=\sqrt{\frac{1}{2M\omega_{0}}}$ is the
zero-point oscillation amplitude of a quantum particle with mass $M$.
Considering that $\lambda$ is defined by the EPC-induced renormalization of
electron energy spectrum around the Fermi surface, $\lambda\sim\delta
E(\textbf{k})/[E(\emph{k})-E(\textbf{k}_{F})]$, we roughly determine the
electronic energy shift around the Fermi surface by applying a simple second-
order perturbation:
$\Delta
E_{\textbf{k}}\approx\sum_{\textbf{q},\nu}\frac{|g_{\textbf{k}+\textbf{q}}^{\textbf{q}\nu}|^{2}}{E_{\textbf{k}+\textbf{q}}^{0}-E_{\textbf{k}}^{0}}\delta(E_{\textbf{k}+\textbf{q}}^{0}-E_{\textbf{k}}^{0}-\omega_{0})\propto\frac{1}{{\omega_{0}}^{2}}$
(7)
And again, we arrive at $\lambda\sim{\omega_{0}}^{-2}$. Now, we see this
relation has two origins. One ${\omega_{0}}^{-1}$ factor comes from the zero-
point oscillation amplitude, i.e. softer phonons inducing larger deformation;
the other ${\omega_{0}}^{-1}$ factor comes from the energy denominator in the
perturbation theory, i.e. softer phonons inducing stronger mixing between
different electronic states around the Fermi surface. The former is reflected
by the increased shaded peak value in the Eliashberg spectral function under
strain (Fig. 2); the latter corresponds exactly to the $1/\omega$ scaling of
$\lambda$($\omega$) (Eq. (4)).
Figure 3: (Color online) 3D plot of $\lambda(n,\varepsilon)$ calculated using
Eq. (8), and selected data (stars) calculated from first-prinsiples.
Combine Eq. (1) and Eq. (2), we derive an empirical function of
$\lambda(n,\varepsilon)$ for the p-type graphene,
$\lambda(n,\varepsilon)=\frac{\sqrt{n}}{11.93-2.19\sqrt{n}}\cdot\frac{0.5}{(1-0.007\varepsilon-0.002\varepsilon^{2})^{2}}$
(8)
which is applicable to a wide range of doping and strain. Eq. (8) underlines
explicitly the combined effects of doping and strain that greatly enhance the
EPC in graphene. Figure 3 shows the 3D plot of $\lambda(n,\varepsilon)$ using
Eq. (8). In comparison, some $\lambda$ values directly calculated from the
first-principles (stars) are also shown, and the two agree well.
Figure 4: (Color online) 3D plot of $T_{c}(n,\varepsilon)$ calculated using
Eq. (9), and selected data (stars) calculated from first-prinsiples.
The greatly enhanced $\lambda$ by the combined effects of doping and tensile
strain, as shown in Fig. 3, have some very interesting physical implications,
including formation of exotic polaronic, charge density wave, and
superconducting state. In particular, the possible superconducting state,
which may occur at the limit of high doping (high carrier density) and strain
(strong e-ph pairing) levels, is very appealing. We have calculated the
critical transition temperature for the superconducting state using McMillan-
Allen-Dynes formulaallen1975transition :
$\displaystyle
T_{c}=\frac{\hbar\omega_{\log}}{1.2k_{B}}\exp[\frac{-1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}]$
(9a) $\omega_{\log}=1035.77-38.05\varepsilon-0.70\varepsilon^{2}$ (9b)
which has been widely used to estimate $T_{c}$ of carbon-based BCS
superconductors, such as fullereneoshiyama1992linear , carbon
nanotubesiyakutti2006electronic and intercalated
graphitecalandra2005theoretical . $\omega_{\log}$ is the logarithmically
averaged phonon frequency, and we found that $\omega_{\log}$ is almost
independent of doping but changes with strain following the empirical relation
of Eq. (9b) SI . $\mu^{*}$ is the retarded Coulomb pseudopotential related to
the dimensionless parameter of screened Coulomb potential $\mu$ as
$\mu^{*}=\mu/[1+\mu\ln(\omega_{e}/\omega_{D})]$, where $\omega_{e}$ and
$\omega_{D}$ are the characteristic electron and phonon energy. We have
evaluated $\mu^{*}$ for graphene SI , which falls in the range of $\sim
0.10-0.15$, consistent with the values reported in other carbon-related
materials and most $sp$-electron metals.
Using Eqs. (8) and (9) with $\mu^{*}$=0.115, we plot in Fig. 4 the calculated
$T_{c}$ as a function of doping ($n$) and strain ($\varepsilon$), superimposed
with selected data obtained from first-principles calculations (stars). Most
remarkably, at 16.5% strain, $T_{c}$ reaches as high as 18.6 K, 23.0 K and
30.2 K for the doping level of $1.55\times 10^{14}$, $3.10\times 10^{14}$ and
$4.65\times 10^{14}$ cm-2, respectively. Such a high $T_{c}$ may appear too
surprising at first, but becomes reasonable after one compares with MgB2, a
well-known BCS superconductor with a theoretically predicted $T_{c}$ $\approx$
40 Kliu2001beyond that is in very good agreement with
experimentnagamatsu2001superconductivity . The characteristic values of
$\lambda=1.01$ and $\omega_{\log}=56.2$ meV (453.3 cm-1)liu2001beyond of MgB2
are very comparable to ours for the doped and strained graphene.
Due to the high electron-hole symmetry in graphene about the Dirac point, we
expect the electron and hole doped graphene to have similar superconductivity
transition under tensile strain. We have calculated $\lambda$ and $T_{c}$ at
different tensile strains for both the $4.65\times 10^{14}$ cm-2 electron- and
hole-doped graphene (See Table S1 SI ). Very similar trends in $\lambda$ and
$T_{c}$ are found, except that the $T_{c}$ in the electron-doped graphene is
slightly lower than $T_{c}$ in the hole-doped graphene at the same doping and
strain levels.
Our findings uncover yet another fascinating property of graphene with
promising implications in graphene-based devices, such as superconducting
quantum interference devices and transistors. We reiterate that the
superconductivity transition of doped graphene we discover here is triggered
by the enhanced EPC under tensile strain, which is fundamentally different
from that in metal-decorated grapheneprofeta2012phonon . The enhancement of
$\lambda$ in metal decorated graphene arises from additional metal-related
electronic states around the Fermi level, which couple strongly in part with
the phonon modes of adsorbed metal atoms. In this sense, the superconductivity
in metal-decorated graphene is ”extrinsic”, arising from additional properties
of foreign metal atoms and it is similar to their 3D counterparts, such as the
intercalated graphitecsanyi2005role ; weller2005superconductivity and
MgB2an2001superconductivity ; while the superconductivity in our case is
”intrinsic”, arising solely from the intrinsic graphene properties modified by
doping and strain. Specifically, tensile strain hardly modifies the electronic
structure except changing the slope of $\pi$ band, i.e., Fermi velocity, and
doping is within the $\pi$ band too. So the electrons scattered by the phonons
still consist of $\pi$ electrons of graphene. Apparently, ours is also very
different from the chiral superconducting state of the exceedingly high-doped
graphene up to the van Hove singularity point nandkishore2012chiral .
Finally, it is important to stress that the high $T_{c}$ is achieved within
the experimental accessible doping and strain levels. Either chemical doping
by adsorptionyokota2011carrier or electrical doping by gating in a field
effect transistorefetov2010controlling has shown doping levels above
$10^{14}$ cm-2 in graphene. On the other hand, as the strongest 2D material in
nature, experimentally graphene has been elastically stretched up to $\sim$25%
tensile strain without breakinglee2008measurement . It is also interesting to
note that a recent theoretical work shows that either electron or hole doping
can in fact further strengthen graphene to reach even higher ideal strength
under tensile strainsi2012electronic . Therefore, it is highly reasonable to
anticipate experimental realization of high $T_{c}$ superconducting graphene
of $\sim 30$ K, making graphene a commercially viable superconductor.
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|
arxiv-papers
| 2013-08-12T13:58:36 |
2024-09-04T02:49:49.352978
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chen Si, Zheng Liu, Wenhui Duan and Feng Liu",
"submitter": "Chen Si",
"url": "https://arxiv.org/abs/1308.2566"
}
|
1308.2668
|
# Friedmann equations in braneworld scenarios from emergence of cosmic space
A. Sheykhi 1,[email protected], M. H. Dehghani 1,2
[email protected] and S. E. Hosseini 1 1 Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
###### Abstract
Recently, it was argued that the spacetime dynamics can be understood by
calculating the difference between the degrees of freedom on the boundary and
in the bulk in a region of space. In this Letter, we apply this new idea to
braneworld scenarios and extract the corresponding Friedmann equations of
$(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk with any
spacial curvature. We will also extend our study to the more general Gauss-
Bonnet braneworld with curvature correction terms on the brane and in the
bulk, and derive the dynamical equation in a nonflat Universe.
## I Introduction
The emergence properties of gravity has a long history since the original
proposal made by Sakharov in 1968 Sak . Recent investigations supports the
idea that gravitational field equations in a wide range of theories can be
recast as the first law of thermodynamics on the boundary of space CaiKim ;
SheyW1 ; SheyW2 ; Shey0 ; Pad0 . Among various proposal on the connection
between thermodynamics and gravity, the so called entropic origin of gravity
proposed by Verlinde Ver , has got a lot of attentions Cai4 ; Other ; newref ;
sheyECFE ; Ling ; Modesto ; Yi ; Sheykhi2 . According to Verlinde, gravity can
be identified with an entropic force caused by changes in the information
associated with the positions of material bodies. Verlinde considers the
gravitational field equations as the equations of emergent phenomenon and
leaves the spacetime as a background geometric which has already exist.
A new insight to the origin of spacetime dynamics, was recently suggested by
PadmanabhanPad1 who claimed that the cosmic space is emergent as the cosmic
time progresses. Using this new idea, Padmanabhan Pad1 derived the Friedmann
equation of a flat Friedmann-Robertson-Walker (FRW) Universe. Following Pad1 ,
further investigations have been carried out to extract the Friedmann
equations of a FRW Universe in various gravity theories Cai1 ; Yang ; FQ ;
Shey1 . In these investigations (Cai1 ; Yang ; Shey1 ; FQ ), following Pad1 ,
the authors could only derive the Friedmann equations of a flat FRW Universe
and they failed to obtain the dynamical equations describing the evolution of
the Universe with any spacial curvature in other gravity theories. Very
recently, an interesting modification of Padmanabhan’s proposal, which works
in a nonflat Universe, was suggested by Sheykhi Shey2 . Using this modified
proposal one is able to derive the corresponding dynamical equations governing
the evolution of the Universe with any spacial curvature not only in Einstein
gravity, but also in Gauss-Bonnet and more general Lovelock gravity Shey2 .
See also FF for some application and extension of Shey2 . In this paper, we
will address the question on the connection between the degrees of freedom and
the spacetime dynamics by investigating whether and how the relation can be
found in braneworld models.
Let us briefly review the proposal of Shey2 . According to Padmanabhan in an
infinitesimal interval $dt$ of cosmic time, the increase $dV$ of the cosmic
volume, in a flat Universe, is given by Pad1
$\frac{dV}{dt}=L_{p}^{2}(N_{\mathrm{sur}}-N_{\mathrm{bulk}}).$ (1)
where $L_{p}$ is the Planck length, $N_{\mathrm{sur}}$ is the number of
degrees of freedom on the boundary and $N_{\mathrm{bulk}}$ is the number of
degrees of freedom in the bulk. Through this paper we set $k_{B}=1=c=\hbar$
for simplicity. Inspired by (1), an improved extension for $n\geq
4$-dimensional Universe with spacial curvature was found as Shey2
$\beta\frac{dV}{dt}=L_{p}^{n-2}H\tilde{r}_{A}\left(N_{\mathrm{sur}}-N_{\mathrm{bulk}}\right),$
(2)
where $H=\dot{a}/a$ is the Hubble parameter, $a$ is the scale factor,
$\beta={(n-2)}/{2(n-3)}$ and $\tilde{r}_{A}$ is the apparent horizon radius of
FRW Universe given by
$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (3)
Motivated by the area law of the entropy, we assume the number of degrees of
freedom on the apparent horizon is
$N_{\mathrm{sur}}=\beta\frac{A}{L_{p}^{n-2}},$ (4)
where $A=(n-1)\Omega_{n-1}\tilde{r}_{A}^{n-2}$ is the area of the apparent
horizon with $\Omega_{n-1}$ is the volume of a unit $(n-1)$-sphere. The volume
of the $(n-1)$-sphere with radius $\tilde{r}_{A}$ is
$V=\Omega_{n-1}\tilde{r}_{A}^{n-1}$. We assume the energy content inside the
$n$-dimensional bulk is in the form of Komar energy Cai1
$E_{\mathrm{Komar}}=\frac{(n-3)\rho+(n-1)p}{n-3}V,$ (5)
where $\rho$ and $p$ are the energy density and pressure of the perfect fluid
inside the Universe, respectively. Hence according to the equipartition law of
energy, the bulk degrees of freedom is obtained as
$\displaystyle N_{\mathrm{bulk}}$ $\displaystyle=$
$\displaystyle\frac{2\left|E_{\mathrm{komar}}\right|}{T}$ (6) $\displaystyle=$
$\displaystyle-4\pi\Omega_{n-1}\tilde{r}^{n}_{A}\frac{(n-3)\rho+(n-1)p}{n-3},$
where $T=1/(2\pi\tilde{r}_{A})$ is the Hawking temperature associated with the
apparent horizon. Substituting Eqs. (4) and (6) in relation (11), we arrive at
$H^{-1}\dot{r}_{A}\tilde{r}_{A}^{-3}-\tilde{r}_{A}^{-2}=\frac{8\pi
L_{p}^{n-2}}{(n-1)}\times\frac{(n-3)\rho+(n-1)p}{(n-2)}$ (7)
Multiplying both hand sides of by factor $2\dot{a}a$, and using the
$n$-dimensional continuity equation:
$\dot{\rho}+(n-1)H(\rho+p)=0,$ (8)
we obtain Shey2
$\frac{d}{dt}\left[a^{2}\left(H^{2}+\frac{k}{a^{2}}\right)\right]=\frac{16\pi
L_{p}^{n-2}}{(n-1)(n-2)}\frac{d}{dt}(\rho a^{2}).$ (9)
After integrating and setting the constant of integration equal to zero, we
find
$H^{2}+\frac{k}{a^{2}}=\frac{16\pi L_{p}^{n-2}}{(n-1)(n-2)}\rho.$ (10)
This is the Friedmann equation of $n$-dimensional FRW Universe with any
spacial curvature CaiKim .
## II Emergence of Friedmann equations in RS II braneworld
In the remaining part of paper, we want to extend the study to the branworld
scenarios. Gravity on the brane does not obey Einstein theory, thus the usual
area formula for the holographic boundary get modified on the brane SheyW1 ;
SheyW2 . Two well-known scenarios in braneworld are Randall-Sundrum (RS) II RS
; Bin and Dvali, Gabadadze, Porrati (DGP) DGP ; DG models. In the first
scenario an $(n-1)$-dimensional brane embedded in an $(n+1)$-dimensional AdS
bulk. In this case, the extra dimension has a finite size and the localization
of gravity on the brane occurs due to the negative cosmological constant in
the bulk. In the second scenario which is called DGP model, an
$(n-1)$-dimensional brane is embedded in a spacetime with an infinite-size
extra dimension, with the hope that this picture could shed new light on the
standing problem of the cosmological constant as well as on supersymmetry
breaking DGP . In the original DGP model the bulk was assumed to be a
Minkowskian spacetime with infinite size. In this case the recovery of the
usual gravitational laws on the brane is obtained by adding an Einstein-
Hilbert term to the action of the brane computed with the brane intrinsic
curvature. The so-called warped DGP model corresponds to the case where both
the intrinsic curvature term on the brane and the negative cosmological
constant in the bulk are taken into account.
In order to apply the proposal (11) to braneworld scenarios, we modify it a
little by replacing $L_{p}^{n-2}$ with $G_{n+1}$, namely
$\beta\frac{dV}{dt}=G_{n+1}H\tilde{r}_{A}\left(N_{\mathrm{sur}}-N_{\mathrm{bulk}}\right).$
(11)
First of all, we consider the RS II scenario. The apparent horizon entropy for
an $(n-1)$-brane embedded in an $(n+1)$-dimensional bulk in RS II model is
given by SheyW1
$S=\frac{2\Omega_{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}}\times{}_{2}F_{1}\left(\frac{n-1}{2},\frac{1}{2},\frac{n+1}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right),$
(12)
where ${}_{2}F_{1}(a,b,c,z)$ is a hypergeometric function, and $\ell$ is the
bulk AdS radius,
$\ell^{2}=-\frac{n(n-1)}{16\pi
G_{n+1}\Lambda_{n+1}}\,,\quad\Omega_{n-1}=\frac{\pi^{(n-1)/2}}{\Gamma((n+1)/2)}.$
(13)
In the above relation, $\Lambda_{n+1}$ represents the $(n+1)$-dimensional bulk
cosmological constant. The entropy expression (12) can be written in the form
SheyW1
$S=\frac{(n-1)\ell\Omega_{n-1}}{2G_{n+1}}{\displaystyle\int_{0}^{\tilde{r}_{A}}\frac{\tilde{r}_{A}^{n-2}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}d\tilde{r}_{A}},$
(14)
and hence we define the effective area as
$\widetilde{A}=4G_{n+1}S=2(n-1)\ell\Omega_{n-1}\int_{0}^{\tilde{r}_{A}}\frac{\widetilde{r}_{A}^{n-2}}{\sqrt{\widetilde{r}_{A}^{2}+\ell^{2}}}d\widetilde{r}_{A}$
(15)
Now we calculate the increasing in the effective volume as
$\displaystyle\frac{d\widetilde{V}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}_{A}}{(n-2)}\frac{d\tilde{A}}{dt}$ (16)
$\displaystyle=$ $\displaystyle
2\ell\Omega_{n-1}\frac{(n-1)}{(n-2)}\frac{\tilde{r}_{A}^{n-1}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}\dot{\tilde{r}}_{A}$
$\displaystyle=$
$\displaystyle-2\Omega_{n-1}\frac{(n-1)}{(n-2)}\tilde{r}_{A}^{n+1}\frac{d}{dt}\left(\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}\right)$
(17)
Motivated by (17), we assume the number of degrees of freedom on the boundary
is given by
$\displaystyle N_{\mathrm{sur}}$ $\displaystyle=$
$\displaystyle\frac{4\beta(n-1)\Omega_{n-1}}{(n-2)G_{n+1}}\tilde{r}_{A}^{n}\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}$
(18) $\displaystyle=$
$\displaystyle\frac{2(n-1)\Omega_{n-1}}{(n-3)G_{n+1}}\tilde{r}_{A}^{n}\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}.$
Inserting Eqs. (6), (17) and (18) in relation (11), after multiplying both
hand side by factor $\dot{a}a$, we get
$\displaystyle-\frac{\widetilde{r}_{A}^{-3}\dot{\widetilde{r}}_{A}}{\sqrt{\widetilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}}a^{2}+2\dot{a}a\sqrt{\widetilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}$
(19) $\displaystyle=$ $\displaystyle-4\pi
G_{n+1}\dot{a}a\left(\frac{(n-3)\rho+(n-1)p}{(n-1)}\right).$
Using the continuity equation (8), after some simplification, we arrive at
$\frac{d}{dt}\left(a^{2}\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}\right)=\frac{4\pi
G_{n+1}}{(n-1)}\frac{d}{dt}\left(\rho a^{2}\right).$ (20)
Integrating and dividing by $a^{2}$, we find
$\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}=\frac{4\pi G_{n+1}}{(n-1)}\rho,$
(21)
where we assumed the integration constant to be zero. Substituting the
apparent horizon radius from relation (3), we get
$\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell^{2}}}=\frac{4\pi
G_{n+1}}{(n-1)}\rho.$ (22)
In this way we derive the Friedmann equation of higher dimensional FRW
Universe in RS II braneworld by calculating the difference between the number
of degrees of freedom on the boundary and in the bulk. This coincides with the
result obtained in SheyW1 from the field equations.
## III Friedmann equations in Warped DGP braneworld
Next we consider an $(n-1)$-dimensional warped DGP brane embedded in an
$(n+1)$-dimensional AdS bulk. The entropy associated with the apparent horizon
is given by SheyW1
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{(n-1)\Omega_{n-1}{\tilde{r}_{A}}^{n-2}}{4G_{n}}+\frac{2\Omega_{n-1}{\tilde{r}_{A}}^{n-1}}{4G_{n+1}}$
(23)
$\displaystyle\times{}_{2}F_{1}\left(\frac{n-1}{2},\frac{1}{2},\frac{n+1}{2},-\frac{{\tilde{r}_{A}}^{2}}{\ell^{2}}\right).$
It is important to note that in DGP braneworld, the entropy expression of the
apparent horizon consists two terms. The first term which satisfies the area
formula on the brane is the contribution from the Einstein-Hilbert term on the
brane. The second term is the same as the entropy of RS II braneword and
therefore obeys the $(n+1)$-dimensional area law in the bulk SheyW1 .
One can write the entropy associated with the apparent horizon on the brane as
SheyW1
$S=(n-1)\Omega_{n-1}\int_{0}^{\tilde{r}_{A}}\left(\frac{(n-2)\tilde{r}_{A}^{n-3}}{4G_{n}}+\frac{\ell}{2G_{n+1}}\frac{\tilde{r}_{A}^{n-2}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}\right)d\tilde{r}_{A}$
(24)
We define the effective surface as
$\displaystyle\widetilde{A}$ $\displaystyle=$ $\displaystyle
4G_{n+1}S=4G_{n+1}(n-1)\Omega_{n-1}$
$\displaystyle\times\int_{0}^{\tilde{r}_{A}}\left(\frac{(n-2)\tilde{r}_{A}^{n-3}}{4G_{n}}+\frac{\ell}{2G_{n+1}}\frac{\tilde{r}_{A}^{n-2}}{\sqrt{\tilde{r}_{A}^{2}+\ell^{2}}}\right)d\tilde{r}_{A}.$
We also obtain the rate of increase in the effective volume as
$\displaystyle\frac{d\widetilde{V}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}_{A}}{(n-2)}\frac{d\tilde{A}}{dt}=\Omega_{n-1}\frac{(n-1)}{(n-2)}\dot{\tilde{r}}_{A}\tilde{r}_{A}^{n-2}$
(26)
$\displaystyle\times\left(\frac{(n-2)G_{n+1}}{G_{n}}+\frac{2}{\sqrt{\tilde{r}_{A}^{-2}+\ell^{-2}}}\right)$
$\displaystyle=$
$\displaystyle-2\Omega_{n-1}\frac{(n-1)}{(n-2)}\tilde{r}_{A}^{n+1}$
$\displaystyle\times\frac{d}{dt}\left(\frac{(n-2)G_{n+1}}{4G_{n}}\tilde{r}_{A}^{-2}+\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}\right)$
Inspired by (26), we suppose the number of degrees of freedom on the apparent
horizon in warped DGP model is given by
$\displaystyle
N_{\mathrm{sur}}=\frac{2\Omega_{n-1}}{G_{n+1}}\frac{(n-1)}{(n-3)}\tilde{r}_{A}^{n}$
$\displaystyle\left(\frac{G_{n+1}(n-2)\tilde{r}_{A}^{-2}}{4G_{n}}\right.$ (27)
$\displaystyle\left.+\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}\right).$
Combining Eqs. (6), (26) and (27) with relation (11), it is a matter of
calculation to find
$\displaystyle\frac{d}{dt}\left(a^{2}\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}\right)$
$\displaystyle=$
$\displaystyle-\frac{G_{n+1}}{4G_{n}}(n-2)\frac{d}{dt}\left(\tilde{r}_{A}^{-2}a^{2}\right)$
(28) $\displaystyle+\frac{4\pi G_{n+1}}{(n-1)}\frac{d}{dt}\left(\rho
a^{2}\right).$
Integrating and dividing by $a^{2}$ we obtain
$\sqrt{\tilde{r}_{A}^{-2}+\frac{1}{\ell^{2}}}=-\frac{G_{n+1}}{4G_{n}}(n-2)\tilde{r}_{A}^{-2}+\frac{4\pi
G_{n+1}}{(n-1)}\rho.$ (29)
Substituting the apparent horizon radius from relation (3), we have
$\displaystyle\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell^{2}}}+\frac{G_{n+1}}{4G_{n}}(n-2)\left(H^{2}+\frac{k}{a^{2}}\right)$
(30) $\displaystyle=$ $\displaystyle\frac{4\pi G_{n+1}}{(n-1)}\rho.$
This equation is indeed the Friedmann equation of FRW Universe in warped DGP
braneworld derived in SheyW1 from the field equations. If we define, as
usual, the crossover length scale between the small and large distances in DGP
braneworld as Def
$r_{c}=\frac{G_{n+1}}{2G_{n}},$ (31)
then one can easily check that for $r_{c}\rightarrow\infty$, the standard
Friedmann equation in $n$-dimensional FRW Universe presented in (10) is
recovered. On the other hand, when $r_{c}\rightarrow 0$, Eq. (30) reduces to
the Friedmann equation in RS II braneworld obtained in the previous section.
## IV Emergence of spacetime dynamics in Gauss-Bonnet braneworld
Finally, we apply the method developed in the previous sections to investigate
the emergence properties of the spacetime dynamics in general braneworld with
curvature correction terms including a 4D scalar curvature from induced
gravity on the brane, and a 5D Gauss-Bonnet curvature term in the bulk. With
these correction terms, especially including a Gauss-Bonnet correction to the
5D action, we have the most general action with second-order field equations
in 5D lovelock , which provides the most general models for the braneworld
scenarios. The entropy of apparent horizon in general Gauss-Bonnet braneworld
embedded in a 5D bulk, can be written as SheyW2
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{3\Omega_{3}{\tilde{r}_{A}}^{2}}{4G_{4}}+\frac{2\Omega_{3}{\tilde{r}_{A}}^{3}}{4G_{5}}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},\Phi_{0}{\tilde{r}_{A}}^{2}\right)$
(32)
$\displaystyle+\frac{6{\alpha}\Omega_{3}{\tilde{r}_{A}}^{3}}{G_{5}}\left(\Phi_{0}\times{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},\Phi_{0}{\tilde{r}_{A}}^{2}\right)\right.$
$\displaystyle\left.+\frac{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}{{\tilde{r}_{A}}^{2}}\right),$
where
$\Phi_{0}=\frac{1}{4{\alpha}}\left(-1+\sqrt{1-\frac{8{\alpha}}{\ell^{2}}}\right)$=constant
SheyW2 , and ${\alpha}$ is the Gauss-Bonnet coefficient with dimension
(length)2. When ${\alpha}\rightarrow 0$ we have $\Phi_{0}=-\ell^{-2}$ and the
above expression reduces to the entropy of warped DGP braneworld presented in
(23) for $n=4$. Expression (32) can be written as SheyW2
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{3\Omega_{3}}{2G_{4}}\int_{0}^{\tilde{r}_{A}}\tilde{r}_{A}d\tilde{r}_{A}+\frac{3\Omega_{3}}{2G_{5}}\int_{0}^{\tilde{r}_{A}}\frac{\tilde{r}_{A}^{2}}{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}d\tilde{r}_{A}$
(33)
$\displaystyle+\frac{6{\alpha}\Omega_{3}}{G_{5}}\int_{0}^{\tilde{r}_{A}}\frac{2-\Phi_{0}\tilde{r}_{A}^{2}}{\sqrt{1-\Phi_{0}\tilde{r}_{A}^{2}}}d\tilde{r}_{A},$
We define the effective area of the apparent horizon corresponding to the
above entropy as
$\displaystyle\tilde{A}=4G_{5}S$ $\displaystyle=$
$\displaystyle\frac{6G_{5}\Omega_{3}}{G_{4}}\int_{0}^{\tilde{r}_{A}}\tilde{r}_{A}d\tilde{r}_{A}+6\Omega_{3}\int_{0}^{\tilde{r}_{A}}\frac{\tilde{r}_{A}d\tilde{r}_{A}}{\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}}$
(34)
$\displaystyle+24{\alpha}\Omega_{3}\int_{0}^{\tilde{r}_{A}}\frac{2\tilde{r}_{A}^{-1}-\Phi_{0}\tilde{r}_{A}}{\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}}d\tilde{r}_{A},$
and therefore the increase of the effective volume is obtained as
$\displaystyle\frac{d\widetilde{V}}{dt}=\frac{\tilde{r}_{A}}{2}\frac{d\tilde{A}}{dt}$
$\displaystyle=$
$\displaystyle\frac{3G_{5}\Omega_{3}}{G_{4}}\tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}+3\Omega_{3}\frac{\tilde{r}_{A}^{2}\dot{\tilde{r}}_{A}}{\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}}$
(35)
$\displaystyle+12{\alpha}\Omega_{3}\frac{2-\Phi_{0}\tilde{r}_{A}^{2}}{\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}}\dot{\tilde{r}}_{A}.$
Motivated by (35), we write the number of degrees of freedom on the boundary
in general Gauss-Bonnet braneworld as
$\displaystyle N_{\mathrm{sur}}$ $\displaystyle=$
$\displaystyle\frac{3\Omega_{3}}{G_{4}}\tilde{r}_{A}^{2}+\frac{6\Omega_{3}}{G_{5}}\tilde{r}_{A}^{4}\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}$
(36)
$\displaystyle+\frac{16{\alpha}\Omega_{3}}{G_{5}}\tilde{r}_{A}^{4}\left(\tilde{r}_{A}^{-2}+\frac{\Phi_{0}}{2}\right)\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}.$
Substituting Eqs. (6), (35) and (36) into (11) and setting $n=4$, after some
mathematic simplification, one obtains
$\displaystyle\frac{3G_{5}}{G_{4}}\frac{d}{dt}\left(a^{2}\tilde{r}_{A}^{-2}\right)+6\frac{d}{dt}\left(a^{2}\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}\right)$
(37)
$\displaystyle+\frac{d}{dt}\Bigg{\\{}16{\alpha}a^{2}\left(\tilde{r}_{A}^{-2}+\frac{\Phi_{0}}{2}\right)\sqrt{\tilde{r}_{A}^{-2}-\Phi_{0}}\Bigg{\\}}$
$\displaystyle=$ $\displaystyle 8\pi G_{5}\frac{d}{dt}\left(\rho
a^{2}\right).$
Integrating, dividing by $a^{2}$ and then using the definition (3), we find
$\displaystyle\left[1+\frac{8}{3}{\alpha}\left(H^{2}+\frac{k}{a^{2}}-\frac{1}{2\ell^{2}}\right)\right]\sqrt{H^{2}+\frac{k}{a^{2}}+\frac{1}{\ell^{2}}}$
(38) $\displaystyle=$ $\displaystyle\frac{4\pi
G_{5}}{3}\rho-\frac{G_{5}}{2G_{4}}\left(H^{2}+\frac{k}{a^{2}}\right).$
This is the Friedmann equation governing the evolution of the Universe in
general Gauss-Bonnet braneworld with curvature correction terms on the brane
and in the bulk. This result is exactly the same as one obtains from the field
equation of Gauss-Bonnet braneworld kofin . Here we arrived at the same result
by using the novel proposal of Shey2 . When $\alpha=0$, the above result
reduces to the Friedmann equation of warped DGP model obtained in Eq. (30) for
$n=4$.
## V Summery and discussion
Recently, Padmanabhan Pad1 argued that the spacetime dynamics can be
considered as an emergent phenomena and the cosmic space is emergent as the
cosmic time progresses. An improved version of Padmanabhan proposal which is
applicable to a nonflat Universe was found by one of the present author Shey2
. In this paper, we extended the study to other gravity theory such as
braneworld scenarios. Gravity on the brane does not obey the Einstein theory
and therefore the usual area formula for the entropy does not hold on the
brane. We have discussed several cases including whether there is or not a
Gauss-Bonnet curvature correction term in the bulk and whether there is or not
an intrinsic curvature term on the brane. We found that one can always derive
the Friedmann equations of FRW Universe with any spacial curvature, by
calculating the difference between the horizon degrees of freedom and the bulk
degrees of freedom regardless of the existence of the intrinsic curvature term
on the brane and the Gauss-Bonnet correction term in the bulk.
The result obtained here in RS II, warped DGP and the general Gauss-Bonnet
braneworld scenarios further supports the novel idea of Padmanabhan (1) and
its extension as (11), and show that this approach is powerful enough to
extract the dynamical equations describing the evolution of the Universe in
other gravity theories with any spacial curvature.
###### Acknowledgements.
We thank from the Research Council of Shiraz University. This work has been
supported financially by Research Institute for Astronomy & Astrophysics of
Maragha (RIAAM), Iran.
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|
arxiv-papers
| 2013-08-12T17:39:58 |
2024-09-04T02:49:49.367291
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi, M. H. Dehghani and S. E. Hosseini",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1308.2668"
}
|
1308.2924
|
# An apparatus for studying spallation neutrons in the Aberdeen Tunnel
laboratory
S. C. Blyth Y. L. Chan X. C. Chen M. C. Chu R. L. Hahn T. H. Ho Y. B.
Hsiung B. Z. Hu K. K. Kwan M. W. Kwok T. Kwok [email protected] Y. P. Lau K.
P. Lee J. K. C. Leung K. Y. Leung G. L. Lin Y. C. Lin K. B. Luk W. H.
Luk H. Y. Ngai S. Y. Ngan C. S. J. Pun K. Shih Y. H. Tam R. H. M. Tsang
C. H. Wang C. M. Wong H. L. Wong H. H. C. Wong K. K. Wong M. Yeh
Chemistry Department, Brookhaven National Laboratory, Upton, NY 11973, USA
Department of Electro-Optical Engineering, National United University, Miao-
Li, Taiwan Department of Physics, National Taiwan University, Taipei, Taiwan
Department of Physics, Chinese University of Hong Kong, Hong Kong, China
Department of Physics, University of Hong Kong, Hong Kong, China Department
of Physics, University of California at Berkeley, Berkeley, CA 94720, USA
Institute of Physics, National Chiao-Tung University, Hsinchu, Taiwan
###### Abstract
In this paper, we describe the design, construction and performance of an
apparatus installed in the Aberdeen Tunnel laboratory in Hong Kong for
studying spallation neutrons induced by cosmic-ray muons under a vertical rock
overburden of 611 meter water equivalent (m.w.e.). The apparatus comprises of
six horizontal layers of plastic-scintillator hodoscopes for determining the
direction and position of the incident cosmic-ray muons. Sandwiched between
the hodoscope planes is a neutron detector filled with 650 kg of liquid
scintillator doped with about 0.06% of Gadolinium by weight for improving the
efficiency of detecting the spallation neutrons. Performance of the apparatus
is also presented.
###### keywords:
Aberdeen Tunnel , Hong Kong , underground , cosmic-ray muon , neutron
###### PACS:
25.30.Mr , 29.40.Mc , 29.40.Vj , 95.55.Vj , 96.40.Tv
††journal: Nuclear Instruments and Methods A
## 1 Introduction
Neutrons are an important background for underground experiments studying
neutrino oscillation, dark matter, neutrinoless double beta decay and the
like. Majority of the neutron background is created by the ($\alpha$, $n$)
interaction, where the $\alpha$ particles come from the decays of
radioisotopes in the vicinity.
Neutrons can also be created by interactions of cosmic-ray muons with matter
in the underground laboratories. Contrary to neutrons coming from ($\alpha$,
$n$), these spallation neutrons have a very broad energy distribution that
extends to GeV. They can travel a long distance from the production vertices,
and penetrate into the detector without being vetoed. As a result, they may be
captured after thermalization, or interact with the detector materials to
create fake signals.
Since the spallation process is very complicated, it is highly desirable to
investigate experimentally the production properties of the muon-induced
spallation neutrons in underground environments. Such studies have been
carried out in several underground experiments at different depths, ranging
from 20 m.w.e. to 5,200 m.w.e. [1]-[6]. The lack of experimental information
also compromises the validity of simulation on spallation neutron.
The goal of the Aberdeen Tunnel experiment in Hong Kong is to study the
production of spallation neutrons by cosmic-ray muons at a vertical depth of
235 m of rock (611 m.w.e.) using a tracking detector for tagging the incoming
muons and a neutron detector filled with a liquid scintillator loaded with
Gadolinium (Gd) for detecting the spallation neutrons. In this paper, we
present the details of the Aberdeen Tunnel experiment in Section 2. We will
describe the design and construction features of the muon tracker and the
neutron detector. In Section 3, calibration of the hodoscopes on the muon
tracker and PMTs in the neutron detector will be discussed. The energy scale
calibration of the neutron detector will be described. Data acquisition and
trigger formation of the detectors will be presented. In Section 4, details of
simulations and detector performance will be given.
## 2 The Aberdeen Tunnel Laboratory
### 2.1 Geological information
The underground laboratory (at 22.23∘N and 114.6∘E) was constructed inside the
Aberdeen Tunnel in Hong Kong Island in the early 1980s. The tunnel is a two-
tube vehicle tunnel of 1.9-km long. It lies beneath the saddle-shape valley
between two hills of over 400 m (Fig. 1), namely Mount Cameron on the west and
Mount Nicholson on the east. The saddle-shape terrain provides a rock
overburden of 611 m.w.e. for the laboratory which is located at the mid-point
of the tunnel. It has dimensions of 6.7 m (L) $\times$ 3.2 m (W) $\times$ 2.2
m (H). The entrance is in a cross passage connecting the two traffic tubes.
Access of the laboratory is only possible when all the traffic are diverted to
one of the tubes during tunnel maintenance from 00:00 hr to 05:00 hr,
typically two to four times a week.
Figure 1: Contour map of the hills above the Aberdeen Tunnel underground
laboratory denoted by a white dot at (0, 0). North is up. Figure 2: Geology
in the vicinity of the Aberdeen Tunnel. Major types of rock are granite (gm,
gfm), fine ash vitric tuff (JAC) and some debris flow deposits (Qd) [8]. The
dotted lines show the location of the Aberdeen Tunnel.
The major types of rock covering the Aberdeen Tunnel are granite and vitric
tuff (Fig. 2). In order to find out their chemical composition, rock samples
were collected on the surface near the Aberdeen Tunnel for analysis. These
rock samples were picked up at eight locations on the hiking trails in the
area, as indicated in Fig. 2. Samples of suitable size (from locations 2, 3, 7
and 8) were analyzed by X-ray fluorescence spectroscopy (XRF) at the
University of Hong Kong. The chemical compositions are tabulated in Table 1.
Determination of their physical properties was done at the Lawrence Berkeley
National Laboratory [7]. The results are shown in Table 2. Since these rock
samples have been weathered, their physical properties may not reflect truly
those of the rock inside the underground laboratory.
Oxides | Composition (%) | Error (%)
---|---|---
Silicon | 76.8 | 6.7
Aluminum | 12.3 | 3.5
Iron | 1.2 | 0.3
Sodium | 1.4 | 1.3
Potassium | 3.7 | 2.0
Table 1: Predominant composition of rock samples collected on the surface near the Aberdeen Tunnel. Location | Bulk | Grain | P-wave | S-wave | Young’s | Shear | Porosity
---|---|---|---|---|---|---|---
| density | density | velocity | velocity | modulus | modulus | (%)
| (g cm-3) | (g cm-3) | (km/s) | (km/s) | (GPa) | (GPa) |
2 | 2.38 | 2.61 | 3.27 | 2.11 | 24.4 | 10.7 | 8.86
3 | 2.57 | 2.62 | 5.56 | 3.29 | 68.5 | 27.8 | 1.86
7 | 2.36 | 2.61 | 3.24 | 2.00 | 22.7 | 9.50 | 9.55
8 | 2.47 | 2.59 | 4.28 | 2.68 | 41.9 | 17.8 | 4.58
Table 2: Physical properties of rock samples collected on the surface at
various locations in Fig. 2 near the Aberdeen Tunnel [7].
Using a modified Gaisser’s parametrization [9] for generating the energy of
the cosmic-ray muons on the surface, a digitized three-dimensional
topographical map with a resolution of 10 m and area of 13.6 km by 10.2 km,
and MUSIC [10] for propagating the muons to the location of the underground
laboratory, the mean energy of the muons getting to the laboratory is
estimated to be 120 GeV and the integrated flux is approximately $9.6\times
10^{-6}$ cm-2 s-1.
### 2.2 Laboratory environment
Humidity and temperature of the laboratory are monitored by sensors and the
data can be accessed remotely through the internet. Ambient temperature in the
laboratory is kept between 20∘C and 30∘C with a typical value of about 23∘C,
relative humidity between 35% and 40% throughout the year. A surveillance
camera is installed for monitoring the environment of the underground
laboratory remotely.
The walls and floor are lined with cement and protective paint to reduce dust
and radon emanation. The mean radon concentration in the laboratory111Measured
with a high-sensitivity radon detector developed by the University of Hong
Kong. is 299 $\pm$ 20 Bq m-3, with a range of 250 to 325 Bq m-3. The neutron
ambient dose equivalent [11] is measured to be 0.7 $\pm$ 0.04 nSv/h with a
Helium-3 detector in a polyethylene sphere of 25-cm diameter manufactured by
Berthold Technologies GmbH & Co. KG (LB6411) [12].
Gamma-rays from primordial Uranium (U), Thorium (Th) and Potassium-40 (40K) in
the surrounding rock are the major sources of background in the Aberdeen
Tunnel laboratory. The amount of ambient gamma-rays was measured in situ for
20 hours with an Ortec GEM 35S high-purity germanium detector (HPGe) that has
a cylindrical Ge crystal of 61.5 mm in diameter and 65.9 mm in height. The
measured energy spectrum is shown in Fig. 3.
Figure 3: Energy spectrum of the gamma-ray background measured with a high-
purity germanium detector inside the Aberdeen Tunnel laboratory.
Activities of 40K and radioisotopes of U and Th series in rock were calculated
by simulation. Gamma-rays were generated uniformly from the surrounding rock,
then propagated to the Ge crystal. Attenuation of the gamma-rays was
calculated from the mass attenuation coefficient of the standard rock. By
normalizing the simulation spectrum to match the experimental results, the
simulated gamma-ray events were converted to the corresponding activities of
the isotopes. At the surface of the rock, the flux of gamma-rays with energy
up to 3 MeV was estimated to be 29 $\pm$ 1 cm-2 s-1. With the approximation of
secular equilibrium, activities of 238U and 232Th in the rock samples were
calculated using the gamma-rays and their branching ratios in the same series.
For 40K, its activity was deduced from the intensity of the 1.46-MeV gamma-ray
line in the spectrum shown in Fig. 3. The results are summarized in Table 3.
Isotope | Activity (Bq/kg)
---|---
238U | 85$\pm$2
232Th | 108$\pm$3
40K | 1007$\pm$60
Table 3: Activity of 238U, 232Th and 40K in the rocks surrounding the Aberdeen
Tunnel laboratory.
## 3 Apparatus
The incoming cosmic-ray muons and the spallation neutrons are measured with
two different detectors. A muon tracker (MT) determines the angular
distribution and flux of the muons, while the neutron detector (ND) observes
the spallation neutrons. Augmented with custom-built and commercially
available electronics modules, a MIDAS-based data acquisition system is used
to collect data. Details of these subsystems are presented in this section.
### 3.1 Muon tracker
The muon tracker (MT) consists of 60 plastic scintillator hodoscopes arranged
in three layers as shown in Figs. 4 and 5. The separation between the top and
bottom layer is 198 cm. Each layer is made up of two planes of hodoscopes
orthogonal to each other for determining the (x,y) coordinates of a muon
passage through the layer.
Figure 4: Front view of the MT and the ND. Right-hand side is east. Figure 5:
Side view of the MT and the ND. Right-hand side is north.
#### 3.1.1 Muon tracker frame
The plastic scintillator hodoscopes are supported by a steel frame. This frame
consists of an upper support structure and a base platform. The upper
structure can be slid on two parallel rails, each of which is 372-cm-long
running in the north-south direction. The slidable structure supports two
layers of hodoscope above the ND. When the upper frame is in the south-most
position, the entire ND on the base platform is sandwiched by the three
hodoscope layers. The configuration of the MT can be changed for measuring
muons at larger zenith angles. This also facilitates the installation and
calibration of the ND.
#### 3.1.2 Top hodoscope layer
The top layer is formed by ten 1-m-long hodoscopes and ten 2-m-long
hodoscopes, as illustrated in Figs. 4 and 5. For the 1-m-long hodoscopes, each
plastic scintillator has dimensions of 100 cm (L) $\times$ 10 cm (W) $\times$
2.54 cm (T). A 5-cm-diameter PMT (Amperex XP2230) is attached to a Lucite
light guide at one end. The whole array forms a sensitive area of (100
$\times$ 100) cm2. Underneath the 1-m-long hodoscope layer lies ten 2-m-long
hodoscopes, forming an active area of (93 $\times$ 200) cm2. Each hodoscope is
made of a piece of 200 cm (L) $\times$ 9.3 cm (W) $\times$ 2.54 cm (T) plastic
scintillator. Two Hamamatsu H7826 photomultiplier tubes (SPMTs), each with a
circular photocathode of 1.9 cm in diameter, are coupled to the two ends of
each plastic scintillator with tapered Lucite light guides.
#### 3.1.3 Middle and bottom hodoscope layers
The middle and bottom layers have identical configuration. In each layer, ten
1-m-long plastic scintillators lie in the east-west direction, each has a 5-cm
Amperex XP2230 PMTs at the eastern end. The whole plane has a sensitive area
of (100 $\times$ 100) cm2. Underneath this hodoscope plane are ten 1.5-m-long
hodoscopes, lying orthogonally to the 1-m-long ones, with Amperex XP2230 PMTs
in the north. Dimensions of the 1.5-m-long scintillator are 150 cm (L)
$\times$ 10 cm (W) $\times$ 2.54 cm (T). They form a sensitive area of (100
$\times$ 150) cm2.
### 3.2 Neutron detector
In the space sandwiched by the hodoscope layers, a neutron detector (ND) is
set up for detecting muon-induced neutrons. The detector employs a 2-zone
design (Fig. 6). In the inner zone, an acrylic vessel is filled with 760 L
(650 kg) of liquid scintillator as the target. The liquid scintillator is
loaded with 0.06% of gadolinium (Gd) to enhance neutron-capture. The acrylic
vessel full of Gd-doped liquid scintillator (Gd-LS) is submerged in 1,900 L
(1,630 kg) of mineral oil that serves as the outer shield for suppressing the
amount of ambient gamma-rays and thermal neutrons entering the Gd-LS. The
mineral oil and Gd-LS are sealed in a rectangular stainless steel tank,
keeping them in a light-tight condition and free from oxygen in the
atmosphere.
Figure 6: Schematic drawing of the ND. The cylindrical acrylic vessel and the
stainless steel rectangular tank are shown. Sixteen 20-cm PMTs are located at
the four corners of the stainless steel tank. Top and bottom reflectors are
not shown.
When gadolinium in Gd-LS captures a neutron, it produces a gamma cascade with
a total energy of about 8 MeV. Scintillation photons created by the gamma-rays
are collected with sixteen Hamamatsu R1408 20-cm PMTs in the ND. Reflectors
are installed at the top and at the bottom of the acrylic vessel for improving
the light-collection efficiency and to have a more uniform energy response.
#### 3.2.1 Acrylic vessel
The Gd-LS is held in a cylindrical acrylic vessel manufactured by Nakano
International Co., Ltd. in Taiwan [13]. UV-transmitting acrylic is chosen for
its compatibility with Gd-LS and higher transmittance in the UV-visible
region. For the curved surface of the vessel, which is 1-cm-thick, the optical
transmittance is above 86% for wavelength longer than 350 nm. This is crucial
for achieving high efficiency for detecting scintillation photons with the
Hamamatsu R1408 PMTs that have good quantum efficiency between 350 and 480 nm.
On the top surface of the acrylic vessel, three calibration ports are opened
for the deployment of calibration sources (Fig. 7). The calibration ports are
located at different radial distances from the center (0 cm, 25 cm and 45 cm).
There is also an overflow port leading to an overflow tank for accommodating
the thermal expansion of the Gd-LS in the vessel.
Figure 7: Acrylic vessel in the ND (unit in millimeters). Left: Side view of
the vessel along the calibration ports. Right: Top view showing the locations
of three calibration ports and one overflow port (46 cm from center).
Both the top and bottom plates of the acrylic vessel are 1.5 cm thick. As
shown in Fig. 7, each plate has eight 1-cm-wide ribs, extending radially for
structural reinforcement. The bottom ribs raise the Gd-LS such that mineral
oil can fill the 13.5-cm space below the acrylic vessel. The top ribs provide
support to the reflector above the acrylic vessel.
#### 3.2.2 Stainless steel tank
The stainless steel tank is responsible for holding all elements of the ND
intact and keeping them in a light- and air-tight environment. The inner
dimensions of the tank are 160 cm (L) $\times$ 160 cm (W) $\times$ 117.3 cm
(H). The interior of the ND is painted with a black fluoropolymer paint which
is compatible with the mineral oil.
On the top lid of the stainless steel tank, three flanges are installed for
the calibration ports of the acrylic vessel. There are gate valves on the
flanges such that the port can be opened for deployment of calibration
sources.
At the four corners of the stainless steel top lid, there are patch panels for
deploying the 20-cm PMTs. The patch panels have hermetic feedthroughs for
connecting the signal cables and high-voltage cables of the ND PMTs.
The relative position of the acrylic vessel and the stainless steel tank is
fixed by an anchor welded to the bottom of the stainless steel tank. This
hemispherical anchor has a size and shape matching the hollow space at the
center of the bottom ribs of the acrylic vessel. Once the acrylic vessel is
placed in the stainless steel tank, it couples to the anchor. Only a small
rotation of the acrylic vessel is possible afterwards.
### 3.3 Gadolinium-doped liquid scintillator
The Gd-LS based on the recipe described in [14] was synthesized in Hong Kong.
To increase the scintillation efficiency and to shift the emission spectrum to
the sensitive region of the Hamamatsu R1408 PMTs, 1.3 g/L of 2,5
diphenyloxazole (PPO) and 6.7 mg/L of $p$-bis-($o$-methylstyryl)-benzene (bis-
MSB) were added to the Gd-LS as primary and secondary fluors. The major
solvent is linear alkylbenzene (LAB), which is sold under the commercial name
of Petrelab 550-Q, by Petresa, Canada [15]. LAB is chemically less active and
has a higher flash point than the other liquid scintillators like
pseudocumene, while the emission spectrum and light yield are comparable [16].
The molecule of Petrelab 550-Q has a benzene ring attached to an alkyl
derivative that contains 10 to 13 carbon atoms. The carbon and hydrogen atoms
can also capture thermal neutrons with cross-sections of 0.00337 barns and
0.332 barns respectively [14]. The capture time of neutron on hydrogen is
about 200 $\mu$s. The energy of the gamma-ray emitted in the subsequent
nuclear de-excitation is 4.95 MeV for carbon and 2.22 MeV for hydrogen. Doping
Gd in liquid scintillator enhances neutron capture significantly because two
of the isotopes of Gd have much larger thermal neutron-capture cross-section
of 60,900 barns (155Gd) and 254,000 barns (157Gd) at 0.0253 eV [17]. With
0.06% of Gd, by weight, added to the liquid scintillator, the neutron capture
time is shortened to about 50 $\mu$s. Moreover, the total energy of the
emitted gamma-rays is about 8 MeV. This provides a powerful criterion for
discriminating the neutron-capture signals against the ambient gamma-rays that
have energies below 2.6 MeV.
#### 3.3.1 Photomultiplier tubes
Sixteen Hamamatsu R1408 20-cm PMTs are used in the ND. R1408 is the
predecessor of the newer model R5912. The R1408 PMT has a hemispherical
photocathode. The high-voltage divider in the PMT base was designed and made
by the MACRO collaboration. Stainless steel PMT mounts are used to hold the
PMTs in place and seal the PMT base from the mineral oil. In addition, the PMT
mounts have wheels for deploying the PMTs into the mineral oil by sliding
along the vertical rails below the patch panels of the stainless steel tank
(Fig. 8). The alignment of the PMT rails was adjusted with a laser pointer. To
reduce the amount of light scattering, the PMT mounts and rails are all coated
with black fluoropolymer paint. The PMTs were installed in four columns at the
corners of the stainless steel tank. All the PMTs are equally spaced, and
point to the vertical central axis of the acrylic vessel at the same radial
distance.
Figure 8: Drawings of PMT rails and PMT mounts inside the stainless steel tank
(unit in millimeters).
#### 3.3.2 Reflectors
Four panels of white diffuse reflectors are mounted to the inner walls of the
stainless steel tank. They are made of DuPont Tyvek 1085D mounted on 8-mm-
thick acrylic plates. DuPont Tyvek 1085D is selected for its appropriate
thickness, reflectivity and availability [18]. The four corners of the Tyvek
panels are hung on the PMT rails (Fig. 9). Two circular reflectors of 140-cm-
diameter are put on the top and at the bottom of the ribs of the acrylic
vessel. They increase light collection of the PMTs by specular reflection. The
size of the reflectors is maximized such that they would not hinder the
deployment of the PMTs through the patch panels. The circular reflectors were
made by gluing reflective Miro-Silver pieces [19] to an acrylic backing plate
with DP810 glue from 3M. The Miro-Silver is a 0.2-mm-thick aluminum sheet
coated with super reflective oxide-layer which has a reflectivity of over 96%
$\pm$ 1% in air and 95% $\pm$ 1% in mineral oil at a wavelength of 532 nm.
Figure 9: Reflectors inside the neutron detector. Tyvek sheets are mounted on
the walls in the stainless steel tank. The specular bottom reflector is under
the acrylic vessel. The top specular reflector is not shown here. Figure 10:
Gd-LS overflow tank of the ND (unit in millimeters).
#### 3.3.3 The overflow tank
A 15-L overflow tank is connected to the acrylic vessel (Fig. 10). As the
thermal expansion of Gd-LS in the ND equals 0.59 L K-1 [16], this reservoir is
able to hold the extra volume of Gd-LS for a temperature rise of 25∘C.
The overflow tank is made of stainless steel lined with an inner acrylic
layer. Air inside the tank is displaced with a slow flow of nitrogen gas.
Liquid level in the overflow tank is revealed by a hollow acrylic indicator
floating in the Gd-LS inside the overflow tube. An observing window can be
opened to check the position of the indicator. An EVOH bag is connected to the
overflow tank as a pressure relief for the expanding Gd-LS and nitrogen gas
when the temperature rises.
To take care of the thermal expansion of the mineral oil, a gap below the
stainless steel top lid filled with nitrogen gas serves as a buffer. The gas
gap is 23-mm-tall, yielding a volume of more than 50 L. This can hold the
additional volume of mineral oil for a temperature rise of 25∘C.
### 3.4 Calibration system
A deployment box is used to place calibration sources at different positions
through the three ports in the ND. The box has a flange on its bottom surface,
which fits to the gate valves of the calibration ports. Inside the box, the
source holder and all the wetted parts are made of acrylic or covered by
materials compatible with Gd-LS to prevent etching.
Each of the radioactive source holder and LED light sources are attached to a
thin wire wound around an acrylic winch. The wire is protected by
fluoropolymer heat-shrink tubing. The winch is engaged to a stepping motor
that controls the vertical position of the calibration source. A separate
winch is made for the radioactive sources and each of the light sources.
Sources can be swapped easily by replacing the whole winch assembly. An
acrylic holder (Fig. 11) is used to seal and to keep the radioactive source
away from the Gd-LS. The screw-cap can be opened to replace the radioactive
source inside.
Radioactive sources are used for calibrating the energy scale of the ND.
Cobalt-60 (60Co) providing 1.17-MeV and 1.33-MeV gamma-rays, Cesium-137
(137Cs) emitting 0.66-MeV gamma-rays, and Americium-Beryllium (Am-Be) neutron
source are used. The construction of the Am-Be source is shown in Fig. 12.
Prominent ambient gamma-rays (1.46 MeV $\gamma$ from Potassium-40, and 2.61
MeV $\gamma$ from Thallium-208) are also used for real-time calibration.
LED sources are used for checking the linear response of the ND. Two kinds of
light sources are available: an isotropic source and a planar source. For the
isotropic source, a Teflon ball is used as a diffuser for the two LEDs
embedded inside. The calibration system sends triggers at different time to
the Sheffield pulser [20] of each LED to control the flashing sequence. The
bias voltage of the LED is also adjustable for producing different light
intensities. The planar source employs the same electronics as the isotropic
source. The Teflon diffuser of this source is partly hidden behind a
horizontal slit of adjustable size such that a horizontal plane of light is
generated. It is used to cross-check the relative gain of the PMTs in the same
horizontal plane.
The deployment box is connected to a computer. Motion of the stepping motor,
the bias voltage and pulsing frequency of each LED are controlled through the
computer. An infrared camera and infrared LEDs are also installed in the
light-tight deployment box to view the passage of the sources through the
narrow calibration port. All infrared LEDs are switched off during data
taking.
The deployment box is sealed with o-rings and draw-latches to prevent oxygen
in air from entering the ND. Moreover, a purging system is connected to it.
Nitrogen is used to purge the box after it is opened for changing the source.
After purging with nitrogen of three volumes of the box, the gate valve is
then opened for deploying the source.
Figure 11: Acrylic holder for deploying radioactive calibration sources into
Gd-LS (unit in millimeters). Figure 12: Geometry of the Am-Be source (unit in
millimeters). Shaded region is the active region, encapsulated in a plastic
shell, and an outer Molybdenum shell.
### 3.5 Data acquisition system
The DAQ setup is shown in Fig. 13. The Front-End Electronics (FEE) is used to
process the PMT output signals. The essential functions of FEE are as follows:
* 1.
Provide fast information to the trigger system
* 2.
Provide precision timing information of each trigger to correlate events
* 3.
Provide hit information of each hodoscope to determine the trajectory of
incoming muons
* 4.
Provide the charge information of each PMT output signal to determine the
energy deposited inside the ND
Figure 13: Block diagram of the data acquisition system.
The FEE for the MT consists of ten custom-designed 6U VME eight-channel
discriminator boards, and a coincidence and pattern register module. Each
discriminator board employs a modular design, with a mother board housing four
daughter boards. The mother board provides power filtering, input and output
connectors, and four 10-bit digital-to-analog converters (DACs) for setting
the threshold and output pulse width remotely. Each daughter board has two
input channels (Fig. 14), each of which has its own amplifier, comparator
(Fig. 15) and a monostable circuit (Fig. 16) at the last stage. A typical
value of the threshold is -35 mV. An ECL pulse of 100-ns wide is generated
when the threshold is crossed.
Figure 14: Schematic diagram of the muon tracker front-end daughter board
(overview). Figure 15: Schematic diagrams of the amplifier (upper) and
comparator (lower) on the muon tracker front-end daughter board. Figure 16:
Schematic diagram of the monostable circuit on the muon tracker front-end
daughter board.
The coincidence and pattern register for the MT are processed by using a CAEN
VME V1495 module with on-board Field-Programmable Gate Array (FPGA) running at
a clock of 100 MHz. This module receives the ECL signals from every MT FEE and
processes the signals according to a preset trigger condition. Channels with
signals can be masked individually at the input. The hit pattern of the MT is
constructed inside the FPGA from the ECL signals and a preset mapping. If the
hit pattern satisfies the trigger condition, that event is then latched and
passed to an event builder inside the FPGA. The event builder adds a header to
the event. This serves as a unique identifier for the start of the event and a
redundant trigger state for cross-checking. Up to 500 events can be stored in
the FIFO implemented in the FPGA.
For the ND, each PMT signal is duplicated with a CAEN V925 linear fan-in/fan-
out module. One copy goes directly into a 12-bit QDC (CAEN V792N) for charge
measurement. The other two copies go into a CAEN V895 leading-edge
discriminator and an analog energy-sum (ESUM) module (Fig. 17) respectively.
The discriminator threshold of each channel is set in 1-mV-steps via the CAEN
V1718 VME interface. An N-hit trigger will be generated if the signals
received by the majority of PMTs exceed the threshold. The output of the ESUM
module goes into the discriminator channel of the CAEN V925 that determines
the energy threshold of the ESUM trigger. The logic signals of the N-hit,
ESUM, and LED trigger from the ND calibration system are input to the CAEN
V1495 module which serves as the Master Trigger Board (MTB) for the final
trigger decision. A CAEN A395D I/O mezzanine board for handling the input of
the logic signals is mounted on the MTB.
The FPGA firmware of the MTB includes a trigger forming logic for the ND. An
optional periodic trigger of the ND can be generated inside the MTB to monitor
the pedestals and the background. The MTB also accepts the busy signals from
the DAQ sub-systems of the MT and the ND respectively. Busy signals are
generated during event building, charge conversion or when the event buffer is
full. Thus a busy signal represents the presence of an accepted trigger. The
MTB time-stamps the falling and rising edges of the busy signals with 10-ns
resolution and records the corresponding event type (MT or ND). Events are
then correlated with the time-stamps in the off-line analysis. Width of the
busy signal is used to determine the dead-time of the DAQ.
Figure 17: Schematic diagram of the energy-sum trigger module.
#### 3.5.1 Triggers
The passage of an energetic charged particle is identified by the temporal
coincidence of signals from different hodoscope planes. It is flexible to form
MT triggers with various coincidence combinations of the top (T), middle (M)
and bottom (B) hodoscope layers. The 2-out-of-3 coincidence $H_{2/3}$ of
either the x- or y-oriented hodoscope planes is defined as:
$H_{2/3}=T\bullet M+T\bullet B+M\bullet B$ (1)
where the outputs of the ten hodoscopes in each plane are logically OR-ed:
$\displaystyle T$ $\displaystyle=\sum_{i=0}^{9}T_{i}$ (2) $\displaystyle M$
$\displaystyle=\sum_{i=0}^{9}M_{i}$ (3) $\displaystyle B$
$\displaystyle=\sum_{i=0}^{9}B_{i}$ (4)
Similarly, the 3-out-of-3 coincidence $H_{3/3}$ for either the x- or y-plane
is defined as:
$H_{3/3}=T\bullet M\bullet B$ (5)
In order to reconstruct a muon track, at least two coordinates are required in
both x- and y-direction. Therefore, “2/3-x and 2/3-y” ($H_{2/3-X}\bullet
H_{2/3-Y}$) is required for muon track reconstructions. The trigger logic is
implemented inside the FPGA for the MT.
For the ND, a N-hit (multiplicity) trigger is used as the primary trigger
instead of an energy-sum (ESUM) trigger. The N-hit threshold can be set via
the VME bus from 1 to 16 out of the total 16 PMTs. Triggers of the ND are
accepted by the MTB only when the QDC is not busy, and when the buffers of the
QDC and the MTB are not full. These criteria ensure the number of events
registered by the QDC and the MTB to be identical. A trigger time window
following an event from the MT can be imposed on the ND to reduce background
events in the muon-induced neutron measurements. The trigger source (N-hit,
ESUM, LED, periodic, or any combinations of the above) of each event is
recorded in the MTB along with the time-stamp to facilitate event selections
in off-line analysis.
#### 3.5.2 Data acquisition
An open source data acquisition system called MIDAS (Maximum Integrated Data
Acquisition System) [21] is used as a skeleton of the DAQ firmwares. The
system consists of a library and several applications which can run under
major operating systems and can be ported easily to others.
The MIDAS library is written in the C programming language. The library
contains routines for buffer management, a message system, a history system
and an on-line database (ODB) [22]. The buffers between the producers and the
consumers are FIFOs. The data transfer rate between a producer and a consumer
over a standard 100BASE-TX Ethernet is on the order of 10 MB/s and can be
higher if both run on the same computer. The history system is used to store
slow control data and periodic events, which include event rate, high-voltage
values, environmental variables or any data fields defined in the ODB, and
produce time series plots. The on-line database is a central data storage for
all relevant experiment variables such as run status, run information, front-
end parameters, slow control variables, logging channel information and
calibration constants. The ODB can be viewed and changed locally by using
ODBEdit or remotely through a Web interface which is served by the MIDAS HTTP
server. The password-protected Web interface provides a status overview of the
experiment. Data taking can be controlled remotely through any internet
browsers.
The implementation of MIDAS has one front-end computer connected to two VME
crates, a CAEN SY1527LC high-voltage system, an environmental
temperature/humidity sensor, a detector temperature sensor, and the
calibration box for the ND. The VME crates and the environmental sensor are
connected to the computer via high-speed USB 2.0. The detector temperature
sensor and the calibration box are connected to the computer via RS232. The
front-end computer communicates with the SY1527LC power supply via TCP/IP.
Three MIDAS front-end programs are written and run on the same LINUX-based
computer simultaneously. They consist of two parts: a system part which is
linked to the MIDAS library for writing events into buffers and accessing ODB,
and a user part which actually performs experiment- and hardware-specific data
acquisition and control. The first one (trigger front-end) uses a polling
scheme to read event responses of the MT and ND. The second one (slow control
front-end) controls the high-voltage and measures temperature and humidity.
The third one (calibration front-end) controls the calibration system of ND.
The front-end computer connects to the front-end electronics via high-speed
USB 2.0. High-speed USB 2.0 hosts schedule transactions within microframes of
125 $\mu$s [23]. This makes an event-by-event polling of trigger events from
the actual hardware inefficient. To accomplish a high event throughput using
the polling scheme with USB 2.0, separate local buffers are maintained within
the trigger front-end program. These local buffers are FIFOs and are managed
by the user part of the program. During the polling cycle, if the local
buffers are empty, the entire content in the buffer of the MTB is copied to a
local buffer using one USB cycle. The trigger front-end program searches the
MTB local buffer and counts the number of MT- and ND-tagged events
respectively. Then the program copies events from the buffers of the
corresponding modules (V1495 FPGA for MT-tagged events and V792N QDC for ND-
tagged events) to their local buffers using one or two more USB cycles
according to the number of tagged events in the MTB. If the local buffers are
not empty during the polling cycle, a data-ready signal is issued and a single
event is read from the local buffers. Data are read from the hardwares again
when all the local buffers are empty.
The front-end computer connects to the back-end computer through a 100BASE-TX
Ethernet. The back-end computer running under LINUX stores data to disks,
performs on-line data analysis and hosts the MIDAS Web interface. Data files
are written to a 500 GB disk in the back-end computer. Data can be transferred
using the SCP protocol from the Aberdeen Tunnel laboratory to a 6-TB RAID 5
disk array in the University of Hong Kong for archive. To reduce the loading
on the limited network bandwidth, data are also transferred by using external
hard disk drives during access to the Aberdeen Tunnel laboratory.
In the Aberdeen Tunnel experiment three basic on-line run types are defined:
1. 1.
pedestal run;
2. 2.
calibration run;
3. 3.
physics run.
The pedestal runs measure and update the QDC pedestal values for all channels
of the ND. In a pedestal run, the N-hit and ESUM triggers of the ND and the
trigger of the MT are disabled. Only the periodic trigger of 500 Hz will be
sent to the ND to measure the QDC pedestal values. Pedestal subtraction and
software gain correction routines in the on-line analysis are disabled and the
raw pedestal values are filled to histograms. At the end of the pedestal run,
the histograms are automatically fitted with Gaussian functions to obtain the
mean pedestal values for all detector channels. The new pedestal values are
updated to the ODB.
The calibration runs acquire data for calculating the calibration constants of
the ND in off-line analysis. In a calibration run, the trigger of the MT is
disabled, and the N-hit trigger of the ND with N being set to 16 is enabled,
with a discriminator threshold of -11 mV for each channel. This threshold
enables the ND to be triggered by relatively low-energy gamma-rays such as the
0.66 MeV ones from 137Cs. A calibration run is often carried in two parts. The
first one is a background run with no source in the ND. The trigger rate of a
background run is typically about 5 kHz. After the background run, the
calibration source is deployed to the designated position for a data run. When
a calibration source is deployed at the ND center, the trigger rate is about
15 kHz for 60Co, and about 7 kHz for 137Cs.
During physics runs, the physics mode acquires events from the MT and ND. The
type of events (MT or ND) is tagged and is stored together with the event.
Data are calibrated on-line and are monitored through an on-line histogram
manipulation tool Roody [24]. Muon tracks and the visible energy deposited in
the ND are reconstructed. The reconstructed events can be visualized on-line
using AbtViz as described in Section 3.6. A typical trigger rate of the MT is
0.013 Hz.
#### 3.5.3 Data model
Events are stored in data files using the MIDAS binary format [25]. For this
experiment, besides the default event headers, several data banks are defined
in the trigger events for storing the data from the MT and ND; these includes
a raw ADC bank, a raw muon hit pattern bank, and a calibrated ADC bank. The
MIDAS logger compresses the data streams in the GNU-zipped format to reduce
the size of the data files by about 50%, while it takes about 20% more CPU
time [22]. The average size of a compressed event is 27 kB. The raw data files
are processed event-by-event in the MIDAS analyzer through user-specific
modules, with analyzer parameters and calibration constants stored in the ODB.
Then the calibrated data and reconstructed events are written into a ROOT file
with a TTree object container provided by a ROOT class. The data can be
accessed through more than thirty ROOT classes with the interactive C++
interpreter CINT embedded in ROOT. With the ROOT data files, the users can
analyze the raw data, calibrated data, reconstructed events, detector
modeling, data calibration, energy calibration, event reconstruction, and run
information.
### 3.6 Event visualization
Figure 18: GUI controls and OpenGL display of an event in AbtViz
The Aberdeen Event Visualization (AbtViz) is developed using the Event
Visualization Environment [26], a high-level visualization library using
ROOT’s [27] data-processing, GUI and OpenGL interfaces that emerged from the
development of the event display of the ALICE experiment at the Large Hadron
Collider. The program is written in the iPython [28] environment to take
advantage of the enhanced interactivity for visual debugging the simulation
and reconstruction algorithms.
The geometries are exported directly from a GEANT4 simulation [29] using the
Virtual Geometry Model [30] package with TGeo objects, which is accessible by
the ROOT-based visualization. As shown in Fig. 18, the MT hodoscopes are
represented as boxes, ND PMTs as cones, reconstructed muon tracks as lines,
and the reconstructed neutron vertex as a sphere. The detector responses are
visualized by highlighting the triggered hodoscopes and each individual PMT.
Their color is determined by mapping its ADC signal value to the RGBAPalette
class.
Event-data are stored as ROOT TObject. They are sent via pika 0.9.5 [31], a
pure-Python implementation of the AMQP 0-9-1 protocol [32], directly from the
Aberdeen Tunnel DAQ machine to a RabbitMQ [33] server situated at the Chinese
University of Hong Kong. By subscribing remote AbtViz to the message queue,
the event-data are received in real-time for on-line analysis.
## 4 Performance of Apparatus
### 4.1 Muon tracker
#### 4.1.1 Hodoscope efficiency
The efficiency of a hodoscope of the MT is determined by using muons acquired
with the MT. Only muons that go almost vertically through the MT are selected.
In these events, the overlapping area of all the fired hodoscopes forms a
rectangle conveniently defined by the widths of the plastic scintillators. In
the analysis, for a particular section $(i,j)$ of a hodoscope in plane $k$
amongst the $m$ planes, its efficiency is determined by the number of 6-fold
coincidences ($N_{i,j;6-fold}$) and 5-fold coincidences that exclude the
hodoscope of interest ($N_{i,j;m\neq k}$) as
$\frac{N_{i,j;m\neq k}}{N_{i,j;6-fold}}=\frac{\prod_{m=1;m\neq
k}^{6}\epsilon(i,j;m)R_{\mu}t}{\prod_{m=1}^{6}\epsilon(i,j;m)R_{\mu}t}=\epsilon(i,j;k)$
(6)
where $R_{\mu}$ is the muon rate and $t$ is the measurement time. Following
the configuration shown in Fig. 4, the hodoscopes aligned in the east-west
direction are divided into $j$ sections, those running along north-south are
partitioned into $i$ sections. The efficiency of each section is obtained with
Eq. 6. The average efficiency of the hodoscopes on the top or in the middle
layer is 95% $\pm$ 4%. For the bottom layer, the efficiency of the hodoscopes
was determined by sandwiching each of them with two reference hodoscopes. The
efficiency of the individual hodoscope in the bottom layer is about 96% $\pm$
4%, with an exception of four 1.5-m-long hodoscopes of which the efficiency
does not depend on the hit position along the length of the plastic
scintillator. Fig. 19 shows the efficiency of the whole MT after integrating
over the solid angle subtended by an area covered by the 2-out-of-3
coincidence of the hodoscopes running in the east-west direction and the
2-out-of-3 coincidence of the hodoscopes along north-south ($H_{2/3-X}\bullet
H_{2/3-Y}$).
Figure 19: Efficiency of the MT plastic-scintillator hodoscopes as a function
of the zenith and azimuthal angles.
### 4.2 Neutron detector
#### 4.2.1 Gamma-ray background in the ND
GEANT4 toolkit [29] is used to estimate the gamma-ray background coming from
rock and detector materials. In the simulation, gamma-rays from rock are
generated based on the results obtained with a HPGe detector as discussed in
Section 2.2. In addition, radioactivities of a sample of the steel frame and a
Hamamatsu R1408 PMT were measured with an Ortec GEM 35S HPGe detector placed
inside a 10-cm-thick low-background lead shield (Canberra 767). The results
are summarized in Table 4. Using detailed descriptions of the detector
geometry and components of the ND, gamma-rays from the steel frame of the MT,
the sixteen Hamamatsu R1408 PMTs, and the two retroreflectors for monitoring
the optical transmittance of the mineral oil are also simulated.
Isotope | R1408 | MT frame | Retroreflector
---|---|---|---
238U | 118$\pm$5 | 7$\pm$6 | 1.7$\pm$ 0.2
232Th | 36$\pm$9 | 13$\pm$10 | 1.1$\pm$0.3
40K | 2820$\pm$80 | 41$\pm$6 | 520$\pm$2
Unit weight | 0.68 kg | 1 kg | 1 kg
Table 4: Activity (Bq/kg) of 238U, 232Th and 40K in different detector
materials.
A time window of 50 $\mu$s is implemented in the code to simulate coincidence
of gamma-rays. Fig. 20 is the simulated energy spectrum of the coincident
gamma-rays that enter the ND. With a threshold of 5.3 MeV, the coincident
gamma-ray background can be suppressed to the level of 0.6 Hz, down by more
than 3 orders of magnitude. In fact, the observed background rate is only 0.24
Hz $\pm$ 0.04 Hz.
Figure 20: Simulated gamma-ray coincidence background in a time window of 50
$\mu$s for the ND in the Aberdeen Tunnel laboratory.
### 4.3 Energy calibration of the neutron detector
Fig. 21 shows the energy spectra before and after background subtraction
obtained with the 137Cs and 60Co sources. A typical energy distribution for
the Am-Be source is illustrated in Fig. 22. The peaks with energies less than
3 MeV are due to gamma-rays coming from natural radioactivity in the vicinity
of the ND whereas the peaks greater than 3 MeV are related to a sequence of
events leading to neutron capture on Gd.
In the off-line analysis, after subtracting the background obtained from the
background run, the peaks in the ADC spectra of the 137Cs and Am-Be sources
are fitted to Gaussian distributions. For the 60Co source, the distribution is
fitted with a Crystal Ball function [34] plus a Gaussian in the low-energy
side of the peak (Fig. 23). The Crystal Ball function is given by:
$f(x;\alpha,n,\bar{x},\sigma)=N\begin{cases}\rm{exp}(-\frac{(x-\bar{x})^{2}}{2\sigma^{2}}),&\text{if}\
\frac{x-\bar{x}}{\sigma}>-\alpha\\\
A(B-\frac{x-\bar{x}}{\sigma})^{-n}&\text{if}\
\frac{x-\bar{x}}{\sigma}\leq-\alpha\end{cases}$ (7)
where
$\displaystyle A$
$\displaystyle=\bigg{(}\frac{n}{\left|\alpha\right|}\bigg{)}^{n}\centerdot\rm{exp}\bigg{(}-\frac{\left|\alpha\right|^{2}}{2}\bigg{)}$
(8) $\displaystyle B$
$\displaystyle=\frac{n}{\left|\alpha\right|}-\left|\alpha\right|$ (9)
and $N$ is the normalization factor, $n$, $\alpha$, $\bar{x}$ and $\sigma$ are
the fitting parameters. The Gaussian component of the Crystal Ball function is
used to model the peak resulting from the two gamma-rays with similar energies
in the 60Co spectrum. The additional Gaussian is used to take care of the
contribution of one of the gamma-rays from 60Co when the other one cannot
deposit all of its energy in the fiducial volume of the ND. The average
calibration constants determined with the radioactive sources from April 2011
to November 2012 are tabulated in Table 5. The results are consistent with
each other within two standard deviations.
Figure 21: ADC distribution obtained with and without 60Co and 137Cs sources. The dark gray spectrum is generated by the 60Co and 137Cs positioned at the center of the Gd-LS. The light gray spectrum is due to background gamma-rays detected by the ND. The black histogram is the background-subtracted distribution of the calibration sources. The peak on the left corresponds to the 0.66-MeV gamma-ray of 137Cs. The peak on the right is the total energy deposited by the (1.17+1.33)-MeV gamma-rays of 60Co. Figure 22: ADC distribution obtained with and without the Am-Be source. The dark gray spectrum is for Am-Be at the center of Gd-LS volume. The light gray spectrum is the background gamma-rays seen by the detector. The black histogram is background-subtracted spectrum of the Am-Be source. The peak near channel 16,000 is the total energy of about 8 MeV released by the gamma-rays when a Gd nucleus captures a neutron. Figure 23: 60Co spectrum fit by a Crystal Ball function and a Gaussian. Source | Average energy scale
---|---
| (ADC counts/MeV)
137Cs | 2300 $\pm$ 34
60Co | 2306 $\pm$ 35
Am-Be neutron | 2243 $\pm$ 20
Table 5: Average energy calibration constants determined with calibration
sources located at the ND center.
### 4.4 Monitoring energy scale of the neutron detector
Energy calibration (Section 4.3) is performed regularly by deploying the
calibration sources (137Cs, 60Co, Am-Be) to the center of the ND. This serves
as a regular check of the operational stability of the apparatus that includes
the DAQ, optical quality of the liquids and PMT gain. The CAEN SY1527LC high-
voltage power supply mainframe and the CAEN A1535SN high-voltage supply
modules are found to be very sensitive to the dusty environment in the
Aberdeen Tunnel laboratory. In about four months of operation, as demonstrated
in Fig. 24, the high-voltage supplied to the PMTs can drop by as much as 16%
from the read-back values, resulting in a drop of PMT gain. Cleaning the high-
voltage system with dry compressed air or perform a factory calibration can
reduce the discrepancies.
In order to compensate for the fluctuation in the detector response, a
correction factor is introduced to each PMT based on its own output charge
distributions. This factor is taken to be a ratio of the endpoint positions of
the singles spectra of the same PMT. The average position of the endpoint
obtained in the first month since the installation of the high-voltage system
is taken as the reference. The fitting range used for determining the endpoint
is between 0.4 MeV and 1 MeV, in which the spectral distortion due to a change
in the trigger condition is not observed. Before determining the endpoints of
the spectra using an exponential function, the distributions in the fitting
range are normalized. The corrected energy scale of the ND as a function of
time is plotted in Fig. 25. Each PMT has its own correction factor. The
overall effect is that the corrected energy scale is about 1.26 times of the
value before correction. It is interesting to note that an increase in
temperature from about 22∘C to 40∘C in the underground laboratory for about
two months due to a failure of the air-conditioning unit did not degrade the
performance of the apparatus, in particular, the Gd-LS.
Figure 24: Charge distributions of PMT NW4 (north-west corner, ring 4) before
(left) and after (right) cleaning the HV module. The gray lines are the fitted
exponential functions for determining the endpoints. Figure 25: Corrected
calibration constant of the neutron detector for the 60Co source as a function
of time. The variation of the temperature in the underground laboratory is
also shown. The rise in temperature in August and September of 2011 was due to
a failure of the air-conditioning unit.
#### 4.4.1 Response of neutron detector to gamma-ray sources
GEANT4 allows implementation of the detector response, in particular the PMT
optical model from GLG4Sim. Response of the ND to the gamma-ray calibration
sources at different positions has been simulated. Gamma-rays of the
corresponding energies are generated isotropically inside the active volume of
the calibration source. The following electromagnetic processes related to
gamma-rays are included: ionization, bremsstrahlung, photoelectric effect,
fluorescence, Rayleigh scattering, Compton scattering and pair production.
Besides, attenuation length of the Gd-LS, and optical properties of the ND
such as the reflectivity of the top, bottom, and side reflectors is
considered. The charge resolution and gain of individual PMT are also
implemented in the simulation.
Fig. 26 shows the simulated and observed energy spectra for the 137Cs and 60Co
source placed at the center of the Gd-LS volume. There is a good agreement
between the experimental and simulated results. From simulation, the energy
resolution at the center of the ND at 0.66 MeV and 2.5 MeV are 13.5% and 9.3%,
in agreement with the experimental measurements of 14.3% $\pm$ 0.2% and 9.5%
$\pm$ 0.2% respectively.
Figure 26: Comparison of measured energy spectrum of 137Cs and 60Co with the
simulated spectrum obtained with GEANT4.
#### 4.4.2 Response of neutron detector to the Am-Be source
The Am-Be source (Fig. 12) used for calibration emits neutrons and gamma-rays
through the following channels:
$\begin{split}{}^{241}\rm{Am}\rightarrow\ ^{237}\rm{Np}+\alpha\\\ n_{b}:\
^{9}\rm{Be}+\alpha\rightarrow&\ {}^{9}\rm{Be}^{*}+\alpha^{\prime}\rightarrow
n+\ ^{8}\rm{Be}+\alpha^{\prime}\\\ n_{0}:\ ^{9}\rm{Be}+\alpha\rightarrow&\ n+\
^{12}\rm{C}\\\ n_{1}:\ ^{9}\rm{Be}+\alpha\rightarrow&\ n+\ ^{12}\rm{C}^{*}\
(4.4\ MeV\ \gamma)\\\ n_{2}:\ ^{9}\rm{Be}+\alpha\rightarrow&\ n+\
^{12}\rm{C}^{*}\ (4.4+3.2\ MeV\ \gamma s)\end{split}$ (10)
The energy spectrum obtained with the Am-Be source in the ND is a combination
of the energies deposited by the neutrons and gamma-rays through various
interactions, including thermalization and capture of neutron in the Gd-LS,
and energy released in the active volume by the gamma-rays from the decay of
12C∗ (for $n_{1}$ and $n_{2}$). In addition, the decay of 12C∗ has a short
half-life of femtoseconds. The gamma-rays from the 12C∗ de-excitation are
detected in coincidence with the signal due to neutron thermalization. Thus,
it is necessary to implement the energy- and time-correlation of neutrons and
associated gamma-rays of the Am-Be source in the simulation in order to
reproduce the observed energy spectrum.
The energy spectrum of the neutrons emitted by an Am-Be source, and the
neutron energy distribution of each channel in Eq. 10 have been reported in
Refs. [35][36]. These published results are used in the simulation. From the
picked neutron energy, the corresponding production channel is identified, and
the energy of the associated gamma-rays is generated accordingly.
The arrival times of the optical photons at the PMTs are recorded for
simulating realistic temporal correlation of events. The ADC sum of the
sixteen ND PMTs are plotted in Fig. 27. In the figure, the peak near channel
5,000 corresponds to the gamma-rays released from neutron capture on protons,
whereas the distribution peaked near channel 18,000 is due to the gamma-rays
generated from neutron capture on Gd nuclei in the Gd-LS. The broad
distribution with a peak around channel 12,000 is the sum of energy deposited
by the 4.4-MeV gamma-ray ($n_{1}$, $n_{2}$) and proton recoil during
thermalization of the energetic neutrons from the Am-Be source. Again, the
simulated energy distribution is consistent with the observed spectrum.
Figure 27: Comparison of the observed (solid) and simulated (dotted) ADC
spectra for the case with the Am-Be source deployed at the center of the
neutron detector. The peaks around the ADC channels 18,000, 12,000, and 5,000
correspond to energies of 8 MeV, 5.3 MeV, and 2.2 MeV respectively.
#### 4.4.3 Detection of spallation neutrons produced by cosmic-ray muons
To demonstrate the capability of the ND to observe spallation neutrons induced
by cosmic-ray muons, the detected energy (in units of ADC channel) of events
seen in the ND versus the first occurrence of the ND trigger after the last MT
trigger is shown in Fig. 28. The data depicted in the plot were collected in
128 days and without any event selection requirement imposed. Most of the
events between ADC channels 0 and 10,000 are accidental gamma rays that are
uncorrelated with the cosmic-ray muons. However, the events clustered around
ADC channel 16,000, corresponding to an energy of 8 MeV, are correlated with
the MT trigger between 0 $\mu$s and 200 $\mu$s. These are events due to
neutrons captured by Gd in the Gd-LS, hence providing the evidence that muon-
induced neutrons have been observed in the ND.
Figure 28: ND visible energy (in ADC channels) versus ND trigger time after
the last MT trigger. Neutron-capture events of muon-induced neutrons clustered
around ADC channel 16,000, from 0 $\mu$s to 200 $\mu$s.
## 5 Conclusion
We have successfully constructed a plastic-scintillator tracker and a neutron
detector for studying spallation neutrons produced by cosmic-ray muons in the
Aberdeen Tunnel laboratory in Hong Kong. The equipment has been in routine
operation for about a year. The average efficiency of the scintillator
hodoscopes is better than 95%. The energy response of the neutron detector
containing 650 kg of 0.06%-Gd-LS has been studied and monitored with gamma-
rays emitted by radioactive sources placed at the center of the detector. The
capability of the neutron detector in detecting low-energy neutrons has been
demonstrated with an Am-Be source. In general, the performance of the
apparatus is consistent with expectation based on comparisons with simulation.
## 6 Acknowledgement
This work is partially supported by grants from the Research Grant Council of
the Hong Kong Special Administrative Region, China (Project nos. HKU703307P,
HKU704007P, CUHK 1/07C and CUHK3/CRF/10), University Development Fund of The
University of Hong Kong, and the Office of Nuclear Physics, Office of High
Energy Physics, Office of Science, US Department of Energy under the Contract
no. DE-AC-02-05CH11231, as well as the National Science Council in Taiwan and
MOE program for Research of Excellence at National Taiwan University and
National Chiao-Tung University.
The authors would like to thank the Commissioner for Transport, The Government
of the Hong Kong Special Administrative Region, for providing the underground
facilities, and Serco Group plc, for their cooperation and support in the
Aberdeen Tunnel.
References
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|
arxiv-papers
| 2013-08-13T17:36:14 |
2024-09-04T02:49:49.381908
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. C. Blyth, Y. L. Chan, X. C. Chen, M. C. Chu, R. L. Hahn, T. H. Ho,\n Y. B. Hsiung, B. Z. Hu, K. K. Kwan, M. W. Kwok, T. Kwok, Y. P. Lau, K. P.\n Lee, J. K. C. Leung, K. Y. Leung, G. L. Lin, Y. C. Lin, K. B. Luk, W. H. Luk,\n H. Y. Ngai, S. Y. Ngan, C. S. J. Pun, K. Shih, Y. H. Tam, R. H. M. Tsang, C.\n H. Wang, C. M. Wong, H. L. Wong, H. H. C. Wong, K. K. Wong, M. Yeh",
"submitter": "Talent Kwok",
"url": "https://arxiv.org/abs/1308.2924"
}
|
1308.3048
|
# Reactivity Boundaries to Separate the Fate of a Chemical Reaction Associated
with Multiple Saddles
Yutaka Nagahata Graduate School of Life Science, Hokkaido University, Kita
12, Nishi 6,Kita-ku, Sapporo 060-0812, Japan Hiroshi Teramoto Graduate
School of Life Science, Hokkaido University, Kita 12, Nishi 6,Kita-ku, Sapporo
060-0812, Japan Molecule and Life Nonlinear Sciences Laboratory, Research
Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-
ku, Sapporo 001-0020, Japan Chun-Biu Li Molecule and Life Nonlinear Sciences
Laboratory, Research Institute for Electronic Science, Hokkaido University,
Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan Graduate School of
Science, Department of Mathematics, Hokkaido University, Kita 12, Nishi
6,Kita-ku, Sapporo 060-0812, Japan Research Center for Integrative
Mathematics, Hokkaido University, Kita 20, Nishi 10, Kita-Ku, Sapporo,
Hokkaido, 001-0020, Japan Shinnosuke Kawai Graduate School of Life Science,
Hokkaido University, Kita 12, Nishi 6,Kita-ku, Sapporo 060-0812, Japan
Molecule and Life Nonlinear Sciences Laboratory, Research Institute for
Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo
001-0020, Japan Tamiki Komatsuzaki [email protected] Graduate School
of Life Science, Hokkaido University, Kita 12, Nishi 6,Kita-ku, Sapporo
060-0812, Japan Molecule and Life Nonlinear Sciences Laboratory, Research
Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-
ku, Sapporo 001-0020, Japan Research Center for Integrative Mathematics,
Hokkaido University, Kita 20, Nishi 10, Kita-Ku, Sapporo, Hokkaido, 001-0020,
Japan
###### Abstract
Reactivity boundaries that divide the origin and destination of trajectories
are crucial of importance to reveal the mechanism of reactions, which was
recently found to exist robustly even at high energies for index-one saddles
[Phys. Rev. Lett. 105, 048304 (2010)]. Here we revisit the concept of the
reactivity boundary and propose a more general definition that can involve a
single reaction associated with a bottleneck made up of higher index saddles
and/or several saddle points with different indices, where the normal form
theory, based on expansion around a single stationary point, does not work. We
numerically demonstrate the reactivity boundary by using a reduced model
system of the $\mathrm{H}_{5}^{+}$ cation where the proton exchange reaction
takes place through a bottleneck made up of two index-two saddle points and
two index-one saddle points. The cross section of the reactivity boundary in
the reactant region of the phase space reveals which initial conditions are
effective in making the reaction happen, and thus sheds light on the reaction
mechanism.
###### pacs:
05.45.-a,34.10.+x,45.20.Jj,82.20.Db
## I Introduction
Studies of chemical reaction dynamics aims for understanding of how and why a
system proceeds from its initial state to the final state in the process of
reaction. Special interest lies in the question of what initial conditions
make the reaction happen. Classically, the process of chemical reaction can be
regarded as motion of a point in the phase space propagating from a region
corresponding to the reactant to another region corresponding to the product.
Some phase space points in the reactant region may go into the product region
after time propagation, whereas other phase space points stay in the reactant
region without undergoing the reaction. In between these reactive initial
conditions and non-reactive ones lies a boundary which we simply call here
reactivity boundary that was previously described by various words, such as
“boundary trajectories” Pechukas (1976); Pechukas and Pollak (1977); Sverdlik
and Koeppl (1978); Pollak and Pechukas (1978); Pechukas and Pollak (1979);
Pollak and Pechukas (1979); Child and Pollak (1980); Pollak _et al._ (1980);
Pollak and Levine (1980); Pollak and Child (1980); Pechukas (1981); Pollak
(1981a, b, c) asymptotic to periodic orbit dividing surface (pods) Child and
Pollak (1980); Pollak _et al._ (1980); Pollak and Levine (1980); Pollak and
Child (1980); Pechukas (1981); Pollak (1981a, b, c), “boundary of” Pechukas
(1976); Pechukas and Pollak (1977); Sverdlik and Koeppl (1978); Pollak and
Pechukas (1978); Pechukas and Pollak (1979); Pollak and Pechukas (1979); Child
and Pollak (1980); Pollak _et al._ (1980); Pollak and Levine (1980); Pollak
and Child (1980); Pechukas (1981); Pollak (1981a, b, c) reactivity bandsWall
_et al._ (1958, 1961); Wall and Porter (1963); Wright _et al._ (1975); Wright
(1976); Wright and Tan (1977); Laidler _et al._ (1977); Tan _et al._ (1977);
Wright (1978); Andrews and Chesnavich (1984); Grice _et al._ (1987), “tube”de
Almeida _et al._ (1990), “cylindrical manifold”de Almeida _et al._ (1990),
“impenetrable barriers”Wiggins _et al._ (2001), “stable/unstable manifold” of
normally hyperbolic invariant manifolds (NHIM)Wiggins _et al._ (2001),
“reaction boundaries”Kawai and Komatsuzaki (2010), and also described on
certain sections, such as “reactivity bands”Wall _et al._ (1958, 1961); Wall
and Porter (1963); Wright _et al._ (1975); Wright (1976); Wright and Tan
(1977); Laidler _et al._ (1977); Tan _et al._ (1977); Wright (1978); Andrews
and Chesnavich (1984); Grice _et al._ (1987), “reactivity map” Wright _et
al._ (1975); Wright (1976); Wright and Tan (1977); Laidler _et al._ (1977);
Tan _et al._ (1977); Wright (1978); Andrews and Chesnavich (1984); Grice _et
al._ (1987), “reactive island”de Almeida _et al._ (1990). The general
definition of the reactivity boundary is the main subject of this paper.
The reactivity boundary is often discussed in relation to saddle points. A
saddle point on a multi-dimensional potential energy surface is defined as a
stationary point at which the Hessian matrix does not have zero eigenvalues
and, at least, one of the eigenvalues is negative. Saddle points are
classified by the number of the negative eigenvalues and a saddle that has $n$
negative eigenvalues is called an index-$n$ saddle. Especially the index-one
saddle on a potential surface has long been considered to make bottleneck of
reactionsGlasstone _et al._ (1941); Steinfeld _et al._ (1989), with the sole
unstable direction corresponding to the “reaction coordinate.” This is because
index-one saddle is considered to be the lowest energy stationary point
connecting two potential minima, of which one corresponds to the reactant and
the other to the product, and the system must traverse the vicinity of the
index-one saddle from the reactant to the product Zhang _et al._ (2006);
Skodje _et al._ (2000); Shiu _et al._ (2004).
Such reactivity boundaries have been investigated from early period of the
study of reaction dynamics. Especially reactivity boundaries of atom-diatom
reactions were extensively studied by Wright et al.Wright _et al._ (1975);
Wright (1976); Wright and Tan (1977); Tan _et al._ (1977); Wright (1978);
Laidler _et al._ (1977) and Pechukas et al.Pollak and Levine (1980); Pollak
(1981c, a); Pechukas (1976); Sverdlik and Koeppl (1978); Pollak and Pechukas
(1979); Pollak and Child (1980); Child and Pollak (1980); Pollak and Pechukas
(1978); Pollak (1981b); Pollak _et al._ (1980); Pechukas and Pollak (1979,
1977); Pechukas (1981). At very early period, Wigner introduced asymptotic
reactant and product regions to calculate reaction rate in the line of his
achievement of the transition state theoryWigner (1937). Independently, Wright
et al. showed reactive bands, which had been found by Wall et al.Wall _et
al._ (1961); Wall and Porter (1963); Wall _et al._ (1958), in the reactivity
maps of $\mathrm{H}+\mathrm{H}_{2}$ and its isotopic variantsWright _et al._
(1975); Wright (1976); Wright and Tan (1977); Tan _et al._ (1977); Wright
(1978); Laidler _et al._ (1977) that consist of bands of nonreactive regions
and reactive regions of each product. The approach was initiated by Ref.
Wright _et al._ , 1975 to see the origin of continuous shift of peak in graph
of initial relative translational (kinetic) energy versus time spent in
“reaction shell” for given initial vibrational phase angles. After Ref. Wright
_et al._ , 1975, a series of study was reported for collinear (1D)Wright
(1976), isotopeWright and Tan (1977), coplanar (2D)Tan _et al._ (1977)
reactions and 3DWright (1978) reaction and also on an improved potential
energy surfaceLaidler _et al._ (1977) with the plot of initial relative
translational energy versus initial phase angle $\theta$. Chesnavich et al.
observed boundary trajectories of the collision-induced dissociation of
$\mathrm{H}+\mathrm{H}_{2}$ reactionAndrews and Chesnavich (1984); Grice _et
al._ (1987) that divide reactive ($\mathrm{H}_{2}+\mathrm{H}$), non-reactive
($\mathrm{H}+\mathrm{H}_{2}$) and dissociative
($\mathrm{H}+\mathrm{H}+\mathrm{H}$) regions in phase space.
Pechukas et al. revealed the role of periodic orbit dividing surface in two-
dimensional collinear atom-diatom reaction systems Pollak and Levine (1980);
Pollak (1981c, a); Pechukas (1976); Sverdlik and Koeppl (1978); Pollak and
Pechukas (1979); Pollak and Child (1980); Child and Pollak (1980); Pollak and
Pechukas (1978); Pollak (1981b); Pollak _et al._ (1980); Pechukas and Pollak
(1979, 1977); Pechukas (1981). The importance of periodic orbit around
interaction region was first recognized by PechukasPechukas (1976). The series
of research can be described by his words at very beginning.
> Somewhere between these two trajectories is a “dividing” trajectory that
> falls away, neither to reactant nor to product; this is the required
> “vibration,” across the saddle point region but not necessarily through the
> saddle point, and the curve executed on the plane by the vibration is the
> best transition state at that energy.
Pechukas and PollakPechukas and Pollak (1977) and Sverdlik and KoepplSverdlik
and Koeppl (1978) started to observe such trajectories in the region of index-
one saddles of two dimensional systems and recognized as “unstable invariant
classical manifold”Pechukas (1981) and call them periodic orbit dividing
surface (pods)Pollak _et al._ (1980). The pods can be identified as the best
transition statePechukas and Pollak (1979) when there is only one pods at
given energy. Pechukas and Pollak investigated the advantage of pods against
variational TSTPollak and Pechukas (1978) and unified statistical theoryPollak
and Pechukas (1979). They also revealed its role in the application of
statistical theories to reaction dynamicsPollak and Pechukas (1979); Pollak
and Levine (1980); Child and Pollak (1980) and provided an iterative method to
calculate reaction probabilityPollak and Child (1980). After the series of
classical investigation they started to look at adiabatic motion perpendicular
to podsPollak (1981b) and quantum correspondencePollak (1981a) and
experimental correspondencePollak (1981c) were elucidated. Those studies were
mostly done on two degrees of freedom (DoFs) systems. The problem one of high
dimension in the region of index-one saddles was later overcome Komatsuzaki
and Nagaoka (1996, 1997); Komatsuzaki and Berry (1999, 2001); Wiggins _et
al._ (2001); Uzer _et al._ (2002); Bartsch _et al._ (2005); Li _et al._
(2006); Kawai and Komatsuzaki (2010); Hernandez _et al._ (2010); Kawai and
Komatsuzaki (2011a); Waalkens _et al._ (2008); Kawai and Komatsuzaki (2011b,
c); Ünver Çiftçi and Waalkens (2012); Teramoto _et al._ (2011); Toda _et
al._ (2005); Komatsuzaki _et al._ (2011); Jaffé _et al._ (2005); Martens
(2002); Komatsuzaki and Berry (2003); de Almeida _et al._ (1990).
The dynamics around the saddle point is recently investigated extensively in
terms of nonlinear dynamics Komatsuzaki and Berry (1999, 2001); Wiggins _et
al._ (2001); Uzer _et al._ (2002); Bartsch _et al._ (2005); Li _et al._
(2006); Kawai and Komatsuzaki (2010); Hernandez _et al._ (2010); Kawai and
Komatsuzaki (2011a); Waalkens _et al._ (2008); Kawai and Komatsuzaki (2011b,
c); Ünver Çiftçi and Waalkens (2012); Teramoto _et al._ (2011); Toda _et
al._ (2005); Komatsuzaki _et al._ (2011); Jaffé _et al._ (2005); Martens
(2002); Komatsuzaki and Berry (2003); de Almeida _et al._ (1990), in the
context of transition state (TS) theory Glasstone _et al._ (1941); Steinfeld
_et al._ (1989) in molecular science. Among them, particularly relevant to the
present work is the finding of the “tube”de Almeida _et al._ (1990)
structures in phase space to conduct the reacting trajectories from the
reactant to the product across an index-one saddle. These studies revealed the
firm theoretical ground for the robust existence of the reactivity boundaries
emanating from the saddle region as well as the no-return TS in the phase
spaceToda _et al._ (2005); Komatsuzaki _et al._ (2011). The scope of the
dynamical reaction theoryKomatsuzaki _et al._ (2011) is not limited to only
chemical reactions, but also includes, for example, ionization of a hydrogen
atom under electromagnetic fields Wiggins _et al._ (2001); Uzer _et al._
(2002), isomerization of clusters Komatsuzaki and Berry (1999, 2001), orbit
designs in solar systems Jaffé _et al._ (2002), and so forth. Recently, these
approaches have been generalized to dissipative multidimensional Langevin
equations Bartsch _et al._ (2005); Hernandez _et al._ (2010); Kawai and
Komatsuzaki (2011a), based on a seminal work by MartensMartens (2002), laser-
controlled chemical reactions with quantum effects Waalkens _et al._ (2008);
Kawai and Komatsuzaki (2011b), systems with rovibrational couplings Kawai and
Komatsuzaki (2011c); Ünver Çiftçi and Waalkens (2012), and showed the robust
existence of reaction boundaries even while a no-return TS ceases to exist
Kawai and Komatsuzaki (2010).
For complex systems, the potential energy surface becomes more complicated,
and transitions from a potential basin to another involve not only index-one
saddles but also higher index saddles Shida (2005); Sicardy (2010); Minyaev
_et al._ (1997, 2004); Getmanskii and Minyaev (2008); Huang _et al._ (2006);
Shank _et al._ (2009); Xie _et al._ (2005); Bowman and Shepler (2011).
Recently the role of index-two saddle was revealed several dynamical aspects.
For example, a simulation study on “phase transitions” from solid-like phase
to liquid-like phase in a seven-atomic cluster Shida (2005) showed that
trajectories spend more time in the region of higher index saddles as the
total energy of the system increases. Under the onset of “melting”, its
occupation ratio around the index-two saddles correlates to its Lindemann’s
$\delta$ and the configuration entropy that are well-known indices of phase
transition. Another example is a systematical survey of global stability of
the triangular Lagrange points L4 and L5 under the condition that the
secondary mass $\mu$ is larger than the Gascheau’s value $\mu_{G}$ (also known
as the Routh value) in the restricted planar circular three-body problem
Sicardy (2010). Those Lagrange points become index-two saddle points when the
condition $\mu>\mu_{G}$ is met, and the range of $\mu$ was identified where
the Lagrange points have global stability and periodic stable orbits around
them.
Chemical reactions associated with index-two saddles were also reported in
several molecular systems Minyaev _et al._ (1997, 2004); Getmanskii and
Minyaev (2008); Huang _et al._ (2006); Shank _et al._ (2009); Xie _et al._
(2005); Bowman and Shepler (2011) by using several searching algorithms
(section 6.3 p. 298 of Ref. Wales, 2004 and references therein). However
index-two saddles have got less interests than index-one saddles. This may be
because of the Murrell-Laidler theorem Murrell and Laidler (1968) that states
the minimum energy path does not pass through any index-two saddle points.
However one can still find many studies such as aminoboraneMinyaev _et al._
(1997), $\text{PF}_{3}$Minyaev _et al._ (1997), $\text{NH}_{5}$Minyaev _et
al._ (2004), $\text{NF}_{2}\text{H}_{3}$Getmanskii and Minyaev (2008), water
dimerHuang _et al._ (2006); Shank _et al._ (2009), $\text{H}_{5}^{+}$Xie
_et al._ (2005), $\text{H}_{2}\text{CO}$Bowman and Shepler (2011) that
identify a variety of index-two saddles in molecular isomerization reactions.
Significant difference between reactions associated with a bottleneck made of
an index-one saddle and those through higher index saddle is that a single
higher index saddle does not necessarily serve as a bottleneck from one
potential basin to another since index-$n$ $(>1)$ saddles are almost always
accompanied with saddles of index less than $n$. Therefore, reactions
associated with higher index saddle(s) are dominated by a bottleneck made up
of multiple saddles, and so are its phase space structures. This non-local
property of the bottleneck is an essential difficulty in treating a reaction
associated with higher index saddles.
To reveal the fundamental mechanism of the passage through a saddle with index
greater than one, the phase space structure was recently studied on the basis
of normal form (NF) theory Ezra and Wiggins (2009); Collins _et al._ (2011);
Haller _et al._ (2010, 2011). For example the pioneering studies to extend
the dynamical reaction theory into higher index saddles were reported Ezra and
Wiggins (2009) for concerted reactions. A dividing surface to separate the
reactant and the product was proposed for higher index saddles Collins _et
al._ (2011) and the associated phase space structure was also discussed Haller
_et al._ (2010, 2011). Those studies are based on NF theory, and therefore
relies on two assumptions: One is that no linear “resonance” is postulated
between more than one reactive modes and the other is that the local dynamics
around the index-two or higher index saddle plays a dominant role in
determining the destination of the trajectory. For the former assumption,
TodaToda (2008) addressed that linear resonance between two reactive modes may
introduce breakdown of the reactivity boundary. As for the latter assumption,
Nagahata et al.Nagahata _et al._ (2013) reported recently that the reactivity
boundary extracted by normal form does not necessarily give the barrier
separating the reactivity in the original coordinate space for higher index
saddles. Moreover, as described above, an index-two saddle often coexists with
index-one saddles and therefore the reaction dynamics or the “bottleneck”
should be determined through interplay among multiple saddle points.
Additionally, current theory for invariant manifold that may dominate
reactions associated with an index-two saddle and a higher index saddle are
only for the largest repulsive directionHaller _et al._ (2010, 2011).
However, for example, Minyaev et al. Minyaev _et al._ (1997) showed that
aminoborane has internal rotation associated with an index-two saddle and
index-one saddles, and that the weaker repulsive direction around the index-
two saddle, corresponding to the hindered internal rotation, connects the two
minima.
Most studies for reaction associated with higher index saddles are based on
NF, a perturbation theory around a single stationary point, and assume that NF
can capture those reaction dynamics. To validate those studies, however, it is
needed first to clarify the concept of reactivity boundaries in reactions
associated with a bottleneck made up of multiple saddles. The reactivity
mapWright _et al._ (1975); Wright (1976); Wright and Tan (1977); Laidler _et
al._ (1977); Tan _et al._ (1977); Wright (1978); Andrews and Chesnavich
(1984); Grice _et al._ (1987) and Pechukas’ foresightPechukas (1976) are
still important to generalize the concept to make it applicable when the
reaction dynamics is not dominated by a single saddle point.
In the present paper, we first review the concept of reactivity boundaries for
the linear system in Sec.II.1. Then we generalize the concept to the reactions
associated with a bottleneck possibly made up of multiple saddle points in
SecII.2. In Sec.III we demonstrate the numerical extraction of reactivity
boundaries in a chemical system with a bottleneck made up of multiple saddle
points including both index-one and index-two saddles. The investigation
reveals what initial condition should be prepared to make the reaction happen,
and why such initial conditions lead to reactions.
## II Theory
In this section, we revisit the concept of reactivity boundaries developed
previously (Sec. II.1) and propose a more general definition that can involve
a single reaction associated with a bottleneck made up of higher index saddles
and/or several saddle points with different indices, where the normal form
theory, based on expansion around a single stationary point, does not work
(Sec. II.2).
### II.1 Linearized Hamiltonian
In this subsection we review the concept of reactivity boundaries developed
previously based on the theory of dynamical systems Komatsuzaki and Berry
(1999, 2001); Wiggins _et al._ (2001); Uzer _et al._ (2002); Bartsch _et
al._ (2005); Li _et al._ (2006); Kawai and Komatsuzaki (2010); Hernandez _et
al._ (2010); Kawai and Komatsuzaki (2011a); Waalkens _et al._ (2008); Kawai
and Komatsuzaki (2011b, c); Ünver Çiftçi and Waalkens (2012); Teramoto _et
al._ (2011); Toda _et al._ (2005); Komatsuzaki _et al._ (2011); Jaffé _et
al._ (2005); Martens (2002); Komatsuzaki and Berry (2003); de Almeida _et
al._ (1990). One of the simplest example of reactivity boundaries can be seen
in the normal mode (NM) approximation. If the total energy of the system is
just slightly above a stationary point, the $n$-DoFs Hamiltonian $H$ can well
be approximated by a NM Hamiltonian $H_{0}$
$H({\bm{p}},{\bm{q}})\approx
H_{0}({\bm{p}},{\bm{q}})=\sum_{j=1}^{n}\frac{1}{2}(p_{j}^{2}+k_{j}q_{j}^{2})$
(1)
with NM coordinates $\bm{q}$=$(q_{1},\dots,q_{n})$ and their conjugate momenta
$\bm{p}$=$(p_{1},\dots,p_{n})$, where $k_{j}\in\mathbb{R}$ is the “spring
constant” or the curvature of the potential energy surface along the $j$th
direction. The constants $k_{j}$ can be positive or negative. If $k_{j}<0$,
the potential energy is maximum along the $j$th direction. Then the direction
exhibits an unstable motion corresponding to “sliding down the barrier,” and
can be regarded as “reaction coordinate.” The index of the saddle corresponds
to the number of negative $k_{j}$’s. Phase space flow of the DoF with negative
$k_{j}$ is depicted in Fig. 1.
Figure 1: (color online). Phase space flow of the normal mode with negative
curvature (hyperbolic degree of freedom). Reactant and product are defined by
the sign of $q_{1}$. $\eta_{1}=0$(or $\xi_{1}$-axis) divides the destination
of trajectories; Trajectories in $\eta_{1}>0$ go into the product side
($q_{1}>0$) as $t\rightarrow+\infty$ and those in $\eta_{1}<0$ go into the
reactant side ($q_{1}<0$). Similarly, $\xi_{1}=0$(or $\eta_{1}$-axis) divides
the origin of trajectories; Trajectories in $\xi_{1}>0$ originate from the
reactant side and those in $\eta_{1}<0$ from the product side.
Here one can introduce the following coordinates
$\displaystyle\eta_{j}=$
$\displaystyle(p_{j}+\lambda_{j}q_{j})/(\lambda_{j}\sqrt{2}),~{}~{}~{}\xi_{j}=$
$\displaystyle(p_{j}-\lambda_{j}q_{j})/\sqrt{2},$ (2)
corresponding to eigenvectors of the coefficient matrix of the linear
differential equation (Eq. 1) with eigenvalue $\lambda_{j}=\pm\sqrt{-k_{j}}$.
Here one can also introduce another set of coordinates
$\displaystyle I_{j}=$ $\displaystyle\xi_{j}\eta_{j},~{}~{}~{}\theta_{j}=$
$\displaystyle\ln|\lambda_{j}\eta_{j}/\xi_{j}|/2,$ (3)
called “action” and “angle” variables. When Eq. (1) holds, the action variable
is an integral of motion, and trajectories run along the hyperbolas given by
$I_{j}=\mathrm{const.}$ shown by gray lines in Fig. 1. The $\eta_{j}$\- and
$\xi_{j}$-axes run along the asymptotic lines of the hyperbolas in Fig. 1. The
Hamiltonian equation of motion can be written as
$\dot{{\bm{\zeta}}}_{j}\approx-
L_{H_{0}}{\bm{\zeta}}_{j}=-\lambda_{j}L_{I_{j}}{\bm{\zeta}}_{j},=\begin{pmatrix}-\lambda_{j}&0\cr
0&\lambda_{j}\end{pmatrix}{\bm{\zeta}}_{j},$ (4)
where ${\bm{\zeta}}_{j}=(\xi_{j},\eta_{j})^{\mathrm{T}}$, and the Lie
derivative $L_{F}$ is defined as
$L_{F}{\bm{\zeta}}_{k}=\\{F,{\bm{\zeta}}_{k}\\}=\sum_{j=1}^{n}\frac{\partial
F}{\partial\eta_{j}}\frac{\partial{\bm{\zeta}}_{k}}{\partial\xi_{j}}-\frac{\partial
F}{\partial\xi_{j}}\frac{\partial{\bm{\zeta}}_{k}}{\partial\eta_{j}}$. One can
tell the destination region and the origin region of trajectories from the
signs of $\eta_{j}$ and $\xi_{j}$ as follows: If $\eta_{j}>0$, the trajectory
goes into $q_{j}>0$ and if $\eta_{j}<0$, then the trajectory goes into
$q_{j}<0$. Therefore one can tell the destination of trajectories from the
sign of $\eta_{j}$. Similarly, the origin of trajectories can be told from the
sign of $\xi_{j}$. Hereafter we call the set $\eta_{j}=0$ “destination-
dividing set,” and $\xi_{j}=0$ “origin-dividing set,” and each of these two
sets constitute “reactivity boundaries”.
When the NM picture dominates the dynamics around the stationary point, the
form of Eq. (4) enables us to identify the fate of reaction. This is also
generally the case if one can achieve a canonical transformation to turn the
Hamiltonian into the form of $H=H({\bm{I}})$, even though $\lambda_{j}$s are
depends on initial ${I_{j}}$s . This transformation has been mostly achieved
by the normal form theory based on expansion around a single stationary point.
The theory has been applied and developed to elucidate the mechanism of
several reaction dynamics about a decade Komatsuzaki and Berry (1999, 2001);
Wiggins _et al._ (2001); Uzer _et al._ (2002); Bartsch _et al._ (2005); Li
_et al._ (2006); Kawai and Komatsuzaki (2010); Hernandez _et al._ (2010);
Kawai and Komatsuzaki (2011a); Waalkens _et al._ (2008); Kawai and
Komatsuzaki (2011b, c); Ünver Çiftçi and Waalkens (2012); Teramoto _et al._
(2011); Toda _et al._ (2005); Komatsuzaki _et al._ (2011). For practical
applications the Lie canonical perturbation theory, developed by a Japanese
astrophysicist Gen-Ichiro HoriHori (1966, 1967) (and equivalent theory was
independently developed by DepritCampbell and Jefferys (1970); Deprit (1969)),
has been frequently used.
### II.2 Reactivity boundary
For complex molecular systems, the potential energy surface becomes more
complicated, and a single transition from a potential basin to another
involves not only index-one saddles but also higher index saddles. The normal
form theory shown in Sec. II.1, based on expansion around a single stationary
point, may not work well for such complex systems, where the fate of the
reaction may not be dominated solely by the local property of the potential
around the point. Therefore the definition of the reactivity boundaries should
not be based on perturbation theory. In this subsection, we seek for a more
general definition of reactivity boundaries, so that the definition can
describe invariant objects previously studied (such as impenetrable
barriersWiggins _et al._ (2001) and reactive islandde Almeida _et al._
(1990)), to analyze more complicated reactions by following the Pechukas’
foresightPechukas (1976).
A “state,” which may refer to reactant or product, forms a certain region in
the phase space $\Omega$. Let us denote the states by $S_{1},\dots,S_{N}$,
which are disjoint subsets of $\Omega$ ($S_{j}\subset\Omega$ and $S_{i}\cap
S_{j}=\emptyset$ where $i,j=1,2,\dots,N$ and $i\neq j$). In between the
regions corresponding to the states, there can be intermediate an region
$\Omega_{0}$ that do not belong to any of the states (Fig. 2):
$\Omega=S_{1}\sqcup\dots\sqcup S_{N}\sqcup\Omega_{0}$. Most of the
trajectories in the intermediate region eventually go into either of the
states as time proceeds. Likewise, when propagated backward in time, most of
them turn out to originate from either of the states. Consider a set of
trajectories that originate from the state $S_{1}$ and go into the state
$S_{2}$ ($\mathrm{r}_{12}$ in Fig. 2), and another set consisting of
trajectories that originate from $S_{1}$ and go back into the same state
$S_{1}$ ($\mathrm{n}_{1}$ in Fig. 2). In between these two sets of
trajectories there may lie a boundary which consists of trajectories that do
not go into either of the states ($\mathrm{d}_{1}$ in Fig. 2). In the cases
discussed in Sec. II.1, such trajectories were seen to asymptotically approach
into some invariant set(s) in the intermediate region. Suppose there exist
such an invariant set $\Omega_{S}$, which is a co-dimension two subset of
$\Omega_{0}$. We then consider co-dimension one subset
$\Omega_{OD},\Omega_{DD}\subset\Omega_{0}$ satisfying
$\lim_{T\rightarrow\infty}\phi^{T}(\Omega_{OD})=\Omega_{S}$ and
$\lim_{T\rightarrow-\infty}\phi^{T}(\Omega_{DD})=\Omega_{S}$ as follows:
* •
Destination-dividing set $\Omega_{DD}$ ($\mathrm{d}_{1}$ and $\mathrm{d}_{2}$
in Fig. 2)
A set of trajectories whose origin belongs to a certain state but whose
destination does not belong to any state.
* •
Origin-dividing set $\Omega_{OD}$ ($\mathrm{o}_{1}$ and $\mathrm{o}_{2}$ in
Fig. 2)
A set of trajectories whose destination belongs to a certain state but whose
origin does not belong to any state.
The former set constitutes a boundary dividing the destination regions of
trajectories, whereas the latter constitutes a boundary dividing the origin
regions of trajectories. The set of the trajectories (invariant set) that
satisfy one of the above conditions will be called reactivity boundary in the
following. The asymptotic limit $\Omega_{S}$ of the reactivity boundary, which
belongs to neither reactant nor product, will be called “seed” of reactivity
boundaries. The definition of the reactivity boundary (the destination- or the
origin-dividing set) can apply systems with multiple states, since the
definition of the reactivity boundaries are only based on a single state. This
definition of the reactivity boundaries and their seed is a generalization of
the previous invariant objects (the stable and unstable manifolds of NHIM, and
the NHIM, respectively) studied in literature Komatsuzaki and Berry (1999,
2001); Wiggins _et al._ (2001); Uzer _et al._ (2002); Bartsch _et al._
(2005); Li _et al._ (2006); Kawai and Komatsuzaki (2010); Hernandez _et al._
(2010); Kawai and Komatsuzaki (2011a); Waalkens _et al._ (2008); Kawai and
Komatsuzaki (2011b, c); Ünver Çiftçi and Waalkens (2012); Teramoto _et al._
(2011); Toda _et al._ (2005); Komatsuzaki _et al._ (2011) and summarized in
Sec. II.1.
Figure 2: (color online). Blue large circles represent states $S_{1}$ and
$S_{2}$. Arrows represent particular sorts of trajectories; blue arrows
($\mathrm{n}_{1}$ and $\mathrm{n}_{2}$) represent non-reactive trajectories,
while red ones ($\mathrm{r}_{12}$ and $\mathrm{r}_{21}$) represent reactive
trajectories. Black arrows ($\mathrm{d}_{1}$ and $\mathrm{d}_{2}$) and gray
arrows ($\mathrm{o}_{1}$ and $\mathrm{o}_{2}$) represent trajectories in the
destination-dividing set, and those in the origin-dividing set, respectively.
## III Numerical Demonstrations
### III.1 Three DoFs model of $\text{H}_{5}^{+}$
We demonstrate here a numerical calculation of the reactivity boundary defined
in Sec. II with a model $\mathrm{H}_{5}^{+}$ system. This cation plays an
important role in interstellar chemistry, especially because of the proton
exchange reaction
$\mathrm{H}_{3}^{+}+\mathrm{HD}\rightleftharpoons\mathrm{H}_{2}+\mathrm{H}_{2}\mathrm{D}^{+}$
occurring through the $\mathrm{H}_{4}\mathrm{D}^{+}$ intermediate. As shown in
the previous ab initio calculation Xie _et al._ (2005), the most stable
structure of the $\mathrm{H}_{5}^{+}$ system is a weakly bound cluster of
$\mathrm{H}_{2}$ and $\mathrm{H}_{3}^{+}$ moieties, with the $\mathrm{H}_{2}$
standing perpendicular to the $\mathrm{H}_{3}^{+}$ molecular plane. Being a
multi-body system, the $\mathrm{H}_{5}^{+}$ cation undergoes various
isomerization reactions. Taking the four lowest stationary points (one
minimum, two index-one saddle points, and one index-two saddle), we have two
reaction directions. One is a torsional isomerization where the
$\mathrm{H}_{2}$ flips by 180∘ with the planar structure corresponding to the
saddle point. The other is the proton exchange between the two moieties
$\mathrm{H}_{2}+\mathrm{H}_{3}^{+}\rightleftharpoons\mathrm{H}_{3}^{+}+\mathrm{H}_{2}$.
Figure 3: (color online).
$\text{H}_{3}^{+}+\text{H}_{2}\rightarrow\text{H}_{2}+\text{H}_{3}^{+}$
reaction can be written by three coordinate $\varphi,R,z$ depicted on picture.
In the present investigation, we treat the dynamics of $\mathrm{H}_{5}^{+}$ by
confining it into a three degrees-of-freedom system. The dynamical variables
are the center-of-mass distance $R$ between the two $\mathrm{H}_{2}$ moieties,
the position $z$ of the central hydrogen atom along the center-of-mass axis,
and the torsional angle $\varphi$ of the two $\mathrm{H}_{2}$ as shown in Fig.
3. The coordinate $z$ corresponds to the proton exchange reaction between the
two moieties, while the angle $\varphi$ corresponds to the torsional
isomerization. We calculated the potential energy surface at the CCSD(T) level
which is the same level with the previous calculation Xie _et al._ (2005).
The ab initio calculations were performed at 439 points in the range
$0\leq|z|\leq 0.4~{}\text{\AA}$ and $2.09~{}\text{\AA}\leq R\leq
2.51~{}\text{\AA}$, with the $\mathrm{H}_{2}$ bond lengths optimized for each
given value of $(z,R,\varphi)$. By checking the energy value, this region was
confirmed to be sufficient to describe the motion with total energy below 200
cm-1. The potential energy values were then fitted to a cubic order polynomial
in $(z^{2},R,\cos 2\varphi)$. The maximum fitting error was 0.8 cm-1,
sufficiently small considering the total energy 170 cm-1 of the trajectories
run in the present investigation. The structures and energies of the four
lowest stationary points of the fitted surface are listed in Table 1 and
compared with the literature values.Xie _et al._ (2005) The mathematical
expression of the fitted potential energy surface is available as the
supporting information to this article.
Table 1: Structures and energies of four lowest stationary points of $\mathrm{H}_{5}^{+}$. The energies are given relative to the first equilibrium point. | $\varphi$ | $R$ / Å | $z$ / Å | Energy / cm-1 | Ref. Xie _et al._ ,2005 |
---|---|---|---|---|---|---
1 | $\pi/2$ | 2.18 | 0.19 | (ref.) | (ref.) | global minimum
2 | $\pi/2$ | 2.11 | 0 | 48.6 | 48.4 | index-one saddle
3 | 0 | 2.19 | 0.21 | 95.9 | 96.4 | index-one saddle
4 | 0 | 2.12 | 0 | 162.7 | 162.8 | index-two saddle
We use this three-dimensional system as an illustrative model to demonstrate
the concepts introduced in Sec. II. Note however that the real
$\mathrm{H}_{5}^{+}$ system has larger DoFs (nine internal modes and three
rotational modes). Quantum effects must also be considered for the complete
treatment of this system. We here briefly mention that the concept of
reactivity boundaries around the index-one saddle point has recently been
extended to incorporate ro-vibrational couplingsKawai and Komatsuzaki (2011c);
Ünver Çiftçi and Waalkens (2012) and quantum effectsWaalkens _et al._ (2008);
Kawai and Komatsuzaki (2011b). It will be an important future work to combine
these studies with the generalized reactivity boundaries proposed in the
present paper. In the present numerical calculation we confine the system
configuration into the three-dimensional subspace mentioned above for the sake
of simplicity. We still note the global minimum, the three lowest saddle
points and their unstable directions are all included in this subspace, while
the motions transverse to this subspace are bath mode oscillations. This
three-dimensional model is therefore expected to capture some of the essential
properties of the isomerization and the proton exchange processes in the real
$\mathrm{H}_{5}^{+}$ system with low energies.
There are two index-one saddle points, denoted as 2 and 3, that correspond to
the proton exchange and the torsional isomerization, respectively. The highest
of these four stationary points is an index-two saddle point, denoted as 4,
representing a concerted reaction of the proton exchange and the torsion.
Figure 4 depicts the two-dimensional potential energy surface in $z$ and
$\varphi$ where the $R$ is relaxed to the minimum energy for each given value
of $(z,\varphi)$. There are four symmetrically equivalent points corresponding
to the global minimum 1. Similarly the saddle points 2, 3, and 4 have two,
four, and two equivalent points, respectively.
The dynamical calculations of the present three-dimensional model of
$\mathrm{H}_{5}^{+}$ are performed by integrating the equation of motion given
by the following Hamiltonian
$\displaystyle
H=\frac{1}{I_{\varphi}}{p_{\varphi}}^{2}+\frac{1}{2\mu_{R}}{p_{R}}^{2}+\frac{1}{2\mu_{z}}{p_{z}}^{2}+V(\varphi,R,z),$
(5)
where $p_{\varphi}$ is the angular momentum conjugate to the torsional angle
$\varphi$, and $p_{R}$ and $p_{z}$ are the linear momenta conjugate to $R$ and
$z$, respectively. The reduced masses are
$\displaystyle\mu_{z}=\frac{4}{5}m_{\mathrm{H}},$
$\displaystyle\mu_{R}=m_{\mathrm{H}},$ (6)
where $m_{\mathrm{H}}$ is the mass of the hydrogen atom, and $I_{\varphi}$ is
the moment of inertia of H2.
Figure 4: The potential energy surface as a function of $z$ and $\varphi$,
representing the proton exchange and the torsional motion, where the other
coordinate $R$ is optimized at each point $(z,\varphi)$. Each number
corresponds to each stationary point listed in Table 1. Blue points, red bars,
and red cross denote the potential minima, index-one saddles, and index-two
saddle, respectively. Contours are spaced with 10 cm-1. The initial condition
for the calculation of the reactivity boundaries are prepared at $z=0$, where
index-one saddle points 2 and index-two saddle points 4 are located (pink
dashed line).
### III.2 Reactivity boundary in $\text{H}_{5}^{+}$
As described in Sec. II.2, the reactivity boundary typically consists of
trajectories emanating from an invariant manifold in the intermediate region.
It is calculated by propagating the system, either forward or backward in
time, from the close vicinity of the invariant manifold. In the present
investigation we focus on the proton exchange reaction from
$\mathrm{H}_{2}+\mathrm{H}_{3}^{+}$ to $\mathrm{H}_{3}^{+}+\mathrm{H}_{2}$, to
demonstrate the extraction of reactivity boundary. The configuration
$\mathrm{H}_{2}+\mathrm{H}_{3}^{+}$ corresponds to a region with $z>0$ and
$\mathrm{H}_{3}^{+}+\mathrm{H}_{2}$ with $z<0$. The intermediate region thus
lies on some region around $z=0$. In this case the surface defined by $z=0$
and $p_{z}=0$ serves as an invariant manifold due to the symmetry of the
system. This means that, once the system stays on that surface, it does
perpetually irrespective of what values the other variables take. This
invariant manifold is unstable in that any infinitesimally small deviation
from the surface of $z=0$ and $p_{z}=0$ makes the system depart from the
surface and fall down into one of the four well regions shown in Fig. 4.
Therefore the reactivity boundaries are stable and unstable manifolds of
$z=0,p_{z}=0$ in this case. The extraction of reactivity boundary can be
carried out as follows: we first uniformly sample phase space points
$(p_{z}=0,p_{R},p_{\varphi},z=0,R,\varphi)$ at a given total energy in that
invariant manifold (see also Appendix for details). Second, we give the system
a small positive deviations in $p_{z}$, and propagate it forward in time
(corresponding to the origin-dividing set o2 in Fig. 2). Those trajectories
correspond to the generalization of $\xi_{1}=0$ with positive $\eta_{1}$ to
divide the origin of trajectories for normal mode approximation in Fig. 1.
Likewise, the propagation of the system backward in time results in
trajectories that divide the destination of trajectories, corresponding to the
set d2 in Fig. 2 (Compare also with $\eta_{1}=0$ with negative $\xi_{1}$ for
normal mode Hamiltonian in Fig. 1). Note here again that the generalization
involves two essential differences from the normal mode picture: one is the
generalization to nonlinear Hamiltonian systems in which normal mode
approximation does not hold, and the other is that the invariant manifold from
which reactivity boundaries emanate can be associated not only with a single
saddle point but with multiple saddle points with different indices.
Figure 5: (color online). The reactivity boundaries of
$\text{H}_{3}^{+}+\text{H}_{2}\rightarrow\text{H}_{2}+\text{H}_{3}^{+}$
reaction. (a) randomly sampled fifty trajectories from the destination-
dividing set (red) and the origin-dividing set (blue), both constituting the
reactivity boundaries, initiated from the section of $z=0$ and $p_{z}\simeq 0$
projected on the $z-R$ space. The normal mode coordinates $\tilde{q}_{1}$ and
$\tilde{q}_{2}$ at the potential minimum are shown by gray lines. (b) a
schematic picture of reactivity boundaries depicted as “tubesde Almeida _et
al._ (1990)” departing from $z=0$ and $p_{z}\simeq 0$. Note here that the
invariant manifold of $z=0$ and $p_{z}=0$ can involve multiple saddle points.
Figure 6: (color online). The reactivity of
$\text{H}_{3}^{+}+\text{H}_{2}\rightarrow\text{H}_{2}+\text{H}_{3}^{+}$
reaction on the section of $\tilde{q}_{1}=0,\tilde{p}_{1}<0$. 100,000
trajectories are uniformly sampled on the surface of $z=0$ with positive
momentum $p_{z}\simeq 0$ and evolved forward in time until they cross a
surface defined by $\tilde{q}_{1}=0$ and $\tilde{p}_{1}<0$ by the normal mode
coordinate $\tilde{{\bm{q}}}$ and its conjugate momentum $\tilde{{\bm{p}}}$ at
the potential minimum (see also Appendix for details). The trajectories
forming the origin-dividing set are shown by blue dots. Likewise, 100,000
trajectories are similarly sampled on that surface with negative $p_{z}\simeq
0$ and propagated backward in time until they cross the surface. The
trajectories forming the destination-dividing set are shown by red dots. (a)
and (b): the projections of the first intersections of the destination-
dividing set (red dots) and the origin-dividing set (blue dots) crossing the
surface of $\tilde{q}_{1}=0$ and $\tilde{p}_{1}<0$ on the section,
respectively, onto the $\tilde{q}_{2}$-$\tilde{p}_{2}$ space and the
$\varphi$-$p_{\varphi}$ space. The gray lines denote the boundaries of
energetically inaccessible region. The values are defined by maximum and
minimum of $p_{\varphi}$ at each $\varphi$. The cross symbols (dw1, dw2,…)
represent the initial positions (on that place) of the trajectories shown in
7. (c), (d), (e), (f): the projections of the phase space points that are
going into the product side (red dots) and those that have come from the
product side (blue dots) are depicted to conform “inside” of reactivity
boundaries and to check validity of the extraction of the reactivity
boundaries. Orange lines in (c) and cyan lines (e) are maximum/minimum
$\tilde{p}_{2}$ of the sets of the reactive points. Brown lines in (c) and
purple lines (e) are maximum/minimum $\tilde{p}_{2}$ of the reactivity
boundaries. Similarly, Orange lines in (d) and cyan lines (f) are
maximum/minimum $p_{\varphi}$ of the sets of the reactive points. Brown lines
in (d) and purple lines (f) are maximum/minimum $p_{\varphi}$ of the
reactivity boundaries.
Figure 5 shows randomly chosen fifty samples from the origin-dividing set
(blue) and the destination-dividing set (red) depicted on the $R$-$z$ space.
The reactivity boundaries are only drawn until they first cross the section
defined by $\tilde{q}_{1}=0$ and $\tilde{p}_{1}<0$ by the normal mode
coordinate $\tilde{{\bm{q}}}$ and its conjugate momentum $\tilde{{\bm{p}}}$ at
the potential minimum (the normal mode coordinates are shown by the gray
arrows in Fig. 5 (a)). The reactivity boundaries are four dimensional surfaces
in an equienergy shell which divide reactive and non-reactive trajectories as
schematically shown in Fig. 5(b). Figures 6(a)(b) show the origin dividing set
(blue) and the destination dividing set (red) on the
$\tilde{q}_{1}=0,\tilde{p}_{1}<0$ section depicted by using 100,000
trajectories whose initial conditions are uniformly sampled on the
$z=0,p_{z}\simeq 0$ section (see also Appendix for details). Let us look into
how reaction selectivity existing in the phase space can be rationalized or
visualized in these projections. In Fig. 6(a), one can find few fingerprints
of the reaction selectivity existing in the phase space with respect to the
signs of the normal mode coordinate and momentum. The reaction path is curved
in the $R$-$z$ space as shown in Fig. 5(a) and the saddle points exist on the
negative side of $\tilde{q}_{1}$. Because Fig. 6(a) is the projection of the
first intersections of the reaction boundaries across the surface of
$\tilde{q}_{1}=0$ and $\tilde{p}_{1}<0$ (i.e., all dots on Fig. 6(a) are
moving towards the surface of $z=0$), one may expect that $\tilde{q}_{2}<0$ or
$\tilde{p}_{2}<0$ on that surface should enhance the reaction probability,
resulting in a nonuniform distribution of the reaction boundaries on the
$\tilde{p}_{2}$-$\tilde{q}_{2}$ space. However, as seen in Fig. 6(a), the
reaction boundaries are distributed rather uniformly in this space (e.g., no
preference in the sign of $\tilde{p}_{2}$). This implies that preparing
$\tilde{q}_{2}<0$ or $\tilde{p}_{2}<0$ on that surface does not increase the
ability of the system to climb the reaction barrier. As shown in Fig. 5 (a),
the trajectories oscillate rapidly in the $\tilde{q}_{2}$-direction and the
bath mode coordinate change its sign many time before coming close to $z=0$,
where the saddle points 2 and 4 for the proton transfer reaction are located,
while they slowly adapt to the curved reaction pathway. The dynamics near the
index-one and index-two saddle points, thus, seems not to be sensitive to the
initial vibrational phase prepared in the well region.
Next let us turn to the $p_{\varphi}$-$\varphi$ projection in Fig. 6(b). The
reaction boundaries, both the destination dividing set (red points in the
figure) and the origin dividing set (blue), are confined in smaller values of
$|p_{\varphi}|$ compared to energetically accessible values. This is because
the energy is more distributed into the reactive mode when the momentum in the
$\varphi$-direction is smaller. In contrast to the
$\tilde{p}_{2}$-$\tilde{q}_{2}$ space, the reaction selectivity existing in
the phase space manifests nonuniformity of the distribution of these reaction
boundaries in the $p_{\varphi}$-$\varphi$ space. The confinement of the
destination-dividing set (red) in smaller $|p_{\varphi}|$ is more pronounced
in $\varphi\approx 0$ than in $\varphi\approx\pi/2$, while the range of
$|p_{\varphi}|$ of the origin-dividing set (blue) is more uniform in
$\varphi$. Note that $\varphi=0$ corresponds to the planar configurations that
involve both the index-one saddle points 3 and the index two saddle points 4
(see Table 1) and the reaction must proceed over the index-two saddle when
$\varphi\approx 0$ (Fig. 4). The relative barrier height through the index-two
saddle 4 for the proton transfer with $\varphi=0$ is $162.7-95.9=66.8\
\mathrm{cm}^{-1}$ which is higher than the barrier height through the index-
one saddle 2 with $\varphi=\pi/2$, 48.6 cm-1 as seen from Table 1. In order to
put sufficient energy into the reactive mode to overcome the barrier,
therefore, the momentum $p_{\varphi}$ in the $\varphi$-direction must be
confined into much smaller values $|p_{\varphi}|$ for $\varphi\approx 0$ than
for $\varphi\approx\pi/2$ due to the conservation of total energy of the
system. This interpretation, done by the relative barrier height with constant
$\varphi$, is consistent with the plot of the sample trajectory (ds1) for
small initial $|p_{\varphi}|$ in Fig. 7. shows some representative sample
trajectories in the $\varphi$-$z$ and $R$-$z$ spaces, whose locations in the
$\tilde{p}_{2}$-$\tilde{q}_{2}$ and the $p_{\varphi}$-$\varphi$ spaces are
also indicated as symbols in Figs. 6(a)(b). It is seen that the motions along
the reactive direction (approximately the $z$-direction) take place more
rapidly than that along the $\varphi$-direction and the value of $\varphi$
does not change much during the course of the reaction.
On the other hand, the trajectories approaching to the surface of $z=0$ and
$p_{z}<0$ with large values of $|p_{\varphi}|$ at $\varphi\approx\pi/2$ at the
section correspond to the motion starting from the well region and approach to
the index-two saddle 4, as shown in the $z-\varphi$ plane in Fig. 7 (dw2).
This is contrasted with the trajectories starting with small $|p_{\varphi}|$
at $\varphi\approx\pi/2$ and approaching to the index-one saddle 2 (dw1). If
we regard $(p_{\varphi},\varphi)$ as roughly corresponding to the nonreactive
mode, this situation seems to be counter-intuitive in that when the
nonreactive degree of freedom is more excited (i.e., larger $|p_{\varphi}|$)
the system is more likely to approach to the higher index-saddle with larger
barrier height. This arises from the fact that the “reaction direction” for
proton transfer through the index-two saddle is not simply along $z$ but runs
diagonal in the $z$-$\varphi$ plane as the system goes from the well directly
to the index-two saddle 4. The large momentum $|p_{\varphi}|$ is also used for
approaching to the higher barrier of the index-two saddle 4 and, therefore,
the large initial value $|p_{\varphi}|$ is favored for the reaction over the
index-two saddle. All the above discussions explain the nonuniformity of the
range of $p_{\varphi}$ with respect to $\varphi$ for the destination-dividing
set (red) in Fig. 6 (b).
Compared to the destination-dividing set, the origin-dividing set is more
uniformly distributed along $\varphi$ (see blue dots in Fig. 6 (b)). This
arises from the choice of the cross section for observing the reaction
boundaries. We chose the section of $\tilde{q}_{1}=0$ with $\tilde{p}_{1}<0$
that is located at the potential minimum. With this choice, we are observing
the origin-dividing set after it is bounced by the potential wall in the
large-$z$ region (Fig. 5 (a)). As seen in the sample trajectories
(os1),(os2),(ow1),(ow2) in Fig. 7, the value of $\varphi$ changes during the
stay in the well region. The change of $p_{\varphi}$ due to the energy
exchange between the $\varphi$-mode and the others can also be seen by the
direction of the trajectories. Therefore the longer time between the
preparation (at $z=0$) and the observation ($\tilde{q}_{1}=0$ with
$\tilde{p}_{1}<0$) of the destination-dividing set than the origin-dividing
set causes some further “mixing” in ($\varphi,p_{\varphi})$ and the reaction
selectivity is lost compared to the direct cross section as observed for the
destination-dividing set in Fig. 6.
Figure 7: (color online). The representative sample trajectories forming the
reactivity boundaries in Fig. 6(a) and (b) on the $\varphi$-$z$ space and the
$R$-$z$ space. The gray points denote the locations in these spaces when those
sample trajectories intersect the section of $\tilde{q}_{1}=0$ with
$\tilde{p}_{1}<0$. The symbol $+$ denotes the location of the index-one point
2 or index-two saddle point 4. The difference of the location of the two
saddle points is invisible in the $R$-$z$ projection with this resolution. The
magenta and orange colored trajectories are of the destination dividing set.
The blue and green colored trajectories are of the origin dividing set. The
color grade represents the time course of trajectories obeying the
Hamiltonian: time goes from the light to the dark grade, and the light and
dark correspond to before and after the intersection of the section of
$\tilde{q}_{1}=0$ with $\tilde{p}_{1}<0$. For instance, trajectory (dw2)
indicates that of the destination-dividing set in the well region with ‘large’
$|p_{\varphi}|$. Trajectory (os1) indicates that of the origin-dividing set at
the index-one saddle region with ‘small’ $|p_{\varphi}|$.
Reactivity boundaries are four dimensional objects, and we cannot capture
their full characteristics by the two dimensional projections. In contrast to
normal mode approximation or normal form theory locally expanded in the
vicinity of a single saddle point, for our present general footing, the
analytic formula of the underlying reaction coordinate is hard to derive and
the invariant manifold locally extracted in the vicinity of a single point or
a collection of multiple saddle points with different indices might not
necessarily provide the boundary to divide the fates of the reactions
originated from the well region far apart from the saddles.Nagahata _et al._
(2013)
To check the validity of our numerical extraction of reactivity boundaries, we
note the fact that both reactive and non-reactive trajectories must exist in
the vicinity of the reactivity boundaries. We therefore check the reactivity
of trajectories in the vicinity of each sampled point
$(\tilde{p}_{2},\tilde{q}_{2},\varphi,p_{\varphi})$ on the reactivity
boundaries on the section. Sampling was made of phase space points
$(\tilde{p}_{2}^{\prime},\tilde{q}_{2}^{\prime},\varphi^{\prime},p_{\varphi}^{\prime})$
that satisfy
$\displaystyle\left|\frac{\tilde{p}_{2}^{\prime}-\tilde{p}_{2}}{0.02\text{\AA}\mathrm{u^{1/2}fs^{-1}}}\right|^{2}+\left|\frac{\tilde{q}_{2}^{\prime}-\tilde{q}_{2}}{0.06\text{\AA}\mathrm{u^{1/2}}}\right|^{2}$
(7)
$\displaystyle+\left|\frac{\varphi^{\prime}-\varphi}{\pi}\right|^{2}+\left|\frac{p_{\varphi}^{\prime}-p_{\varphi}}{0.8\hbar}\right|^{2}$
$\displaystyle=10^{-20}$ (8)
for all the sampled points $(\tilde{p}_{2},\tilde{q}_{2},\varphi,p_{\varphi})$
of the reactivity boundaries. As expected, both reactive and non-reactive
trajectories were found from this sampling (data not shown).
To give more visual representation for the validity of our numerical
extraction of reactivity boundaries, we uniformly sampled 1,000,000 points on
the $\tilde{q}_{1}=0,\tilde{p}_{1}<0$ section in the well region and
propagated them forward and backward in time. The phase space points that
turned out to go into the other well region in the forward time propagation
are shown in (c) and (d) in Fig. 6 by projection on the
$\tilde{q}_{2}-\tilde{p}_{2}$ space and the $\varphi-p_{\varphi}$ space. Those
that turned out to have come from the other well in the backward propagation
are shown in (e) and (f). Of the total 1,000,000 sampled points, about 100,000
were found to be reactive trajectories. As can be seen in Fig. 6(c)-(f), a
good coincidence was observed in the maximum/minimum $\tilde{p}_{2}$ and
$p_{\varphi}$ at each $\tilde{q}_{2}$ and $\varphi$ between the reactive
trajectories (corresponding to the inside of “tubesde Almeida _et al._
(1990)” in Fig. 5(b)) and the reactivity boundaries. In any neighborhood of
the reactivity boundaries extracted from the surface of $z=0$ and $p_{z}\simeq
0$ apart from the well regions, reactive trajectories exist in the projected
space. The results, therefore, also give some support (necessary condition) to
the validity of the reactivity boundaries calculated in the present
investigation.
## IV Conclusion and Perspectives
In this article, the concept of reactivity boundary, which is an invariant
manifold lying between reacting and non-reacting trajectories in the phase
space, was revisited and generalized. It is defined as a set of trajectories
that converge into a seed of reactivity boundaries. The latter is located
between the reactant and the product regions, and goes neither into the
reactant or the product, in either forward or backward time propagation. When
only one saddle point controls the reaction dynamics and the energy is not
very high above the saddle point, the reactivity boundaries are readily
extracted analytically by normal form theory. The definition given here is,
however, not limited to such cases but generalized to a single reaction
passing through multiple saddle points including higher index saddles.
The reactivity boundaries constitute a skeleton of the phase space of the
reaction system. Observation of their locations in certain cross sections
tells which initial conditions can lead to chemical reactions. We applied the
concept of the reactivity boundaries to the three-dimensional model system of
the proton exchange reaction associated with a bottleneck made up of two
index-one saddles (2) and two index-two saddles (4) in $\mathrm{H}_{5}^{+}$
cation. The bath mode vibration represented by the normal mode
$(\tilde{p}_{2},\tilde{q}_{2})$ was found to be almost separate from the
reactive mode, and the fast change of its vibrational phase masked the
reaction selectivity existing in the phase space.
On the other hand, the reaction selectivity in the phase space manifested high
degree of selectivity for the torsional motion, related to the existence of
multiple types of saddle points for different values of the torsion angle. In
addition to the reaction through the index-one saddle 2 of the proton
exchange, two limiting behaviors of reacting trajectories were identified. In
one group, the trajectories go from the index-one saddle 3 of the torsion
isomerization to the index-two saddle 4. Small initial values of the torsional
angular momentum $|p_{\varphi}|$ is favored for this reaction pathway because
of the high energy difference between the index-two saddle point 4 and the
index-one torsion saddle point 3. The other group of the reacting trajectories
is those going directly from the well region to the index-two saddle 4. For
this group, high initial values of $|p_{\varphi}|$ is favored because the
reaction pathway runs diagonal in the $z$-$\varphi$ plane rather than parallel
to the $z$-direction. These pictures of the reaction dynamics were obtained
with the help of the concept of reactivity boundaries stated in the present
paper.
In this article we have focused on the first intersection of the reactivity
boundaries across the section of $\tilde{q}_{1}=0$ with $\tilde{p}_{1}<0$
located in the well region. This corresponds to the fast stage of the reaction
process, that is, “before leaving from that well” and “after entering with one
reflection back by the potential wall in that well.” Reactivity boundaries
also enable us to quantify the slow stage of the process by the projection of
the second, third, fourth intersections of the boundaries onto, e.g., the
$\varphi-p_{\varphi}$ space. Distributions of such intersections on some
projected spaces can trace how statistical properties may emerge for slower
timescales (yielding a more uniform distribution), making conventional
statistical rate theories applicable. Note that as demonstrated in this
article the first intersection corresponding to the reactive initial
conditions are distributed in a nonuniform manner, to which conventional
statistical rate theories are not applicable. The essential understanding of
reactions requires reactivity boundaries that enable us to predict the fate of
reactions independent of which timescale to be considered.
In the extraction scheme of reaction boundary presented in Sec. II.2, we have
not restricted the definition of states to a local equilibrium state in which
highly-developed chaos is implicitly postulated. As known, at least for two
DoFs systems in Ref. Davis and Gray, 1986; Mackay _et al._ , 1984, there may
exist several dynamic states within a single potential well whose number and
the reaction rate constants among them are energy-dependent. The definition of
states in Sec. II.2 can involve such nonergodic states. In addition, as
discussed in the text, the seed of reactivity boundaries existing in between
the states involves not necessarily only one single saddle point but also
several saddle points with different indices.
The practical methods for extracting the reactivity boundaries, however, need
still much to be considered. When only one saddle point plays a dominant role
in determining the occurrence of the reaction, normal form theory readily
extracts the seed of reactivity boundaries in the analytical way. In contrast,
there is still no practical method applicable to general cases where more than
one saddle points are involved in the reaction process. In the present
investigation, because of the preknowledge concerning the existence of
symmetry, we can identify the seed of reactivity boundaries in the
intermediate region. When the symmetry cannot be exploited easily, it is still
a challenging future work to devise convenient methods to extract seeds of
reactivity boundariesNagahata _et al._ (2013).
## V Acknowledgment
TK has greatly benefited and been inspired from many discussions with Prof.
Oka and his enthusiasm on how new concepts emerge more than we may expect when
two different disciplines meet with each other such as chemistry and astronomy
in nature. TK would like to dedicate this article using the concept of
chemistry and celestial mechanics to him in token of his gratitude for Prof.
Oka’s insightful thought. This work has been partially supported by JSPS,
Research Center for Computational Science, Okazaki, Japan, Grant-in-Aid for
Young Scientists (B) (to SK), Grant-in-Aid for challenging Exploratory
Research (to TK), and Grant-in-Aid for Scientific Research (B) (to TK) from
the Ministry of Education, Culture, Sports, Science and Technology.
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## VI Appendix: Uniform sampling
Here we explain how we sample the uniform distributions under constraints to
depict the reactivity boundaries and the sets of reacted/reacting
trajectories, i.e., those having just crossed the surface of $z=0$ from the
product well and those being about to cross the surface, in the reactant well
described in Sec. IIIB. To depict reactivity boundaries, we sample the
position coordinate $(R,\varphi)$ according to the following distributions.
$\displaystyle\rho(R,\varphi;z=0,p_{z}=0,H=E)$
$\displaystyle\propto\int\delta(E-H(\mathbf{p},\mathbf{q}))\delta(z)\delta(p_{z})dp_{R}dp_{z}dp_{\varphi}dz$
$\displaystyle\propto\sqrt{E-V(R,\varphi;z=0)}.$ (9)
Here we define
$\bar{\rho}_{\text{sd}}(R,\varphi)=\sqrt{\frac{E-V(R,\varphi;z=0)}{E-V_{0}}}$
yielding $0<\bar{\rho}_{\text{sd}}<1$, where
$V_{0}=\min_{R,\varphi}V(R,\varphi;z=0)$. We employ the rejection method Press
_et al._ (2007) to sample phase space points with the distribution
$\bar{\rho}_{\text{sd}}$. We first sample points uniformly in the range of
$\varphi\in[-\pi,\pi]$ and $R\in[2\text{\AA},2.2\text{\AA}]$ which include the
whole energetically accessible region. The point is accepted or rejected by
the following criterion:
$\begin{cases}\text{accept}&\bar{\rho}_{\text{sd}}(R,\varphi)>\text{RAND},\\\
\text{reject}&\text{otherwise},\end{cases}$ (10)
where $\mathrm{RAND}$ is a uniform random number from $0$ to $1$. Then we
perform sampling of the momentum for each sampled configuration as follows:
$\displaystyle p_{R}=\sqrt{2(E-V)}\sin\theta/m_{R},$ $\displaystyle
p_{\varphi}=\sqrt{2(E-V)}\cos\theta/(I_{\varphi}/2),$
where $\theta$ is a uniform random number from $-\pi$ to $\pi$.
Similarly, to depict the sets of reacted/reacting trajectories in the reactant
well, we sample phase space points according to the following distribution:
$\displaystyle\rho(\tilde{q}_{2},\varphi;\tilde{q}_{1}=0,\tilde{p}_{1}<0,H=E)$
$\displaystyle\propto\int\delta(E-H(\mathbf{p},\mathbf{q}))\Theta(-\tilde{p}_{1})\delta(\tilde{q}_{1})d\tilde{p}_{1}d\tilde{p}_{2}d\tilde{p}_{\varphi}d\tilde{q}_{1}$
$\displaystyle\propto E-V(\tilde{q}_{2},\varphi;\tilde{q}_{1}=0).$ (11)
Here $\Theta(x)$ is the Heaviside step function, and we define
$\bar{\rho}_{\text{wl}}(\tilde{q}_{2},\varphi)=(E-V(\tilde{q}_{2},\varphi;\tilde{q}_{1}=0))/(E-V_{0})$,
yielding $0<\bar{\rho}_{\text{wl}}<1$, where
$V_{0}=\min_{\tilde{q}_{2},\varphi}V(\tilde{q}_{2},\varphi;\tilde{q}_{1}=0)$.
We sample points uniformly in the range of $\varphi\in[-\pi,\pi]$ and
$\tilde{q}_{2}\in[-0.15\text{\AA}\mathrm{u^{1}/2},0.15\text{\AA}\mathrm{u^{1}/2}]$
which include the whole energetically accessible region on this section. We
apply the same rejection method Press _et al._ (2007) to construct
$\bar{\rho}_{\text{wl}}$ distribution
$\begin{cases}\text{accept}&\bar{\rho}_{\text{wl}}(\tilde{q}_{2},\varphi)>\text{RAND},\\\
\text{reject}&\text{otherwise}.\end{cases}$ (12)
Then we perform sampling of the momentum for each sampled configuration as
follows:
$\begin{cases}\text{accept}&\sin\theta_{1}>\text{RAND},\\\
\text{reject}&\text{otherwise}.\end{cases}$ (13)
$\displaystyle\tilde{p}_{\varphi}$ $\displaystyle=$
$\displaystyle\sqrt{2(E-V)}\sin\theta_{1}\sin\theta_{2},$
$\displaystyle\tilde{p}_{1}$ $\displaystyle=$
$\displaystyle-\sqrt{2(E-V)}\cos\theta_{1},$ $\displaystyle\tilde{p}_{2}$
$\displaystyle=$ $\displaystyle\sqrt{2(E-V)}\sin\theta_{1}\cos\theta_{2},$
since coordinate transformation to polar coordinates introduces phase space
Jacobian $J=2(E-V)\sin\theta_{1}$, where $\theta_{1},\theta_{2}$ are uniform
random numbers from $0$ to $\pi/2$ and from $-\pi$ to $\pi$, respectively.
|
arxiv-papers
| 2013-08-14T06:53:47 |
2024-09-04T02:49:49.396812
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yutaka Nagahata, Hiroshi Teramoto, Chun-Biu Li, Shinnosuke Kawai, and\n Tamiki Komatsuzaki",
"submitter": "Yutaka Nagahata",
"url": "https://arxiv.org/abs/1308.3048"
}
|
1308.3089
|
# LAN property for discretely observed solutions to Lévy driven SDE’s
D. O. Ivanenko Kyiv National Taras Shevchenko University, Volodymyrska, 64,
Kyiv, 01033, Ukraine [email protected] and A. M. Kulik Institute of
Mathematics, Ukrainian National Academy of Sciences, 01601 Tereshchenkivska,
3, Kyiv, Ukraine [email protected]
###### Abstract.
The LAN property is proved in the statistical model based on discrete-time
observations of a solution to a Lévy driven SDE. The proof is based on a
general sufficient condition for a statistical model based on a discrete
observations of a Markov process to possess the LAN property, and involves
substantially the Malliavin calculus-based integral representations for
derivatives of log-likelihood of the model.
###### Key words and phrases:
LAN property, Likelihood function, Lévy driven SDE, Regular statistical
experiment
## 1\. Introduction
Consider stochastic equation of the form
(1) $dX_{t}^{\theta}=a_{\theta}(X_{t}^{\theta})dt+dZ_{t},$
where $a:\Theta\times\mathbb{R}\to\mathbb{R}$ is a measurable function,
$\Theta=(\theta_{1},\theta_{2})\in\mathbb{R}$ is a parametric set. For a given
$\theta\in\Theta$, assuming that the drift term $a_{\theta}$ satisfies the
standard local Lipschitz and linear growth conditions, Eq. (1) uniquely
defines a Markov process $X$. The aim of this paper is to establish the _local
asymptotic normality_ property (LAN in the sequel) in a model, where the
process $X$ is discretely observed with a fixed time discretization value
$h>0$, and a number of observation $n\to\infty$.
The LAN property provides a convenient and powerful tool for establishing
lower efficiency bounds in a statistical model, e.g. [6], [17], [18]. Such a
property for statistical models, based on discrete observations of processes
with Lévy noise, was studied mostly in the cases, where the likelihood
function (or, at least its “main part”) is explicit, in a sense, e.g. [1],
[2], [7], [12], [13]. In the above references the models are linear in the
sense that the process under the observation is either a Lévy process, or a
solution of a linear (Ornstein-Uhlenbeck type) SDE driven by a Lévy process.
The general non-linear case remains non-studied to a great extent, and
apparently the main reason for this is that the transition probability density
of the observed Markov process in that case is highly implicit. In this paper
we develop tools, convenient for proving the LAN property in the framework of
discretely observed solutions to SDE’s with a Lévy noise. To make the
exposition reasonably transparent, we confine ourselves to a particular case
of one-dimensional and one-parameter model, and a fixed sample frequency $h$.
Various extensions (general state space, multiparameter model, high frequency
sampling, etc.) are visible, but we postpone their detailed analysis for a
further research.
Our approach consists of two principal parts. On one hand, we design a general
sufficient condition for a statistical model based on a discrete observations
of a Markov process to possess the LAN property, see Theorem 1 below. This
result extends the classical LeCam’s result about the LAN property for i.i.d.
samples, and it close [5, Theorem 13], with some substantial differences in
the basic assumptions, which makes our result well designed to a study of a
model based on observations of a Lévy driven SDE, see Remark 1 below. On the
other hand, we integral representations of derivatives of 1st and 2nd orders
of the log-likelihood are available: our recent papers [11] and [10] we have
derived such representations using the Malliavin calculus tools. The virtue of
this approach is the same with the one developed in [4] in the diffusion
setting, but with substantial changes which comes from non-diffusive structure
of the noise. Combination of these two principal parts leads to a required LAN
property.
The structure of the paper follows the two-stage scheme outlined above. First
we formulate in Section 2.1 (and prove in Section 3) a general sufficient
condition for the LAN property in a Markov model. Then we formulate in Section
2.2 (and prove in Section 4) our main result about the LAN property for a
discretely observed solution to a Lévy driven SDE; here the proof involves
substantially the Malliavin calculus-based integral representations of
derivatives of the log-likelihood from [11] and [10].
## 2\. The main results
### 2.1. LAN property for discretely observed Markov processes
Let $X$ be a Markov process taking its values in a locally compact metric
space $\mathbb{X}$. The law of $X$ is assumed to depend on a real-valued
parameter $\theta$; in what follows, we assume that the parametric set
$\Theta$ is an interval $(\theta_{1},\theta_{2})\in\mathbb{R}$. We denote by
$\mathsf{P}_{x}^{\theta}$ the law of $X$ with $X_{0}=x$, which corresponds to
the parameter value $\theta$; the expectation w.r.t. $\mathsf{P}_{x}^{\theta}$
is denoted by $\mathsf{E}_{x}^{\theta}$. For a given $h>0$, we denote by
$\mathsf{P}_{x,n}^{\theta}$ the law w.r.t. $\mathsf{P}_{x}^{\theta}$ of the
vector $X^{n}=\left\\{X_{hk},k=1,\dots,n\right\\}$ of discrete time
observations of $X$ with the step $h$. Denote by $\mathcal{E}_{n}$ the
statistical experiment generated by the sample $X^{n}$ with $X_{0}=x,$ i.e.
(2)
$\mathcal{E}_{n}=\Big{(}\mathbb{X}^{n},\mathcal{B}(\mathbb{X}^{n}),\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\Big{)};$
we refer to [8] for the notation and terminology. Our aim is to establish the
LAN property for the sequence of experiments $\\{\mathcal{E}_{n}\\}$.
Recall that the sequence of statistical experiments $\\{\mathcal{E}_{n}\\}$
(or, equivalently, the family
$\\{\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\\}$) is said to have _the LAN
property_ at the point $\theta_{0}\in\Theta$ as $n\rightarrow\infty$, if for
some sequence $r(n)>0,n\geq 1$ and all $u\in\mathbb{R}$
$Z_{n,\theta_{0}}(u):=\frac{d\mathsf{P}_{x,n}^{\theta_{0}+r(n)u}}{d\mathsf{P}_{x,n}^{\theta_{0}}}(X^{n})=\exp\left\\{\Delta_{n}(\theta_{0})u-\frac{1}{2}u^{2}+\Psi_{n}(u,\theta_{0})\right\\},$
with
(3) $\mathcal{L}\left(\Delta_{n}(\theta_{0})\ |\
\mathsf{P}_{x,n}^{\theta_{0}}\right)\Rightarrow N(0,1),\quad
n\rightarrow\infty;$ (4)
$\Psi_{n}(u,\theta_{0})\stackrel{{\scriptstyle\mathsf{P}_{x,n}^{\theta_{0}}}}{{\longrightarrow}}0,\quad
n\rightarrow\infty.$
In what follows we assume that $X$ admits a transition probability density
$p_{h}(\theta;x,y)$ w.r.t. some $\sigma$-finite measure $\lambda$.
Furthermore, we assume that the experiment $\mathcal{E}_{1}$ is _regular_ ;
that is, for every $x\in\mathbb{X}$
* (a)
the function $\theta\mapsto p_{h}(\theta;x,y)$ is continuous for
$\lambda$-a.a. $y\in\mathbb{X}$;
* (b)
the function $\sqrt{p_{h}(\theta;x,\cdot)}$ is differentiable in
$L_{2}(\mathbb{X},\lambda)$; that is, there exists $q_{h}(\theta;x,\cdot)\in
L_{2}(\mathbb{X},\lambda)$ such that
$\int_{\mathbb{X}}\left({\sqrt{p_{h}(\theta+\delta;x,y)}-\sqrt{p_{h}(\theta;x,y)}\over\delta}-q_{h}(\theta;x,y)\right)^{2}\lambda(dy)\to
0,\quad\delta\to 0;$
* (c)
the function $q_{h}(\theta;x,\cdot)$ is continuous in
$L_{2}(\mathbb{X},\lambda)$ w.r.t. $\theta$; that is,
$\int_{\mathbb{X}}\Big{(}q_{h}(\theta+\delta;x,y)-q_{h}(\theta;x,y)\Big{)}^{2}\lambda(dy)\to
0,\quad\delta\to 0.$
Denote
(5) $g_{h}(\theta,x,y)=2q_{h}(\theta;x,y)\sqrt{p_{h}(\theta;x,y)};$
note that by the definition of $q_{h}$ the function $g_{h}$ is well defined
and satisfies
(6) $\mathsf{E}_{x}^{\theta}g_{h}(\theta;x,X_{h})=0$
for every $x\in\mathbb{R},\theta\in\Theta$. Furthermore, denote
(7)
$I_{n}(\theta)=\sum_{k=1}^{n}\mathsf{E}_{x}^{\theta}\Big{(}g_{h}(\theta;X_{h(k-1)},X_{hk})\Big{)}^{2}=4\mathsf{E}_{x}^{\theta}\sum_{k=1}^{n}\int_{\mathbb{X}}\left(q_{h}(\theta;X_{h(k-1)},y)\right)^{2}\lambda(dy).$
Assuming that the statistical experiment $\mathcal{E}_{n}$ is regular, the
above integral is finite and defines the _Fisher information_ for
$\mathcal{E}_{n}$.
We fix $\theta_{0}\in\Theta$, and put $r(n)=I_{n}^{-1/2}(\theta_{0})$ for $n$
large enough, assuming that for those $n$ one has $I_{n}(\theta_{0})>0$.
###### Theorem 1.
Suppose the following.
* 1.
Statistical experiment (2) is regular for every $x\in\mathbb{X}$ and $n\geq
1$; for $n$ large enough $I_{n}(\theta_{0})>0$.
* 2.
The sequence
$r(n)\sum_{j=1}^{n}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right),\quad n\geq
1$
is asymptotically normal w.r.t. $P_{x}^{\theta_{0}}$ with parameters $(0,1)$.
* 3.
The sequence
$r^{2}(n)\sum_{j=1}^{n}g_{h}^{2}(\theta_{0};X_{h(j-1)},X_{hj}),\quad n\geq 1$
converges to 1 in $P_{x}^{\theta_{0}}$-probability.
* 4.
There exists a constant $p>2$ such that
(8)
$\lim_{n\rightarrow\infty}r^{p}(n)\mathsf{E}_{x}^{\theta_{0}}\sum_{j=1}^{n}\left|g_{h}(\theta_{0};X_{h(j-1)},X_{hj})\right|^{p}=0.$
* 5.
For every $N>0$
(9)
$\lim_{n\rightarrow\infty}\sup_{|v|<N}r^{2}(n)\mathsf{E}_{x}^{\theta_{0}}\sum_{j=1}^{n}\int_{\mathbb{X}}\left(q_{h}\left(\theta_{0}+r(n)v;X_{h(j-1)},y\right)-q_{h}(\theta_{0};X_{h(j-1)},y)\right)^{2}\lambda(dy)=0.$
Then $\\{\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\\}$ has the LAN property at
the point $\theta_{0}$.
###### Remark 1.
The above theorem is closely related to [5, Theorem 13]. One important
difference is that in [5] the main conditions are formulated in the terms of
the functions
$\sqrt{p_{h}(\theta+t;x,y)/p_{h}(\theta;x,y)}-1,$
while within our approach the main assumptions are imposed on the log-
likelihood derivative $g_{h}(\theta;x,y)$, and can be verified efficiently
e.g. in a model where $X$ is defined by an SDE with jumps; see Section 2.2
below. Another important difference is that the whole approach in [5] is
developed under the assumption that the log-likelihood function smoothly
depends on the parameter $\theta$. For a model where $X$ is defined by an SDE
with jumps, such an assumption may be very restrictive, see the detailed
discussion in [11]. This is the reason why we use instead the assumption of
regularity of the experiments, which both is much milder and is easily
verifiable, see [11].
Let us note briefly two possible extensions of the above result, which can be
obtained without any essential changes in the proof. We do not expose them
here in details, because they will not be used in the current paper.
###### Remark 2.
The statement of Theorem 1 still holds true if $h$ is allowed to depend on
$n$, with conditions 1 – 5 respectively changed.
###### Remark 3.
The statement of Theorem 1 still holds true if, instead of one $\theta_{0}$, a
sequence $\theta_{n}\to\theta_{0}$ is considered, with conditions 2 – 5
respectively changed. Moreover, in that case relation (3) and (4) would still
hold true if instead of a fixed $u$ a sequence $u_{n}\to u$ is considered.
That is, under the uniform version of conditions 2 – 5 the _uniform asymptotic
normality_ would hold true; see [8, Definition 2.2].
### 2.2. LAN property for families of distributions of solutions to Lévy
driven SDE’s
We assume that $Z$ in the SDE (1) is a Lévy process without a diffusion
component; that is,
$Z_{t}=ct+\int_{0}^{t}\int_{|u|>1}u\nu(ds,du)+\int_{0}^{t}\int_{|u|\leq
1}u\tilde{\nu}(ds,du),$
where $\nu$ is a Poisson point measure with the intensity measure $ds\mu(du)$,
and $\tilde{\nu}(ds,du)=\nu(ds,du)-ds\mu(du)$ is respective compensated
Poisson measure. In the sequel, we assume the Lévy measure $\mu$ to satisfy
the following.
H. (i) for some $\beta>0$,
$\int_{|u|\geq 1}u^{4+\beta}\mu(du)<\infty;$
(ii) for some $u_{0}>0$, the restriction of $\mu$ on $[-u_{0},u_{0}]$ has a
positive density $m\in
C^{2}\left(\left[-u_{0},0\right)\cup\left(0,u_{0}\right]\right)$;
(iii) there exists $C_{0}$ such that
$|m^{\prime}(u)|\leq C_{0}|u|^{-1}m(u),\quad|m^{\prime\prime}(u)|\leq
C_{0}u^{-2}m(u),\quad|u|\in(0,u_{0}];$
(iv)
$\left(\log\frac{1}{\varepsilon}\right)^{-1}\mu\Big{(}\\{u:|u|\geq\varepsilon\\}\Big{)}\to\infty,\quad\varepsilon\to
0.$
One particularly important class of Lévy processes satisfying H consists of
_tempered $\alpha$-stable processes_ (see [21]), which arise naturally in
models of turbulence [20], economical models of stochastic volatility [3],
etc.
Denote by $C^{k,m}(\mathbb{R}\times\Theta),k,m\geq 0$ the class of functions
$f:\mathbb{R}\times\Theta\to\mathbb{R}$ which has continuous derivatives
$\frac{\partial^{i}}{\partial x^{i}}\frac{\partial^{j}}{\partial\
\theta^{j}}f,\quad i\leq k,\quad j\leq m.$
About the coefficient $a_{\theta}(x)$ in Eq. (1) we assume the following.
A. (i) $a\in C^{3,2}(\mathbb{R}\times\Theta)$ have bounded derivatives
$\partial_{x}a$, $\partial^{2}_{xx}a$, $\partial^{2}_{x\theta}a$,
$\partial^{3}_{xxx}a$, $\partial^{3}_{x\theta\theta}a$,
$\partial^{3}_{xx\theta}a$, $\partial^{4}_{xxx\theta}a$ and
(10)
$|a_{\theta}(x)|+|\partial_{\theta}a_{\theta}(x)|+|\partial^{2}_{\theta\theta}a_{\theta}(x)|\leq
C(1+|x|),\quad\theta\in\Theta,\quad x\in\mathbb{R}.$
(ii) For a given $\theta_{0}\in\Theta$, there exists a neighbourhood
$(\theta_{-},\theta_{+})\subset\Theta$ of $\theta_{0}$ such that
$\limsup_{|x|\rightarrow\infty}\frac{a_{\theta}(x)}{x}<0\quad\hbox{uniformly
by }\theta\in(\theta_{-},\theta_{+}).$
It is proved in [11] that, under conditions A(i) and H, the following
properties hold:
* •
the Markov process $X$ given by (1) has a transition probability density
$p_{t}^{\theta}$ w.r.t. the Lebesgue measure;
* •
this density has a derivative $\partial_{\theta}p_{t}^{\theta}(x,y),$ and the
statistical experiment (2) is regular;
* •
the function $g_{t}^{\theta}$, given by (5) satisfies (6).
Hence all the pre-requisites for Theorem 1, given in Section 2.1, are
available with $\lambda(dx)=dx$ (the Lebesgue measure).
Furthermore, under conditions A and H, for $\theta=\theta_{0}$ corresponding
Markov process $X$ is ergodic, i.e. there exists unique invariant probability
measure $\varkappa_{inv}^{\theta_{0}}$ for $X$. One can verify this easily,
using conditions, sufficient for ergodicity of solutions to Lévy driven SDE’s,
given in [19] and [14]. Denote by $\\{X^{st,\theta_{0}}_{t},t\in\mathbb{R}\\}$
corresponding stationary version of $X$; that is, a Markov process, defined on
whole axis $\mathbb{R}$, which has the transition probabilities with $X$ and
one-dimensional distributions equal to $\varkappa_{inv}^{\theta_{0}}$.
Clearly, the existence of such a process, on a proper probability space, is
guaranteed by the Kolmogorov consistency theorem. Denote
(11)
$\sigma^{2}(\theta_{0})=\mathsf{E}\Big{(}g_{h}(\theta_{0};X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}})\Big{)}^{2}=\int_{\mathbb{R}}\int_{\mathbb{R}}\Big{(}g_{h}(\theta_{0};x,y)\Big{)}^{2}p_{h}(\theta_{0};x,y)\,dy\varkappa_{inv}^{\theta_{0}}(dx).$
The following theorem performs the main result of this paper. Its proof is
given in Section 4 below.
###### Theorem 2.
Let conditions A and H hold true, and
$\sigma^{2}(\theta_{0})>0.$
Then the family $\\{\mathsf{P}_{x,n}^{\theta},\theta\in\Theta\\}$ possesses
the LAN property at the point $\theta=\theta_{0}$.
## 3\. Proof of Theorem 1
The method of proof goes back to LeCam’s proof of the LAN property for i.i.d.
samples, see e.g. Theorem II.1.1 and Theorem II.3.1 in [8]. In the Markov
setting, the dependence in the observations lead to some additional
technicalities; see e.g. (19). The possible ways to overcome these additional
difficulties can be found, in a slightly different setting, in the proof of
[5, Theorem 13]. In order to keep the exposition transparent and self-
sufficient, we prefer to give a complete proof of Theorem 1 explicitly, rather
than to give a chain of partly relevant references.
We separate the proof into several lemmas; in all the lemmas in this section
we assume the conditions of Theorem 1 to be fulfilled. Values $x,\theta_{0},$
and $u$ are fixed; we assume that $n$ is large enough, so that
$\theta_{0}+r(n)u\in\Theta$. In order to simplify the notation below we write
$\theta$ instead of $\theta_{0}$.
Denote
$\zeta^{\theta}_{jn}(u)=\left(\left(\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}\right)^{1/2}-1\right)I\left\\{p_{h}(\theta;X_{h(j-1)},X_{hj})\neq
0\right\\}.$
###### Lemma 1.
One has
(12)
$\limsup_{n\rightarrow\infty}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}\leq\frac{1}{4}u^{2}$
and
(13)
$\lim_{n\rightarrow\infty}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}=0.$
###### Proof.
By the regularity of $\mathcal{E}_{1}$ and the Cauchy inequality we have
$\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}\\\
=\mathsf{E}_{x}^{\theta}\int\limits_{\\{y:p^{\theta}_{h}(z,y)\neq
0\\}}\left(\sqrt{p_{h}\left(\theta+r(n)u;X_{h(j-1)},y\right)}\right.\\\
\left.-\sqrt{p_{h}\left(\theta;X_{h(j-1)},y\right)}-r(n)uq_{h}(\theta;X_{h(j-1)},y)\right)^{2}\lambda(dy)\\\
\leq(r(n)u)^{2}\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\left(\int_{0}^{1}q_{h}\left(\theta+r(n)uv,X_{h(j-1)},y\right)-q_{h}\left(\theta;X_{h(j-1)},y\right)dv\right)^{2}\lambda(dy)\\\
\leq(r(n)u)^{2}\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\lambda(dy)\int_{0}^{1}\left(q_{h}\left(\theta+r(n)uv;X_{h(j-1)},y\right)-q_{h}\left(\theta;X_{h(j-1)},y\right)\right)^{2}dv.$
This and (9) yield (13). To deduce (12) from (13), recall an elementary
inequality
(14) $|AB|\leq\frac{\alpha}{2}A^{2}+\frac{1}{2\alpha}B^{2},\quad\alpha>0,$
and write
$\zeta^{\theta}_{jn}(u)=\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})+\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)=:A+B.$
Then
$\displaystyle\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}$
$\displaystyle\leq(1+\alpha){1\over
4}u^{2}r^{2}(n)\mathsf{E}_{x}^{\theta}\left(g_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}$
$\displaystyle+\left(1+{1\over\alpha}\right)\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}.$
Because by the construction
(15)
$\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(g_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}=I_{n}(\theta)=r^{-2}(n),$
this leads to the bound
$\limsup_{n\rightarrow\infty}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}\leq\frac{1+\alpha}{4}u^{2}.$
Since $\alpha>0$ is arbitrary, this completes the proof. ∎
###### Lemma 2.
One has
(16) $\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}\to\frac{u^{2}}{4},\quad
n\to\infty$
in $\mathsf{P}_{x}^{\theta}$-probability.
###### Proof.
By the Chebyshev inequality,
$\mathsf{P}_{x}^{\theta}\left\\{\left|\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}-\frac{1}{4}r^{2}(n)u^{2}\sum_{j=1}^{n}(g_{h}(\theta;X_{h(j-1)},X_{hj}))^{2}\right|>\varepsilon\right\\}\\\
\leq\frac{1}{\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left|(\zeta^{\theta}_{jn}(u))^{2}-\frac{1}{4}r^{2}(n)u^{2}(g_{h}(\theta;X_{h(j-1)},X_{hj}))^{2}\right|\\\
=\frac{1}{\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left|\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right|\left|\zeta^{\theta}_{jn}(u)+\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right|$
which by (14), for a given $\alpha>0$, is dominated by
$\frac{1}{2\alpha\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}\\\
+\frac{\alpha}{2\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)+\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2}.$
By (13) the first item of this expression tends to zero as
$n\rightarrow\infty$. Furthermore, the Cauchy inequality together with (12)
and (15) imply that for the second one the upper limit does not exceed
$\limsup_{n\to\infty}\left(\frac{\alpha}{\varepsilon}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}+\frac{\alpha
u^{2}}{2\varepsilon}r^{2}(n)\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(g_{h}\left(\theta;X_{h(j-1)},X_{hj}\right))^{2}\right)\leq\frac{3\alpha
u^{2}}{2\varepsilon}.$
Since $\alpha>0$ is arbitrary, this proves that the difference
$\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}-\frac{1}{4}r^{2}(n)u^{2}\sum_{j=1}^{n}(g_{h}(\theta;X_{h(j-1)},X_{hj}))^{2}$
tends to $0$ in $\mathsf{P}_{x}^{\theta}$-probability. Combined with the
condition 3 of Theorem 1, this gives the required statement. ∎
###### Lemma 3.
One has
(17) $\max_{1\leq j\leq n}|\zeta^{\theta}_{jn}(u)|\to 0,\quad n\to\infty$
in $\mathsf{P}_{x}^{\theta}$-probability.
###### Proof.
We have
$\mathsf{P}_{x}^{\theta}\left\\{\max_{1\leq j\leq
n}|\zeta^{\theta}_{jn}(u)|>\varepsilon\right\\}\leq\sum_{j=1}^{n}\mathsf{P}_{x}^{\theta}\left\\{|\zeta^{\theta}_{jn}(u)|>\varepsilon\right\\}\\\
\leq\sum_{j=1}^{n}\mathsf{P}_{x}^{\theta}\left\\{\left|\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right|>\frac{\varepsilon}{2}\right\\}\\\
+\sum_{j=1}^{n}\mathsf{P}_{x}^{\theta}\left\\{\left|g_{h}(\theta;X_{h(j-1)},X_{hj})\right|>\frac{\varepsilon}{4r(n)|u|}\right\\}.$
The first sum in the r.h.s. of this inequality vanishes as
$n\rightarrow\infty$ because of (13), the second sum vanishes because of the
condition 4 of Theorem 1. ∎
###### Corollary 1.
By Lemma 3 and Lemma 2, we have
(18) $\sum_{j=1}^{n}|\zeta^{\theta}_{jn}(u)|^{3}\to 0,\quad n\to\infty$
in $\mathsf{P}_{x}^{\theta}$-probability.
Because of the Markov structure of the sample, in addition to Lemma 2 we will
need the following statement. Denote
$\mathcal{F}_{j}=\sigma(X_{hi},i\leq
j),\quad\mathsf{E}_{x,j}^{\theta}=\mathsf{E}_{x}^{\theta}[\cdot|\mathcal{F}_{j}].$
###### Lemma 4.
One has
(19)
$\sum_{j=1}^{n}\mathsf{E}_{x,j-1}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}\to\frac{u^{2}}{4},\quad
n\to\infty$
in $\mathsf{P}_{x}^{\theta}$-probability.
###### Proof.
Denote
$\chi_{jn}=(\zeta^{\theta}_{jn}(u))^{2}-\mathsf{E}_{x,j-1}^{\theta}(\zeta^{\theta}_{jn}(u))^{2},\quad
S_{n}=\sum_{j=1}^{n}\chi_{jn},$
then by (16) it us enough to prove that $S_{n}\to 0$ in
$\mathsf{P}_{x}^{\theta}$-probability. Fix $\varepsilon>0$, and put
$\chi_{jn}^{\varepsilon}=(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|\leq\varepsilon}-\mathsf{E}_{x,j-1}^{\theta}\Big{(}(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|\leq\varepsilon}\Big{)},\quad
S_{n}^{\varepsilon}=\sum_{j=1}^{n}\chi_{jn}^{\varepsilon}.$
By the construction $\\{\chi_{j}^{\varepsilon},j=1,\dots,n\\}$ is a martingale
difference, hence
$\displaystyle\mathsf{E}_{x}^{\theta}(S_{n}^{\varepsilon})^{2}$
$\displaystyle=\sum_{k=1}^{n}\mathsf{E}_{x}^{\theta}(\chi_{jn}^{\varepsilon})^{2}\leq\sum_{k=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{4}1_{|\zeta^{\theta}_{jn}(u)|\leq\varepsilon}\leq\varepsilon^{2}\mathsf{E}_{x}^{\theta}\sum_{k=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}.$
Hence by (12) and the Cauchy inequality,
(20)
$\limsup_{n\to\infty}\mathsf{E}_{x}^{\theta}|S_{n}^{\varepsilon}|\leq{\varepsilon|u|\over
2}$
Now, let us estimate the difference $S_{n}-S_{n}^{\varepsilon}$. Note that,
using the first statement in Lemma 1, one can improve the statement of Lemma 2
and show that the convergence (16) holds true in
$L_{1}(\mathsf{P}_{x}^{\theta})$; see e.g. Theorem A.I.4 in [8]. In
particular, this means that the sequence
$\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2},\quad n\geq 1$
is uniformly integrable. Hence, because by Lemma 3 the probabilities of the
sets
(21) $\Omega_{n}^{\varepsilon}=\left\\{\max_{j\leq
n}|\zeta_{jn}|>\varepsilon\right\\}$
tend to zero as $n\to\infty$, we have
$\mathsf{E}_{x}^{\theta}\left(1_{\Omega_{n}^{\varepsilon}}\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}\right)\to
0.$
One has
$\chi_{jn}-\chi_{jn}^{\varepsilon}=(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|>\varepsilon}-\mathsf{E}_{x,j}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|>\varepsilon},$
hence
$\displaystyle\mathsf{E}_{x}^{\theta}|S_{n}-S_{n}^{\varepsilon}|$
$\displaystyle\leq
2\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}(\zeta^{\theta}_{jn}(u))^{2}1_{|\zeta^{\theta}_{jn}(u)|>\varepsilon}\leq
2\mathsf{E}_{x}^{\theta}\left(1_{\Omega_{n}^{\varepsilon}}\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}\right)\to
0.$
Together with (20) this gives
$\limsup_{n\to\infty}\mathsf{E}_{x}^{\theta}|S_{n}|\leq{\varepsilon|u|\over
2},$
which completes the proof because $\varepsilon>0$ is arbitrary. ∎
The final preparatory result we require is the following.
###### Lemma 5.
One has
(22)
$2\sum_{j=1}^{n}\zeta^{\theta}_{jn}(u)-r(n)u\sum_{j=1}^{n}g_{h}(\theta;X_{h(j-1)},X_{hj})\to-\frac{u^{2}}{4},\quad
n\to\infty$
in $\mathsf{P}_{x}^{\theta}$-probability.
###### Proof.
We have the equality
$(\zeta^{\theta}_{jn}(u))^{2}=\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}-1-2\zeta^{\theta}_{jn}(u)$
valid $\mathsf{P}_{x}^{\theta}$-a.s. Note that by the Markov property of $X$
one has
$\displaystyle\mathsf{E}_{x,j-1}^{\theta}$
$\displaystyle\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}$
$\displaystyle=\int_{\mathbb{X}}\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},y)}{p_{h}(\theta;X_{h(j-1)},y)}p_{h}(\theta;X_{h(j-1)},y)\lambda(dy)=1;$
hence by Lemma 4 one has that
$\sum\limits_{j=1}^{n}\mathsf{E}_{x,j-1}^{\theta}\zeta^{\theta}_{jn}(u)\to-\frac{u^{2}}{8}$
in $\mathsf{P}_{x}^{\theta}$-probability. Therefore, what we have to prove in
fact is that
$V_{n}:=2\sum_{j=1}^{n}\left(\zeta^{\theta}_{jn}(u)-\mathsf{E}_{x,j-1}^{\theta}\zeta^{\theta}_{jn}(u)\right)-r(n)u\sum_{j=1}^{n}g_{h}(\theta;X_{h(j-1)},X_{hj})\to
0$
in $\mathsf{P}_{x}^{\theta}$-probability. By (6) the sequence
$\zeta^{\theta}_{jn}(u)-\mathsf{E}_{x,j-1}^{\theta}\zeta^{\theta}_{jn}(u)-r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj}),\quad
j=1,\dots n$
is a martingale difference, hence
$\mathsf{E}_{x}^{\theta}V_{n}^{2}\leq
4\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\left(\zeta^{\theta}_{jn}(u)-\frac{1}{2}r(n)ug_{h}(\theta;X_{h(j-1)},X_{hj})\right)^{2},$
which tends to zero as $n\to\infty$ by (13). ∎
Now, we can finalize the proof of Theorem 1. Fix $\varepsilon\in(0,1)$ and
consider the sets $\Omega_{n}^{\varepsilon}$ defined by (21); by Lemma 3 we
have $\mathsf{P}_{x}^{\theta}(\Omega_{n}^{\varepsilon})\to 0$. Using the
Taylor expansion for the function $\log(1+x)$, we obtain that there exist a
constant $C_{\varepsilon}$ and random variables $\alpha_{jn}$ such that
$|\alpha_{jn}|<C_{\varepsilon}$, for which the following identity holds true
outside of the set $\Omega_{n}^{\varepsilon}$:
$\sum_{j=1}^{n}\log\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}=2\sum_{j=1}^{n}\zeta^{\theta}_{jn}(u)-\sum_{j=1}^{n}(\zeta^{\theta}_{jn}(u))^{2}+\sum_{j=1}^{n}\alpha_{jn}|\zeta^{\theta}_{jn}(u)|^{3}.$
Then by Lemma 2, Lemma 5, and Corollary 1 we have
$\displaystyle\log Z_{n,\theta}(u)$
$\displaystyle=\sum_{j=1}^{n}\log\frac{p_{h}(\theta+r(n)u;X_{h(j-1)},X_{hj})}{p_{h}(\theta;X_{h(j-1)},X_{hj})}$
$\displaystyle\hskip
28.45274pt=r(n)u\sum_{j=1}^{n}g_{h}(\theta;X_{h(j-1)},X_{hj})-\frac{u^{2}}{4}-\frac{u^{2}}{4}+\Psi_{n},$
where $\Psi_{n}\to 0$ in $\mathsf{P}_{x}^{\theta}$-probability. By the
asymptotic normality condition 2, this completes the proof. ∎
## 4\. Proof of Theorem 2
To prove Theorem 2, we verify the conditions of Theorem 1. First, let us give
an auxiliary result, which will be used repeatedly in the proof.
###### Lemma 6.
Under conditions A and H for every $p\in(2,4+\beta)$ there exists a constant
$C$ such that for all $x\in\mathbb{R}$, $\theta\in(\theta_{-},\theta_{+})$,
and $t\geq 0$
(23) $\mathsf{E}_{x}^{\theta}\Big{|}g_{h}(\theta;x,X_{h})\Big{|}^{p}\leq
C(1+|x|)^{p},\quad\mathsf{E}_{x}^{\theta}|X_{t}|^{p}\leq C(1+|x|^{p}).$
###### Proof.
The first inequality is proved in Lemma 1 [11]. One can prove the second
inequality, using a standard argument based on the Lyapunov condition for the
function $V(x)=|x|^{p};$ e.g. Proposition 4.1 [14]. ∎
Recall (e.g. [14], Section 3.2) that one standard way to construct the
invariant measure $\varkappa^{\theta_{0}}_{inv}$ is to take a weak limit point
(as $T\to\infty$) for the family of _Khas’minskii’s averages_
$\varkappa^{\theta_{0}}_{T}(dy)={1\over
T}\int_{0}^{T}\mathsf{P}_{x}^{\theta_{0}}(X_{t}\in dy)\,dt.$
Then, by the Fatou lemma, the second relation in (23) implies the following
moment bound for $\varkappa^{\theta_{0}}_{inv}$.
###### Corollary 2.
For every $p\in(2,4+\beta)$,
$\int_{\mathbb{R}}|y|^{p}\varkappa^{\theta_{0}}_{inv}(dy)<\infty.$
Everywhere below we assume conditions of Theorem 2 to hold true.
###### Lemma 7.
The sequence
${1\over\sqrt{n}}\sum_{j=1}^{n}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right),\quad
n\geq 1$
is asymptotically normal w.r.t. $P_{x}^{\theta_{0}}$ with parameters
$(0,\sigma^{2}(\theta_{0}))$, where $\sigma^{2}(\theta_{0})$ is defined in
(11).
###### Proof.
The idea of the proof is similar to the one of the proof of Theorem 3.3 [16].
Denote
$Q_{n}(\theta_{0},X)={1\over\sqrt{n}}\sum_{j=1}^{n}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right).$
By Theorem 2.2 [19] (see also Theorem 1.2 [14]), the $\alpha$-mixing
coefficient $\alpha(t)$ for the stationary version of the process $X$ does not
exceed $C_{3}e^{-C_{4}t}$, where $C_{3},C_{4}$ are some positive constants.
Then by CLT for stationary sequences (Theorem 18.5.3 [9]) and (23) we have
$Q_{n}(\theta_{0},X^{st,\theta_{0}})\Rightarrow\mathcal{N}\left(0,\widetilde{\sigma}^{2}(\theta_{0})\right),\quad
n\rightarrow\infty$
with
$\widetilde{\sigma}^{2}(\theta_{0})=\sum_{k=-\infty}^{+\infty}\mathsf{E}\left(g_{h}\left(\theta_{0};X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}}\right)g_{h}\left(\theta;X_{h(k-1)}^{st,\theta_{0}},X_{hk}^{st,\theta_{0}}\right)\right).$
Furthermore, under conditions of Theorem 2 there exists an exponential
coupling for the process $X$; that is, a two-component process
$Y=(Y^{1},Y^{2})$, possibly defined on another probability space, such that
$Y^{1}$ has the distribution $\mathsf{P}_{x}^{\theta_{0}}$, $Y^{2}$ has the
same distribution with $X^{st,\theta_{0}}$, and for all $t>0$
(24) $\mathsf{P}\Big{(}Y_{t}^{1}\neq Y_{t}^{2}\Big{)}\leq C_{1}e^{-C_{2}t}$
with some constants $C_{1}$, $C_{2}$. The proof of this fact can be found in
[15] (Theorem 2.2). Then for any Lipschitz continuous function
$f:\mathbb{R}\to\mathbb{R}$ we have
(25)
$|\mathsf{E}_{x}^{\theta}f(Q_{n}(\theta_{0},X))-\mathsf{E}f(Q_{n}(\theta_{0},X^{st,\theta_{0}}))|=|\mathsf{E}f(Q_{n}(\theta_{0},Y^{1}))-\mathsf{E}f(Q_{n}(\theta_{0},Y^{2}))|\\\
=\mathrm{Lip}(f)\mathsf{E}|Q_{n}(\theta_{0},Y^{1})-Q_{n}(\theta_{0},Y^{2})|\\\
\leq\frac{\mathrm{Lip}(f)}{\sqrt{n}}\sum_{k=1}^{n}\mathsf{E}\left|g_{h}(\theta_{0};Y^{1}_{h(k-1)},Y^{1}_{hk})-g_{h}(\theta_{0};Y^{2}_{h(k-1)},Y^{2}_{hk})\right|1_{(Y^{1}_{h(k-1)},Y^{1}_{hk})\neq(Y^{2}_{h(k-1)},Y^{2}_{hk})}\\\
\leq\frac{2\mathrm{Lip}(f)}{\sqrt{n}}\sum_{k=1}^{n}\left(\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{1}_{h(k-1)},Y^{1}_{hk}\right)\right|^{p}+\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{2}_{h(k-1)},Y^{2}_{hk}\right)\right|^{p}\right)^{1/p}\\\
\times\left(P\Big{(}Y^{1}_{h(k-1)}\neq
Y^{2}_{h(k-1)}\Big{)}+P\Big{(}Y^{1}_{hk}\neq Y^{2}_{hk}\Big{)}\right)^{1/q},$
where $p,q>1$ are such that $1/p+1/q=1$. Since $Y^{1}$ has the distribution
$\mathsf{P}_{x}^{\theta_{0}}$, by (23) we have for $p\in n(2,4+\beta)$
(26)
$\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{1}_{h(k-1)},Y^{1}_{hk}\right)\right|^{p}=\mathsf{E}_{x}^{\theta_{0}}\left|g_{h}\left(\theta_{0};X_{h(k-1)},X_{hk}\right)\right|^{p}\\\
\leq C\mathsf{E}_{x}^{\theta_{0}}\Big{(}1+|X_{h(k-1)}|^{p})\Big{)}\leq
C+C^{2}(1+|x|^{p}).$
Similarly,
(27)
$\mathsf{E}\left|g_{h}\left(\theta_{0};Y^{2}_{h(k-1)},Y^{2}_{hk}\right)\right|^{p}=\mathsf{E}\left|g_{h}\left(\theta_{0};X_{h(k-1)}^{st,\theta_{0}},X_{hk}^{st,\theta_{0}}\right)\right|^{p}\\\
\leq
C\mathsf{E}\Big{(}1+\Big{|}X_{h(k-1)}^{st,\theta_{0}}\Big{|}^{p})\Big{)}=C+C\int_{\mathbb{R}}|y|^{p}\varkappa^{\theta_{0}}_{inv}(dy),$
and the constant in the right hand side is finite by Corollary 2. Hence (24)
and (25) yield that
$\mathsf{E}_{x}^{\theta}f(Q_{n}(\theta_{0},X))\to\mathsf{E}f(\xi),\quad
n\to\infty,\quad\xi\sim\mathcal{N}\left(0,\widetilde{\sigma}^{2}(\theta_{0})\right)$
for every Lipschitz continuous function $f:\mathbb{R}\to\mathbb{R}$. This
means that the sequence $Q_{n}(\theta_{0},X),n\geq 1$ is asymptotically normal
w.r.t. $P_{x}^{\theta_{0}}$ with parameters
$(0,\tilde{\sigma}^{2}(\theta_{0}))$.
To conclude the proof, it remains to show that
$\widetilde{\sigma}^{2}(\theta_{0})={\sigma}^{2}(\theta_{0})$. This follows
easily from (6) because, by the Markov property of $X^{st,\theta_{0}}$,
$\widetilde{\sigma}^{2}(\theta_{0})=\sigma^{2}(\theta_{0})+2\sum_{k=1}^{\infty}\mathsf{E}\left(g_{h}\left(\theta_{0};X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}}\right)g_{h}\left(\theta;X_{h(k-1)}^{st,\theta_{0}},X_{hk}^{st,\theta_{0}}\right)\right)\\\
=\sigma^{2}(\theta_{0})+2\sum_{k=1}^{\infty}\mathsf{E}\left[g_{h}(\theta;X_{0}^{st,\theta_{0}},X_{h}^{st,\theta_{0}})\Big{(}\mathsf{E}_{x}^{\theta}g_{h}(\theta_{0};x,X_{h})\Big{)}_{x=X_{h(k-1)}^{st,\theta_{0}}}\right].$
∎
Similarly, one can prove that
${1\over{n}}\sum_{j=1}^{n}\Big{(}g_{h}\left(\theta_{0};X_{h(j-1)},X_{hj}\right)\Big{)}^{2}\to\sigma^{2}(\theta_{0}),\quad
n\to\infty$
in $L_{1}(\mathsf{P}_{x}^{\theta_{0}})$; the argument is completely the same,
with the CLT for a stationary sequence replaced by the Birkhoff-Khinchin
ergodic theorem (we omit the details). Hence
(28) $I_{n}(\theta_{0})\sim n\sigma^{2}(\theta_{0}),\quad
r(n)\sim{1\over\sqrt{n}\sigma(\theta_{0})},\quad n\to\infty.$
Therefore conditions 2 – 4 of Theorem 1 are verified. Condition 1 of Theorem 1
also holds true: regularity property is proved in [11], and positivity of
$I_{n}(\theta)$ follows from (28).
Let us prove (9), which then would allow us to apply Theorem 1. It is proved
in [10] that, under the conditions of Theorem 2, the function
$q_{h}(\theta,x,y)$ is $L_{2}$-differentiable w.r.t. $\theta$, and
$\partial_{\theta}q_{h}=\frac{1}{2}(\partial_{\theta}g_{h})\sqrt{p_{h}}+\frac{1}{4}(g_{h})^{2}\sqrt{p_{h}}.$
In addition, it is proved therein that for every $\gamma\in[1,2+\beta/2)$
(29)
$\mathsf{E}_{x}^{\theta}\Big{|}\partial_{\theta}g_{h}(\theta;x,X_{h})\Big{|}^{\gamma}\leq
C(1+|x|)^{\gamma}.$
Then
$\displaystyle\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\left(q_{h}\left(\theta+r(n)v,X_{h(j-1)},y\right)-q_{h}(\theta,X_{h(j-1)},y)\right)^{2}dy$
$\displaystyle\hskip 56.9055pt\leq
r(n)v\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}dy\int_{0}^{r(n)v}\left(\partial_{\theta}q_{h}\left(\theta+s,X_{h(j-1)},y\right)\right)^{2}ds$
$\displaystyle\hskip
56.9055pt\leq\frac{r(n)v}{4}\mathsf{E}_{x}^{\theta}\int_{0}^{r(n)v}ds\int_{\mathbb{R}}\left(\partial_{\theta}g_{h}\left(\theta+s;X_{h(j-1)},y\right)+\frac{1}{2}g_{h}\left(\theta+s;X_{h(j-1)},y\right)^{2}\right)^{2}$
$\displaystyle\hskip 199.16928pt\times p_{h}^{s}(X_{h(j-1)},y)dy$
$\displaystyle\hskip 56.9055pt\leq
Cr(n)^{2}v^{2}\mathsf{E}_{x}^{\theta}\left(1+(X_{h(j-1)})^{4}\right);$
in the last inequality we have used (29) and the first relation in (23). Using
the second relation in (23), we get then
$\sup_{|v|<N}r(n)^{2}\mathsf{E}_{x}^{\theta}\sum_{j=1}^{n}\mathsf{E}_{x}^{\theta}\int_{\mathbb{R}}\left(q_{h}\left(\theta+r(n)v,X_{h(j-1)},y\right)-q_{h}(\theta,X_{h(j-1)},y)\right)^{2}dy\leq
CN^{2}nr(n)^{4}$
with a constant $C$ that depends only on $x$. This relation together with (28)
completes the proof.
## Acknowledgements
The authors are deeply grateful to H. Masuda for a valuable bibliographic help
and useful discussion.
## References
* [1] Aït-Sahalia, Y. and Jacod, J. (2007), Volatility estimators for discretely sampled Lévy processes. Ann. Statist. 35, 355 – 392.
* [2] M. G. Akritas and R. A. Johnson. Asymptotic inference in Lévy processes of the discontinuous type. Ann. Statist., 9:604 – 614, 1981
* [3] P. Carr, H. Geman, D.B. Madan, M. Yor. Stochastic volatility for Lévy processes. Math. Finance, 13:345-382, 2003.
* [4] E. Gobet. Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli, 7(6):899-912, 2001.
* [5] P.E. Greenwood, A.N. Shiryayev. Contiguity and the statistical invariance principle. London, Th. and Appl. of Stoch., Proc., 1985.
* [6] J. Hajek, Local asymptotic minimax admissibility in estimation, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Berkeley and Los Angeles, Univ. of California Press, 175 - 194, 1971.
* [7] R. Höpfner. Two comments on parameter estimation in stable processes. Mathematical methods of statistics , 6:125 – 134, 1997.
* [8] I. A. Ibragimov and R. Z. Hasminskii. Statistical estimation: asymptotic theory. New York, Springer-Verlag, 1981.
* [9] I.A. Ibragimov and Yu.V. Linnik, Independents and stationary associated variables. Moscow, Nauka, 1965 (In Russian).
* [10] D.O. Ivanenko. Stohastic derivatives of solution to Lévy driven SDE’s. To appear in Visnyk Kyiv Nat. Univ.
* [11] D.O. Ivanenko and A.M. Kulik. Malliavin calculus approach to statistical inference for Lévy driven SDE’s. To appear in Meth. and Comp. in Appl. Prob., preprint available at arXiv:1301.5141
* [12] R. Kawai and H. Masuda. Local asymptotic normality for normal inverse Gaussian Lévy pro- cesses with high-frequency sampling. ESAIM: Probability and Statistics 17, 2013.
* [13] A. Kohatsu-Higa, E. Nualart, and Ngoc Khue Tran. LAN property for a linear model with jumps arXiv:1402.4956
* [14] A.M. Kulik. Exponential ergodicity of the solutions to SDE’s with a jump noise. Stoch. Proc. and their Appl., 119:602-632, 2009\.
* [15] A.M. Kulik. Asymptotic and spectral properties of exponentially $\phi$-ergodic Markov processes. Stoch. Proc. and their Appl., 121:1044-1075, 2011.
* [16] A.M. Kulik and N.N. Leonenko. Ergodicity and mixing bounds for the Fisher-Snedecor diffusion. Bernulli, In Press, 2011.
* [17] L. Le Cam, Limits of experiments, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Berkeley and Los Angeles, Univ. of California Press, 245 261, 1971.
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* [19] H. Masuda. Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps. Stoch. Proc. Appl., 117:35-56, 2007.
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|
arxiv-papers
| 2013-08-14T11:33:00 |
2024-09-04T02:49:49.409786
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dmytro Ivanenko, Alexey Kulik",
"submitter": "Dmitry Ivanenko Alexandrovich",
"url": "https://arxiv.org/abs/1308.3089"
}
|
1308.3113
|
# Pseudorandomness in 0’s and 2’s distribution in the iterated absolute
differences of primes
Raffaele Salvia
###### Abstract
Be $d_{m,n}$ a generic element in the infinite matrix $D$, with $d_{1,n}$
defined as the $n^{th}$ prime number and, for any $m>1$,
$d_{m,n}=|d_{m-1,n}-d_{m-1,n+1}|$ (1)
When $n\neq 1$, after the first few terms the columns in the matrix appear to
be constituted entirely by 0s and 2s. Here is reported a computation over
about $4.55\cdot 10^{8}$ elements of $D$, which suggests a pseudo-random
distribution of these two values.
## 1 Introduction
###### Definition 1.
Let $d_{m,n}$ be an element in the infinite matrix $D$, such as
$d_{1,n}=p_{n},$ (2)
where $p_{n}$ denotes the $n^{th}$ prime number, and
$d_{m,n}=|d_{m-1,n}-d_{m-1,n+1}|$ (3)
In 1878, Proth [1] tried in vain to prove that $d_{m,1}=1$ for every $m\geq
1$. This conjecture was reproposed in 1958 by Gilbreath [2] (hence it is known
as _Gilbreath’s Conjecture_). Kilgrove and Ralston (1959 [2]) checked the
conjecture for $m<63\>419$, and Odlyzko (1993 [3]) expanded the verification
up to $m\approx 3.4\cdot 10^{11}$. The vast majority of $D$’s entries is
apparently equal to either 0 or 2. The results reported here encourage the
supposition that the occurrences of these two values have got pseudorandom
properties.
All the programs written for this article were made in C. The computation
covered a total of 454 526 325 items of $D$, i.e., the ones for which
$m+n\leq 30151$ (4)
## 2 Preliminary considerations
### 2.1 Apparent confinement of values greater than 2
Some elements of $D$ are shown in table 1. If $m,n>1$, $d_{m,n}$ must be even;
it is also expected to be equal to 0 or 2 for a sufficiently large $m$.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83
1 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83
2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 6 | 2 | 6 | 4 | 2 | 4 | 6 | 6 | 2 | 6 | 4 | 2 | 6 | 4 | 6
3 | 1 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 0 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2
4 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 4 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 2
5 | 1 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 4 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0
6 | 1 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2
7 | 1 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0
8 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2
9 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 4
10 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2
11 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0
12 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0
13 | 1 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 2
14 | 1 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 0
15 | 1 | 0 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2
16 | 1 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0
17 | 1 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0
18 | 1 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0
19 | 1 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2
20 | 1 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 2
21 | 1 | 0 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0
22 | 1 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2
23 | 1 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2
24 | 1 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 2
25 | 1 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 0
26 | 1 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0
27 | 1 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0
28 | 1 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2
29 | 1 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0
30 | 1 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 2
31 | 1 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 0
32 | 1 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2
33 | 1 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2
34 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 0
35 | 1 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2
36 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 2
37 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 0
38 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0
39 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 2
40 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0
41 | 1 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 2 | 0 | 2
42 | 1 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 0
43 | 1 | 0 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 2
44 | 1 | 2 | 0 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 0
45 | 1 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2
46 | 1 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0
47 | 1 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 2
48 | 1 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 2
49 | 1 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 0
50 | 1 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0
51 | 1 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 0
52 | 1 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 2
53 | 1 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 2
54 | 1 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 2 | 2
55 | 1 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0
56 | 1 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0
57 | 1 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2
58 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0
59 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
60 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
61 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
62 | 1 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2
63 | 1 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2
64 | 1 | 0 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0
65 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2
66 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 2
67 | 1 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2
68 | 1 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 2 | 0
69 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2
70 | 1 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2
Table 1: Values of $d_{m,n}$ for $m\leq 70$ and $n\leq 23$
###### Definition 2.
Given a positive integer $i$, $B(i)$ is the value $k$, if it exists, such as
$d_{k,i}>2$ (5)
and
$d_{m,i}=0\lor d_{m,i}=2$ (6)
for any $m>k$.
A proof of the exitence of any $B(n)$ would likely facilitate the validation
of Gilbreath’s conjecture. In the region of $D$ here analyzed, however, no
value greater than 2 has been found with $m>69$.
### 2.2 Conjectured values of $B(n)$
The first hypothetic values of $B(n)$ are presented in table 2.
$n$ | $B(n)$ | | $n$ | $B(n)$ | | $n$ | $B(n)$ | | $n$ | $B(n)$ | | $n$ | $B(n)$ | | $n$ | $B(n)$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | - | | 46 | 6 | | 91 | 3 | | 136 | 5 | | 181 | 12 | | 226 | 2
2 | 1 | | 47 | 7 | | 92 | 3 | | 137 | 13 | | 182 | 13 | | 227 | 2
3 | 1 | | 48 | 2 | | 93 | 3 | | 138 | 11 | | 183 | 13 | | 228 | 3
4 | 2 | | 49 | 1 | | 94 | 3 | | 139 | 11 | | 184 | 13 | | 229 | 2
5 | 1 | | 50 | 2 | | 95 | 4 | | 140 | 9 | | 185 | 13 | | 230 | 3
6 | 2 | | 51 | 9 | | 96 | 4 | | 141 | 9 | | 186 | 5 | | 231 | 8
7 | 1 | | 52 | 8 | | 97 | 4 | | 142 | 12 | | 187 | 6 | | 232 | 10
8 | 2 | | 53 | 5 | | 98 | 15 | | 143 | 12 | | 188 | 5 | | 233 | 7
9 | 3 | | 54 | 5 | | 99 | 14 | | 144 | 12 | | 189 | 9 | | 234 | 7
10 | 3 | | 55 | 5 | | 100 | 9 | | 145 | 12 | | 190 | 8 | | 235 | 6
11 | 2 | | 56 | 8 | | 101 | 9 | | 146 | 12 | | 191 | 2 | | 236 | 5
12 | 2 | | 57 | 8 | | 102 | 9 | | 147 | 9 | | 192 | 2 | | 237 | 4
13 | 1 | | 58 | 10 | | 103 | 9 | | 148 | 9 | | 193 | 4 | | 238 | 4
14 | 2 | | 59 | 9 | | 104 | 9 | | 149 | 9 | | 194 | 2 | | 239 | 5
15 | 5 | | 60 | 11 | | 105 | 8 | | 150 | 8 | | 195 | 2 | | 240 | 3
16 | 3 | | 61 | 10 | | 106 | 7 | | 151 | 9 | | 196 | 4 | | 241 | 3
17 | 3 | | 62 | 5 | | 107 | 11 | | 152 | 9 | | 197 | 5 | | 242 | 6
18 | 2 | | 63 | 5 | | 108 | 8 | | 153 | 9 | | 198 | 2 | | 243 | 2
19 | 2 | | 64 | 6 | | 109 | 8 | | 154 | 9 | | 199 | 5 | | 244 | 2
20 | 3 | | 65 | 3 | | 110 | 8 | | 155 | 9 | | 200 | 11 | | 245 | 7
21 | 2 | | 66 | 10 | | 111 | 8 | | 156 | 9 | | 201 | 11 | | 246 | 8
22 | 2 | | 67 | 8 | | 112 | 8 | | 157 | 9 | | 202 | 11 | | 247 | 5
23 | 9 | | 68 | 8 | | 113 | 7 | | 158 | 9 | | 203 | 13 | | 248 | 9
24 | 9 | | 69 | 1 | | 114 | 6 | | 159 | 9 | | 204 | 14 | | 249 | 7
25 | 9 | | 70 | 2 | | 115 | 8 | | 160 | 9 | | 205 | 13 | | 250 | 7
26 | 10 | | 71 | 2 | | 116 | 1 | | 161 | 7 | | 206 | 13 | | 251 | 2
27 | 5 | | 72 | 2 | | 117 | 2 | | 162 | 10 | | 207 | 12 | | 252 | 3
28 | 5 | | 73 | 2 | | 118 | 4 | | 163 | 2 | | 208 | 12 | | 253 | 1
29 | 5 | | 74 | 2 | | 119 | 10 | | 164 | 2 | | 209 | 11 | | 254 | 7
30 | 6 | | 75 | 2 | | 120 | 13 | | 165 | 5 | | 210 | 10 | | 255 | 6
31 | 2 | | 76 | 2 | | 121 | 12 | | 166 | 4 | | 211 | 12 | | 256 | 6
32 | 7 | | 77 | 8 | | 122 | 2 | | 167 | 5 | | 212 | 14 | | 257 | 13
33 | 6 | | 78 | 7 | | 123 | 2 | | 168 | 5 | | 213 | 13 | | 258 | 9
34 | 4 | | 79 | 6 | | 124 | 2 | | 169 | 2 | | 214 | 13 | | 259 | 9
35 | 4 | | 80 | 6 | | 125 | 2 | | 170 | 12 | | 215 | 13 | | 260 | 9
36 | 2 | | 81 | 5 | | 126 | 2 | | 171 | 11 | | 216 | 15 | | 261 | 13
37 | 2 | | 82 | 4 | | 127 | 2 | | 172 | 14 | | 217 | 15 | | 262 | 11
38 | 2 | | 83 | 3 | | 128 | 2 | | 173 | 13 | | 218 | 9 | | 263 | 17
39 | 6 | | 84 | 2 | | 129 | 2 | | 174 | 12 | | 219 | 8 | | 264 | 16
40 | 5 | | 85 | 2 | | 130 | 2 | | 175 | 16 | | 220 | 8 | | 265 | 15
41 | 10 | | 86 | 2 | | 131 | 8 | | 176 | 15 | | 221 | 8 | | 266 | 14
42 | 9 | | 87 | 6 | | 132 | 7 | | 177 | 20 | | 222 | 12 | | 267 | 13
43 | 8 | | 88 | 6 | | 133 | 6 | | 178 | 12 | | 223 | 12 | | 268 | 12
44 | 8 | | 89 | 4 | | 134 | 6 | | 179 | 12 | | 224 | 2 | | 269 | 12
45 | 7 | | 90 | 7 | | 135 | 6 | | 180 | 12 | | 225 | 1 | | 270 | 11
Table 2: Supposed values of $B(n)$ for $n\leq 270$
Table 3 and figure 1 show the overall distribution of the conjectured values
up to 30 050. The maximum power of 2 which divides $k-1$ seemingly plays a
role in determining how many times $B(n)=k$.
1 | 40 | | 13 | 1728 | | 25 | 619 | | 37 | 200 | | 49 | 113 | | 61 | 9
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
2 | 379 | | 14 | 1079 | | 26 | 162 | | 38 | 59 | | 50 | 9 | | 62 | 4
3 | 464 | | 15 | 1328 | | 27 | 254 | | 39 | 128 | | 51 | 39 | | 63 | 9
4 | 812 | | 16 | 881 | | 28 | 131 | | 40 | 43 | | 52 | 10 | | 64 | 2
5 | 1424 | | 17 | 1179 | | 29 | 319 | | 41 | 175 | | 53 | 44 | | 65 | 31
6 | 1821 | | 18 | 590 | | 30 | 104 | | 42 | 25 | | 54 | 12 | | 66 | 3
7 | 2051 | | 19 | 850 | | 31 | 163 | | 43 | 60 | | 55 | 16 | | 67 | 4
8 | 2064 | | 20 | 420 | | 32 | 98 | | 44 | 28 | | 56 | 5 | | 68 | 0
9 | 2295 | | 21 | 711 | | 33 | 353 | | 45 | 100 | | 57 | 14 | | 69 | 4
10 | 1838 | | 22 | 284 | | 34 | 84 | | 46 | 20 | | 58 | 4 | | $\geq 70$ | 0
11 | 1874 | | 23 | 451 | | 35 | 152 | | 47 | 48 | | 59 | 8 | | |
12 | 1501 | | 24 | 227 | | 36 | 82 | | 48 | 8 | | 60 | 3 | | |
Table 3: Number of times in which the conjectured $B(n)$ happens to be equal
to a given number, for $n\leq 30050$ Figure 1: Distribution of $B(n)$.
$\gamma_{1}\approx 1.532$; $\gamma_{2}\approx 1.050$
### 2.3 Differences between consecutive $B(n)$
Provided that the supposed values above mentioned are correct, in our sample
$B(n)<B(n+1)$ 11 428 times, $B(n)=B(n+1)$ 10 498 times and ${B(n)>B(n+1)}$ 8
123 times. Absolute frequencies of each possible difference value are reported
in table 4 and figures 2 and 3.
-68 | 0 | | -45 | 1 | | -22 | 9 | | 1 | 2307 | | 24 | 9 | | 47 | 0
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
-67 | 0 | | -44 | 0 | | -21 | 15 | | 2 | 1771 | | 25 | 0 | | 48 | 0
-66 | 0 | | -43 | 1 | | -20 | 11 | | 3 | 1172 | | 26 | 7 | | 49 | 1
-65 | 0 | | -42 | 1 | | -19 | 12 | | 4 | 815 | | 27 | 5 | | 50 | 0
-64 | 0 | | -41 | 0 | | -18 | 17 | | 5 | 469 | | 28 | 4 | | 51 | 0
-63 | 0 | | -40 | 1 | | -17 | 16 | | 6 | 377 | | 29 | 6 | | 52 | 0
-62 | 0 | | -39 | 1 | | -16 | 31 | | 7 | 266 | | 30 | 4 | | 53 | 0
-61 | 0 | | -38 | 4 | | -15 | 25 | | 8 | 202 | | 31 | 0 | | 54 | 0
-60 | 0 | | -37 | 3 | | -14 | 32 | | 9 | 142 | | 32 | 3 | | 55 | 1
-59 | 0 | | -36 | 1 | | -13 | 31 | | 10 | 118 | | 33 | 1 | | 56 | 0
-58 | 2 | | -35 | 1 | | -12 | 52 | | 11 | 76 | | 34 | 2 | | 57 | 0
-57 | 0 | | -34 | 0 | | -11 | 61 | | 12 | 78 | | 35 | 2 | | 58 | 0
-56 | 0 | | -33 | 5 | | -10 | 73 | | 13 | 52 | | 36 | 2 | | 59 | 0
-55 | 2 | | -32 | 1 | | -9 | 97 | | 14 | 49 | | 37 | 1 | | 60 | 0
-54 | 0 | | -31 | 5 | | -8 | 161 | | 15 | 26 | | 38 | 3 | | 61 | 0
-53 | 0 | | -30 | 4 | | -7 | 203 | | 16 | 30 | | 39 | 4 | | 62 | 0
-52 | 0 | | -29 | 5 | | -6 | 247 | | 17 | 30 | | 40 | 0 | | 63 | 0
-51 | 1 | | -28 | 6 | | -5 | 423 | | 18 | 15 | | 41 | 2 | | 64 | 0
-50 | 0 | | -27 | 6 | | -4 | 701 | | 19 | 9 | | 42 | 0 | | 65 | 0
-49 | 0 | | -26 | 4 | | -3 | 1019 | | 20 | 23 | | 43 | 1 | | 66 | 0
-48 | 1 | | -25 | 7 | | -2 | 2202 | | 21 | 14 | | 44 | 2 | | 67 | 0
-47 | 0 | | -24 | 9 | | -1 | 5905 | | 22 | 14 | | 45 | 0 | | 68 | 0
-46 | 1 | | -23 | 12 | | 0 | 10498 | | 23 | 8 | | 46 | 0 | | |
Table 4: Number of times in which, supposedly, $B(n+1)-B(n)$ takes a
determined value, for $n\leq 30050$ Figure 2: Distribution of the differences
between consecutive values of $B$ function. $\gamma_{1}\approx 7.654$;
$\gamma_{2}\approx 64.76$ Figure 3: Distribution of figure 2 in the central
interval $[-25,+25]$. In this range, $\gamma_{1}\approx 4.472$ and
$\gamma_{2}\approx 21.33$.
## 3 Two-valued sequences in $D$
Here we are concerned with the region of $D$ which contains entries $d_{m,n}$
such that
$m>B(n)$ (7)
Putatively, conditions 4 and 7 are simuoltaneosly fulfilled by 454 070 791
elements of $D$. Of these, 227 020 108 are zeros and 227 050 683 are twos.
###### Definition 3.
An _orizzontal sequence_ in $D$ of lenght $l$ is a sequence whose $n^{th}$
term is given by $d_{m,n+k}$, with $m,n$ positive integers and $0\leq k<l$.
###### Definition 4.
A _vertical sequence_ in $D$ of lenght $l$ is a sequence whose $n^{th}$ term
is given by $d_{m+k,n}$, with $m,n$ positive integers and $0\leq k<l$.
Abundances of the $2^{l}$ possible sequences in $\\{0;2\\}^{*}$ of lenght $l$,
both orizzontal and vertical, are listed hereunder up to $l=6$. Only sequences
made of $d_{m,n}$ for which exists no known $d_{k,n}>2,\>k>m$ are taken into
account.
### 3.1 Sequences of lenght 2
#### 3.1.1 Orizzontal sequences of lenght 2
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
00 | 113 495 530 | 24.9985 %
02 | 113 503 199 | 25.0002 %
20 | 113 498 664 | 24.9992 %
22 | 113 511 915 | 25.0021 %
Table 5: Vertical sequences with $l=2$
#### 3.1.2 Orizzontal sequences of lenght 2
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
00 | 113 497 370 | 24.9972 %
02 | 113 507 723 | 24.9995 %
20 | 113 515 298 | 25.0011 %
22 | 113 520 268 | 25.0022 %
Table 6: Orizzonatal sequences with $l=2$
### 3.2 Sequences of lenght 3
#### 3.2.1 Orizzontal sequences of lenght 3
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
000 | 56 735 185 | 12.4980 %
002 | 56 750 289 | 12.5013 %
020 | 56 747 553 | 12.5007 %
022 | 56 739 392 | 12.4989 %
200 | 56 748 814 | 12.5010 %
202 | 56 739 590 | 12.4990 %
220 | 56 737 491 | 12.4985 %
222 | 56 756 389 | 12.5027 %
Table 7: Orizzontal sequences with $l=3$
#### 3.2.2 Vertical sequences of lenght 3
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
000 | 56 736 545 | 12.4967 %
002 | 56 753 286 | 12.5004 %
020 | 56 756 042 | 12.5010 %
022 | 56 744 040 | 12.4984 %
200 | 56 758 993 | 12.5017 %
202 | 56 748 830 | 12.4995 %
220 | 56 749 197 | 12.4995 %
222 | 56 763 596 | 12.5027 %
Table 8: Vertical sequences with $l=3$
### 3.3 Sequences of lenght 4
#### 3.3.1 Orizzontal sequences of lenght 4
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
0000 | 28 354 868 | 6.24689 %
0002 | 28 375 392 | 6.25141 %
0020 | 28 370 048 | 6.25023 %
0022 | 28 372 913 | 6.25086 %
0200 | 28 371 071 | 6.25045 %
0202 | 28 371 712 | 6.25060 %
0220 | 28 361 809 | 6.24841 %
0222 | 28 370 048 | 6.25023 %
2000 | 28 374 851 | 6.25129 %
2002 | 28 369 113 | 6.25002 %
2020 | 28 371 744 | 6.25060 %
2022 | 28 360 094 | 6.24804 %
2200 | 28 371 421 | 6.25053 %
2202 | 28 361 024 | 6.24824 %
2220 | 28 369 154 | 6.25003 %
2222 | 28 378 846 | 6.25217 %
Table 9: Orizzontal sequences with $l=4$
#### 3.3.2 Vertical sequences of lenght 4
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
0000 | 28 355 671 | 6.24601 %
0002 | 28 377 063 | 6.25072 %
0020 | 28 375 071 | 6.25029 %
0022 | 28 374 364 | 6.25013 %
0200 | 28 375 433 | 6.25037 %
0202 | 28 376 924 | 6.25069 %
0220 | 28 366 930 | 6.24849 %
0222 | 28 373 364 | 6.24991 %
2000 | 28 380 415 | 6.25146 %
2002 | 28 374 850 | 6.25024 %
2020 | 28 378 302 | 6.25100 %
2022 | 28 366 739 | 6.24845 %
2200 | 28 379 063 | 6.25116 %
2202 | 28 366 344 | 6.24836 %
2220 | 28 376 273 | 6.25055 %
2222 | 28 383 595 | 6.25216 %
Table 10: Vertical sequences with $l=4$
### 3.4 Sequences of lenght 5
#### 3.4.1 Orizzontal sequences of lenght 5
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
00000 | 14 166 536 | 3.12137 %
00002 | 14 185 912 | 3.12564 %
00020 | 14 183 929 | 3.12520 %
00022 | 14 187 985 | 3.12609 %
00200 | 14 182 309 | 3.12484 %
00202 | 14 185 450 | 3.12554 %
00220 | 14 182 611 | 3.12491 %
00222 | 14 186 947 | 3.12587 %
02000 | 14 188 885 | 3.12629 %
02002 | 14 179 877 | 3.12431 %
02020 | 14 188 841 | 3.12628 %
02022 | 14 179 517 | 3.12423 %
02200 | 14 187 478 | 3.12598 %
02202 | 14 171 930 | 3.12256 %
02220 | 14 179 032 | 3.12412 %
02222 | 14 187 481 | 3.12598 %
20000 | 14 185 699 | 3.12559 %
20002 | 14 186 732 | 3.12582 %
20020 | 14 183 518 | 3.12511 %
20022 | 14 182 142 | 3.12481 %
20200 | 14 186 003 | 3.12566 %
20202 | 14 183 390 | 3.12508 %
20220 | 14 176 448 | 3.12355 %
20222 | 14 180 026 | 3.12434 %
22000 | 14 182 962 | 3.12499 %
22002 | 14 186 080 | 3.12567 %
22020 | 14 180 016 | 3.12434 %
22022 | 14 177 330 | 3.12375 %
22200 | 14 180 951 | 3.12454 %
22202 | 14 185 750 | 3.12560 %
22220 | 14 187 053 | 3.12589 %
22222 | 14 187 747 | 3.12604 %
Table 11: Orizzontal sequences with $l=5$
#### 3.4.2 Vertical sequences of lenght 5
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
00000 | 14 166 969 | 3.12082 %
00002 | 14 186 785 | 3.12518 %
00020 | 14 185 922 | 3.12499 %
00022 | 14 189 193 | 3.12572 %
00200 | 14 184 821 | 3.12475 %
00202 | 14 188 416 | 3.12554 %
00220 | 14 184 965 | 3.12478 %
00222 | 14 187 563 | 3.12536 %
02000 | 14 191 647 | 3.12626 %
02002 | 14 181 907 | 3.12411 %
02020 | 14 192 188 | 3.12637 %
02022 | 14 182 859 | 3.12432 %
02200 | 14 190 442 | 3.12599 %
02202 | 14 174 602 | 3.12250 %
02220 | 14 182 003 | 3.12413 %
02222 | 14 189 527 | 3.12579 %
20000 | 14 188 591 | 3.12558 %
20002 | 14 189 930 | 3.12588 %
20020 | 14 188 467 | 3.12556 %
20022 | 14 184 480 | 3.12468 %
20200 | 14 189 368 | 3.12575 %
20202 | 14 187 083 | 3.12525 %
20220 | 14 180 534 | 3.12381 %
20222 | 14 184 295 | 3.12464 %
22000 | 14 186 756 | 3.12518 %
22002 | 14 190 458 | 3.12599 %
22020 | 14 183 398 | 3.12444 %
22022 | 14 181 035 | 3.12392 %
22200 | 14 185 857 | 3.12498 %
22202 | 14 188 512 | 3.12557 %
22220 | 14 191 090 | 3.12613 %
22222 | 14 190 611 | 3.12603 %
Table 12: Vertical sequences with $l=5$
### 3.5 Sequences of lenght 6
#### 3.5.1 Orizzontal sequences of lenght 6
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
000000 | 7 075 353 | 1.55910 %
000002 | 7 089 941 | 1.56231 %
000020 | 7 092 765 | 1.56293 %
000022 | 7 091 446 | 1.56264 %
000200 | 7 089 677 | 1.56225 %
000202 | 7 093 181 | 1.56303 %
000220 | 7 092 454 | 1.56287 %
000222 | 7 093 932 | 1.56319 %
002000 | 7 096 709 | 1.56380 %
002002 | 7 084 511 | 1.56112 %
002020 | 7 094 638 | 1.56335 %
002022 | 7 089 230 | 1.56216 %
002200 | 7 096 453 | 1.56375 %
002202 | 7 084 995 | 1.56122 %
002220 | 7 087 401 | 1.56175 %
002222 | 7 097 854 | 1.56406 %
020000 | 7 093 715 | 1.56314 %
020002 | 7 093 980 | 1.56320 %
020020 | 7 089 308 | 1.56217 %
020022 | 7 088 950 | 1.56209 %
020200 | 7 096 149 | 1.56368 %
020202 | 7 091 517 | 1.56266 %
020220 | 7 089 293 | 1.56217 %
020222 | 7 088 600 | 1.56202 %
022000 | 7 093 531 | 1.56310 %
022002 | 7 092 793 | 1.56294 %
022020 | 7 086 632 | 1.56158 %
022022 | 7 083 610 | 1.56092 %
022200 | 7 087 464 | 1.56177 %
022202 | 7 090 423 | 1.56242 %
022220 | 7 090 612 | 1.56246 %
022222 | 7 095 092 | 1.56345 %
200000 | 7 089 887 | 1.56230 %
200002 | 7 094 666 | 1.56335 %
200020 | 7 089 844 | 1.56229 %
200022 | 7 095 213 | 1.56347 %
200200 | 7 091 371 | 1.56263 %
200202 | 7 090 963 | 1.56254 %
200220 | 7 088 860 | 1.56207 %
200222 | 7 091 653 | 1.56269 %
202000 | 7 090 842 | 1.56251 %
202002 | 7 093 984 | 1.56320 %
202020 | 7 092 827 | 1.56295 %
202022 | 7 088 939 | 1.56209 %
202200 | 7 089 729 | 1.56227 %
202202 | 7 085 533 | 1.56134 %
202220 | 7 090 215 | 1.56237 %
202222 | 7 088 133 | 1.56191 %
220000 | 7 090 520 | 1.56244 %
220002 | 7 091 269 | 1.56260 %
220020 | 7 092 821 | 1.56295 %
220022 | 7 091 613 | 1.56268 %
220200 | 7 088 511 | 1.56200 %
220202 | 7 090 389 | 1.56241 %
220220 | 7 085 669 | 1.56137 %
220222 | 7 089 866 | 1.56230 %
222000 | 7 087 976 | 1.56188 %
222002 | 7 091 801 | 1.56272 %
222020 | 7 091 891 | 1.56274 %
222022 | 7 092 080 | 1.56278 %
222200 | 7 092 019 | 1.56277 %
222202 | 7 093 790 | 1.56316 %
222220 | 7 094 900 | 1.56340 %
222222 | 7 090 896 | 1.56252 %
Table 13: Orizzontal sequences with $l=6$
#### 3.5.2 Vertical sequences of lenght 6
Sequence | Absolute frequency | Percentage (6 s.f.)
---|---|---
000000 | 7 075 585 | 1.55877 %
000002 | 7 090 398 | 1.56204 %
000020 | 7 092 020 | 1.56239 %
000022 | 7 093 788 | 1.56278 %
000200 | 7 090 933 | 1.56215 %
000202 | 7 094 132 | 1.56286 %
000220 | 7 094 668 | 1.56298 %
000222 | 7 093 620 | 1.56275 %
002000 | 7 097 736 | 1.56365 %
002002 | 7 086 146 | 1.56110 %
002020 | 7 098 725 | 1.56387 %
002022 | 7 088 739 | 1.56167 %
002200 | 7 098 190 | 1.56375 %
002202 | 7 085 870 | 1.56104 %
002220 | 7 090 386 | 1.56203 %
002222 | 7 096 237 | 1.56332 %
020000 | 7 092 780 | 1.56256 %
020002 | 7 097 912 | 1.56369 %
020020 | 7 091 059 | 1.56218 %
020022 | 7 089 917 | 1.56193 %
020200 | 7 095 586 | 1.56318 %
020202 | 7 095 633 | 1.56319 %
020220 | 7 091 282 | 1.56223 %
020222 | 7 090 613 | 1.56208 %
022000 | 7 094 230 | 1.56288 %
022002 | 7 095 292 | 1.56311 %
022020 | 7 088 502 | 1.56162 %
022022 | 7 085 167 | 1.56088 %
022200 | 7 091 833 | 1.56235 %
022202 | 7 089 238 | 1.56178 %
022220 | 7 092 263 | 1.56245 %
022222 | 7 096 301 | 1.56334 %
200000 | 7 091 346 | 1.56225 %
200002 | 7 096 314 | 1.56334 %
200020 | 7 093 743 | 1.56277 %
200022 | 7 095 216 | 1.56310 %
200200 | 7 093 551 | 1.56273 %
200202 | 7 093 939 | 1.56282 %
200220 | 7 089 948 | 1.56194 %
200222 | 7 093 601 | 1.56274 %
202000 | 7 093 323 | 1.56268 %
202002 | 7 095 105 | 1.56307 %
202020 | 7 092 758 | 1.56256 %
202022 | 7 093 400 | 1.56270 %
202200 | 7 091 596 | 1.56230 %
202202 | 7 087 958 | 1.56150 %
202220 | 7 090 859 | 1.56214 %
202222 | 7 092 542 | 1.56251 %
220000 | 7 094 851 | 1.56302 %
220002 | 7 090 966 | 1.56216 %
220020 | 7 096 180 | 1.56331 %
220022 | 7 093 306 | 1.56268 %
220200 | 7 092 489 | 1.56250 %
220202 | 7 090 027 | 1.56195 %
220220 | 7 087 799 | 1.56146 %
220222 | 7 092 290 | 1.56245 %
222000 | 7 091 181 | 1.56221 %
222002 | 7 093 747 | 1.56277 %
222020 | 7 093 357 | 1.56269 %
222022 | 7 094 177 | 1.56287 %
222200 | 7 092 481 | 1.56250 %
222202 | 7 097 637 | 1.56363 %
222220 | 7 097 153 | 1.56352 %
222222 | 7 092 527 | 1.56251 %
Table 14: Vertical sequences with $l=6$
## 4 Comparison with the pseudorandomness hypothesis
If the region of the matrix $D$ satisfying equation 7 has got pseudo-random
characteristics, we should expect all the sequences of same lenght to be
equally likely. The $2^{l}$ sequences of lenght $l$ should appear with
probability $2^{-l}$, therefore with an absolute frequency $n$ close to
$\operatorname{E}[n]=2^{-l}N$ (8)
, where $N$ is the cumulative frequency of all the sequencies. The density of
the population of frequencies should reflect the binomial distribution
$\mathcal{B}(N,2^{-l})$ (9)
The expected standard deviation is given by
$\sigma^{\prime}=\sqrt{2^{-l}N(1-2^{-l})}=2^{-l}\sqrt{(2^{l}-1)N}$ (10)
Table 15 compares $\sigma^{\prime}$ with the effective standard deviation
$\sigma$ of the collected data. Apart from the case $l=1$ -in which $\sigma$
is calculated from only two values-, $\sigma$ and $\sigma^{\prime}$ never
differ by a factor grater than $\sqrt{2}$.
$l$ | Orientation | $N$ | $\sigma$ (6 s.f.) | $\sigma^{\prime}$ (6 s.f.) | $\sigma/\sigma^{\prime}$ (5 s.f.)
---|---|---|---|---|---
1 | | 454070791 | 21619.8 | 10654.5 | 2.0292
2 | Orizzontal | 454009308 | 7125.23 | 9226.42 | 0.77226
Vertical | 454040659 | 9968.18 | 9226.73 | 1.0804
3 | Orizzontal | 453954703 | 7451.53 | 7046.37 | 1.0575
Vertical | 454010529 | 8600.20 | 7046.80 | 1.2204
4 | Orizzontal | 453904108 | 6371.66 | 5157.13 | 1.2355
Vertical | 453980401 | 6838.03 | 5157.56 | 1.3258
5 | Orizzontal | 453856567 | 5019.11 | 3706.72 | 1.3541
Vertical | 453950274 | 5204.61 | 3707.11 | 1.4040
6 | Orizzontal | 453810879 | 3617.03 | 2641.97 | 1.3691
Vertical | 453920148 | 3696.80 | 2642.29 | 1.3991
Table 15: Expected and real standard deviations of samples
We can approximate, with a negligible error, the predicted distribution 9 to
the normal distribution
$\mathcal{N}\left(2^{-l}N,\>(2^{-l}-2^{-2l})N\right)$ (11)
Therefore, we expect to find a number of frequencies
$n_{i}<\operatorname{E}[n]+x\sigma^{\prime}$ (with $1\leq i\leq 2^{l}$)
approximately equal to
$2^{l}\Phi(x)$ (12)
, where $\Phi$ is the cumulative distribution function of the standard normal
distribution. Table 16 reports how many $n_{i}$ fall in a certain interval of
deviation from the mean $\operatorname{E}[n]$, measured in terms of
$\sigma^{\prime}$.
| | Interval of deviation from the expected value
---|---|---
$l$ | Orientation | $(-\infty,-2\sigma^{\prime})$ | $[-2\sigma^{\prime},-\sigma^{\prime})$ | $[-\sigma^{\prime},0)$ | $[0,\sigma^{\prime})$ | $[\sigma^{\prime},2\sigma^{\prime})$ | $[2\sigma^{\prime},+\infty)$
1 | | 0 | 1 | 0 | 0 | 1 | 0
2 | Orizzontal | 0 | 0 | 2 | 1 | 1 | 0
Vertical | 0 | 1 | 1 | 1 | 1 | 0
3 | Orizzontal | 0 | 1 | 3 | 3 | 1 | 0
Vertical | 1 | 1 | 2 | 2 | 2 | 0
4 | Orizzontal | 1 | 3 | 0 | 9 | 3 | 0
Vertical | 1 | 3 | 1 | 8 | 3 | 0
5 | Orizzontal | 2 | 3 | 9 | 9 | 9 | 0
Vertical | 2 | 4 | 8 | 11 | 7 | 0
6 | Orizzontal | 4 | 7 | 19 | 19 | 11 | 4
Vertical | 4 | 5 | 20 | 21 | 11 | 3
Table 16: Distribution of observed frequencies $n_{i}$ around the average
value $N/2^{l}$
A $\chi^{2}$ test has been performed to confront data in table 16 with the
hypothesized distribution 11. Since there are several small values, Yates’
correction for continuity [4] was applied. Thus, the $\chi^{2}$ statistic has
been computed as
$\chi^{2}=\sum_{-2\leq j\leq
2;\>j\to\infty}\frac{(|O_{j}-2^{l}\Phi(j)|-0.5)^{2}}{2^{l}\Phi(j)},\>O_{j}=\\#\\{n_{i}:n_{i}<\operatorname{E}[n]+j\sigma^{\prime}\\}$
(13)
Obtained values of $\chi^{2}$ are listed in table 17, togheter with the
corresponding $p$-value under the null hypothesis.
$l$ | Orientation | $\chi^{2}$ (6 s.f.) | Estimated $p$-value (5 s.f.)
---|---|---|---
1 | | 5.14450 | 0.39851
2 | Orizzontal | 2.10121 | 0.83497
Vertical | 2.10121 | 0.83497
3 | Orizzontal | 0.710874 | 0.98237
Vertical | 0.710874 | 0.98237
4 | Orizzontal | 1.96188 | 0.85439
Vertical | 1.21187 | 0.94373
5 | Orizzontal | 1.43792 | 0.92012
Vertical | 1.07793 | 0.95604
6 | Orizzontal | 3.37179 | 0.64287
Vertical | 3.33488 | 0.64851
Total | 23.1649 | 0.99995
Table 17: Results of the $\chi^{2}$ test
The distribution of 0s and 2s in $D$ seems reasonably presumable as pseudo-
random.
## References
* [1] François Proth (1878). _”Sur la série des nombres premiers”_ , Nouvelle Correspondence Mathématique 4: 236–240.
* [2] R. B. Killgrove; K. E. Ralston (1959). _”On a conjecture concerning the primes”_ , Mathematics of Computation (Mathematical Tables and Other Aids to Computation) 13: 121–122.
* [3] Andrew M. Odlyzko (1993). _”Iterated absolute values of differences of consecutive primes”_ , Mathematics of Computation 61: 373–380
* [4] Frank Yates (1934). _”Contingency table involving small numbers and the $\chi^{2}$ test”_, Supplement to the Journal of the Royal Statistical Society 1(2): 217–235
RAFFAELE SALVIA
_E-mail address: [email protected]_
|
arxiv-papers
| 2013-08-14T13:16:08 |
2024-09-04T02:49:49.418207
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Raffaele Salvia",
"submitter": "Raffaele Salvia",
"url": "https://arxiv.org/abs/1308.3113"
}
|
1308.3189
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-151 LHCb-PAPER-2013-041 15 August 2013
Model-independent search for $C\\!P$ violation in $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$ and
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
A search for $C\\!P$ violation in the phase-space structures of $D^{0}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to the final states
$K^{-}K^{+}\pi^{-}\pi^{+}$ and $\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ is presented.
The search is carried out with a data set corresponding to an integrated
luminosity of 1.0$\mbox{\,fb}^{-1}$ collected in 2011 by the LHCb experiment
in $pp$ collisions at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$. For the $K^{-}K^{+}\pi^{-}\pi^{+}$ final
state, the four-body phase space is divided into 32 bins, each bin with
approximately 1800 decays. The $p$-value under the hypothesis of no $C\\!P$
violation is 9.1%, and in no bin is a $C\\!P$ asymmetry greater than 6.5%
observed. The phase space of the $\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ final state is
partitioned into 128 bins, each bin with approximately 2500 decays. The
$p$-value under the hypothesis of no $C\\!P$ violation is 41%, and in no bin
is a $C\\!P$ asymmetry greater than 5.5% observed. All results are consistent
with the hypothesis of no $C\\!P$ violation at the current sensitivity.
Submitted to Phys. Lett. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, E.
Cowie45, D.C. Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8,
P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O.
Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M.
Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M. Fontana10, F.
Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M.
Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c,
M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra
Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53,
T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C.
Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H.
Gordon37, C. Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado
Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S.
Gregson46, P. Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu.
Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C.
Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N.
Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, T. Head37, V.
Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van
Herwijnen37, M. Hess60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C.
Hombach53, P. Hopchev4, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54,
D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R.
Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M.
John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S.
Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T.
Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41,
P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11,
M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V.
Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44,
R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J.
Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li
Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I.
Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J.
Luisier38, H. Luo49, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O.
Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U.
Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M.
Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D.
Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40,
F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, T. Palczewski27, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
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Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen61,
K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35, B.
Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8,
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P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
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E. Smith54,48, J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
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P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37, A. Ukleja27, D. Urner53, A.
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D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C.
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Xie49,37, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M.
Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3,
A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
Standard Model predictions for the magnitude of $C\\!P$ violation (CPV) in
charm meson decays are generally of $\mathcal{O}$($10^{-3}$) [1, 2], although
values up to $\mathcal{O}$($10^{-2}$) cannot be ruled out [3, 4]. The size of
CPV can be significantly enhanced in new physics models [5, 6], making charm
transitions a promising area to search for new physics. Previous searches for
CPV in charm decays caused a large interest in the community [7, 8, 9] and
justify detailed searches for CPV in many different final states. Direct CPV
can occur when at least two amplitudes interfere with strong and weak phases
that each differ from one another. Singly-Cabibbo-suppressed charm hadron
decays, where both tree processes and electroweak loop processes can
contribute, are promising channels with which to search for CPV. The rich
structure of interfering amplitudes makes four-body decays ideal to perform
such searches.
The phase-space structures of the $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$ and
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays111Unless otherwise
specified, inclusion of charge-conjugate processes is implied. are
investigated for localised CPV in a manner that is independent of an amplitude
model of the $D^{0}$ meson decay. The Cabibbo-favoured $D^{0}\\!\rightarrow
K^{-}\pi^{+}\pi^{+}\pi^{-}$ decay, where direct CPV can not occur in the
Standard Model, is used as a control channel. A model-dependent search for CPV
in $D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ was previously carried out
by the CLEO collaboration [10] with a data set of approximately 3000 signal
decays, where no evidence for CPV was observed. This analysis is carried out
on a data set of approximately $5.7\times 10^{4}$ $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$ decays and $3.3\times 10^{5}$
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays. The data set is
based on an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ of $pp$ collisions
with a centre-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$, recorded by
the LHCb experiment during 2011. The analysis is based on $D^{0}$ mesons
produced in $D^{*+}\\!\rightarrow D^{0}\pi^{+}$ decays. The charge of the soft
pion ($\pi^{+}$) identifies the flavour of the meson at production. The phase
space is partitioned into $N_{\rm bins}$ bins, and the significance of the
difference in population between $C\\!P$ conjugate decays for each bin is
calculated as
$S_{C\\!P}^{i}=\frac{N_{i}(D^{0})-\alpha N_{i}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})}{\sqrt{\alpha\left(\sigma_{i}^{2}(D^{0})+\sigma_{i}^{2}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})\right)}},~{}~{}~{}\alpha=\frac{\sum_{i}{N_{i}(D^{0})}}{\sum_{i}{N_{i}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})}},$ (1)
where $N_{i}$ is the number of signal decays in bin $i$, and $\sigma_{i}$ is
the associated uncertainty in the number of signal decays in bin $i$ [11]. The
normalisation constant $\alpha$ removes global production and detection
differences between $D^{*+}$ and $D^{*-}$ decays.
In the absence of any asymmetry, $S_{C\\!P}$ is Gaussian distributed with a
mean of zero and a width of one. A significant variation from a unit Gaussian
distribution indicates the presence of an asymmetry. The sum of squared
$S_{C\\!P}$ values is a $\chi^{2}$ statistic,
$\chi^{2}=\sum_{i}{\left(S_{C\\!P}^{i}\right)^{2}},$
with $N_{\mathrm{bins}}-1$ degrees of freedom, from which a $p$-value is
calculated. Previous analyses of three-body $D$ meson decays have employed
similar analysis techniques [12, 13].
## 2 Detector
The LHCb detector [14] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a vertically oriented magnetic field and bending power of
about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw
drift tubes placed downstream. To alleviate the impact of charged particle-
antiparticle detection asymmetries, the magnetic field polarity is switched
regularly, and data are taken in each polarity. The two magnet polarities are
henceforth referred to as “magnet up” and “magnet down”. The combined tracking
system provides momentum measurement with relative uncertainty that varies
from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20$\,\upmu\rm m$ for tracks with high transverse momentum. Charged hadrons are
identified with two ring-imaging Cherenkov (RICH) detectors [15]. Photon,
electron, and hadron candidates are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter, and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage [16]. Events are
required to pass both hardware and software trigger levels. The software
trigger optimised for the reconstruction of four-body hadronic charm decays
requires a four-track secondary vertex with a scalar sum of the transverse
momenta, $p_{\rm T}$, of the tracks greater than
$2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. At least two tracks are required to
have $\mbox{$p_{\rm T}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
momentum, $p$, greater than $5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
remaining two tracks are required to have $\mbox{$p_{\rm
T}$}>250{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and
$\mbox{$p$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. A requirement is also
imposed on the $\chi^{2}$ of the impact parameter ($\chi^{2}_{\rm IP}$) of the
remaining two tracks with respect to any primary interaction to be greater
than 10, where $\chi^{2}_{\rm IP}$ is defined as the difference in $\chi^{2}$
of a given primary vertex reconstructed with and without the considered track.
## 3 Selection
Candidate $D^{0}$ decays are reconstructed from combinations of pion and kaon
candidate tracks. The $D^{0}$ candidates are required to have $\mbox{$p_{\rm
T}$}>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The $D^{0}$ decay products are
required to have $\mbox{$p$}>3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\mbox{$p_{\rm T}$}>350{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The $D^{0}$
decay products are required to form a vertex with a $\chi^{2}$ per degree of
freedom ($\chi^{2}/\mathrm{ndf}$) less than 10 and a maximum distance of
closest approach between any pair of $D^{0}$ decay products less than
0.12$\rm\,mm$. The RICH system is used to distinguish between kaons and pions
when reconstructing the $D^{0}$ candidate. The $D^{*+}$ candidates are
reconstructed from $D^{0}$ candidates combined with a track with
$\mbox{$p_{\rm T}$}>120{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Decays are
selected with candidate $D^{0}$ mass, $m(hhhh)$, of
$1804<m(hhhh)<1924{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the
notation $m(hhhh)$ denotes the invariant mass of any of the considered final
states; specific notations are used where appropriate. The difference, $\Delta
m$, in the reconstructed $D^{*+}$ mass and $m(hhhh)$ for candidate decays is
required to be $137.9<\Delta m<155.0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The decay vertex of the $D^{*}$ is constrained to coincide with the primary
vertex [17].
Differences in $D^{*+}$ and $D^{*-}$ meson production and detection
efficiencies can introduce asymmetries across the phase-space distributions of
the $D^{0}$ decay. To ensure that the soft pion is detected in the central
region of the detector, fiducial cuts on its momentum are applied, as in Ref.
[9]. The $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
candidates are weighted by removing events so that they have same transverse
momentum and pseudorapidity distributions. To further cancel detection
asymmetries the data set is selected to contain equal quantities of data
collected with each magnetic field polarity. Events are randomly removed from
the largest subsample of the two magnetic field polarity configurations.
Each data sample is investigated for background contamination. The
reconstructed $D^{0}$ mass is searched for evidence of backgrounds from
misreconstructed $D^{0}$ decays in which $K$/$\pi$ misidentification has
occurred. Candidates in which only a single final-state particle is
misidentified are reconstructed outside the $m(hhhh)$ signal range. No
evidence for candidates with two, three, or four $K$/$\pi$ misidentifications
is observed. Charm mesons from $b$-hadron decays are strongly suppressed by
the requirement that the $D^{0}$ candidate originates from a primary vertex.
This source of background is found to have a negligible contribution.
## 4 Method
LABEL:sub@fig:mass:KKPiPi:D
(a)
LABEL:sub@fig:mass:KKPiPi:DELTAM
(b)
LABEL:sub@fig:mass:FourPi:D
(c)
LABEL:sub@fig:mass:FourPi:DeltaM
(d)
LABEL:sub@fig:mass:KThreePi:D
(e)
LABEL:sub@fig:mass:KThreePi:DELTAM
(f)
Figure 1: Distributions of
(LABEL:sub@fig:mass:KKPiPi:D,LABEL:sub@fig:mass:FourPi:D,LABEL:sub@fig:mass:KThreePi:D)
$m(hhhh)$ and
(LABEL:sub@fig:mass:KKPiPi:DELTAM,LABEL:sub@fig:mass:FourPi:DeltaM,LABEL:sub@fig:mass:KThreePi:DELTAM)
$\Delta m$ for (LABEL:sub@fig:mass:KKPiPi:D,LABEL:sub@fig:mass:KKPiPi:DELTAM)
$D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$,
(LABEL:sub@fig:mass:FourPi:D,LABEL:sub@fig:mass:FourPi:DeltaM)
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$, and
(LABEL:sub@fig:mass:KThreePi:D,LABEL:sub@fig:mass:KThreePi:DELTAM)
$D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ candidates for magnet up
polarity. Projections of the two-dimensional fits are overlaid, showing the
contributions for signal, combinatorial background, and random soft pion
background. The contributions from $D^{0}\\!\rightarrow
K^{-}\pi^{+}\pi^{-}\pi^{+}\pi^{0}$ and $D_{s}^{+}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}\pi^{+}$ contamination are also shown for the
$D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ sample.
Figure 1 shows the $m(hhhh)$ and $\Delta m$ distributions for $D^{0}$
candidate decays to the final states $K^{-}K^{+}\pi^{-}\pi^{+}$,
$\pi^{-}\pi^{+}\pi^{+}\pi^{-}$, and $K^{-}\pi^{+}\pi^{+}\pi^{-}$, for data
taken with magnet up polarity. The distributions for $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidates and data taken with
magnet down polarity are consistent with the distributions shown. Two-
dimensional unbinned likelihood fits are made to the $m(hhhh)$ and $\Delta m$
distributions to separate signal and background contributions. Each two-
dimensional $[m(hhhh),\Delta m]$ distribution includes contributions from the
following sources: signal $D^{0}$ mesons from $D^{*+}$ decays, which peak in
both $m(hhhh)$ and $\Delta m$; combinatorial background candidates, which do
not peak in either $m(hhhh)$ or $\Delta m$; background candidates from an
incorrect association of a soft pion with a real $D^{0}$ meson, which peak in
$m(hhhh)$ and not in $\Delta m$; incorrectly reconstructed
$D_{s}^{+}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}\pi^{+}$ decays, which peak
at low values of $m(hhhh)$ but not in $\Delta m$; and misreconstructed
$D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}\pi^{0}$ decays, which have
broad distributions in both $m(hhhh)$ and $\Delta m$. The signal distribution
is described by a Johnson function [18] in $\Delta m$ and a Crystal Ball
function [19] plus a Gaussian function, with a shared peak value, in
$m(hhhh)$. The combinatorial background is modelled with a first-order
polynomial in $m(hhhh)$, and the background from $D^{0}$ candidates each
associated with a random soft pion is modelled by a Gaussian distribution in
$m(hhhh)$. Both combinatorial and random soft pion backgrounds are modelled
with a function of the form
$f(\Delta m)=\left[\left(\Delta m-\Delta m_{0}\right)+p_{1}\left(\Delta
m-\Delta m_{0}\right)^{2}\right]^{a}$ (2)
in $\Delta m$, where $\Delta m_{0}$ is the kinematic threshold (fixed to the
pion mass), and the parameters $p_{1}$ and $a$ are allowed to float.
Partially reconstructed $D_{s}^{+}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}\pi^{+}$ decays, where a single pion is not
reconstructed, are investigated with simulated decays. This background is
modelled with a Gaussian distribution in $m(hhhh)$ and with a function
$f(\Delta m)$ as defined in Eq. 2. Misreconstructed $D^{0}\\!\rightarrow
K^{-}\pi^{+}\pi^{-}\pi^{+}\pi^{0}$ decays where a single $K$/$\pi$
misidentification has occurred and where the $\pi^{0}$ is not reconstructed
are modelled with a shape from simulated decays. Other potential sources of
background are found to be negligible.
For each two-dimensional $[m(hhhh),\Delta m]$ distribution a fit is first
performed to the background region, $139<\Delta
m<143{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or $149<\Delta
m<155{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, to obtain the shapes of the
combinatorial and soft pion backgrounds. The $\Delta m$ components of these
shapes are fixed and a two-dimensional fit is subsequently performed
simultaneously over four samples ($D^{0}$ magnet up, $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ magnet up, $D^{0}$ magnet down,
and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ magnet down). The peak
positions and widths of the signal shapes and all yields are allowed to vary
independently for each sample, whilst all other parameters are shared among
the four samples. Signal and background distributions are separated with the
sPlot statistical method [20]. The data sets contain $5.7\times 10^{4}$
$D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$, $3.3\times 10^{5}$
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$, and $2.9\times 10^{6}$
$D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ signal decays.
(a)
(b)
(c)
(d)
(e)
Figure 2: Invariant mass-squared distributions for $D^{0}$ meson (black,
closed circles) and $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ meson
(red, open squares) decays to the final state $K^{-}K^{+}\pi^{-}\pi^{+}$. The
invariant mass-squared combinations s(1,2), s(2,3), s(1,2,3), s(2,3,4), and
s(3,4) correspond to s($K^{-}$, $K^{+}$), s($K^{+}$, $\pi^{-}$), s($K^{-}$,
$K^{+}$, $\pi^{-}$), s($K^{+}$, $\pi^{-}$, $\pi^{+}$), and s($\pi^{-}$,
$\pi^{+}$), respectively for the $D^{0}$ mode. The charge conjugate is taken
for the $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ mode. The phase-
space distribution of the $D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ decay
is expected to be dominated by the quasi-two-body decay
$D^{0}\\!\rightarrow\phi\rho^{0}$ with additional contributions from
$D^{0}\\!\rightarrow K_{1}(1270)^{\pm}K^{\mp}$ and $D^{0}\\!\rightarrow
K^{*}(1410)^{\pm}K^{\mp}$ decays [10].
(a)
(b)
(c)
Figure 3: Invariant mass-squared distributions for $D^{0}$ meson (black,
closed circles) and $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ meson
(red, open squares) decays to the final state $\pi^{-}\pi^{+}\pi^{+}\pi^{-}$.
The invariant mass-squared combinations s(1,2), s(2,3), s(1,2,3), s(2,3,4),
and s(3,4) correspond to s($\pi^{-}$, $\pi^{+}$), s($\pi^{+}$, $\pi^{+}$),
s($\pi^{-}$, $\pi^{+}$, $\pi^{+}$), s($\pi^{+}$, $\pi^{+}$, $\pi^{-}$), and
s($\pi^{+}$, $\pi^{-}$), respectively for the $D^{0}$ mode. The charge
conjugate is taken for the $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$
mode. Owing to the randomisation of the order of identical final-state
particles the invariant mass-squared distributions s(2,3,4) and s(3,4) are
statistically compatible with the invariant mass-squared distributions
s(1,2,3) and s(1,2), respectively. As such the invariant mass-squared
distributions s(2,3,4) and s(3,4) are not shown. The phase-space distribution
of the $D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decay is expected to
be dominated by contributions from $D^{0}\\!\rightarrow
a_{1}(1260)^{+}\pi^{-}$ and $D^{0}\\!\rightarrow\rho^{0}\rho^{0}$ decays [21].
The phase space of a spin-0 decay to four pseudoscalars can be described with
five invariant mass-squared combinations: s(1,2), s(2,3), s(1,2,3), s(2,3,4),
and s(3,4), where the indices 1, 2, 3, and 4 correspond to the decay products
of the $D^{0}$ meson following the ordering of the decay definitions. The
ordering of identical final-state particles is randomised.
The rich amplitude structures are visible in the invariant mass-squared
distributions for $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to the final states
$K^{-}K^{+}\pi^{-}\pi^{+}$ and $\pi^{-}\pi^{+}\pi^{+}\pi^{-}$, shown in Figs.
2 and 3, respectively. The momenta of the final-state particles are calculated
with the decay vertex of the $D^{*}$ constrained to coincide with the primary
vertex and the mass of the $D^{0}$ candidates constrained to the world average
value of 1864.86${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [22].
An adaptive binning algorithm is devised to partition the phase space of the
decay into five-dimensional hypercubes. The bins are defined such that each
contains a similar number of candidates, resulting in fine bins around
resonances and coarse bins across sparsely populated regions of phase space.
For each phase-space bin, $S_{C\\!P}^{i}$, defined in Eq. 1, is calculated.
The number of signal events in bin $i$, $N_{i}$, is calculated as the sum of
the signal weights in bin $i$ and $\sigma_{i}^{2}$ is the sum of the squared
weights. The normalisation factor, $\alpha$, is calculated as the ratio of the
sum of the weights for $D^{0}$ candidates and the sum of the weights for
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ candidates and is $1.001\pm
0.008$, $0.996\pm 0.003$, and $0.998\pm 0.001$ for the final states
$K^{-}K^{+}\pi^{-}\pi^{+}$, $\pi^{-}\pi^{+}\pi^{+}\pi^{-}$, and
$K^{-}\pi^{+}\pi^{+}\pi^{-}$, respectively.
## 5 Production and instrumental asymmetries
Checks for remaining production or reconstruction asymmetries are carried out
by comparing the phase-space distributions from a variety of data sets
designed to test particle/antiparticle detection asymmetries and “left/right”
detection asymmetries. The “left” direction is defined as the bending
direction of a positively charged particle with the magnet up polarity.
Asymmetries in the background are studied with weighted background candidates
and mass sidebands.
Left/right asymmetries in detection efficiencies are investigated by comparing
the phase-space distributions of $D^{0}$ candidates in data taken with
opposite magnet polarities, thus investigating the same flavour particles in
opposite sides of the detector. Particle/antiparticle asymmetries are studied
with the control channel $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$. The
weighting based on $p_{\rm T}$ and pseudorapidity of the $D^{0}$ candidate and
the normalisation across the phase space of the $D^{0}$ decay cancel the
$K^{+}$/$K^{-}$ detection asymmetry in this control channel. The phase-space
distribution of $D^{0}$ decays from data taken with one magnet polarity is
compared with that of $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
decays from data taken with the opposite magnet polarity, for any sources of
particle/antiparticle detection asymmetry, localised across the phase space of
the $D^{0}$ decay.
The weighted distributions for each of the background components in the two-
dimensional fits are investigated for asymmetries in $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$, $D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$,
and $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ candidates. The $\Delta
m$ and $m(hhhh)$ sidebands are also investigated to identify sources of
asymmetry.
The sensitivity to asymmetries is limited by the sample size, so $S_{C\\!P}$
is calculated only with statistical uncertainties.
## 6 Sensitivity studies
Pseudo-experiments are carried out to investigate the dependence of the
sensitivity on the number of bins. Each pseudo-experiment is generated with a
sample size comparable to that available in data.
LABEL:sub@fig:Toy:SCP:NoCPV
(a)
LABEL:sub@fig:Toy:SCP:CPV
(b)
Figure 4: Distributions of $S_{C\\!P}$ for LABEL:sub@fig:Toy:SCP:NoCPV a
typical pseudo-experiment with generated
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays without CPV and for
LABEL:sub@fig:Toy:SCP:CPV a typical pseudo-experiment with a generated 10∘
phase difference between $D^{0}\rightarrow a_{1}(1260)^{+}\pi^{-}$ and $\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\rightarrow a_{1}(1260)^{-}\pi^{+}$
resonant decays. The points show the data distribution and the solid line is a
reference Gaussian distribution corresponding to the no CPV hypothesis. The
corresponding $p$-values under the hypothesis of no asymmetry for
LABEL:sub@fig:Toy:SCP:NoCPV decays without CPV and LABEL:sub@fig:Toy:SCP:CPV
decays with a 10∘ phase difference between $D^{0}\rightarrow
a_{1}(1260)^{+}\pi^{-}$ and $\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}\rightarrow a_{1}(1260)^{-}\pi^{+}$
resonant components are 85.6% and $1.1\times 10^{-16}$, respectively.
Decays are generated with MINT, a software package for amplitude analysis of
multi-body decays that has also been used by the CLEO collaboration [10]. A
sample of $D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ decays is generated
according to the amplitude model reported by CLEO [10], and
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays are generated
according to the amplitude model from the FOCUS collaboration [21]. Phase and
magnitude differences between $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays are introduced. Figure 4
shows the $S_{C\\!P}$ distributions for a typical pseudo-experiment in which
no CPV is present and for a typical pseudo-experiment with a phase difference
of $10^{\circ}$ between $D^{0}\\!\rightarrow a_{1}(1260)^{+}\pi^{-}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow
a_{1}(1260)^{-}\pi^{+}$ decays.
Based on the results of the sensitivity study, a partition with 32 bins, with
approximately 1800 signal events, is chosen for $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$ decays while a partition with 128 bins, with
approximately 2500 signal events is chosen for
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays. The $p$-values for
the pseudo-experiments are uniformly distributed for the case of no CPV. The
average $p$-value for a pseudo-experiment with a phase difference of 10∘ or a
magnitude difference of 10$\%$ between $D^{0}\\!\rightarrow\phi\rho^{0}$ and
$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow{\phi}{\rho}^{0}$
decays for the $D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ mode and between
$D^{0}\\!\rightarrow a_{1}(1260)^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow
a_{1}(1260)^{-}\pi^{+}$ decays for the
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ mode is below $10^{-3}$.
## 7 Results
Table 1: The $\chi^{2}/\mathrm{ndf}$ and $p$-values under the hypothesis of no CPV for the control channel $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$. The $p$-values are calculated separately for data samples taken with magnet up polarity, magnet down polarity, and the two polarities combined. | $p$-value (%) ($\chi^{2}/\mathrm{ndf}$) | $p$-value (%) ($\chi^{2}/\mathrm{ndf}$) | $p$-value (%) ($\chi^{2}/\mathrm{ndf}$)
---|---|---|---
Bins | Magnet down | Magnet up | Combined sample
16 | 80.8 (10.2/15) | 21.2 (19.1/15) | 34.8 (16.5/15)
128 | 62.0 (121.5/127) | 75.9 (115.5/127) | 80.0 (113.4/127)
1024 | 27.5 (1049.6/1023) | 9.9 (1081.6/1023) | 22.1 (1057.5/1023)
Table 2: The $\chi^{2}/\mathrm{ndf}$ and $p$-values under the hypothesis of no
CPV with three different partitions for $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$ decays and
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays. The $p$-values are
calculated for a combined data sample with both data taken with magnet up
polarity and data taken with magnet down polarity.
$D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$
---
Bins | $p$-value (%) | $\chi^{2}/\mathrm{ndf}$
16 | 9.1 | 22.7/15
32 | 9.1 | 42.0/31
64 | 13.1 | 75.7/63
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$
---
Bins | $p$-value (%) | $\chi^{2}/\mathrm{ndf}$
64 | 28.8 | 68.8/63
128 | 41.0 | 130.0/127
256 | 61.7 | 247.7/255
LABEL:sub@fig:SCP:KKPiPi
(a)
LABEL:sub@fig:RAW:KKPiPi
(b)
LABEL:sub@fig:SCP:FourPi
(c)
LABEL:sub@fig:RAW:FourPi
(d)
LABEL:sub@fig:control:SCP
(e)
LABEL:sub@fig:control:RAW
(f)
Figure 5: Distributions of
(LABEL:sub@fig:SCP:KKPiPi,LABEL:sub@fig:SCP:FourPi,LABEL:sub@fig:control:SCP)
$S_{C\\!P}$ and
(LABEL:sub@fig:RAW:KKPiPi,LABEL:sub@fig:RAW:FourPi,LABEL:sub@fig:control:RAW)
local $C\\!P$ asymmetry per bin for
(LABEL:sub@fig:SCP:KKPiPi,LABEL:sub@fig:RAW:KKPiPi) $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$ decays partitioned with 32 bins, for
(LABEL:sub@fig:SCP:FourPi,LABEL:sub@fig:RAW:FourPi)
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays partitioned with 128
bins, and for (LABEL:sub@fig:control:SCP,LABEL:sub@fig:control:RAW) the
control channel $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ partitioned
with 128 bins. The points show the data distribution and the solid line is a
reference Gaussian distribution corresponding to the no CPV hypothesis.
Asymmetries are searched for in the $D^{0}\\!\rightarrow
K^{-}\pi^{+}\pi^{+}\pi^{-}$ control channel. The distributions of $S_{C\\!P}$
and local $C\\!P$ asymmetry, defined as
$A^{i}_{C\\!P}=\frac{N_{i}(D^{0})-\alpha N_{i}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})}{N_{i}(D^{0})+\alpha N_{i}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0})},$
are shown in Fig. 5 for the $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$
control channel. The data set is also studied to identify sources of asymmetry
with two alternative partitions and by separating data taken with each magnet
polarity. The results, displayed in Table 1, show that no asymmetry is
observed in $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ decays.
Furthermore, the data sample is split into 10 time-ordered samples of
approximately equal size, for each polarity. The $p$-values under the
hypothesis of no asymmetry are uniformly distributed across the data taking
period. No evidence for a significant asymmetry in any bin is found.
The $S_{C\\!P}$ and local $C\\!P$ asymmetry distributions for
$D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ decays for a partition
containing 32 bins and for $D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$
decays with a partition containing 128 bins are shown in Fig. 5. The
$p$-values under the hypothesis of no $C\\!P$ violation for the decays
$D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ and
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ are 9.1% and 41%,
respectively. The consistency of the results is checked with alternative
partitions and the $p$-values are displayed in Table 2.
The stability of the results is checked for each polarity in 10 approximately
equal-sized, time-ordered data samples. The $p$-values are uniformly
distributed across the 2011 data taking period and are consistent with the no
CPV hypothesis.
## 8 Conclusions
A model-independent search for CPV in $5.7\times 10^{4}$ $D^{0}\\!\rightarrow
K^{-}K^{+}\pi^{-}\pi^{+}$ decays and $3.3\times 10^{5}$
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays is presented. The
analysis is sensitive to CPV that would arise from a phase difference of
$\mathcal{O}$(10∘) or a magnitude difference of $\mathcal{O}$(10$\%$) between
$D^{0}\\!\rightarrow\phi\rho^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow{\phi}{\rho^{0}}$
decays for the $D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ mode and between
$D^{0}\\!\rightarrow a_{1}(1260)^{+}\pi^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\\!\rightarrow
a_{1}(1260)^{-}\pi^{+}$ decays for the
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ mode. For none of the 32
bins, each with approximately 1800 signal events, is an asymmetry greater than
6.5% observed for $D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ decays, and
for none of the 128 bins, each with approximately 2500 signal events, is an
asymmetry greater than 5.5% observed for
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ decays. Assuming $C\\!P$
conservation, the probabilities to observe local asymmetries across the phase-
space of the $D^{0}$ meson decay as large or larger than those in data for the
decays $D^{0}\\!\rightarrow K^{-}K^{+}\pi^{-}\pi^{+}$ and
$D^{0}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ are 9.1% and 41%,
respectively. All results are consistent with $C\\!P$ conservation at the
current sensitivity.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-08-14T17:42:43 |
2024-09-04T02:49:49.437898
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, B. Couturier, G.A. Cowan, E. Cowie, D.C. Craik, S.\n Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De\n Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, C. Gotti, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. Hess, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, P. Hopchev, W.\n Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V.\n Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A.\n Jawahery, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G.\n Mancinelli, J. Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe,\n C. Matteuzzi, E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, T. Palczewski, M. Palutan, J. Panman, A. Papanestis, M.\n Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A.\n Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk,\n T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, M. Sannino, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n K. Senderowska, I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I.\n Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko,\n V. Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P.\n Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P.\n Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S.\n Swientek, V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin,\n U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez,\n P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M.\n Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss},\n H. Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, S.A.\n Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Matthew Coombes Mr",
"url": "https://arxiv.org/abs/1308.3189"
}
|
1308.3191
|
Near-field heat transfer between gold nanoparticle arrays
Anh D. Phan^1,2, The-Long Phan^3, and Lilia M. Woods^1
$^{1}$Department of Physics, University of South Florida, Tampa, Florida 33620, USA
$^{2}$Institute of Physics, 10 Daotan, Badinh, Hanoi, Vietnam
$^{3}$Department of Physics, Chungbuk National University, Cheongju 361-763, Korea
The radiative heat transfer between gold nanoparticle layers is presented using the coupled dipole method. Gold nanoparticles are modelled as effective electric and magnetic dipoles interacting via electromagnetic fluctuations. The effect of higher-order multipoles is implemented in the expression of electric polarizability to calculate the interactions at short distances. Our findings show that the near-field radiation reduces as the radius of the nanoparticles is increased. Also, the magnetic dipole contribution to the heat exchange becomes more important for larger particles. When one layer is displayed in parallel with respect to the other layer, the near-field heat transfer exhibits oscillatory-like features due to the influence of the individual nanostructures. Further details about the effect of the nanoparticles size are also discussed.
§ INTRODUCTION
Noble metallic nanoparticles (MNPs) have been exploited in a wide range of technological applications due to their unique properties. In particular, their strong absorption of radiation together with the ability of control of localized surface plasmon resonances have been key factors in a number of optical devices [1, 2]. For many targeted uses and perspectives, periodic two- or three-dimensional MNP arrays have been utilized [1, 3, 4]. It was shown that many-body effects enhance the electromagnetic behavior of the system compared to the one of the individual particles. As two nanoplasmonic arrays are brought at small separations and maintained at different temperatures, radiative heat transfer occurs. The origin of this exchange process originates from the electromagnetic fluctuations between the objects [5]. Since the electric properties of MNs are sensitive to the external fields [6], it is possible to employ these fields to change the heat radiation. Much experimental [7, 8, 9] and theoretical [10, 11, 12, 13, 14, 15] efforts have been devoted in understanding this phenomenon and finding ways for efficient control.
§ THEORETICAL BACKGROUND
In this work we focus on the radiative heat transfer between two gold MNP layers. Each nanoparticle is modelled as a dipole. Each layer consists of $20\times 20$ identical particles separated by $1$ nm, as shown in Fig.<ref>. It is assumed that each nanoparticle has a spherical shape with radius $R$ and the dielectric and magnetic properties are described via a dipolar model. The radiative heat exchange $P_{i\rightarrow j}(\omega)$ between the $i$-th and $j$-th dipoles consists of electric $P^e_{i\rightarrow j}(\omega)$ and magnetic $P^m_{i\rightarrow j}(\omega)$ contributions [16, 17], as follows:
\begin{eqnarray}
P^e_{i\rightarrow j}(\omega)=\frac{\omega\varepsilon_0}{\pi}\ce{Im}\alpha^e_j({\omega})\left\langle |\mathbf{E}_{ji}|^2\right\rangle,\nonumber\\
P^m_{i\rightarrow j}(\omega)=\frac{\omega\mu_0}{\pi}\ce{Im}\alpha^m_j({\omega})\left\langle |\mathbf{H}_{ji}|^2\right\rangle,
\label{eq:1}
\end{eqnarray}
where $\alpha^e_j({\omega})$ and $\alpha^m_j({\omega})$ are the electric and magnetic polarizabilities, respectively, of the j-th dipole with an electric $p_j$ and magnetic $m_j$ components.
Also, $\mathbf{E}_{ji}$ and $\mathbf{H}_{ji}$ are the electric and magnetic fields, respectively, at position $\mathbf{r}_j$ due to the fluctuations of dipole. $\varepsilon_0$ is the vacuum permittivity and $\mu_0$ is the permeability of free space. The relation between $\mathbf{E}_{ji}$ and the electric dipole moment $\mathbf{p}_{j}$ is given $\mathbf{E}_{ji}(\omega)=\mu_0\omega^2 \mathbf{G}(\mathbf{r}_j,\mathbf{r}_i,\omega)\mathbf{p}_i$ [5, 18]. Here $\mathbf{G}(\mathbf{r}_j,\mathbf{r}_i,\omega)$ is the dyadic Green tensor [18]. Using the fluctuation dissipation theorem [5], one finds
\begin{eqnarray}
\left\langle \mathbf{E}_{ji}(\omega)\mathbf{E}^*_{ji}(\omega ')\right\rangle &=&\mu_0^2\omega^2\omega '^2\sum_{k,l,t}G_{kl}(\mathbf{r}_j,\mathbf{r}_i,\omega)\nonumber\\
&\times & G^\dagger_{kt}(\mathbf{r}_j,\mathbf{r}_i,\omega ')\left\langle p_{i,l}(\omega)p^*_{i,t}(\omega ')\right\rangle, \nonumber\\
\left\langle p_{i,l}(\omega)p^*_{i,t}(\omega ')\right\rangle &=& \frac{2\varepsilon_0}{\omega}\ce{Im}\alpha^e_i(\omega)\Theta(\omega,T_i)\delta_{lt}\delta(\omega - \omega '),\nonumber\\
\Theta(\omega,T_i) &=& \frac{\hbar\omega}{e^{\hbar\omega/k_BT_i}-1},
\label{eq:2}
\end{eqnarray}
where $k, l, t = x, y, z$; $\hbar$ is the Planck constant, $k_B$ is the Boltzmann constant, $T_i$ is the temperature of dipole i.
(Color online) Schematic representation of two layers of gold MNPs kept at temperature $T$ and 0 $K$ on the top and bottom, respectively. The two surfaces are separated by a distance $a$. The separation between centers of adjacent MNPs is $d = 2R + 1$ nm.
Solving Eq.(<ref>) and Eq.(<ref>) together, the exchanged power caused by the electric dipoles is found to be:
\begin{eqnarray}
P^e_{i\rightarrow j}(\omega)&=&\frac{2}{\pi}\frac{\omega^4}{ c^4}\ce{Im}\alpha^e_j(\omega)\ce{Im}\alpha^e_i(\omega)\Theta(\omega,T_i)\nonumber\\
&\times & \ce{Tr}\left(\mathbf{G}(\mathbf{r}_j,\mathbf{r}_i,\omega)\mathbf{G}(\mathbf{r}_j,\mathbf{r}_i,\omega)^\dagger \right),
\label{eq:3}
\end{eqnarray}
where c is the speed of light.
Similar considerations apply for the magnetic dipole moments and the magnetic fields, yielding $\mathbf{H}_{ji}(\omega)=(\omega/c)^2 \mathbf{G}(\mathbf{r}_j,\mathbf{r}_i,\omega)\mathbf{m}_i$ [19].Consequently, the correlation functions for the magnetic dipoles is expressed as [16]
\begin{eqnarray}
\left\langle m_{i,l}(\omega)m^*_{i,t}(\omega ')\right\rangle &=& \frac{2\delta_{lt}}{\omega\mu_0}\ce{Im}\alpha^m_i(\omega)\Theta(\omega,T_i)\delta(\omega - \omega ').\nonumber\\
\label{eq:4}
\end{eqnarray}
Thus the exchanged power due to the magnetic field fluctuations becomes
\begin{eqnarray}
P^m_{i\rightarrow j}(\omega)&=&\frac{2}{\pi}\frac{\omega^4}{ c^4}\ce{Im}\alpha^m_j(\omega)\ce{Im}\alpha^m_i(\omega)\Theta(\omega,T_i)\nonumber\\
&\times & \ce{Tr}\left(\mathbf{G}(\mathbf{r}_j,\mathbf{r}_i,\omega)\mathbf{G}(\mathbf{r}_j,\mathbf{r}_i,\omega)^\dagger \right).
\label{eq:5}
\end{eqnarray}
Since particles are taken to be identical, one has $\alpha^{e,m}_1=\alpha^{e,m}_2=...=\alpha^{e,m}_N=\alpha^{e,m}$. It is important to note that since the separation distance between two adjacent gold NPs is not much larger than their radius, the influence of higher-order multipoles (quadrupole in our calculation) on the polarizability of MNPs should be taken into account. We can introduce the effective electric and magnetic polarizabilities for MNPs (R less than the skin-depth) derived from the Mie scattering theory [20, 21]
\begin{eqnarray}
\alpha^e(\omega) &=&4\pi R^3\left[\frac{\varepsilon-1}{\varepsilon+2} +\frac{1}{12}\left(\frac{\omega R}{c}\right)^2\frac{\varepsilon - 1}{\varepsilon + 3/2}\right], \nonumber\\
\alpha^m(\omega) &=& \frac{2\pi}{15}R^3\left(\frac{\omega R}{c}\right)^2(\varepsilon-1),
\label{eq:7}
\end{eqnarray}
where $\varepsilon(\omega)$ is the dielectric function of gold NPs. The first and second term in the expression of $\alpha^e(\omega)$ correspond to the dipole and quadrupole contributions, respectively. Authors in Ref.[22] used the dipole term and indicated that the distance between centers of MNPs should be at least few times greater than their radius $R$ to ensure the validity of the model for $\alpha^e(\omega)$. The quadrupole term added in Eq.(<ref>) allows us to calculate the near-field heat transfer between nanoparticles at shorter distances than calculations from other models [16, 18, 22].
The heat interchange between two particles is calculated [5]
\begin{eqnarray}
Q_{ij}^{TE,TM}(\omega)=\int_0^{\infty}d\omega\left[P^{e,m}_{i\rightarrow j}(\omega)- P^{e,m}_{j\rightarrow i}(\omega) \right],
\label{eq:6}
\end{eqnarray}
The heat transfer per unit area from the top array to the bottom array is calculated
\begin{eqnarray}
Q = \sum_{i=1}^{N_1}\sum_{j=N_1+1}^{N_1+N_1}\left(Q^{TE}_{ij}+Q^{TM}_{ij}\right)/S
\label{eq:8}
\end{eqnarray}
where $N_1 = 400$ is the number of NPs in top and bottom object, $S$ is the area of an array, $Q_{TE}$ and $Q_{TM}$ are the radiative heat transfer of electric and magnetic contribution in NPs, respectively. The first and second sum correspond to the summation of nanoparticles in the bottom and top layer.
§ NUMERICAL RESULTS AND DISCUSSIONS
Increasing the distance $d$ leads to the increase of center-center distance between particles in the systems. The importance of the many-particle effect significantly reduces. Therefore, in our paper, we chose $d = 2R + 1$ nm to be suitable with pervious experiments [4] and clearly exhibit the many-body effects.
(Color online) The radiative heat transfer between two gold MNP layers as a function of separation distance $a$ at different temperatures $T$ using the Lorentz-Drude and Drude model for the dielectric function.
The dielectric function of gold NPs is modelled by the Lorentz-Drude (LD) model [23]
\begin{eqnarray}
\varepsilon(\omega)=1-\frac{f_0\omega_p^2}{\omega(\omega +i\Gamma_0)}+\sum_{j}\frac{f_j\omega_p^2}{\omega_j^2-i\omega\Gamma_j-\omega^2},
\label{eq:9}
\end{eqnarray}
where $f_0$ and and $\omega_p$ are $0.845$ and $9.01$ eV, respectively. Also, $f_j$ are the oscillator strengths corresponding to characteristic frequencies $\omega_j$ and damping parameters $\Gamma_j$ given in [23]. These parameters were fitted from data set that was measured for gold nanostructure. The first two terms in Eq.(<ref>) describe the contribution of a free electron gas to the response, while the other terms represent interband transitions. In previous studies, authors used the Drude model $\varepsilon(\omega)=1-\omega_p^2/\omega(\omega +i\Gamma_0)$ for the dielectric function of gold. The model is suitable for the dielectric response of bulk, however. The inclusion of the Lorentz oscillators accounts for the localized surface plasmon modes of MNPs with wavelengths $\sim$ 500 nm. Note that the finite spherical size of the nanoparticles affects the damping parameter $\Gamma_0$ for gold. Here we take that $\Gamma_0\rightarrow \Gamma_0 + Av_f/R$ [24]. For gold, the parameter $A \approx 1$ and $v_f$ is the Fermi velocity of gold [24].
We note that the finite size of the nanoparticles, taken via the modification in $\Gamma_0$, can play an important role in the heat exchange process. Fig. <ref> shows a comparison between the heat transfer between two MNP arrays using the LD and Drude model. The bottom layer is kept at $T_0=0$ $K$, while the top layer is maintained at a finite temperature $T$. [25]. For the two chosen temperatures, $Q$ is much larger for the LD model. The huge difference for two models shows that it is impossible to obtain correct value with the Drude model because of the neglect of the bound electron contribution in the polarizability.
(Color online) The heat flux between two gold nanoparticle layers as a function of $\omega$ with a variety of $R$ and $T$ at $a = 10$ nm.
To investigate the radiative heat transfer, we have to know the frequency range that is important for the thermal conductance through the heat flux as a function of frequency. The expression of the heat transfer between two arrays versus $\omega$ is given
\begin{eqnarray}
P(\omega) = \sum_{s=e,m}\sum_{i=1}^{N_1}\sum_{j=N_1+1}^{N_1+N_1}\left[ P^{s}_{i\rightarrow j}(\omega)- P^{s}_{j\rightarrow i}(\omega)\right]
\end{eqnarray}
where $N_1 = 400$ is the number of nanoparticles in a layer. The first sum corresponds to the two modes (TE, TM), the second one - to the number of particles in the top layer, and the third one - to number of particles in the bottom layer. Figure <ref> shows the heat transfer versus frequencies with different sizes of NPs. The radiative heat transfer is contributed significantly by frequencies ranging from $2\times 10^{13}$ to $6\times 10^{14}$ rad/s. The position of the peak of $P(\omega)$ shifts from left to right when enlarging the nanoparticle's radius.
(Color online) The heat exchange due to magnetic dipole $Q_{TM}$ and electric dipole $Q_{TE}$ contribution at temperature $T = 300$ and 500 $K$ are calculated for different MNPs with different radii.
We also investigate how $Q$ is affected by the $TE$ and $TM$ modes of the system. Fig.<ref> shows that for spheres with smaller $R$, $Q_{TE}$ is dominant. As the radius is increased, the contribution from $Q_{TM}$ becomes more significant. The role of the quadrupole term in the electric polarizability in the absorption and scattering spectrum of MNPs becomes considerable when the NP radius is large [21] because of the proportionality of the term to $R^5$. The higher-order multipole terms are found to be proportional to $R^{2l+1}/[\varepsilon + (l+1)/l]$ with the integer $l \ge 3$. Nevertheless, the quadrupole and higher-order multipole contribution to the heat transfer for the studied structures are small. This is due to the large denominator $[\varepsilon + (l+1)/l]$ and small radius $R$. The contribution of the magnetic polarizability to the heat radiation surpasses that of the quadrupole term. Using Eq. (<ref>), (<ref>) and (<ref>), one finds that $Q_{TM}\sim R^{10}$ and $Q_{TE}\sim R^{6}$. Thus increasing the MNP radius enhances the effect of the magnetic polarizability and reduces the influence of the electric polarizability in the near-field radiation. At certain temperature $T$ and separation distance $a$, the heat radiation between the two nanoparticles is amplified as $R$ increases. In the layered systems, however, the heat flux $Q_{TM}$ and $Q_{TE}$ dramatically decreases because the distance from a particle to particle located in different layers, except for the nearest neighbors, increases. In comparison with bulk material and thin film systems, the near-field radiation of the nanoparticle arrays is weaker. The main reason is that the layer systems have a thin thickness and spacing between among MNPs in the same array. The total heat flux $Q$ is $1.92, 1.82$ and $1.78$ times greater than the heat flux of $400$ nearest neighbor pairs of particles placed two arrays at $a = 2$ nm for $R = 5, 9, 12$ nm, respectively. The ratios decrease when the separation $a$ is expanded since the many-body effects are strengthened if ${r}_{ij}/a$ is smaller, here $\mathbf{r}_{i}$ and $\mathbf{r}_{j}$ are the positions of particles in different layers, and $\mathbf{r}_{ij} = \mathbf{r}_{i}-\mathbf{r}_{j}$.
(Color online) The radiative heat transfer $Q_{TE}$ and $Q_{TM}$ at $a = 10$ nm as a function of displacement along $x$ axis of the top gold MNP layers with $R$ = 12 nm shown in (a), (b), (c) and (d) at $300$ and $500$ $K$. The net heat flux versus $x$ with $R$ = $5$ and $9$ nm described in (e) and (f), respectively, at $300$ and $500$ $K$.
In Fig.<ref>, we show results for the heat transfer for the $TE$ and $TM$ modes when there is relative translational displacement along the $x$ axis between the two MNP layers. It is found that the maximum heat is transferred when the layers are completely overlapping $(x=0)$. As the relative displacement between the layers is increased, $Q_{TE}$, and $Q_{TM}$ decrease at an oscillatory-like fashion. One finds that the period of oscillations of 25 nm for the $R=12$ nm spheres corresponds to distance separation between two neighboring nanoparticles in a layer.
Combining the contributions from both modes, it is found that the oscillatory-like behavior of $Q$ vs $x$ is not as pronounced, although some oscillations are seen for the the nanoparticles with radius $R=9$ nm (Fig.<ref> $e$ and $f$). Our calculations indicate that the heat transfer depends strongly on the overlap between the two layers when sliding one array along $x$ axis with respect to each other. The oscillatory trends of $Q_{TE}$ and $Q_{TM}$ for NPs $R = 12$ nm are observed by means of the couple dipole method in Fig.<ref> (a), (b), (c) and (d). It is very easily to see that the period of this oscillatory behavior between two neighboring peaks is approximately $25$ nm, which relatively corresponds to the distance $d$ between two nearest NPs at the same array. It suggests that the oscillatory feature depends on how well the horizontal plane projections of the top gold NP array and the bottom one matches each other. For $R = 9$ nm, Fig.<ref> (c) and (d) still show the periodic oscillation in the heat transfer band although this behavior is quite small. Thus one can conclude that when $a \gg R$, the actual distribution of the nanoparticles is not important, however, the overlap between the layers can change $Q$ by several orders of magnitude.
§ CONCLUSIONS
This paper has presented theoretical calculations for the near-field radiation in systems involving gold MNPs. Our method can investigate the discrete nanostructures with arbitrary geometries and consider the size effect of NPs including in the dielectric response. We have considered the role of the structure of MNP layers on the heat transfer when these two arrays are displaced with respect to each other along parallel and perpendicular directions. These results can provide guidelines for designing thermal devices utilizing electromagnetic radiation.
Lilia M. Woods acknowledges the Department of Energy under contract DE-FG02-06ER46297.
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|
arxiv-papers
| 2013-08-14T17:52:06 |
2024-09-04T02:49:49.446898
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anh D. Phan, The-Long Phan, and Lilia M. Woods",
"submitter": "Anh Phan Mr.",
"url": "https://arxiv.org/abs/1308.3191"
}
|
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